Variogram Calculation in
Geostatistics
1 PROBLEM 1: VARIOGRAM CALCULATION
Question: Given the following paired data points (distance, semivariance), calculate the
experimental variogram and fit a spherical model to it:
(10, 2.1), (20, 3.5), (30, 4.2), (40, 4.8), (50, 5.1), (60, 5.3)
Solution:
Step 1: Plot the experimental variogram using the given data points.
Step 2: Estimate the sill, range, and nugget effect from the plot:
• Sill (C): Approximately 5.3
• Range (a): Approximately 50
• Nugget effect (C0): Approximately 0
Step 3: Use the spherical model equation:
𝛾(ℎ)={𝐶0+𝐶[3ℎ
2𝑎−1
2(ℎ
𝑎)3]for ℎ≤𝑎
𝐶0+𝐶 for ℎ>𝑎
Step 4: Substitute the estimated values into the equation:
𝛾(ℎ)={5.3[3ℎ
100−1
2(ℎ
50)3]for ℎ≤50
5.3 for ℎ>50
Step 5: Plot the fitted model alongside the experimental variogram to assess the fit.
2 PROBLEM 2: KRIGING ESTIMATION
Question: Using simple kriging, estimate the value at an unsampled location (5, 5) given the
following data points and their values:
(0, 0): 10 (0, 10): 15 (10, 0): 12 (10, 10): 18
Assume an exponential variogram model with a sill of 20 and a range of 8.
Solution:
Step 1: Calculate the distances between the unsampled location and the data points:
• d1 = sqrt((5-0)^2 + (5-0)^2) = 7.07
• d2 = sqrt((5-0)^2 + (5-10)^2) = 7.07
• d3 = sqrt((5-10)^2 + (5-0)^2) = 7.07
• d4 = sqrt((5-10)^2 + (5-10)^2) = 7.07
Step 2: Calculate the semivariances using the exponential model:
𝛾(ℎ)=20(1−𝑒−3ℎ
8)
• γ1 = γ2 = γ3 = γ4 = 20 * (1 - exp(-3 * 7.07 / 8)) = 15.8
Step 3: Set up the kriging system:
[015.8 15.8 15.8
15.8015.8 15.8
15.8 15.8 0 15.8
15.8 15.8 15.8 0 ][𝑤1
𝑤2
𝑤3
𝑤4]=[15.8
15.8
15.8
15.8]
Step 4: Solve for the weights (w1, w2, w3, w4).
Step 5: Calculate the estimated value:
𝑍∗=𝑤1∗10+𝑤2∗15+𝑤3∗12+𝑤4∗18
3 PROBLEM 3: SPATIAL AUTOCORRELATION
Question: Calculate Moran’s I for the following spatial data:
10
12
8
15
9
11
7
13
14
Assume rook’s case adjacency for the weights matrix.
Solution:
Step 1: Calculate the mean (μ) of the values: μ = (10 + 12 + 8 + 15 + 9 + 11 + 7 + 13 + 14)
/ 9 = 11
Step 2: Create the weights matrix (W) using rook’s case adjacency:
𝑊=
[
0 1 0 1 0 0 0 0 0
1 0 1 0 1 0 0 0 0
0 1 0 0 0 1 0 0 0
1 0 0 0 1 0 1 0 0
0 1 0 1 0 1 0 1 0
0 0 1 0 1 0 0 0 1
0 0 0 1 0 0 0 1 0
0 0 0 0 1 0 1 0 1
0 0 0 0 0 1 0 1 0
]
Step 3: Calculate the numerator of Moran’s I:
𝑁=∑∑𝑤𝑖𝑗
𝑛
𝑗=1
𝑛
𝑖=1 (𝑥𝑖−𝜇)(𝑥𝑗−𝜇)
Step 4: Calculate the denominator of Moran’s I:
𝐷=∑(𝑥𝑖−𝜇)2
𝑛
𝑖=1
Step 5: Calculate Moran’s I:
𝐼=𝑛
𝑊⋅𝑁
𝐷
where n is the number of observations (9) and W is the sum of all weights (24).
Step 6: Interpret the result:
• I > 0: Positive spatial autocorrelation
• I < 0: Negative spatial autocorrelation
• I ≈ 0: No spatial autocorrelation
4 PROBLEM 4: TREND SURFACE ANALYSIS
Question: Perform a first-order trend surface analysis on the following data:
X
Y
Z
1
1
10
1
2
12
2
1
15
2
2
18
Solution:
Step 1: Set up the equation for a first-order trend surface:
𝑍=𝑏0+𝑏1𝑋+𝑏2𝑌
Step 2: Create the design matrix:
𝑋=[1 1 1
1 1 2
1 2 1
1 2 2]
Step 3: Set up the normal equations: (𝑋𝑇𝑋)𝑏=𝑋𝑇𝑍
Step 4: Solve for the coefficients (b0, b1, b2):
𝑏=(𝑋𝑇𝑋)−1𝑋𝑇𝑍
Step 5: Interpret the results:
• b0: Intercept
• b1: Coefficient for X (east-west trend)
• b2: Coefficient for Y (north-south trend)
Step 6: Write the equation of the fitted trend surface:
𝑍=𝑏0+𝑏1𝑋+𝑏2𝑌
5 PROBLEM 5: SPATIAL INTERPOLATION
Question: Using inverse distance weighting (IDW) with a power of 2, estimate the value at
point (2.5, 2.5) given the following data points:
(1, 1): 10 (1, 4): 15 (4, 1): 12 (4, 4): 18
Solution:
Step 1: Calculate the distances from the estimation point to each data point:
• d1 = sqrt((2.5-1)^2 + (2.5-1)^2) = 2.12
• d2 = sqrt((2.5-1)^2 + (2.5-4)^2) = 2.69
• d3 = sqrt((2.5-4)^2 + (2.5-1)^2) = 2.69
• d4 = sqrt((2.5-4)^2 + (2.5-4)^2) = 2.12
Step 2: Calculate the weights using the inverse distance formula:
𝑤𝑖=1
𝑑𝑖2
• w1 = 1 / 2.12^2 = 0.222
• w2 = 1 / 2.69^2 = 0.138
• w3 = 1 / 2.69^2 = 0.138
• w4 = 1 / 2.12^2 = 0.222
Step 3: Normalize the weights: 𝑤𝑖′= 𝑤𝑖
∑𝑤𝑗
𝑛
𝑗=1
Step 4: Calculate the estimated value:
𝑍∗=∑𝑤𝑖
𝑛
𝑖=1 ′𝑍𝑖
where Z_i are the known values at the data points.
Step 5: Interpret the result and discuss the limitations of IDW interpolation.
6 PROBLEM 6: VARIOGRAM CLOUD ANALYSIS
Question: Given the following spatial data:
X
Y
Z
1
1
10
2
1
12
1
2
15
2
2
18
3
3
20
Create a variogram cloud and interpret the results.
Solution:
Step 1: Calculate the distances and squared differences for all pairs of points:
For each pair (i, j):
• Distance: h_ij = sqrt((x_i - x_j)^2 + (y_i - y_j)^2)
• Squared difference: γ_ij = (z_i - z_j)^2 / 2
Step 2: Create a scatter plot with distance (h) on the x-axis and semivariance (γ) on the y-axis.
Step 3: Analyze the variogram cloud:
• Look for trends or patterns in the distribution of points
• Identify any outliers or anomalies
• Observe the overall shape and spread of the cloud
Step 4: Interpret the results:
• Discuss the presence or absence of spatial correlation
• Comment on the variability of the data at different distances
• Identify any potential anisotropy or directional effects
Step 5: Consider next steps, such as binning the data to create an experimental variogram.
7 PROBLEM 7: CROSS-VALIDATION FOR KRIGING
Question: Perform leave-one-out cross-validation for ordinary kriging using the following data:
X
Y
Z
1
1
10
1
3
15
3
1
12
3
3
18
2
2
14
Assume an exponential variogram model with a sill of 20 and a range of 2.5.
Solution:
Step 1: For each data point, remove it from the dataset and estimate its value using ordinary
kriging with the remaining points.
Step 2: Calculate the kriging equations for each iteration:
[𝛾(ℎ11)⋯ 𝛾(ℎ1𝑛)1
⋮ ⋱ ⋮ ⋮
𝛾(ℎ𝑛1)⋯ 𝛾(ℎ𝑛𝑛)1
1 ⋯ 1 0][𝑤1
⋮
𝑤𝑛
𝜇]=[𝛾(ℎ10)
⋮
𝛾(ℎ𝑛0)
1]
where γ(h) is the exponential variogram model:
𝛾(ℎ)=20(1−𝑒−3ℎ
2.5)
Step 3: Solve the kriging equations for each iteration to obtain the weights and Lagrange
multiplier.
Step 4: Calculate the estimated value for each removed point:
𝑍𝑖∗=∑𝑤𝑗
𝑛−1
𝑗=1 𝑍𝑗
Step 5: Calculate the cross-validation statistics:
• Mean Error (ME): 𝑀𝐸=1
𝑛∑(𝑍𝑖∗−𝑍𝑖)
𝑛
𝑖=1
• Mean Squared Error (MSE): 𝑀𝑆𝐸=1
𝑛∑(𝑍𝑖∗−𝑍𝑖)2
𝑛
𝑖=1
• Root Mean Squared Error (RMSE): 𝑅𝑀𝑆𝐸=√𝑀𝑆𝐸
• Mean Standardized Error (MSE): 𝑀𝑆𝐸=1
𝑛∑𝑍𝑖∗−𝑍𝑖
𝜎𝑖
𝑛
𝑖=1
• Root Mean Squared Standardized Error (RMSSE): 𝑅𝑀𝑆𝑆𝐸=√1
𝑛∑(𝑍𝑖∗−𝑍𝑖
𝜎𝑖)2
𝑛
𝑖=1
Step 6: Interpret the results:
• ME should be close to 0 for unbiased estimation
• RMSE should be as small as possible
• MSE should be close to 0
• RMSSE should be close to 1
Step 7: Based on the cross-validation results, discuss the performance of the kriging model and
suggest potential improvements.
8 PROBLEM 8: INDICATOR KRIGING
Question: Perform indicator kriging to estimate the probability that the value at location (2.5,
2.5) exceeds a threshold of 15, given the following data:
X
Y
Z
1
1
10
1
4
18
4
1
12
4
4
20
Assume a spherical variogram model for the indicator variable with a sill of 0.25 and a range of
3.
Solution:
Step 1: Transform the data into indicator variables:
𝐼(𝑥)={1if 𝑍(𝑥)>15
0if 𝑍(𝑥)≤15
Step 2: Calculate the distances between the estimation point and the data points:
• d1 = sqrt((2.5-1)^2 + (2.5-1)^2) = 2.12
• d2 = sqrt((2.5-1)^2 + (2.5-4)^2) = 2.69
• d3 = sqrt((2.5-4)^2 + (2.5-1)^2) = 2.69
• d4 = sqrt((2.5-4)^2 + (2.5-4)^2) = 2.12
Step 3: Calculate the semivariances using the spherical model:
𝛾(ℎ)={0.25[3ℎ
2(3)−1
2(ℎ
3)3]for ℎ≤3
0.25 for ℎ>3
Step 4: Set up the kriging system:
[
𝛾(𝑑11)𝛾(𝑑12)𝛾(𝑑13)𝛾(𝑑14)1
𝛾(𝑑21)𝛾(𝑑22)𝛾(𝑑23)𝛾(𝑑24)1
𝛾(𝑑31)𝛾(𝑑32)𝛾(𝑑33)𝛾(𝑑34)1
𝛾(𝑑41)𝛾(𝑑42)𝛾(𝑑43)𝛾(𝑑44)1
1 1 1 1 0
]
[
𝑤1
𝑤2
𝑤3
𝑤4
𝜇
]
=
[
𝛾(𝑑1)
𝛾(𝑑2)
𝛾(𝑑3)
𝛾(𝑑4)
1
]
Step 5: Solve for the weights (w1, w2, w3, w4) and Lagrange multiplier (μ).
Step 6: Calculate the estimated probability:
𝑃∗=𝑤1𝐼(𝑥1)+𝑤2𝐼(𝑥2)+𝑤3𝐼(𝑥3)+𝑤4𝐼(𝑥4)
Step 7: Interpret the result as the probability that the value at (2.5, 2.5) exceeds the threshold
of 15.
9 PROBLEM 9: BAYESIAN GEOSTATISTICS
Question: Using a Bayesian approach, estimate the mean and variance of a spatial random
field given the following observations:
X
Y
Z
1
1
10
1
2
12
2
1
15
2
2
18
Assume a Gaussian prior with mean μ_0 = 13 and variance σ_0^2 = 16, and a measurement
error variance of σ_ε^2 = 1.
Solution:
Step 1: Define the prior distribution: 𝜇∼𝑁(𝜇0,𝜎02)
Step 2: Define the likelihood function: 𝑍𝑖|𝜇∼𝑁(𝜇,𝜎𝜀2)
Step 3: Calculate the sample mean and variance:
𝑍‾=1
𝑛∑𝑍𝑖
𝑛
𝑖=1
𝑠2=1
𝑛−1∑(𝑍𝑖−𝑍‾)2
𝑛
𝑖=1
Step 4: Calculate the posterior mean:
𝜇𝑛=𝑛
𝜎𝜀2𝑍‾+1
𝜎02𝜇0
𝑛
𝜎𝜀2+1
𝜎02
Step 5: Calculate the posterior variance:
𝜎𝑛2=1
𝑛
𝜎𝜀2+1
𝜎02
Step 6: Calculate the 95
𝐶𝐼95%=𝜇𝑛±1.96√𝜎𝑛2
Step 7: Interpret the results:
• Discuss the posterior mean and how it compares to the prior mean and sample mean
• Explain the meaning of the posterior variance and credible interval
• Comment on the influence of the prior and the data on the posterior distribution
10 PROBLEM 10: MULTIVARIATE GEOSTATISTICS
Question: Perform a linear model of coregionalization (LMC) for two variables, Z1 and Z2,
given the following cross-variogram and direct variogram models:
• γ11(h) = 10 Sph(h/100) + 5 Exp(h/50)
• γ22(h) = 15 Sph(h/100) + 8 Exp(h/50)
• γ12(h) = 8 Sph(h/100) + 4 Exp(h/50)
Where Sph and Exp denote spherical and exponential models, respectively.
Solution:
Step 1: Identify the basic structures and their ranges:
• Structure 1: Spherical model with range a1 = 100
• Structure 2: Exponential model with range a2 = 50
Step 2: Set up the coregionalization matrices for each structure:
For the spherical model:
𝐵1=[𝑏11
(1)𝑏12
(1)
𝑏21
(1)𝑏22
(1)]
For the exponential model:
𝐵2=[𝑏11
(2)𝑏12
(2)
𝑏21
(2)𝑏22
(2)]
Step 3: Solve for the elements of B1 and B2 using the sill values:
• b11^(1) = 10, b22^(1) = 15, b12^(1) = b21^(1) = 8
• b11^(2) = 5, b22^(2) = 8, b12^(2) = b21^(2) = 4
Step 4: Verify that B1 and B2 are positive semi-definite matrices:
• Calculate the determinants: det(B1) ≥ 0 and det(B2) ≥ 0
• Check that the diagonal elements are non-negative
Step 5: Write the final LMC model:
[𝛾11(ℎ)𝛾12(ℎ)
𝛾21(ℎ)𝛾22(ℎ)]=𝐵1Sph(ℎ/100)+𝐵2Exp(ℎ/50)
Step 6: Interpret the results:
• Discuss the spatial correlation structure for each variable
• Explain the cross-correlation between the variables
• Comment on the different ranges and their implications for spatial prediction
Step 7: Discuss potential applications of the LMC model, such as cokriging for improved spatial
prediction.
Given the following paired data points (distance, semivariance), calculate the experimental
variogram and fit a spherical model to it:
(10, 2.1), (20, 3.5), (30, 4.2), (40, 4.8), (50, 5.1), (60, 5.3)
Solution:
Step 1: Plot the experimental variogram using the given data points.
Step 2: Estimate the sill, range, and nugget effect from the plot:
• Sill (C): Approximately 5.3
• Range (a): Approximately 50
• Nugget effect (C0): Approximately 0
Step 3: Use the spherical model equation:
𝛾(ℎ)={𝐶0+𝐶[3ℎ
2𝑎−1
2(ℎ
𝑎)3]for ℎ≤𝑎
𝐶0+𝐶 for ℎ>𝑎
Step 4: Substitute the estimated values into the equation:
𝛾(ℎ)={5.3[3ℎ
100−1
2(ℎ
50)3]for ℎ≤50
5.3 for ℎ>50
Step 5: Plot the fitted model alongside the experimental variogram to assess the fit.
11 PROBLEM 2: KRIGING ESTIMATION
Question: Using simple kriging, estimate the value at an unsampled location (5, 5) given the
following data points and their values:
(0, 0): 10 (0, 10): 15 (10, 0): 12 (10, 10): 18
Assume an exponential variogram model with a sill of 20 and a range of 8.
Solution:
Step 1: Calculate the distances between the unsampled location and the data points:
• d1 = sqrt((5-0)^2 + (5-0)^2) = 7.07
• d2 = sqrt((5-0)^2 + (5-10)^2) = 7.07
• d3 = sqrt((5-10)^2 + (5-0)^2) = 7.07
• d4 = sqrt((5-10)^2 + (5-10)^2) = 7.07
Step 2: Calculate the semivariances using the exponential model:
𝛾(ℎ)=20(1−𝑒−3ℎ
8)
• γ1 = γ2 = γ3 = γ4 = 20 * (1 - exp(-3 * 7.07 / 8)) = 15.8
Step 3: Set up the kriging system:
[015.8 15.8 15.8
15.8015.8 15.8
15.8 15.8 0 15.8
15.8 15.8 15.8 0 ][𝑤1
𝑤2
𝑤3
𝑤4]=[15.8
15.8
15.8
15.8]
Step 4: Solve for the weights (w1, w2, w3, w4).
Step 5: Calculate the estimated value:
𝑍∗=𝑤1∗10+𝑤2∗15+𝑤3∗12+𝑤4∗18
12 PROBLEM 3: SPATIAL AUTOCORRELATION
Question: Calculate Moran’s I for the following spatial data:
10
12
8
15
9
11
7
13
14
Assume rook’s case adjacency for the weights matrix.
Solution:
Step 1: Calculate the mean (μ) of the values: μ = (10 + 12 + 8 + 15 + 9 + 11 + 7 + 13 + 14)
/ 9 = 11
Step 2: Create the weights matrix (W) using rook’s case adjacency:
𝑊=
[
0 1 0 1 0 0 0 0 0
1 0 1 0 1 0 0 0 0
0 1 0 0 0 1 0 0 0
1 0 0 0 1 0 1 0 0
0 1 0 1 0 1 0 1 0
0 0 1 0 1 0 0 0 1
0 0 0 1 0 0 0 1 0
0 0 0 0 1 0 1 0 1
0 0 0 0 0 1 0 1 0
]
Step 3: Calculate the numerator of Moran’s I:
𝑁=∑∑𝑤𝑖𝑗
𝑛
𝑗=1
𝑛
𝑖=1 (𝑥𝑖−𝜇)(𝑥𝑗−𝜇)
Step 4: Calculate the denominator of Moran’s I:
𝐷=∑(𝑥𝑖−𝜇)2
𝑛
𝑖=1
Step 5: Calculate Moran’s I:
𝐼=𝑛
𝑊⋅𝑁
𝐷
where n is the number of observations (9) and W is the sum of all weights (24).
Step 6: Interpret the result:
• I > 0: Positive spatial autocorrelation
• I < 0: Negative spatial autocorrelation
• I ≈ 0: No spatial autocorrelation
13 PROBLEM 4: TREND SURFACE ANALYSIS
Question: Perform a first-order trend surface analysis on the following data:
X
Y
Z
1
1
10
1
2
12
2
1
15
2
2
18
Solution:
Step 1: Set up the equation for a first-order trend surface:
𝑍=𝑏0+𝑏1𝑋+𝑏2𝑌
Step 2: Create the design matrix:
𝑋=[1 1 1
1 1 2
1 2 1
1 2 2]
Step 3: Set up the normal equations: (𝑋𝑇𝑋)𝑏=𝑋𝑇𝑍
Step 4: Solve for the coefficients (b0, b1, b2):
𝑏=(𝑋𝑇𝑋)−1𝑋𝑇𝑍
Step 5: Interpret the results:
• b0: Intercept
• b1: Coefficient for X (east-west trend)
• b2: Coefficient for Y (north-south trend)
Step 6: Write the equation of the fitted trend surface:
𝑍=𝑏0+𝑏1𝑋+𝑏2𝑌
14 PROBLEM 5: SPATIAL INTERPOLATION
Question: Using inverse distance weighting (IDW) with a power of 2, estimate the value at
point (2.5, 2.5) given the following data points:
(1, 1): 10 (1, 4): 15 (4, 1): 12 (4, 4): 18
Solution:
Step 1: Calculate the distances from the estimation point to each data point:
• d1 = sqrt((2.5-1)^2 + (2.5-1)^2) = 2.12
• d2 = sqrt((2.5-1)^2 + (2.5-4)^2) = 2.69
• d3 = sqrt((2.5-4)^2 + (2.5-1)^2) = 2.69
• d4 = sqrt((2.5-4)^2 + (2.5-4)^2) = 2.12
Step 2: Calculate the weights using the inverse distance formula:
𝑤𝑖=1
𝑑𝑖2
• w1 = 1 / 2.12^2 = 0.222
• w2 = 1 / 2.69^2 = 0.138
• w3 = 1 / 2.69^2 = 0.138
• w4 = 1 / 2.12^2 = 0.222
Step 3: Normalize the weights: 𝑤𝑖′= 𝑤𝑖
∑𝑤𝑗
𝑛
𝑗=1
Step 4: Calculate the estimated value:
𝑍∗=∑𝑤𝑖
𝑛
𝑖=1 ′𝑍𝑖
where Z_i are the known values at the data points.
Step 5: Interpret the result and discuss the limitations of IDW interpolation.
15 PROBLEM 6: VARIOGRAM CLOUD ANALYSIS
Question: Given the following spatial data:
X
Y
Z
1
1
10
2
1
12
1
2
15
2
2
18
3
3
20
Create a variogram cloud and interpret the results.
Solution:
Step 1: Calculate the distances and squared differences for all pairs of points:
For each pair (i, j):
• Distance: h_ij = sqrt((x_i - x_j)^2 + (y_i - y_j)^2)
• Squared difference: γ_ij = (z_i - z_j)^2 / 2
Step 2: Create a scatter plot with distance (h) on the x-axis and semivariance (γ) on the y-axis.
Step 3: Analyze the variogram cloud:
• Look for trends or patterns in the distribution of points
• Identify any outliers or anomalies
• Observe the overall shape and spread of the cloud
Step 4: Interpret the results:
• Discuss the presence or absence of spatial correlation
• Comment on the variability of the data at different distances
• Identify any potential anisotropy or directional effects
Step 5: Consider next steps, such as binning the data to create an experimental variogram.
16 PROBLEM 7: CROSS-VALIDATION FOR KRIGING
Question: Perform leave-one-out cross-validation for ordinary kriging using the following data:
X
Y
Z
1
1
10
1
3
15
X
Y
Z
3
1
12
3
3
18
2
2
14
Assume an exponential variogram model with a sill of 20 and a range of 2.5.
Solution:
Step 1: For each data point, remove it from the dataset and estimate its value using ordinary
kriging with the remaining points.
Step 2: Calculate the kriging equations for each iteration:
[𝛾(ℎ11)⋯ 𝛾(ℎ1𝑛)1
⋮ ⋱ ⋮ ⋮
𝛾(ℎ𝑛1)⋯ 𝛾(ℎ𝑛𝑛)1
1 ⋯ 1 0][𝑤1
⋮
𝑤𝑛
𝜇]=[𝛾(ℎ10)
⋮
𝛾(ℎ𝑛0)
1]
where γ(h) is the exponential variogram model:
𝛾(ℎ)=20(1−𝑒−3ℎ
2.5)
Step 3: Solve the kriging equations for each iteration to obtain the weights and Lagrange
multiplier.
Step 4: Calculate the estimated value for each removed point:
𝑍𝑖∗=∑𝑤𝑗
𝑛−1
𝑗=1 𝑍𝑗
Step 5: Calculate the cross-validation statistics:
• Mean Error (ME): 𝑀𝐸=1
𝑛∑(𝑍𝑖∗−𝑍𝑖)
𝑛
𝑖=1
• Mean Squared Error (MSE): 𝑀𝑆𝐸=1
𝑛∑(𝑍𝑖∗−𝑍𝑖)2
𝑛
𝑖=1
• Root Mean Squared Error (RMSE): 𝑅𝑀𝑆𝐸=√𝑀𝑆𝐸
• Mean Standardized Error (MSE): 𝑀𝑆𝐸=1
𝑛∑𝑍𝑖∗−𝑍𝑖
𝜎𝑖
𝑛
𝑖=1
• Root Mean Squared Standardized Error (RMSSE): 𝑅𝑀𝑆𝑆𝐸=√1
𝑛∑(𝑍𝑖∗−𝑍𝑖
𝜎𝑖)2
𝑛
𝑖=1
Step 6: Interpret the results:
• ME should be close to 0 for unbiased estimation
• RMSE should be as small as possible
• MSE should be close to 0
• RMSSE should be close to 1
Step 7: Based on the cross-validation results, discuss the performance of the kriging model and
suggest potential improvements.
17 PROBLEM 8: INDICATOR KRIGING
Question: Perform indicator kriging to estimate the probability that the value at location (2.5,
2.5) exceeds a threshold of 15, given the following data:
X
Y
Z
1
1
10
1
4
18
4
1
12
4
4
20
Assume a spherical variogram model for the indicator variable with a sill of 0.25 and a range of
3.
Solution:
Step 1: Transform the data into indicator variables:
𝐼(𝑥)={1if 𝑍(𝑥)>15
0if 𝑍(𝑥)≤15
Step 2: Calculate the distances between the estimation point and the data points:
• d1 = sqrt((2.5-1)^2 + (2.5-1)^2) = 2.12
• d2 = sqrt((2.5-1)^2 + (2.5-4)^2) = 2.69
• d3 = sqrt((2.5-4)^2 + (2.5-1)^2) = 2.69
• d4 = sqrt((2.5-4)^2 + (2.5-4)^2) = 2.12
Step 3: Calculate the semivariances using the spherical model:
𝛾(ℎ)={0.25[3ℎ
2(3)−1
2(ℎ
3)3]for ℎ≤3
0.25 for ℎ>3
Step 4: Set up the kriging system:
[
𝛾(𝑑11)𝛾(𝑑12)𝛾(𝑑13)𝛾(𝑑14)1
𝛾(𝑑21)𝛾(𝑑22)𝛾(𝑑23)𝛾(𝑑24)1
𝛾(𝑑31)𝛾(𝑑32)𝛾(𝑑33)𝛾(𝑑34)1
𝛾(𝑑41)𝛾(𝑑42)𝛾(𝑑43)𝛾(𝑑44)1
1 1 1 1 0
]
[
𝑤1
𝑤2
𝑤3
𝑤4
𝜇
]
=
[
𝛾(𝑑1)
𝛾(𝑑2)
𝛾(𝑑3)
𝛾(𝑑4)
1
]
Step 5: Solve for the weights (w1, w2, w3, w4) and Lagrange multiplier (μ).
Step 6: Calculate the estimated probability:
𝑃∗=𝑤1𝐼(𝑥1)+𝑤2𝐼(𝑥2)+𝑤3𝐼(𝑥3)+𝑤4𝐼(𝑥4)
Step 7: Interpret the result as the probability that the value at (2.5, 2.5) exceeds the threshold
of 15.
18 PROBLEM 9: BAYESIAN GEOSTATISTICS
Question: Using a Bayesian approach, estimate the mean and variance of a spatial random
field given the following observations:
X
Y
Z
1
1
10
1
2
12
2
1
15
2
2
18
Assume a Gaussian prior with mean μ_0 = 13 and variance σ_0^2 = 16, and a measurement
error variance of σ_ε^2 = 1.
Solution:
Step 1: Define the prior distribution: 𝜇∼𝑁(𝜇0,𝜎02)
Step 2: Define the likelihood function: 𝑍𝑖|𝜇∼𝑁(𝜇,𝜎𝜀2)
Step 3: Calculate the sample mean and variance:
𝑍‾=1
𝑛∑𝑍𝑖
𝑛
𝑖=1
𝑠2=1
𝑛−1∑(𝑍𝑖−𝑍‾)2
𝑛
𝑖=1
Step 4: Calculate the posterior mean:
𝜇𝑛=𝑛
𝜎𝜀2𝑍‾+1
𝜎02𝜇0
𝑛
𝜎𝜀2+1
𝜎02
Step 5: Calculate the posterior variance:
𝜎𝑛2=1
𝑛
𝜎𝜀2+1
𝜎02
Step 6: Calculate the 95
𝐶𝐼95%=𝜇𝑛±1.96√𝜎𝑛2
Step 7: Interpret the results:
• Discuss the posterior mean and how it compares to the prior mean and sample mean
• Explain the meaning of the posterior variance and credible interval
• Comment on the influence of the prior and the data on the posterior distribution
19 PROBLEM 10: MULTIVARIATE GEOSTATISTICS
Question: Perform a linear model of coregionalization (LMC) for two variables, Z1 and Z2,
given the following cross-variogram and direct variogram models:
• γ11(h) = 10 Sph(h/100) + 5 Exp(h/50)
• γ22(h) = 15 Sph(h/100) + 8 Exp(h/50)
• γ12(h) = 8 Sph(h/100) + 4 Exp(h/50)
Where Sph and Exp denote spherical and exponential models, respectively.
Solution:
Step 1: Identify the basic structures and their ranges:
• Structure 1: Spherical model with range a1 = 100
• Structure 2: Exponential model with range a2 = 50
Step 2: Set up the coregionalization matrices for each structure:
For the spherical model:
𝐵1=[𝑏11
(1)𝑏12
(1)
𝑏21
(1)𝑏22
(1)]
For the exponential model:
𝐵2=[𝑏11
(2)𝑏12
(2)
𝑏21
(2)𝑏22
(2)]
Step 3: Solve for the elements of B1 and B2 using the sill values:
• b11^(1) = 10, b22^(1) = 15, b12^(1) = b21^(1) = 8
• b11^(2) = 5, b22^(2) = 8, b12^(2) = b21^(2) = 4
Step 4: Verify that B1 and B2 are positive semi-definite matrices:
• Calculate the determinants: det(B1) ≥ 0 and det(B2) ≥ 0
• Check that the diagonal elements are non-negative
Step 5: Write the final LMC model:
[𝛾11(ℎ)𝛾12(ℎ)
𝛾21(ℎ)𝛾22(ℎ)]=𝐵1Sph(ℎ/100)+𝐵2Exp(ℎ/50)
Step 6: Interpret the results:
• Discuss the spatial correlation structure for each variable
• Explain the cross-correlation between the variables
• Comment on the different ranges and their implications for spatial prediction
Step 7: Discuss potential applications of the LMC model, such as cokriging for improved spatial
prediction.
Given the following paired data points (distance, semivariance), calculate the experimental
variogram and fit a spherical model to it:
(10, 2.1), (20, 3.5), (30, 4.2), (40, 4.8), (50, 5.1), (60, 5.3)
Solution:
Step 1: Plot the experimental variogram using the given data points.
Step 2: Estimate the sill, range, and nugget effect from the plot:
• Sill (C): Approximately 5.3
• Range (a): Approximately 50
• Nugget effect (C0): Approximately 0
Step 3: Use the spherical model equation:
𝛾(ℎ)={𝐶0+𝐶[3ℎ
2𝑎−1
2(ℎ
𝑎)3]for ℎ≤𝑎
𝐶0+𝐶 for ℎ>𝑎
Step 4: Substitute the estimated values into the equation:
𝛾(ℎ)={5.3[3ℎ
100−1
2(ℎ
50)3]for ℎ≤50
5.3 for ℎ>50
Step 5: Plot the fitted model alongside the experimental variogram to assess the fit.
20 PROBLEM 2: KRIGING ESTIMATION
Question: Using simple kriging, estimate the value at an unsampled location (5, 5) given the
following data points and their values:
(0, 0): 10 (0, 10): 15 (10, 0): 12 (10, 10): 18
Assume an exponential variogram model with a sill of 20 and a range of 8.
Solution:
Step 1: Calculate the distances between the unsampled location and the data points:
• d1 = sqrt((5-0)^2 + (5-0)^2) = 7.07
• d2 = sqrt((5-0)^2 + (5-10)^2) = 7.07
• d3 = sqrt((5-10)^2 + (5-0)^2) = 7.07
• d4 = sqrt((5-10)^2 + (5-10)^2) = 7.07
Step 2: Calculate the semivariances using the exponential model:
𝛾(ℎ)=20(1−𝑒−3ℎ
8)
• γ1 = γ2 = γ3 = γ4 = 20 * (1 - exp(-3 * 7.07 / 8)) = 15.8
Step 3: Set up the kriging system:
[015.8 15.8 15.8
15.8015.8 15.8
15.8 15.8 0 15.8
15.8 15.8 15.8 0 ][𝑤1
𝑤2
𝑤3
𝑤4]=[15.8
15.8
15.8
15.8]
Step 4: Solve for the weights (w1, w2, w3, w4).
Step 5: Calculate the estimated value:
𝑍∗=𝑤1∗10+𝑤2∗15+𝑤3∗12+𝑤4∗18
21 PROBLEM 3: SPATIAL AUTOCORRELATION
Question: Calculate Moran’s I for the following spatial data:
10
12
8
15
9
11
7
13
14
Assume rook’s case adjacency for the weights matrix.
Solution:
Step 1: Calculate the mean (μ) of the values: μ = (10 + 12 + 8 + 15 + 9 + 11 + 7 + 13 + 14)
/ 9 = 11
Step 2: Create the weights matrix (W) using rook’s case adjacency:
𝑊=
[
0 1 0 1 0 0 0 0 0
1 0 1 0 1 0 0 0 0
0 1 0 0 0 1 0 0 0
1 0 0 0 1 0 1 0 0
0 1 0 1 0 1 0 1 0
0 0 1 0 1 0 0 0 1
0 0 0 1 0 0 0 1 0
0 0 0 0 1 0 1 0 1
0 0 0 0 0 1 0 1 0
]
Step 3: Calculate the numerator of Moran’s I:
𝑁=∑∑𝑤𝑖𝑗
𝑛
𝑗=1
𝑛
𝑖=1 (𝑥𝑖−𝜇)(𝑥𝑗−𝜇)
Step 4: Calculate the denominator of Moran’s I:
𝐷=∑(𝑥𝑖−𝜇)2
𝑛
𝑖=1
Step 5: Calculate Moran’s I:
𝐼=𝑛
𝑊⋅𝑁
𝐷
where n is the number of observations (9) and W is the sum of all weights (24).
Step 6: Interpret the result:
• I > 0: Positive spatial autocorrelation
• I < 0: Negative spatial autocorrelation
• I ≈ 0: No spatial autocorrelation
22 PROBLEM 4: TREND SURFACE ANALYSIS
Question: Perform a first-order trend surface analysis on the following data:
X
Y
Z
1
1
10
1
2
12
2
1
15
2
2
18
Solution:
Step 1: Set up the equation for a first-order trend surface:
𝑍=𝑏0+𝑏1𝑋+𝑏2𝑌
Step 2: Create the design matrix:
𝑋=[1 1 1
1 1 2
1 2 1
1 2 2]
Step 3: Set up the normal equations: (𝑋𝑇𝑋)𝑏=𝑋𝑇𝑍
Step 4: Solve for the coefficients (b0, b1, b2):
𝑏=(𝑋𝑇𝑋)−1𝑋𝑇𝑍
Step 5: Interpret the results:
• b0: Intercept
• b1: Coefficient for X (east-west trend)
• b2: Coefficient for Y (north-south trend)
Step 6: Write the equation of the fitted trend surface:
𝑍=𝑏0+𝑏1𝑋+𝑏2𝑌
23 PROBLEM 5: SPATIAL INTERPOLATION
Question: Using inverse distance weighting (IDW) with a power of 2, estimate the value at
point (2.5, 2.5) given the following data points:
(1, 1): 10 (1, 4): 15 (4, 1): 12 (4, 4): 18
Solution:
Step 1: Calculate the distances from the estimation point to each data point:
• d1 = sqrt((2.5-1)^2 + (2.5-1)^2) = 2.12
• d2 = sqrt((2.5-1)^2 + (2.5-4)^2) = 2.69
• d3 = sqrt((2.5-4)^2 + (2.5-1)^2) = 2.69
• d4 = sqrt((2.5-4)^2 + (2.5-4)^2) = 2.12
Step 2: Calculate the weights using the inverse distance formula:
𝑤𝑖=1
𝑑𝑖2
• w1 = 1 / 2.12^2 = 0.222
• w2 = 1 / 2.69^2 = 0.138
• w3 = 1 / 2.69^2 = 0.138
• w4 = 1 / 2.12^2 = 0.222
Step 3: Normalize the weights: 𝑤𝑖′= 𝑤𝑖
∑𝑤𝑗
𝑛
𝑗=1
Step 4: Calculate the estimated value:
𝑍∗=∑𝑤𝑖
𝑛
𝑖=1 ′𝑍𝑖
where Z_i are the known values at the data points.
Step 5: Interpret the result and discuss the limitations of IDW interpolation.
24 PROBLEM 6: VARIOGRAM CLOUD ANALYSIS
Question: Given the following spatial data:
X
Y
Z
1
1
10
2
1
12
1
2
15
2
2
18
3
3
20
Create a variogram cloud and interpret the results.
Solution:
Step 1: Calculate the distances and squared differences for all pairs of points:
For each pair (i, j):
• Distance: h_ij = sqrt((x_i - x_j)^2 + (y_i - y_j)^2)
• Squared difference: γ_ij = (z_i - z_j)^2 / 2
Step 2: Create a scatter plot with distance (h) on the x-axis and semivariance (γ) on the y-axis.
Step 3: Analyze the variogram cloud:
• Look for trends or patterns in the distribution of points
• Identify any outliers or anomalies
• Observe the overall shape and spread of the cloud
Step 4: Interpret the results:
• Discuss the presence or absence of spatial correlation
• Comment on the variability of the data at different distances
• Identify any potential anisotropy or directional effects
Step 5: Consider next steps, such as binning the data to create an experimental variogram.
25 PROBLEM 7: CROSS-VALIDATION FOR KRIGING
Question: Perform leave-one-out cross-validation for ordinary kriging using the following data:
X
Y
Z
1
1
10
1
3
15
3
1
12
3
3
18
2
2
14
Assume an exponential variogram model with a sill of 20 and a range of 2.5.
Solution:
Step 1: For each data point, remove it from the dataset and estimate its value using ordinary
kriging with the remaining points.
Step 2: Calculate the kriging equations for each iteration:
[𝛾(ℎ11)⋯ 𝛾(ℎ1𝑛)1
⋮ ⋱ ⋮ ⋮
𝛾(ℎ𝑛1)⋯ 𝛾(ℎ𝑛𝑛)1
1 ⋯ 1 0][𝑤1
⋮
𝑤𝑛
𝜇]=[𝛾(ℎ10)
⋮
𝛾(ℎ𝑛0)
1]
where γ(h) is the exponential variogram model:
𝛾(ℎ)=20(1−𝑒−3ℎ
2.5)
Step 3: Solve the kriging equations for each iteration to obtain the weights and Lagrange
multiplier.
Step 4: Calculate the estimated value for each removed point:
𝑍𝑖∗=∑𝑤𝑗
𝑛−1
𝑗=1 𝑍𝑗
Step 5: Calculate the cross-validation statistics:
• Mean Error (ME): 𝑀𝐸=1
𝑛∑(𝑍𝑖∗−𝑍𝑖)
𝑛
𝑖=1
• Mean Squared Error (MSE): 𝑀𝑆𝐸=1
𝑛∑(𝑍𝑖∗−𝑍𝑖)2
𝑛
𝑖=1
• Root Mean Squared Error (RMSE): 𝑅𝑀𝑆𝐸=√𝑀𝑆𝐸
• Mean Standardized Error (MSE): 𝑀𝑆𝐸=1
𝑛∑𝑍𝑖∗−𝑍𝑖
𝜎𝑖
𝑛
𝑖=1
• Root Mean Squared Standardized Error (RMSSE): 𝑅𝑀𝑆𝑆𝐸=√1
𝑛∑(𝑍𝑖∗−𝑍𝑖
𝜎𝑖)2
𝑛
𝑖=1
Step 6: Interpret the results:
• ME should be close to 0 for unbiased estimation
• RMSE should be as small as possible
• MSE should be close to 0
• RMSSE should be close to 1
Step 7: Based on the cross-validation results, discuss the performance of the kriging model and
suggest potential improvements.
26 PROBLEM 8: INDICATOR KRIGING
Question: Perform indicator kriging to estimate the probability that the value at location (2.5,
2.5) exceeds a threshold of 15, given the following data:
X
Y
Z
1
1
10
1
4
18
4
1
12
4
4
20
Assume a spherical variogram model for the indicator variable with a sill of 0.25 and a range of
3.
Solution:
Step 1: Transform the data into indicator variables:
𝐼(𝑥)={1if 𝑍(𝑥)>15
0if 𝑍(𝑥)≤15
Step 2: Calculate the distances between the estimation point and the data points:
• d1 = sqrt((2.5-1)^2 + (2.5-1)^2) = 2.12
• d2 = sqrt((2.5-1)^2 + (2.5-4)^2) = 2.69
• d3 = sqrt((2.5-4)^2 + (2.5-1)^2) = 2.69
• d4 = sqrt((2.5-4)^2 + (2.5-4)^2) = 2.12
Step 3: Calculate the semivariances using the spherical model:
𝛾(ℎ)={0.25[3ℎ
2(3)−1
2(ℎ
3)3]for ℎ≤3
0.25 for ℎ>3
Step 4: Set up the kriging system:
[
𝛾(𝑑11)𝛾(𝑑12)𝛾(𝑑13)𝛾(𝑑14)1
𝛾(𝑑21)𝛾(𝑑22)𝛾(𝑑23)𝛾(𝑑24)1
𝛾(𝑑31)𝛾(𝑑32)𝛾(𝑑33)𝛾(𝑑34)1
𝛾(𝑑41)𝛾(𝑑42)𝛾(𝑑43)𝛾(𝑑44)1
1 1 1 1 0
]
[
𝑤1
𝑤2
𝑤3
𝑤4
𝜇
]
=
[
𝛾(𝑑1)
𝛾(𝑑2)
𝛾(𝑑3)
𝛾(𝑑4)
1
]
Step 5: Solve for the weights (w1, w2, w3, w4) and Lagrange multiplier (μ).
Step 6: Calculate the estimated probability:
𝑃∗=𝑤1𝐼(𝑥1)+𝑤2𝐼(𝑥2)+𝑤3𝐼(𝑥3)+𝑤4𝐼(𝑥4)
Step 7: Interpret the result as the probability that the value at (2.5, 2.5) exceeds the threshold
of 15.
27 PROBLEM 9: BAYESIAN GEOSTATISTICS
Question: Using a Bayesian approach, estimate the mean and variance of a spatial random
field given the following observations:
X
Y
Z
1
1
10
1
2
12
2
1
15
2
2
18
Assume a Gaussian prior with mean μ_0 = 13 and variance σ_0^2 = 16, and a measurement
error variance of σ_ε^2 = 1.
Solution:
Step 1: Define the prior distribution: 𝜇∼𝑁(𝜇0,𝜎02)
Step 2: Define the likelihood function: 𝑍𝑖|𝜇∼𝑁(𝜇,𝜎𝜀2)
Step 3: Calculate the sample mean and variance:
𝑍‾=1
𝑛∑𝑍𝑖
𝑛
𝑖=1
𝑠2=1
𝑛−1∑(𝑍𝑖−𝑍‾)2
𝑛
𝑖=1
Step 4: Calculate the posterior mean:
𝜇𝑛=𝑛
𝜎𝜀2𝑍‾+1
𝜎02𝜇0
𝑛
𝜎𝜀2+1
𝜎02
Step 5: Calculate the posterior variance:
𝜎𝑛2=1
𝑛
𝜎𝜀2+1
𝜎02
Step 6: Calculate the 95
𝐶𝐼95%=𝜇𝑛±1.96√𝜎𝑛2
Step 7: Interpret the results:
• Discuss the posterior mean and how it compares to the prior mean and sample mean
• Explain the meaning of the posterior variance and credible interval
• Comment on the influence of the prior and the data on the posterior distribution
28 PROBLEM 10: MULTIVARIATE GEOSTATISTICS
Question: Perform a linear model of coregionalization (LMC) for two variables, Z1 and Z2,
given the following cross-variogram and direct variogram models:
• γ11(h) = 10 Sph(h/100) + 5 Exp(h/50)
• γ22(h) = 15 Sph(h/100) + 8 Exp(h/50)
• γ12(h) = 8 Sph(h/100) + 4 Exp(h/50)
Where Sph and Exp denote spherical and exponential models, respectively.
Solution:
Step 1: Identify the basic structures and their ranges:
• Structure 1: Spherical model with range a1 = 100
• Structure 2: Exponential model with range a2 = 50
Step 2: Set up the coregionalization matrices for each structure:
For the spherical model:
𝐵1=[𝑏11
(1)𝑏12
(1)
𝑏21
(1)𝑏22
(1)]
For the exponential model:
𝐵2=[𝑏11
(2)𝑏12
(2)
𝑏21
(2)𝑏22
(2)]
Step 3: Solve for the elements of B1 and B2 using the sill values:
• b11^(1) = 10, b22^(1) = 15, b12^(1) = b21^(1) = 8
• b11^(2) = 5, b22^(2) = 8, b12^(2) = b21^(2) = 4
Step 4: Verify that B1 and B2 are positive semi-definite matrices:
• Calculate the determinants: det(B1) ≥ 0 and det(B2) ≥ 0
• Check that the diagonal elements are non-negative
Step 5: Write the final LMC model:
[𝛾11(ℎ)𝛾12(ℎ)
𝛾21(ℎ)𝛾22(ℎ)]=𝐵1Sph(ℎ/100)+𝐵2Exp(ℎ/50)
Step 6: Interpret the results:
• Discuss the spatial correlation structure for each variable
• Explain the cross-correlation between the variables
• Comment on the different ranges and their implications for spatial prediction
Step 7: Discuss potential applications of the LMC model, such as cokriging for improved spatial
prediction.
Given the following paired data points (distance, semivariance), calculate the experimental
variogram and fit a spherical model to it:
(10, 2.1), (20, 3.5), (30, 4.2), (40, 4.8), (50, 5.1), (60, 5.3)
Solution:
Step 1: Plot the experimental variogram using the given data points.
Step 2: Estimate the sill, range, and nugget effect from the plot:
• Sill (C): Approximately 5.3
• Range (a): Approximately 50
• Nugget effect (C0): Approximately 0
Step 3: Use the spherical model equation:
𝛾(ℎ)={𝐶0+𝐶[3ℎ
2𝑎−1
2(ℎ
𝑎)3]for ℎ≤𝑎
𝐶0+𝐶 for ℎ>𝑎
Step 4: Substitute the estimated values into the equation:
𝛾(ℎ)={5.3[3ℎ
100−1
2(ℎ
50)3]for ℎ≤50
5.3 for ℎ>50
Step 5: Plot the fitted model alongside the experimental variogram to assess the fit.
29 PROBLEM 2: KRIGING ESTIMATION
Question: Using simple kriging, estimate the value at an unsampled location (5, 5) given the
following data points and their values:
(0, 0): 10 (0, 10): 15 (10, 0): 12 (10, 10): 18
Assume an exponential variogram model with a sill of 20 and a range of 8.
Solution:
Step 1: Calculate the distances between the unsampled location and the data points:
• d1 = sqrt((5-0)^2 + (5-0)^2) = 7.07
• d2 = sqrt((5-0)^2 + (5-10)^2) = 7.07
• d3 = sqrt((5-10)^2 + (5-0)^2) = 7.07
• d4 = sqrt((5-10)^2 + (5-10)^2) = 7.07
Step 2: Calculate the semivariances using the exponential model:
𝛾(ℎ)=20(1−𝑒−3ℎ
8)
• γ1 = γ2 = γ3 = γ4 = 20 * (1 - exp(-3 * 7.07 / 8)) = 15.8
Step 3: Set up the kriging system:
[015.8 15.8 15.8
15.8015.8 15.8
15.8 15.8 0 15.8
15.8 15.8 15.8 0 ][𝑤1
𝑤2
𝑤3
𝑤4]=[15.8
15.8
15.8
15.8]
Step 4: Solve for the weights (w1, w2, w3, w4).
Step 5: Calculate the estimated value:
𝑍∗=𝑤1∗10+𝑤2∗15+𝑤3∗12+𝑤4∗18
30 PROBLEM 3: SPATIAL AUTOCORRELATION
Question: Calculate Moran’s I for the following spatial data:
10
12
8
15
9
11
7
13
14
Assume rook’s case adjacency for the weights matrix.
Solution:
Step 1: Calculate the mean (μ) of the values: μ = (10 + 12 + 8 + 15 + 9 + 11 + 7 + 13 + 14)
/ 9 = 11
Step 2: Create the weights matrix (W) using rook’s case adjacency:
𝑊=
[
0 1 0 1 0 0 0 0 0
1 0 1 0 1 0 0 0 0
0 1 0 0 0 1 0 0 0
1 0 0 0 1 0 1 0 0
0 1 0 1 0 1 0 1 0
0 0 1 0 1 0 0 0 1
0 0 0 1 0 0 0 1 0
0 0 0 0 1 0 1 0 1
0 0 0 0 0 1 0 1 0
]
Step 3: Calculate the numerator of Moran’s I:
𝑁=∑∑𝑤𝑖𝑗
𝑛
𝑗=1
𝑛
𝑖=1 (𝑥𝑖−𝜇)(𝑥𝑗−𝜇)
Step 4: Calculate the denominator of Moran’s I:
𝐷=∑(𝑥𝑖−𝜇)2
𝑛
𝑖=1
Step 5: Calculate Moran’s I:
𝐼=𝑛
𝑊⋅𝑁
𝐷
where n is the number of observations (9) and W is the sum of all weights (24).
Step 6: Interpret the result:
• I > 0: Positive spatial autocorrelation
• I < 0: Negative spatial autocorrelation
• I ≈ 0: No spatial autocorrelation
31 PROBLEM 4: TREND SURFACE ANALYSIS
Question: Perform a first-order trend surface analysis on the following data:
X
Y
Z
1
1
10
1
2
12
2
1
15
2
2
18
Solution:
Step 1: Set up the equation for a first-order trend surface:
𝑍=𝑏0+𝑏1𝑋+𝑏2𝑌
Step 2: Create the design matrix:
𝑋=[1 1 1
1 1 2
1 2 1
1 2 2]
Step 3: Set up the normal equations: (𝑋𝑇𝑋)𝑏=𝑋𝑇𝑍
Step 4: Solve for the coefficients (b0, b1, b2):
𝑏=(𝑋𝑇𝑋)−1𝑋𝑇𝑍
Step 5: Interpret the results:
• b0: Intercept
• b1: Coefficient for X (east-west trend)
• b2: Coefficient for Y (north-south trend)
Step 6: Write the equation of the fitted trend surface:
𝑍=𝑏0+𝑏1𝑋+𝑏2𝑌
32 PROBLEM 5: SPATIAL INTERPOLATION
Question: Using inverse distance weighting (IDW) with a power of 2, estimate the value at
point (2.5, 2.5) given the following data points:
(1, 1): 10 (1, 4): 15 (4, 1): 12 (4, 4): 18
Solution:
Step 1: Calculate the distances from the estimation point to each data point:
• d1 = sqrt((2.5-1)^2 + (2.5-1)^2) = 2.12
• d2 = sqrt((2.5-1)^2 + (2.5-4)^2) = 2.69
• d3 = sqrt((2.5-4)^2 + (2.5-1)^2) = 2.69
• d4 = sqrt((2.5-4)^2 + (2.5-4)^2) = 2.12
Step 2: Calculate the weights using the inverse distance formula:
𝑤𝑖=1
𝑑𝑖2
• w1 = 1 / 2.12^2 = 0.222
• w2 = 1 / 2.69^2 = 0.138
• w3 = 1 / 2.69^2 = 0.138
• w4 = 1 / 2.12^2 = 0.222
Step 3: Normalize the weights: 𝑤𝑖′= 𝑤𝑖
∑𝑤𝑗
𝑛
𝑗=1
Step 4: Calculate the estimated value:
𝑍∗=∑𝑤𝑖
𝑛
𝑖=1 ′𝑍𝑖
where Z_i are the known values at the data points.
Step 5: Interpret the result and discuss the limitations of IDW interpolation.
33 PROBLEM 6: VARIOGRAM CLOUD ANALYSIS
Question: Given the following spatial data:
X
Y
Z
1
1
10
2
1
12
1
2
15
2
2
18
3
3
20
Create a variogram cloud and interpret the results.
Solution:
Step 1: Calculate the distances and squared differences for all pairs of points:
For each pair (i, j):
• Distance: h_ij = sqrt((x_i - x_j)^2 + (y_i - y_j)^2)
• Squared difference: γ_ij = (z_i - z_j)^2 / 2
Step 2: Create a scatter plot with distance (h) on the x-axis and semivariance (γ) on the y-axis.
Step 3: Analyze the variogram cloud:
• Look for trends or patterns in the distribution of points
• Identify any outliers or anomalies
• Observe the overall shape and spread of the cloud
Step 4: Interpret the results:
• Discuss the presence or absence of spatial correlation
• Comment on the variability of the data at different distances
• Identify any potential anisotropy or directional effects
Step 5: Consider next steps, such as binning the data to create an experimental variogram.
34 PROBLEM 7: CROSS-VALIDATION FOR KRIGING
Question: Perform leave-one-out cross-validation for ordinary kriging using the following data:
X
Y
Z
1
1
10
1
3
15
3
1
12
3
3
18
2
2
14
Assume an exponential variogram model with a sill of 20 and a range of 2.5.
Solution:
Step 1: For each data point, remove it from the dataset and estimate its value using ordinary
kriging with the remaining points.
Step 2: Calculate the kriging equations for each iteration:
[𝛾(ℎ11)⋯ 𝛾(ℎ1𝑛)1
⋮ ⋱ ⋮ ⋮
𝛾(ℎ𝑛1)⋯ 𝛾(ℎ𝑛𝑛)1
1 ⋯ 1 0][𝑤1
⋮
𝑤𝑛
𝜇]=[𝛾(ℎ10)
⋮
𝛾(ℎ𝑛0)
1]
where γ(h) is the exponential variogram model:
𝛾(ℎ)=20(1−𝑒−3ℎ
2.5)
Step 3: Solve the kriging equations for each iteration to obtain the weights and Lagrange
multiplier.
Step 4: Calculate the estimated value for each removed point:
𝑍𝑖∗=∑𝑤𝑗
𝑛−1
𝑗=1 𝑍𝑗
Step 5: Calculate the cross-validation statistics:
• Mean Error (ME): 𝑀𝐸=1
𝑛∑(𝑍𝑖∗−𝑍𝑖)
𝑛
𝑖=1
• Mean Squared Error (MSE): 𝑀𝑆𝐸=1
𝑛∑(𝑍𝑖∗−𝑍𝑖)2
𝑛
𝑖=1
• Root Mean Squared Error (RMSE): 𝑅𝑀𝑆𝐸=√𝑀𝑆𝐸
• Mean Standardized Error (MSE): 𝑀𝑆𝐸=1
𝑛∑𝑍𝑖∗−𝑍𝑖
𝜎𝑖
𝑛
𝑖=1
• Root Mean Squared Standardized Error (RMSSE): 𝑅𝑀𝑆𝑆𝐸=√1
𝑛∑(𝑍𝑖∗−𝑍𝑖
𝜎𝑖)2
𝑛
𝑖=1
Step 6: Interpret the results:
• ME should be close to 0 for unbiased estimation
• RMSE should be as small as possible
• MSE should be close to 0
• RMSSE should be close to 1
Step 7: Based on the cross-validation results, discuss the performance of the kriging model and
suggest potential improvements.
35 PROBLEM 8: INDICATOR KRIGING
Question: Perform indicator kriging to estimate the probability that the value at location (2.5,
2.5) exceeds a threshold of 15, given the following data:
X
Y
Z
1
1
10
1
4
18
4
1
12
4
4
20
Assume a spherical variogram model for the indicator variable with a sill of 0.25 and a range of
3.
Solution:
Step 1: Transform the data into indicator variables:
𝐼(𝑥)={1if 𝑍(𝑥)>15
0if 𝑍(𝑥)≤15
Step 2: Calculate the distances between the estimation point and the data points:
• d1 = sqrt((2.5-1)^2 + (2.5-1)^2) = 2.12
• d2 = sqrt((2.5-1)^2 + (2.5-4)^2) = 2.69
• d3 = sqrt((2.5-4)^2 + (2.5-1)^2) = 2.69
• d4 = sqrt((2.5-4)^2 + (2.5-4)^2) = 2.12
Step 3: Calculate the semivariances using the spherical model:
𝛾(ℎ)={0.25[3ℎ
2(3)−1
2(ℎ
3)3]for ℎ≤3
0.25 for ℎ>3
Step 4: Set up the kriging system:
[
𝛾(𝑑11)𝛾(𝑑12)𝛾(𝑑13)𝛾(𝑑14)1
𝛾(𝑑21)𝛾(𝑑22)𝛾(𝑑23)𝛾(𝑑24)1
𝛾(𝑑31)𝛾(𝑑32)𝛾(𝑑33)𝛾(𝑑34)1
𝛾(𝑑41)𝛾(𝑑42)𝛾(𝑑43)𝛾(𝑑44)1
1 1 1 1 0
]
[
𝑤1
𝑤2
𝑤3
𝑤4
𝜇
]
=
[
𝛾(𝑑1)
𝛾(𝑑2)
𝛾(𝑑3)
𝛾(𝑑4)
1
]
Step 5: Solve for the weights (w1, w2, w3, w4) and Lagrange multiplier (μ).
Step 6: Calculate the estimated probability:
𝑃∗=𝑤1𝐼(𝑥1)+𝑤2𝐼(𝑥2)+𝑤3𝐼(𝑥3)+𝑤4𝐼(𝑥4)
Step 7: Interpret the result as the probability that the value at (2.5, 2.5) exceeds the threshold
of 15.
36 PROBLEM 9: BAYESIAN GEOSTATISTICS
Question: Using a Bayesian approach, estimate the mean and variance of a spatial random
field given the following observations:
X
Y
Z
1
1
10
1
2
12
2
1
15
2
2
18
Assume a Gaussian prior with mean μ_0 = 13 and variance σ_0^2 = 16, and a measurement
error variance of σ_ε^2 = 1.
Solution:
Step 1: Define the prior distribution: 𝜇∼𝑁(𝜇0,𝜎02)
Step 2: Define the likelihood function: 𝑍𝑖|𝜇∼𝑁(𝜇,𝜎𝜀2)
Step 3: Calculate the sample mean and variance:
𝑍‾=1
𝑛∑𝑍𝑖
𝑛
𝑖=1
𝑠2=1
𝑛−1∑(𝑍𝑖−𝑍‾)2
𝑛
𝑖=1
Step 4: Calculate the posterior mean:
𝜇𝑛=𝑛
𝜎𝜀2𝑍‾+1
𝜎02𝜇0
𝑛
𝜎𝜀2+1
𝜎02
Step 5: Calculate the posterior variance:
𝜎𝑛2=1
𝑛
𝜎𝜀2+1
𝜎02
Step 6: Calculate the 95
𝐶𝐼95%=𝜇𝑛±1.96√𝜎𝑛2
Step 7: Interpret the results:
• Discuss the posterior mean and how it compares to the prior mean and sample mean
• Explain the meaning of the posterior variance and credible interval
• Comment on the influence of the prior and the data on the posterior distribution
37 PROBLEM 10: MULTIVARIATE GEOSTATISTICS
Question: Perform a linear model of coregionalization (LMC) for two variables, Z1 and Z2,
given the following cross-variogram and direct variogram models:
• γ11(h) = 10 Sph(h/100) + 5 Exp(h/50)
• γ22(h) = 15 Sph(h/100) + 8 Exp(h/50)
• γ12(h) = 8 Sph(h/100) + 4 Exp(h/50)
Where Sph and Exp denote spherical and exponential models, respectively.
Solution:
Step 1: Identify the basic structures and their ranges:
• Structure 1: Spherical model with range a1 = 100
• Structure 2: Exponential model with range a2 = 50
Step 2: Set up the coregionalization matrices for each structure:
For the spherical model:
𝐵1=[𝑏11
(1)𝑏12
(1)
𝑏21
(1)𝑏22
(1)]
For the exponential model:
𝐵2=[𝑏11
(2)𝑏12
(2)
𝑏21
(2)𝑏22
(2)]
Step 3: Solve for the elements of B1 and B2 using the sill values:
• b11^(1) = 10, b22^(1) = 15, b12^(1) = b21^(1) = 8
• b11^(2) = 5, b22^(2) = 8, b12^(2) = b21^(2) = 4
Step 4: Verify that B1 and B2 are positive semi-definite matrices:
• Calculate the determinants: det(B1) ≥ 0 and det(B2) ≥ 0
• Check that the diagonal elements are non-negative
Step 5: Write the final LMC model:
[𝛾11(ℎ)𝛾12(ℎ)
𝛾21(ℎ)𝛾22(ℎ)]=𝐵1Sph(ℎ/100)+𝐵2Exp(ℎ/50)
Step 6: Interpret the results:
• Discuss the spatial correlation structure for each variable
• Explain the cross-correlation between the variables
• Comment on the different ranges and their implications for spatial prediction
Step 7: Discuss potential applications of the LMC model, such as cokriging for improved spatial
prediction.
Given the following paired data points (distance, semivariance), calculate the experimental
variogram and fit a spherical model to it:
(10, 2.1), (20, 3.5), (30, 4.2), (40, 4.8), (50, 5.1), (60, 5.3)
Solution:
Step 1: Plot the experimental variogram using the given data points.
Step 2: Estimate the sill, range, and nugget effect from the plot:
• Sill (C): Approximately 5.3
• Range (a): Approximately 50
• Nugget effect (C0): Approximately 0
Step 3: Use the spherical model equation:
𝛾(ℎ)={𝐶0+𝐶[3ℎ
2𝑎−1
2(ℎ
𝑎)3]for ℎ≤𝑎
𝐶0+𝐶 for ℎ>𝑎
Step 4: Substitute the estimated values into the equation:
𝛾(ℎ)={5.3[3ℎ
100−1
2(ℎ
50)3]for ℎ≤50
5.3 for ℎ>50
Step 5: Plot the fitted model alongside the experimental variogram to assess the fit.
38 PROBLEM 2: KRIGING ESTIMATION
Question: Using simple kriging, estimate the value at an unsampled location (5, 5) given the
following data points and their values:
(0, 0): 10 (0, 10): 15 (10, 0): 12 (10, 10): 18
Assume an exponential variogram model with a sill of 20 and a range of 8.
Solution:
Step 1: Calculate the distances between the unsampled location and the data points:
• d1 = sqrt((5-0)^2 + (5-0)^2) = 7.07
• d2 = sqrt((5-0)^2 + (5-10)^2) = 7.07
• d3 = sqrt((5-10)^2 + (5-0)^2) = 7.07
• d4 = sqrt((5-10)^2 + (5-10)^2) = 7.07
Step 2: Calculate the semivariances using the exponential model:
𝛾(ℎ)=20(1−𝑒−3ℎ
8)
• γ1 = γ2 = γ3 = γ4 = 20 * (1 - exp(-3 * 7.07 / 8)) = 15.8
Step 3: Set up the kriging system:
[015.8 15.8 15.8
15.8015.8 15.8
15.8 15.8 0 15.8
15.8 15.8 15.8 0 ][𝑤1
𝑤2
𝑤3
𝑤4]=[15.8
15.8
15.8
15.8]
Step 4: Solve for the weights (w1, w2, w3, w4).
Step 5: Calculate the estimated value:
𝑍∗=𝑤1∗10+𝑤2∗15+𝑤3∗12+𝑤4∗18
39 PROBLEM 3: SPATIAL AUTOCORRELATION
Question: Calculate Moran’s I for the following spatial data:
10
12
8
15
9
11
7
13
14
Assume rook’s case adjacency for the weights matrix.
Solution:
Step 1: Calculate the mean (μ) of the values: μ = (10 + 12 + 8 + 15 + 9 + 11 + 7 + 13 + 14)
/ 9 = 11
Step 2: Create the weights matrix (W) using rook’s case adjacency:
𝑊=
[
0 1 0 1 0 0 0 0 0
1 0 1 0 1 0 0 0 0
0 1 0 0 0 1 0 0 0
1 0 0 0 1 0 1 0 0
0 1 0 1 0 1 0 1 0
0 0 1 0 1 0 0 0 1
0 0 0 1 0 0 0 1 0
0 0 0 0 1 0 1 0 1
0 0 0 0 0 1 0 1 0
]
Step 3: Calculate the numerator of Moran’s I:
𝑁=∑∑𝑤𝑖𝑗
𝑛
𝑗=1
𝑛
𝑖=1 (𝑥𝑖−𝜇)(𝑥𝑗−𝜇)
Step 4: Calculate the denominator of Moran’s I:
𝐷=∑(𝑥𝑖−𝜇)2
𝑛
𝑖=1
Step 5: Calculate Moran’s I:
𝐼=𝑛
𝑊⋅𝑁
𝐷
where n is the number of observations (9) and W is the sum of all weights (24).
Step 6: Interpret the result:
• I > 0: Positive spatial autocorrelation
• I < 0: Negative spatial autocorrelation
• I ≈ 0: No spatial autocorrelation
40 PROBLEM 4: TREND SURFACE ANALYSIS
Question: Perform a first-order trend surface analysis on the following data:
X
Y
Z
1
1
10
1
2
12
2
1
15
2
2
18
Solution:
Step 1: Set up the equation for a first-order trend surface:
𝑍=𝑏0+𝑏1𝑋+𝑏2𝑌
Step 2: Create the design matrix:
𝑋=[1 1 1
1 1 2
1 2 1
1 2 2]
Step 3: Set up the normal equations: (𝑋𝑇𝑋)𝑏=𝑋𝑇𝑍
Step 4: Solve for the coefficients (b0, b1, b2):
𝑏=(𝑋𝑇𝑋)−1𝑋𝑇𝑍
Step 5: Interpret the results:
• b0: Intercept
• b1: Coefficient for X (east-west trend)
• b2: Coefficient for Y (north-south trend)
Step 6: Write the equation of the fitted trend surface:
𝑍=𝑏0+𝑏1𝑋+𝑏2𝑌
41 PROBLEM 5: SPATIAL INTERPOLATION
Question: Using inverse distance weighting (IDW) with a power of 2, estimate the value at
point (2.5, 2.5) given the following data points:
(1, 1): 10 (1, 4): 15 (4, 1): 12 (4, 4): 18
Solution:
Step 1: Calculate the distances from the estimation point to each data point:
• d1 = sqrt((2.5-1)^2 + (2.5-1)^2) = 2.12
• d2 = sqrt((2.5-1)^2 + (2.5-4)^2) = 2.69
• d3 = sqrt((2.5-4)^2 + (2.5-1)^2) = 2.69
• d4 = sqrt((2.5-4)^2 + (2.5-4)^2) = 2.12
Step 2: Calculate the weights using the inverse distance formula:
𝑤𝑖=1
𝑑𝑖2
• w1 = 1 / 2.12^2 = 0.222
• w2 = 1 / 2.69^2 = 0.138
• w3 = 1 / 2.69^2 = 0.138
• w4 = 1 / 2.12^2 = 0.222
Step 3: Normalize the weights: 𝑤𝑖′= 𝑤𝑖
∑𝑤𝑗
𝑛
𝑗=1
Step 4: Calculate the estimated value:
𝑍∗=∑𝑤𝑖
𝑛
𝑖=1 ′𝑍𝑖
where Z_i are the known values at the data points.
Step 5: Interpret the result and discuss the limitations of IDW interpolation.
42 PROBLEM 6: VARIOGRAM CLOUD ANALYSIS
Question: Given the following spatial data:
X
Y
Z
1
1
10
2
1
12
1
2
15
2
2
18
3
3
20
Create a variogram cloud and interpret the results.
Solution:
Step 1: Calculate the distances and squared differences for all pairs of points:
For each pair (i, j):
• Distance: h_ij = sqrt((x_i - x_j)^2 + (y_i - y_j)^2)
• Squared difference: γ_ij = (z_i - z_j)^2 / 2
Step 2: Create a scatter plot with distance (h) on the x-axis and semivariance (γ) on the y-axis.
Step 3: Analyze the variogram cloud:
• Look for trends or patterns in the distribution of points
• Identify any outliers or anomalies
• Observe the overall shape and spread of the cloud
Step 4: Interpret the results:
• Discuss the presence or absence of spatial correlation
• Comment on the variability of the data at different distances
• Identify any potential anisotropy or directional effects
Step 5: Consider next steps, such as binning the data to create an experimental variogram.
43 PROBLEM 7: CROSS-VALIDATION FOR KRIGING
Question: Perform leave-one-out cross-validation for ordinary kriging using the following data:
X
Y
Z
1
1
10
1
3
15
X
Y
Z
3
1
12
3
3
18
2
2
14
Assume an exponential variogram model with a sill of 20 and a range of 2.5.
Solution:
Step 1: For each data point, remove it from the dataset and estimate its value using ordinary
kriging with the remaining points.
Step 2: Calculate the kriging equations for each iteration:
[𝛾(ℎ11)⋯ 𝛾(ℎ1𝑛)1
⋮ ⋱ ⋮ ⋮
𝛾(ℎ𝑛1)⋯ 𝛾(ℎ𝑛𝑛)1
1 ⋯ 1 0][𝑤1
⋮
𝑤𝑛
𝜇]=[𝛾(ℎ10)
⋮
𝛾(ℎ𝑛0)
1]
where γ(h) is the exponential variogram model:
𝛾(ℎ)=20(1−𝑒−3ℎ
2.5)
Step 3: Solve the kriging equations for each iteration to obtain the weights and Lagrange
multiplier.
Step 4: Calculate the estimated value for each removed point:
𝑍𝑖∗=∑𝑤𝑗
𝑛−1
𝑗=1 𝑍𝑗
Step 5: Calculate the cross-validation statistics:
• Mean Error (ME): 𝑀𝐸=1
𝑛∑(𝑍𝑖∗−𝑍𝑖)
𝑛
𝑖=1
• Mean Squared Error (MSE): 𝑀𝑆𝐸=1
𝑛∑(𝑍𝑖∗−𝑍𝑖)2
𝑛
𝑖=1
• Root Mean Squared Error (RMSE): 𝑅𝑀𝑆𝐸=√𝑀𝑆𝐸
• Mean Standardized Error (MSE): 𝑀𝑆𝐸=1
𝑛∑𝑍𝑖∗−𝑍𝑖
𝜎𝑖
𝑛
𝑖=1
• Root Mean Squared Standardized Error (RMSSE): 𝑅𝑀𝑆𝑆𝐸=√1
𝑛∑(𝑍𝑖∗−𝑍𝑖
𝜎𝑖)2
𝑛
𝑖=1
Step 6: Interpret the results:
• ME should be close to 0 for unbiased estimation
• RMSE should be as small as possible
• MSE should be close to 0
• RMSSE should be close to 1
Step 7: Based on the cross-validation results, discuss the performance of the kriging model and
suggest potential improvements.
44 PROBLEM 8: INDICATOR KRIGING
Question: Perform indicator kriging to estimate the probability that the value at location (2.5,
2.5) exceeds a threshold of 15, given the following data:
X
Y
Z
1
1
10
1
4
18
4
1
12
4
4
20
Assume a spherical variogram model for the indicator variable with a sill of 0.25 and a range of
3.
Solution:
Step 1: Transform the data into indicator variables:
𝐼(𝑥)={1if 𝑍(𝑥)>15
0if 𝑍(𝑥)≤15
Step 2: Calculate the distances between the estimation point and the data points:
• d1 = sqrt((2.5-1)^2 + (2.5-1)^2) = 2.12
• d2 = sqrt((2.5-1)^2 + (2.5-4)^2) = 2.69
• d3 = sqrt((2.5-4)^2 + (2.5-1)^2) = 2.69
• d4 = sqrt((2.5-4)^2 + (2.5-4)^2) = 2.12
Step 3: Calculate the semivariances using the spherical model:
𝛾(ℎ)={0.25[3ℎ
2(3)−1
2(ℎ
3)3]for ℎ≤3
0.25 for ℎ>3
Step 4: Set up the kriging system:
[
𝛾(𝑑11)𝛾(𝑑12)𝛾(𝑑13)𝛾(𝑑14)1
𝛾(𝑑21)𝛾(𝑑22)𝛾(𝑑23)𝛾(𝑑24)1
𝛾(𝑑31)𝛾(𝑑32)𝛾(𝑑33)𝛾(𝑑34)1
𝛾(𝑑41)𝛾(𝑑42)𝛾(𝑑43)𝛾(𝑑44)1
1 1 1 1 0
]
[
𝑤1
𝑤2
𝑤3
𝑤4
𝜇
]
=
[
𝛾(𝑑1)
𝛾(𝑑2)
𝛾(𝑑3)
𝛾(𝑑4)
1
]
Step 5: Solve for the weights (w1, w2, w3, w4) and Lagrange multiplier (μ).
Step 6: Calculate the estimated probability:
𝑃∗=𝑤1𝐼(𝑥1)+𝑤2𝐼(𝑥2)+𝑤3𝐼(𝑥3)+𝑤4𝐼(𝑥4)
Step 7: Interpret the result as the probability that the value at (2.5, 2.5) exceeds the threshold
of 15.
45 PROBLEM 9: BAYESIAN GEOSTATISTICS
Question: Using a Bayesian approach, estimate the mean and variance of a spatial random
field given the following observations:
X
Y
Z
1
1
10
1
2
12
2
1
15
2
2
18
Assume a Gaussian prior with mean μ_0 = 13 and variance σ_0^2 = 16, and a measurement
error variance of σ_ε^2 = 1.
Solution:
Step 1: Define the prior distribution: 𝜇∼𝑁(𝜇0,𝜎02)
Step 2: Define the likelihood function: 𝑍𝑖|𝜇∼𝑁(𝜇,𝜎𝜀2)
Step 3: Calculate the sample mean and variance:
𝑍‾=1
𝑛∑𝑍𝑖
𝑛
𝑖=1
𝑠2=1
𝑛−1∑(𝑍𝑖−𝑍‾)2
𝑛
𝑖=1
Step 4: Calculate the posterior mean:
𝜇𝑛=𝑛
𝜎𝜀2𝑍‾+1
𝜎02𝜇0
𝑛
𝜎𝜀2+1
𝜎02
Step 5: Calculate the posterior variance:
𝜎𝑛2=1
𝑛
𝜎𝜀2+1
𝜎02
Step 6: Calculate the 95
𝐶𝐼95%=𝜇𝑛±1.96√𝜎𝑛2
Step 7: Interpret the results:
• Discuss the posterior mean and how it compares to the prior mean and sample mean
• Explain the meaning of the posterior variance and credible interval
• Comment on the influence of the prior and the data on the posterior distribution
46 PROBLEM 10: MULTIVARIATE GEOSTATISTICS
Question: Perform a linear model of coregionalization (LMC) for two variables, Z1 and Z2,
given the following cross-variogram and direct variogram models:
• γ11(h) = 10 Sph(h/100) + 5 Exp(h/50)
• γ22(h) = 15 Sph(h/100) + 8 Exp(h/50)
• γ12(h) = 8 Sph(h/100) + 4 Exp(h/50)
Where Sph and Exp denote spherical and exponential models, respectively.
Solution:
Step 1: Identify the basic structures and their ranges:
• Structure 1: Spherical model with range a1 = 100
• Structure 2: Exponential model with range a2 = 50
Step 2: Set up the coregionalization matrices for each structure:
For the spherical model:
𝐵1=[𝑏11
(1)𝑏12
(1)
𝑏21
(1)𝑏22
(1)]
For the exponential model:
𝐵2=[𝑏11
(2)𝑏12
(2)
𝑏21
(2)𝑏22
(2)]
Step 3: Solve for the elements of B1 and B2 using the sill values:
• b11^(1) = 10, b22^(1) = 15, b12^(1) = b21^(1) = 8
• b11^(2) = 5, b22^(2) = 8, b12^(2) = b21^(2) = 4
Step 4: Verify that B1 and B2 are positive semi-definite matrices:
• Calculate the determinants: det(B1) ≥ 0 and det(B2) ≥ 0
• Check that the diagonal elements are non-negative
Step 5: Write the final LMC model:
[𝛾11(ℎ)𝛾12(ℎ)
𝛾21(ℎ)𝛾22(ℎ)]=𝐵1Sph(ℎ/100)+𝐵2Exp(ℎ/50)
Step 6: Interpret the results:
• Discuss the spatial correlation structure for each variable
• Explain the cross-correlation between the variables
• Comment on the different ranges and their implications for spatial prediction
Step 7: Discuss potential applications of the LMC model, such as cokriging for improved spatial
prediction.
Given the following paired data points (distance, semivariance), calculate the experimental
variogram and fit a spherical model to it:
(10, 2.1), (20, 3.5), (30, 4.2), (40, 4.8), (50, 5.1), (60, 5.3)
Solution:
Step 1: Plot the experimental variogram using the given data points.
Step 2: Estimate the sill, range, and nugget effect from the plot:
• Sill (C): Approximately 5.3
• Range (a): Approximately 50
• Nugget effect (C0): Approximately 0
Step 3: Use the spherical model equation:
𝛾(ℎ)={𝐶0+𝐶[3ℎ
2𝑎−1
2(ℎ
𝑎)3]for ℎ≤𝑎
𝐶0+𝐶 for ℎ>𝑎
Step 4: Substitute the estimated values into the equation:
𝛾(ℎ)={5.3[3ℎ
100−1
2(ℎ
50)3]for ℎ≤50
5.3 for ℎ>50
Step 5: Plot the fitted model alongside the experimental variogram to assess the fit.
47 PROBLEM 2: KRIGING ESTIMATION
Question: Using simple kriging, estimate the value at an unsampled location (5, 5) given the
following data points and their values:
(0, 0): 10 (0, 10): 15 (10, 0): 12 (10, 10): 18
Assume an exponential variogram model with a sill of 20 and a range of 8.
Solution:
Step 1: Calculate the distances between the unsampled location and the data points:
• d1 = sqrt((5-0)^2 + (5-0)^2) = 7.07
• d2 = sqrt((5-0)^2 + (5-10)^2) = 7.07
• d3 = sqrt((5-10)^2 + (5-0)^2) = 7.07
• d4 = sqrt((5-10)^2 + (5-10)^2) = 7.07
Step 2: Calculate the semivariances using the exponential model:
𝛾(ℎ)=20(1−𝑒−3ℎ
8)
• γ1 = γ2 = γ3 = γ4 = 20 * (1 - exp(-3 * 7.07 / 8)) = 15.8
Step 3: Set up the kriging system:
[015.8 15.8 15.8
15.8015.8 15.8
15.8 15.8 0 15.8
15.8 15.8 15.8 0 ][𝑤1
𝑤2
𝑤3
𝑤4]=[15.8
15.8
15.8
15.8]
Step 4: Solve for the weights (w1, w2, w3, w4).
Step 5: Calculate the estimated value:
𝑍∗=𝑤1∗10+𝑤2∗15+𝑤3∗12+𝑤4∗18
48 PROBLEM 3: SPATIAL AUTOCORRELATION
Question: Calculate Moran’s I for the following spatial data:
10
12
8
15
9
11
7
13
14
Assume rook’s case adjacency for the weights matrix.
Solution:
Step 1: Calculate the mean (μ) of the values: μ = (10 + 12 + 8 + 15 + 9 + 11 + 7 + 13 + 14)
/ 9 = 11
Step 2: Create the weights matrix (W) using rook’s case adjacency:
𝑊=
[
0 1 0 1 0 0 0 0 0
1 0 1 0 1 0 0 0 0
0 1 0 0 0 1 0 0 0
1 0 0 0 1 0 1 0 0
0 1 0 1 0 1 0 1 0
0 0 1 0 1 0 0 0 1
0 0 0 1 0 0 0 1 0
0 0 0 0 1 0 1 0 1
0 0 0 0 0 1 0 1 0
]
Step 3: Calculate the numerator of Moran’s I:
𝑁=∑∑𝑤𝑖𝑗
𝑛
𝑗=1
𝑛
𝑖=1 (𝑥𝑖−𝜇)(𝑥𝑗−𝜇)
Step 4: Calculate the denominator of Moran’s I:
𝐷=∑(𝑥𝑖−𝜇)2
𝑛
𝑖=1
Step 5: Calculate Moran’s I:
𝐼=𝑛
𝑊⋅𝑁
𝐷
where n is the number of observations (9) and W is the sum of all weights (24).
Step 6: Interpret the result:
• I > 0: Positive spatial autocorrelation
• I < 0: Negative spatial autocorrelation
• I ≈ 0: No spatial autocorrelation
49 PROBLEM 4: TREND SURFACE ANALYSIS
Question: Perform a first-order trend surface analysis on the following data:
X
Y
Z
1
1
10
1
2
12
2
1
15
2
2
18
Solution:
Step 1: Set up the equation for a first-order trend surface:
𝑍=𝑏0+𝑏1𝑋+𝑏2𝑌
Step 2: Create the design matrix:
𝑋=[1 1 1
1 1 2
1 2 1
1 2 2]
Step 3: Set up the normal equations: (𝑋𝑇𝑋)𝑏=𝑋𝑇𝑍
Step 4: Solve for the coefficients (b0, b1, b2):
𝑏=(𝑋𝑇𝑋)−1𝑋𝑇𝑍
Step 5: Interpret the results:
• b0: Intercept
• b1: Coefficient for X (east-west trend)
• b2: Coefficient for Y (north-south trend)
Step 6: Write the equation of the fitted trend surface:
𝑍=𝑏0+𝑏1𝑋+𝑏2𝑌
50 PROBLEM 5: SPATIAL INTERPOLATION
Question: Using inverse distance weighting (IDW) with a power of 2, estimate the value at
point (2.5, 2.5) given the following data points:
(1, 1): 10 (1, 4): 15 (4, 1): 12 (4, 4): 18
Solution:
Step 1: Calculate the distances from the estimation point to each data point:
• d1 = sqrt((2.5-1)^2 + (2.5-1)^2) = 2.12
• d2 = sqrt((2.5-1)^2 + (2.5-4)^2) = 2.69
• d3 = sqrt((2.5-4)^2 + (2.5-1)^2) = 2.69
• d4 = sqrt((2.5-4)^2 + (2.5-4)^2) = 2.12
Step 2: Calculate the weights using the inverse distance formula:
𝑤𝑖=1
𝑑𝑖2
• w1 = 1 / 2.12^2 = 0.222
• w2 = 1 / 2.69^2 = 0.138
• w3 = 1 / 2.69^2 = 0.138
• w4 = 1 / 2.12^2 = 0.222
Step 3: Normalize the weights: 𝑤𝑖′= 𝑤𝑖
∑𝑤𝑗
𝑛
𝑗=1
Step 4: Calculate the estimated value:
𝑍∗=∑𝑤𝑖
𝑛
𝑖=1 ′𝑍𝑖
where Z_i are the known values at the data points.
Step 5: Interpret the result and discuss the limitations of IDW interpolation.
51 PROBLEM 6: VARIOGRAM CLOUD ANALYSIS
Question: Given the following spatial data:
X
Y
Z
1
1
10
2
1
12
1
2
15
2
2
18
3
3
20
Create a variogram cloud and interpret the results.
Solution:
Step 1: Calculate the distances and squared differences for all pairs of points:
For each pair (i, j):
• Distance: h_ij = sqrt((x_i - x_j)^2 + (y_i - y_j)^2)
• Squared difference: γ_ij = (z_i - z_j)^2 / 2
Step 2: Create a scatter plot with distance (h) on the x-axis and semivariance (γ) on the y-axis.
Step 3: Analyze the variogram cloud:
• Look for trends or patterns in the distribution of points
• Identify any outliers or anomalies
• Observe the overall shape and spread of the cloud
Step 4: Interpret the results:
• Discuss the presence or absence of spatial correlation
• Comment on the variability of the data at different distances
• Identify any potential anisotropy or directional effects
Step 5: Consider next steps, such as binning the data to create an experimental variogram.
52 PROBLEM 7: CROSS-VALIDATION FOR KRIGING
Question: Perform leave-one-out cross-validation for ordinary kriging using the following data:
X
Y
Z
1
1
10
1
3
15
3
1
12
3
3
18
2
2
14
Assume an exponential variogram model with a sill of 20 and a range of 2.5.
Solution:
Step 1: For each data point, remove it from the dataset and estimate its value using ordinary
kriging with the remaining points.
Step 2: Calculate the kriging equations for each iteration:
[𝛾(ℎ11)⋯ 𝛾(ℎ1𝑛)1
⋮ ⋱ ⋮ ⋮
𝛾(ℎ𝑛1)⋯ 𝛾(ℎ𝑛𝑛)1
1 ⋯ 1 0][𝑤1
⋮
𝑤𝑛
𝜇]=[𝛾(ℎ10)
⋮
𝛾(ℎ𝑛0)
1]
where γ(h) is the exponential variogram model:
𝛾(ℎ)=20(1−𝑒−3ℎ
2.5)
Step 3: Solve the kriging equations for each iteration to obtain the weights and Lagrange
multiplier.
Step 4: Calculate the estimated value for each removed point:
𝑍𝑖∗=∑𝑤𝑗
𝑛−1
𝑗=1 𝑍𝑗
Step 5: Calculate the cross-validation statistics:
• Mean Error (ME): 𝑀𝐸=1
𝑛∑(𝑍𝑖∗−𝑍𝑖)
𝑛
𝑖=1
• Mean Squared Error (MSE): 𝑀𝑆𝐸=1
𝑛∑(𝑍𝑖∗−𝑍𝑖)2
𝑛
𝑖=1
• Root Mean Squared Error (RMSE): 𝑅𝑀𝑆𝐸=√𝑀𝑆𝐸
• Mean Standardized Error (MSE): 𝑀𝑆𝐸=1
𝑛∑𝑍𝑖∗−𝑍𝑖
𝜎𝑖
𝑛
𝑖=1
• Root Mean Squared Standardized Error (RMSSE): 𝑅𝑀𝑆𝑆𝐸=√1
𝑛∑(𝑍𝑖∗−𝑍𝑖
𝜎𝑖)2
𝑛
𝑖=1
Step 6: Interpret the results:
• ME should be close to 0 for unbiased estimation
• RMSE should be as small as possible
• MSE should be close to 0
• RMSSE should be close to 1
Step 7: Based on the cross-validation results, discuss the performance of the kriging model and
suggest potential improvements.
53 PROBLEM 8: INDICATOR KRIGING
Question: Perform indicator kriging to estimate the probability that the value at location (2.5,
2.5) exceeds a threshold of 15, given the following data:
X
Y
Z
1
1
10
1
4
18
4
1
12
4
4
20
Assume a spherical variogram model for the indicator variable with a sill of 0.25 and a range of
3.
Solution:
Step 1: Transform the data into indicator variables:
𝐼(𝑥)={1if 𝑍(𝑥)>15
0if 𝑍(𝑥)≤15
Step 2: Calculate the distances between the estimation point and the data points:
• d1 = sqrt((2.5-1)^2 + (2.5-1)^2) = 2.12
• d2 = sqrt((2.5-1)^2 + (2.5-4)^2) = 2.69
• d3 = sqrt((2.5-4)^2 + (2.5-1)^2) = 2.69
• d4 = sqrt((2.5-4)^2 + (2.5-4)^2) = 2.12
Step 3: Calculate the semivariances using the spherical model:
𝛾(ℎ)={0.25[3ℎ
2(3)−1
2(ℎ
3)3]for ℎ≤3
0.25 for ℎ>3
Step 4: Set up the kriging system:
[
𝛾(𝑑11)𝛾(𝑑12)𝛾(𝑑13)𝛾(𝑑14)1
𝛾(𝑑21)𝛾(𝑑22)𝛾(𝑑23)𝛾(𝑑24)1
𝛾(𝑑31)𝛾(𝑑32)𝛾(𝑑33)𝛾(𝑑34)1
𝛾(𝑑41)𝛾(𝑑42)𝛾(𝑑43)𝛾(𝑑44)1
1 1 1 1 0
]
[
𝑤1
𝑤2
𝑤3
𝑤4
𝜇
]
=
[
𝛾(𝑑1)
𝛾(𝑑2)
𝛾(𝑑3)
𝛾(𝑑4)
1
]
Step 5: Solve for the weights (w1, w2, w3, w4) and Lagrange multiplier (μ).
Step 6: Calculate the estimated probability:
𝑃∗=𝑤1𝐼(𝑥1)+𝑤2𝐼(𝑥2)+𝑤3𝐼(𝑥3)+𝑤4𝐼(𝑥4)
Step 7: Interpret the result as the probability that the value at (2.5, 2.5) exceeds the threshold
of 15.
54 PROBLEM 9: BAYESIAN GEOSTATISTICS
Question: Using a Bayesian approach, estimate the mean and variance of a spatial random
field given the following observations:
X
Y
Z
1
1
10
1
2
12
2
1
15
2
2
18
Assume a Gaussian prior with mean μ_0 = 13 and variance σ_0^2 = 16, and a measurement
error variance of σ_ε^2 = 1.
Solution:
Step 1: Define the prior distribution: 𝜇∼𝑁(𝜇0,𝜎02)
Step 2: Define the likelihood function: 𝑍𝑖|𝜇∼𝑁(𝜇,𝜎𝜀2)
Step 3: Calculate the sample mean and variance:
𝑍‾=1
𝑛∑𝑍𝑖
𝑛
𝑖=1
𝑠2=1
𝑛−1∑(𝑍𝑖−𝑍‾)2
𝑛
𝑖=1
Step 4: Calculate the posterior mean:
𝜇𝑛=𝑛
𝜎𝜀2𝑍‾+1
𝜎02𝜇0
𝑛
𝜎𝜀2+1
𝜎02
Step 5: Calculate the posterior variance:
𝜎𝑛2=1
𝑛
𝜎𝜀2+1
𝜎02
Step 6: Calculate the 95
𝐶𝐼95%=𝜇𝑛±1.96√𝜎𝑛2
Step 7: Interpret the results:
• Discuss the posterior mean and how it compares to the prior mean and sample mean
• Explain the meaning of the posterior variance and credible interval
• Comment on the influence of the prior and the data on the posterior distribution
55 PROBLEM 10: MULTIVARIATE GEOSTATISTICS
Question: Perform a linear model of coregionalization (LMC) for two variables, Z1 and Z2,
given the following cross-variogram and direct variogram models:
• γ11(h) = 10 Sph(h/100) + 5 Exp(h/50)
• γ22(h) = 15 Sph(h/100) + 8 Exp(h/50)
• γ12(h) = 8 Sph(h/100) + 4 Exp(h/50)
Where Sph and Exp denote spherical and exponential models, respectively.
Solution:
Step 1: Identify the basic structures and their ranges:
• Structure 1: Spherical model with range a1 = 100
• Structure 2: Exponential model with range a2 = 50
Step 2: Set up the coregionalization matrices for each structure:
For the spherical model:
𝐵1=[𝑏11
(1)𝑏12
(1)
𝑏21
(1)𝑏22
(1)]
For the exponential model:
𝐵2=[𝑏11
(2)𝑏12
(2)
𝑏21
(2)𝑏22
(2)]
Step 3: Solve for the elements of B1 and B2 using the sill values:
• b11^(1) = 10, b22^(1) = 15, b12^(1) = b21^(1) = 8
• b11^(2) = 5, b22^(2) = 8, b12^(2) = b21^(2) = 4
Step 4: Verify that B1 and B2 are positive semi-definite matrices:
• Calculate the determinants: det(B1) ≥ 0 and det(B2) ≥ 0
• Check that the diagonal elements are non-negative
Step 5: Write the final LMC model:
[𝛾11(ℎ)𝛾12(ℎ)
𝛾21(ℎ)𝛾22(ℎ)]=𝐵1Sph(ℎ/100)+𝐵2Exp(ℎ/50)
Step 6: Interpret the results:
• Discuss the spatial correlation structure for each variable
• Explain the cross-correlation between the variables
• Comment on the different ranges and their implications for spatial prediction
Step 7: Discuss potential applications of the LMC model, such as cokriging for improved spatial
prediction.
Given the following paired data points (distance, semivariance), calculate the experimental
variogram and fit a spherical model to it:
(10, 2.1), (20, 3.5), (30, 4.2), (40, 4.8), (50, 5.1), (60, 5.3)
Solution:
Step 1: Plot the experimental variogram using the given data points.
Step 2: Estimate the sill, range, and nugget effect from the plot:
• Sill (C): Approximately 5.3
• Range (a): Approximately 50
• Nugget effect (C0): Approximately 0
Step 3: Use the spherical model equation:
𝛾(ℎ)={𝐶0+𝐶[3ℎ
2𝑎−1
2(ℎ
𝑎)3]for ℎ≤𝑎
𝐶0+𝐶 for ℎ>𝑎
Step 4: Substitute the estimated values into the equation:
𝛾(ℎ)={5.3[3ℎ
100−1
2(ℎ
50)3]for ℎ≤50
5.3 for ℎ>50
Step 5: Plot the fitted model alongside the experimental variogram to assess the fit.
56 PROBLEM 2: KRIGING ESTIMATION
Question: Using simple kriging, estimate the value at an unsampled location (5, 5) given the
following data points and their values:
(0, 0): 10 (0, 10): 15 (10, 0): 12 (10, 10): 18
Assume an exponential variogram model with a sill of 20 and a range of 8.
Solution:
Step 1: Calculate the distances between the unsampled location and the data points:
• d1 = sqrt((5-0)^2 + (5-0)^2) = 7.07
• d2 = sqrt((5-0)^2 + (5-10)^2) = 7.07
• d3 = sqrt((5-10)^2 + (5-0)^2) = 7.07
• d4 = sqrt((5-10)^2 + (5-10)^2) = 7.07
Step 2: Calculate the semivariances using the exponential model:
𝛾(ℎ)=20(1−𝑒−3ℎ
8)
• γ1 = γ2 = γ3 = γ4 = 20 * (1 - exp(-3 * 7.07 / 8)) = 15.8
Step 3: Set up the kriging system:
[015.8 15.8 15.8
15.8015.8 15.8
15.8 15.8 0 15.8
15.8 15.8 15.8 0 ][𝑤1
𝑤2
𝑤3
𝑤4]=[15.8
15.8
15.8
15.8]
Step 4: Solve for the weights (w1, w2, w3, w4).
Step 5: Calculate the estimated value:
𝑍∗=𝑤1∗10+𝑤2∗15+𝑤3∗12+𝑤4∗18
57 PROBLEM 3: SPATIAL AUTOCORRELATION
Question: Calculate Moran’s I for the following spatial data:
10
12
8
15
9
11
7
13
14
Assume rook’s case adjacency for the weights matrix.
Solution:
Step 1: Calculate the mean (μ) of the values: μ = (10 + 12 + 8 + 15 + 9 + 11 + 7 + 13 + 14)
/ 9 = 11
Step 2: Create the weights matrix (W) using rook’s case adjacency:
𝑊=
[
0 1 0 1 0 0 0 0 0
1 0 1 0 1 0 0 0 0
0 1 0 0 0 1 0 0 0
1 0 0 0 1 0 1 0 0
0 1 0 1 0 1 0 1 0
0 0 1 0 1 0 0 0 1
0 0 0 1 0 0 0 1 0
0 0 0 0 1 0 1 0 1
0 0 0 0 0 1 0 1 0
]
Step 3: Calculate the numerator of Moran’s I:
𝑁=∑∑𝑤𝑖𝑗
𝑛
𝑗=1
𝑛
𝑖=1 (𝑥𝑖−𝜇)(𝑥𝑗−𝜇)
Step 4: Calculate the denominator of Moran’s I:
𝐷=∑(𝑥𝑖−𝜇)2
𝑛
𝑖=1
Step 5: Calculate Moran’s I:
𝐼=𝑛
𝑊⋅𝑁
𝐷
where n is the number of observations (9) and W is the sum of all weights (24).
Step 6: Interpret the result:
• I > 0: Positive spatial autocorrelation
• I < 0: Negative spatial autocorrelation
• I ≈ 0: No spatial autocorrelation
58 PROBLEM 4: TREND SURFACE ANALYSIS
Question: Perform a first-order trend surface analysis on the following data:
X
Y
Z
1
1
10
1
2
12
2
1
15
2
2
18
Solution:
Step 1: Set up the equation for a first-order trend surface:
𝑍=𝑏0+𝑏1𝑋+𝑏2𝑌
Step 2: Create the design matrix:
𝑋=[1 1 1
1 1 2
1 2 1
1 2 2]
Step 3: Set up the normal equations: (𝑋𝑇𝑋)𝑏=𝑋𝑇𝑍
Step 4: Solve for the coefficients (b0, b1, b2):
𝑏=(𝑋𝑇𝑋)−1𝑋𝑇𝑍
Step 5: Interpret the results:
• b0: Intercept
• b1: Coefficient for X (east-west trend)
• b2: Coefficient for Y (north-south trend)
Step 6: Write the equation of the fitted trend surface:
𝑍=𝑏0+𝑏1𝑋+𝑏2𝑌
59 PROBLEM 5: SPATIAL INTERPOLATION
Question: Using inverse distance weighting (IDW) with a power of 2, estimate the value at
point (2.5, 2.5) given the following data points:
(1, 1): 10 (1, 4): 15 (4, 1): 12 (4, 4): 18
Solution:
Step 1: Calculate the distances from the estimation point to each data point:
• d1 = sqrt((2.5-1)^2 + (2.5-1)^2) = 2.12
• d2 = sqrt((2.5-1)^2 + (2.5-4)^2) = 2.69
• d3 = sqrt((2.5-4)^2 + (2.5-1)^2) = 2.69
• d4 = sqrt((2.5-4)^2 + (2.5-4)^2) = 2.12
Step 2: Calculate the weights using the inverse distance formula:
𝑤𝑖=1
𝑑𝑖2
• w1 = 1 / 2.12^2 = 0.222
• w2 = 1 / 2.69^2 = 0.138
• w3 = 1 / 2.69^2 = 0.138
• w4 = 1 / 2.12^2 = 0.222
Step 3: Normalize the weights: 𝑤𝑖′= 𝑤𝑖
∑𝑤𝑗
𝑛
𝑗=1
Step 4: Calculate the estimated value:
𝑍∗=∑𝑤𝑖
𝑛
𝑖=1 ′𝑍𝑖
where Z_i are the known values at the data points.
Step 5: Interpret the result and discuss the limitations of IDW interpolation.
60 PROBLEM 6: VARIOGRAM CLOUD ANALYSIS
Question: Given the following spatial data:
X
Y
Z
1
1
10
2
1
12
1
2
15
2
2
18
3
3
20
Create a variogram cloud and interpret the results.
Solution:
Step 1: Calculate the distances and squared differences for all pairs of points:
For each pair (i, j):
• Distance: h_ij = sqrt((x_i - x_j)^2 + (y_i - y_j)^2)
• Squared difference: γ_ij = (z_i - z_j)^2 / 2
Step 2: Create a scatter plot with distance (h) on the x-axis and semivariance (γ) on the y-axis.
Step 3: Analyze the variogram cloud:
• Look for trends or patterns in the distribution of points
• Identify any outliers or anomalies
• Observe the overall shape and spread of the cloud
Step 4: Interpret the results:
• Discuss the presence or absence of spatial correlation
• Comment on the variability of the data at different distances
• Identify any potential anisotropy or directional effects
Step 5: Consider next steps, such as binning the data to create an experimental variogram.
61 PROBLEM 7: CROSS-VALIDATION FOR KRIGING
Question: Perform leave-one-out cross-validation for ordinary kriging using the following data:
X
Y
Z
1
1
10
1
3
15
3
1
12
3
3
18
2
2
14
Assume an exponential variogram model with a sill of 20 and a range of 2.5.
Solution:
Step 1: For each data point, remove it from the dataset and estimate its value using ordinary
kriging with the remaining points.
Step 2: Calculate the kriging equations for each iteration:
[𝛾(ℎ11)⋯ 𝛾(ℎ1𝑛)1
⋮ ⋱ ⋮ ⋮
𝛾(ℎ𝑛1)⋯ 𝛾(ℎ𝑛𝑛)1
1 ⋯ 1 0][𝑤1
⋮
𝑤𝑛
𝜇]=[𝛾(ℎ10)
⋮
𝛾(ℎ𝑛0)
1]
where γ(h) is the exponential variogram model:
𝛾(ℎ)=20(1−𝑒−3ℎ
2.5)
Step 3: Solve the kriging equations for each iteration to obtain the weights and Lagrange
multiplier.
Step 4: Calculate the estimated value for each removed point:
𝑍𝑖∗=∑𝑤𝑗
𝑛−1
𝑗=1 𝑍𝑗
Step 5: Calculate the cross-validation statistics:
• Mean Error (ME): 𝑀𝐸=1
𝑛∑(𝑍𝑖∗−𝑍𝑖)
𝑛
𝑖=1
• Mean Squared Error (MSE): 𝑀𝑆𝐸=1
𝑛∑(𝑍𝑖∗−𝑍𝑖)2
𝑛
𝑖=1
• Root Mean Squared Error (RMSE): 𝑅𝑀𝑆𝐸=√𝑀𝑆𝐸
• Mean Standardized Error (MSE): 𝑀𝑆𝐸=1
𝑛∑𝑍𝑖∗−𝑍𝑖
𝜎𝑖
𝑛
𝑖=1
• Root Mean Squared Standardized Error (RMSSE): 𝑅𝑀𝑆𝑆𝐸=√1
𝑛∑(𝑍𝑖∗−𝑍𝑖
𝜎𝑖)2
𝑛
𝑖=1
Step 6: Interpret the results:
• ME should be close to 0 for unbiased estimation
• RMSE should be as small as possible
• MSE should be close to 0
• RMSSE should be close to 1
Step 7: Based on the cross-validation results, discuss the performance of the kriging model and
suggest potential improvements.
62 PROBLEM 8: INDICATOR KRIGING
Question: Perform indicator kriging to estimate the probability that the value at location (2.5,
2.5) exceeds a threshold of 15, given the following data:
X
Y
Z
1
1
10
1
4
18
4
1
12
4
4
20
Assume a spherical variogram model for the indicator variable with a sill of 0.25 and a range of
3.
Solution:
Step 1: Transform the data into indicator variables:
𝐼(𝑥)={1if 𝑍(𝑥)>15
0if 𝑍(𝑥)≤15
Step 2: Calculate the distances between the estimation point and the data points:
• d1 = sqrt((2.5-1)^2 + (2.5-1)^2) = 2.12
• d2 = sqrt((2.5-1)^2 + (2.5-4)^2) = 2.69
• d3 = sqrt((2.5-4)^2 + (2.5-1)^2) = 2.69
• d4 = sqrt((2.5-4)^2 + (2.5-4)^2) = 2.12
Step 3: Calculate the semivariances using the spherical model:
𝛾(ℎ)={0.25[3ℎ
2(3)−1
2(ℎ
3)3]for ℎ≤3
0.25 for ℎ>3
Step 4: Set up the kriging system:
[
𝛾(𝑑11)𝛾(𝑑12)𝛾(𝑑13)𝛾(𝑑14)1
𝛾(𝑑21)𝛾(𝑑22)𝛾(𝑑23)𝛾(𝑑24)1
𝛾(𝑑31)𝛾(𝑑32)𝛾(𝑑33)𝛾(𝑑34)1
𝛾(𝑑41)𝛾(𝑑42)𝛾(𝑑43)𝛾(𝑑44)1
1 1 1 1 0
]
[
𝑤1
𝑤2
𝑤3
𝑤4
𝜇
]
=
[
𝛾(𝑑1)
𝛾(𝑑2)
𝛾(𝑑3)
𝛾(𝑑4)
1
]
Step 5: Solve for the weights (w1, w2, w3, w4) and Lagrange multiplier (μ).
Step 6: Calculate the estimated probability:
𝑃∗=𝑤1𝐼(𝑥1)+𝑤2𝐼(𝑥2)+𝑤3𝐼(𝑥3)+𝑤4𝐼(𝑥4)
Step 7: Interpret the result as the probability that the value at (2.5, 2.5) exceeds the threshold
of 15.
63 PROBLEM 9: BAYESIAN GEOSTATISTICS
Question: Using a Bayesian approach, estimate the mean and variance of a spatial random
field given the following observations:
X
Y
Z
1
1
10
1
2
12
2
1
15
2
2
18
Assume a Gaussian prior with mean μ_0 = 13 and variance σ_0^2 = 16, and a measurement
error variance of σ_ε^2 = 1.
Solution:
Step 1: Define the prior distribution: 𝜇∼𝑁(𝜇0,𝜎02)
Step 2: Define the likelihood function: 𝑍𝑖|𝜇∼𝑁(𝜇,𝜎𝜀2)
Step 3: Calculate the sample mean and variance:
𝑍‾=1
𝑛∑𝑍𝑖
𝑛
𝑖=1
𝑠2=1
𝑛−1∑(𝑍𝑖−𝑍‾)2
𝑛
𝑖=1
Step 4: Calculate the posterior mean:
𝜇𝑛=𝑛
𝜎𝜀2𝑍‾+1
𝜎02𝜇0
𝑛
𝜎𝜀2+1
𝜎02
Step 5: Calculate the posterior variance:
𝜎𝑛2=1
𝑛
𝜎𝜀2+1
𝜎02
Step 6: Calculate the 95
𝐶𝐼95%=𝜇𝑛±1.96√𝜎𝑛2
Step 7: Interpret the results:
• Discuss the posterior mean and how it compares to the prior mean and sample mean
• Explain the meaning of the posterior variance and credible interval
• Comment on the influence of the prior and the data on the posterior distribution
64 PROBLEM 10: MULTIVARIATE GEOSTATISTICS
Question: Perform a linear model of coregionalization (LMC) for two variables, Z1 and Z2,
given the following cross-variogram and direct variogram models:
• γ11(h) = 10 Sph(h/100) + 5 Exp(h/50)
• γ22(h) = 15 Sph(h/100) + 8 Exp(h/50)
• γ12(h) = 8 Sph(h/100) + 4 Exp(h/50)
Where Sph and Exp denote spherical and exponential models, respectively.
Solution:
Step 1: Identify the basic structures and their ranges:
• Structure 1: Spherical model with range a1 = 100
• Structure 2: Exponential model with range a2 = 50
Step 2: Set up the coregionalization matrices for each structure:
For the spherical model:
𝐵1=[𝑏11
(1)𝑏12
(1)
𝑏21
(1)𝑏22
(1)]
For the exponential model:
𝐵2=[𝑏11
(2)𝑏12
(2)
𝑏21
(2)𝑏22
(2)]
Step 3: Solve for the elements of B1 and B2 using the sill values:
• b11^(1) = 10, b22^(1) = 15, b12^(1) = b21^(1) = 8
• b11^(2) = 5, b22^(2) = 8, b12^(2) = b21^(2) = 4
Step 4: Verify that B1 and B2 are positive semi-definite matrices:
• Calculate the determinants: det(B1) ≥ 0 and det(B2) ≥ 0
• Check that the diagonal elements are non-negative
Step 5: Write the final LMC model:
[𝛾11(ℎ)𝛾12(ℎ)
𝛾21(ℎ)𝛾22(ℎ)]=𝐵1Sph(ℎ/100)+𝐵2Exp(ℎ/50)
Step 6: Interpret the results:
• Discuss the spatial correlation structure for each variable
• Explain the cross-correlation between the variables
• Comment on the different ranges and their implications for spatial prediction
Step 7: Discuss potential applications of the LMC model, such as cokriging for improved spatial
prediction.
Given the following paired data points (distance, semivariance), calculate the experimental
variogram and fit a spherical model to it:
(10, 2.1), (20, 3.5), (30, 4.2), (40, 4.8), (50, 5.1), (60, 5.3)
Solution:
Step 1: Plot the experimental variogram using the given data points.
Step 2: Estimate the sill, range, and nugget effect from the plot:
• Sill (C): Approximately 5.3
• Range (a): Approximately 50
• Nugget effect (C0): Approximately 0
Step 3: Use the spherical model equation:
𝛾(ℎ)={𝐶0+𝐶[3ℎ
2𝑎−1
2(ℎ
𝑎)3]for ℎ≤𝑎
𝐶0+𝐶 for ℎ>𝑎
Step 4: Substitute the estimated values into the equation:
𝛾(ℎ)={5.3[3ℎ
100−1
2(ℎ
50)3]for ℎ≤50
5.3 for ℎ>50
Step 5: Plot the fitted model alongside the experimental variogram to assess the fit.
65 PROBLEM 2: KRIGING ESTIMATION
Question: Using simple kriging, estimate the value at an unsampled location (5, 5) given the
following data points and their values:
(0, 0): 10 (0, 10): 15 (10, 0): 12 (10, 10): 18
Assume an exponential variogram model with a sill of 20 and a range of 8.
Solution:
Step 1: Calculate the distances between the unsampled location and the data points:
• d1 = sqrt((5-0)^2 + (5-0)^2) = 7.07
• d2 = sqrt((5-0)^2 + (5-10)^2) = 7.07
• d3 = sqrt((5-10)^2 + (5-0)^2) = 7.07
• d4 = sqrt((5-10)^2 + (5-10)^2) = 7.07
Step 2: Calculate the semivariances using the exponential model:
𝛾(ℎ)=20(1−𝑒−3ℎ
8)
• γ1 = γ2 = γ3 = γ4 = 20 * (1 - exp(-3 * 7.07 / 8)) = 15.8
Step 3: Set up the kriging system:
[015.8 15.8 15.8
15.8015.8 15.8
15.8 15.8 0 15.8
15.8 15.8 15.8 0 ][𝑤1
𝑤2
𝑤3
𝑤4]=[15.8
15.8
15.8
15.8]
Step 4: Solve for the weights (w1, w2, w3, w4).
Step 5: Calculate the estimated value:
𝑍∗=𝑤1∗10+𝑤2∗15+𝑤3∗12+𝑤4∗18
66 PROBLEM 3: SPATIAL AUTOCORRELATION
Question: Calculate Moran’s I for the following spatial data:
10
12
8
15
9
11
7
13
14
Assume rook’s case adjacency for the weights matrix.
Solution:
Step 1: Calculate the mean (μ) of the values: μ = (10 + 12 + 8 + 15 + 9 + 11 + 7 + 13 + 14)
/ 9 = 11
Step 2: Create the weights matrix (W) using rook’s case adjacency:
𝑊=
[
0 1 0 1 0 0 0 0 0
1 0 1 0 1 0 0 0 0
0 1 0 0 0 1 0 0 0
1 0 0 0 1 0 1 0 0
0 1 0 1 0 1 0 1 0
0 0 1 0 1 0 0 0 1
0 0 0 1 0 0 0 1 0
0 0 0 0 1 0 1 0 1
0 0 0 0 0 1 0 1 0
]
Step 3: Calculate the numerator of Moran’s I:
𝑁=∑∑𝑤𝑖𝑗
𝑛
𝑗=1
𝑛
𝑖=1 (𝑥𝑖−𝜇)(𝑥𝑗−𝜇)
Step 4: Calculate the denominator of Moran’s I:
𝐷=∑(𝑥𝑖−𝜇)2
𝑛
𝑖=1
Step 5: Calculate Moran’s I:
𝐼=𝑛
𝑊⋅𝑁
𝐷
where n is the number of observations (9) and W is the sum of all weights (24).
Step 6: Interpret the result:
• I > 0: Positive spatial autocorrelation
• I < 0: Negative spatial autocorrelation
• I ≈ 0: No spatial autocorrelation
67 PROBLEM 4: TREND SURFACE ANALYSIS
Question: Perform a first-order trend surface analysis on the following data:
X
Y
Z
1
1
10
1
2
12
2
1
15
2
2
18
Solution:
Step 1: Set up the equation for a first-order trend surface:
𝑍=𝑏0+𝑏1𝑋+𝑏2𝑌
Step 2: Create the design matrix:
𝑋=[1 1 1
1 1 2
1 2 1
1 2 2]
Step 3: Set up the normal equations: (𝑋𝑇𝑋)𝑏=𝑋𝑇𝑍
Step 4: Solve for the coefficients (b0, b1, b2):
𝑏=(𝑋𝑇𝑋)−1𝑋𝑇𝑍
Step 5: Interpret the results:
• b0: Intercept
• b1: Coefficient for X (east-west trend)
• b2: Coefficient for Y (north-south trend)
Step 6: Write the equation of the fitted trend surface:
𝑍=𝑏0+𝑏1𝑋+𝑏2𝑌
68 PROBLEM 5: SPATIAL INTERPOLATION
Question: Using inverse distance weighting (IDW) with a power of 2, estimate the value at
point (2.5, 2.5) given the following data points:
(1, 1): 10 (1, 4): 15 (4, 1): 12 (4, 4): 18
Solution:
Step 1: Calculate the distances from the estimation point to each data point:
• d1 = sqrt((2.5-1)^2 + (2.5-1)^2) = 2.12
• d2 = sqrt((2.5-1)^2 + (2.5-4)^2) = 2.69
• d3 = sqrt((2.5-4)^2 + (2.5-1)^2) = 2.69
• d4 = sqrt((2.5-4)^2 + (2.5-4)^2) = 2.12
Step 2: Calculate the weights using the inverse distance formula:
𝑤𝑖=1
𝑑𝑖2
• w1 = 1 / 2.12^2 = 0.222
• w2 = 1 / 2.69^2 = 0.138
• w3 = 1 / 2.69^2 = 0.138
• w4 = 1 / 2.12^2 = 0.222
Step 3: Normalize the weights: 𝑤𝑖′= 𝑤𝑖
∑𝑤𝑗
𝑛
𝑗=1
Step 4: Calculate the estimated value:
𝑍∗=∑𝑤𝑖
𝑛
𝑖=1 ′𝑍𝑖
where Z_i are the known values at the data points.
Step 5: Interpret the result and discuss the limitations of IDW interpolation.
69 PROBLEM 6: VARIOGRAM CLOUD ANALYSIS
Question: Given the following spatial data:
X
Y
Z
1
1
10
2
1
12
1
2
15
2
2
18
3
3
20
Create a variogram cloud and interpret the results.
Solution:
Step 1: Calculate the distances and squared differences for all pairs of points:
For each pair (i, j):
• Distance: h_ij = sqrt((x_i - x_j)^2 + (y_i - y_j)^2)
• Squared difference: γ_ij = (z_i - z_j)^2 / 2
Step 2: Create a scatter plot with distance (h) on the x-axis and semivariance (γ) on the y-axis.
Step 3: Analyze the variogram cloud:
• Look for trends or patterns in the distribution of points
• Identify any outliers or anomalies
• Observe the overall shape and spread of the cloud
Step 4: Interpret the results:
• Discuss the presence or absence of spatial correlation
• Comment on the variability of the data at different distances
• Identify any potential anisotropy or directional effects
Step 5: Consider next steps, such as binning the data to create an experimental variogram.
70 PROBLEM 7: CROSS-VALIDATION FOR KRIGING
Question: Perform leave-one-out cross-validation for ordinary kriging using the following data:
X
Y
Z
1
1
10
1
3
15
X
Y
Z
3
1
12
3
3
18
2
2
14
Assume an exponential variogram model with a sill of 20 and a range of 2.5.
Solution:
Step 1: For each data point, remove it from the dataset and estimate its value using ordinary
kriging with the remaining points.
Step 2: Calculate the kriging equations for each iteration:
[𝛾(ℎ11)⋯ 𝛾(ℎ1𝑛)1
⋮ ⋱ ⋮ ⋮
𝛾(ℎ𝑛1)⋯ 𝛾(ℎ𝑛𝑛)1
1 ⋯ 1 0][𝑤1
⋮
𝑤𝑛
𝜇]=[𝛾(ℎ10)
⋮
𝛾(ℎ𝑛0)
1]
where γ(h) is the exponential variogram model:
𝛾(ℎ)=20(1−𝑒−3ℎ
2.5)
Step 3: Solve the kriging equations for each iteration to obtain the weights and Lagrange
multiplier.
Step 4: Calculate the estimated value for each removed point:
𝑍𝑖∗=∑𝑤𝑗
𝑛−1
𝑗=1 𝑍𝑗
Step 5: Calculate the cross-validation statistics:
• Mean Error (ME): 𝑀𝐸=1
𝑛∑(𝑍𝑖∗−𝑍𝑖)
𝑛
𝑖=1
• Mean Squared Error (MSE): 𝑀𝑆𝐸=1
𝑛∑(𝑍𝑖∗−𝑍𝑖)2
𝑛
𝑖=1
• Root Mean Squared Error (RMSE): 𝑅𝑀𝑆𝐸=√𝑀𝑆𝐸
• Mean Standardized Error (MSE): 𝑀𝑆𝐸=1
𝑛∑𝑍𝑖∗−𝑍𝑖
𝜎𝑖
𝑛
𝑖=1
• Root Mean Squared Standardized Error (RMSSE): 𝑅𝑀𝑆𝑆𝐸=√1
𝑛∑(𝑍𝑖∗−𝑍𝑖
𝜎𝑖)2
𝑛
𝑖=1
Step 6: Interpret the results:
• ME should be close to 0 for unbiased estimation
• RMSE should be as small as possible
• MSE should be close to 0
• RMSSE should be close to 1
Step 7: Based on the cross-validation results, discuss the performance of the kriging model and
suggest potential improvements.
71 PROBLEM 8: INDICATOR KRIGING
Question: Perform indicator kriging to estimate the probability that the value at location (2.5,
2.5) exceeds a threshold of 15, given the following data:
X
Y
Z
1
1
10
1
4
18
4
1
12
4
4
20
Assume a spherical variogram model for the indicator variable with a sill of 0.25 and a range of
3.
Solution:
Step 1: Transform the data into indicator variables:
𝐼(𝑥)={1if 𝑍(𝑥)>15
0if 𝑍(𝑥)≤15
Step 2: Calculate the distances between the estimation point and the data points:
• d1 = sqrt((2.5-1)^2 + (2.5-1)^2) = 2.12
• d2 = sqrt((2.5-1)^2 + (2.5-4)^2) = 2.69
• d3 = sqrt((2.5-4)^2 + (2.5-1)^2) = 2.69
• d4 = sqrt((2.5-4)^2 + (2.5-4)^2) = 2.12
Step 3: Calculate the semivariances using the spherical model:
𝛾(ℎ)={0.25[3ℎ
2(3)−1
2(ℎ
3)3]for ℎ≤3
0.25 for ℎ>3
Step 4: Set up the kriging system:
[
𝛾(𝑑11)𝛾(𝑑12)𝛾(𝑑13)𝛾(𝑑14)1
𝛾(𝑑21)𝛾(𝑑22)𝛾(𝑑23)𝛾(𝑑24)1
𝛾(𝑑31)𝛾(𝑑32)𝛾(𝑑33)𝛾(𝑑34)1
𝛾(𝑑41)𝛾(𝑑42)𝛾(𝑑43)𝛾(𝑑44)1
1 1 1 1 0
]
[
𝑤1
𝑤2
𝑤3
𝑤4
𝜇
]
=
[
𝛾(𝑑1)
𝛾(𝑑2)
𝛾(𝑑3)
𝛾(𝑑4)
1
]
Step 5: Solve for the weights (w1, w2, w3, w4) and Lagrange multiplier (μ).
Step 6: Calculate the estimated probability:
𝑃∗=𝑤1𝐼(𝑥1)+𝑤2𝐼(𝑥2)+𝑤3𝐼(𝑥3)+𝑤4𝐼(𝑥4)
Step 7: Interpret the result as the probability that the value at (2.5, 2.5) exceeds the threshold
of 15.
72 PROBLEM 9: BAYESIAN GEOSTATISTICS
Question: Using a Bayesian approach, estimate the mean and variance of a spatial random
field given the following observations:
X
Y
Z
1
1
10
1
2
12
2
1
15
2
2
18
Assume a Gaussian prior with mean μ_0 = 13 and variance σ_0^2 = 16, and a measurement
error variance of σ_ε^2 = 1.
Solution:
Step 1: Define the prior distribution: 𝜇∼𝑁(𝜇0,𝜎02)
Step 2: Define the likelihood function: 𝑍𝑖|𝜇∼𝑁(𝜇,𝜎𝜀2)
Step 3: Calculate the sample mean and variance:
𝑍‾=1
𝑛∑𝑍𝑖
𝑛
𝑖=1
𝑠2=1
𝑛−1∑(𝑍𝑖−𝑍‾)2
𝑛
𝑖=1
Step 4: Calculate the posterior mean:
𝜇𝑛=𝑛
𝜎𝜀2𝑍‾+1
𝜎02𝜇0
𝑛
𝜎𝜀2+1
𝜎02
Step 5: Calculate the posterior variance:
𝜎𝑛2=1
𝑛
𝜎𝜀2+1
𝜎02
Step 6: Calculate the 95
𝐶𝐼95%=𝜇𝑛±1.96√𝜎𝑛2
Step 7: Interpret the results:
• Discuss the posterior mean and how it compares to the prior mean and sample mean
• Explain the meaning of the posterior variance and credible interval
• Comment on the influence of the prior and the data on the posterior distribution
73 PROBLEM 10: MULTIVARIATE GEOSTATISTICS
Question: Perform a linear model of coregionalization (LMC) for two variables, Z1 and Z2,
given the following cross-variogram and direct variogram models:
• γ11(h) = 10 Sph(h/100) + 5 Exp(h/50)
• γ22(h) = 15 Sph(h/100) + 8 Exp(h/50)
• γ12(h) = 8 Sph(h/100) + 4 Exp(h/50)
Where Sph and Exp denote spherical and exponential models, respectively.
Solution:
Step 1: Identify the basic structures and their ranges:
• Structure 1: Spherical model with range a1 = 100
• Structure 2: Exponential model with range a2 = 50
Step 2: Set up the coregionalization matrices for each structure:
For the spherical model:
𝐵1=[𝑏11
(1)𝑏12
(1)
𝑏21
(1)𝑏22
(1)]
For the exponential model:
𝐵2=[𝑏11
(2)𝑏12
(2)
𝑏21
(2)𝑏22
(2)]
Step 3: Solve for the elements of B1 and B2 using the sill values:
• b11^(1) = 10, b22^(1) = 15, b12^(1) = b21^(1) = 8
• b11^(2) = 5, b22^(2) = 8, b12^(2) = b21^(2) = 4
Step 4: Verify that B1 and B2 are positive semi-definite matrices:
• Calculate the determinants: det(B1) ≥ 0 and det(B2) ≥ 0
• Check that the diagonal elements are non-negative
Step 5: Write the final LMC model:
[𝛾11(ℎ)𝛾12(ℎ)
𝛾21(ℎ)𝛾22(ℎ)]=𝐵1Sph(ℎ/100)+𝐵2Exp(ℎ/50)
Step 6: Interpret the results:
• Discuss the spatial correlation structure for each variable
• Explain the cross-correlation between the variables
• Comment on the different ranges and their implications for spatial prediction
Step 7: Discuss potential applications of the LMC model, such as cokriging for improved spatial
prediction.
Given the following paired data points (distance, semivariance), calculate the experimental
variogram and fit a spherical model to it:
(10, 2.1), (20, 3.5), (30, 4.2), (40, 4.8), (50, 5.1), (60, 5.3)
Solution:
Step 1: Plot the experimental variogram using the given data points.
Step 2: Estimate the sill, range, and nugget effect from the plot:
• Sill (C): Approximately 5.3
• Range (a): Approximately 50
• Nugget effect (C0): Approximately 0
Step 3: Use the spherical model equation:
𝛾(ℎ)={𝐶0+𝐶[3ℎ
2𝑎−1
2(ℎ
𝑎)3]for ℎ≤𝑎
𝐶0+𝐶 for ℎ>𝑎
Step 4: Substitute the estimated values into the equation:
𝛾(ℎ)={5.3[3ℎ
100−1
2(ℎ
50)3]for ℎ≤50
5.3 for ℎ>50
Step 5: Plot the fitted model alongside the experimental variogram to assess the fit.
74 PROBLEM 2: KRIGING ESTIMATION
Question: Using simple kriging, estimate the value at an unsampled location (5, 5) given the
following data points and their values:
(0, 0): 10 (0, 10): 15 (10, 0): 12 (10, 10): 18
Assume an exponential variogram model with a sill of 20 and a range of 8.
Solution:
Step 1: Calculate the distances between the unsampled location and the data points:
• d1 = sqrt((5-0)^2 + (5-0)^2) = 7.07
• d2 = sqrt((5-0)^2 + (5-10)^2) = 7.07
• d3 = sqrt((5-10)^2 + (5-0)^2) = 7.07
• d4 = sqrt((5-10)^2 + (5-10)^2) = 7.07
Step 2: Calculate the semivariances using the exponential model:
𝛾(ℎ)=20(1−𝑒−3ℎ
8)
• γ1 = γ2 = γ3 = γ4 = 20 * (1 - exp(-3 * 7.07 / 8)) = 15.8
Step 3: Set up the kriging system:
[015.8 15.8 15.8
15.8015.8 15.8
15.8 15.8 0 15.8
15.8 15.8 15.8 0 ][𝑤1
𝑤2
𝑤3
𝑤4]=[15.8
15.8
15.8
15.8]
Step 4: Solve for the weights (w1, w2, w3, w4).
Step 5: Calculate the estimated value:
𝑍∗=𝑤1∗10+𝑤2∗15+𝑤3∗12+𝑤4∗18
75 PROBLEM 3: SPATIAL AUTOCORRELATION
Question: Calculate Moran’s I for the following spatial data:
10
12
8
15
9
11
7
13
14
Assume rook’s case adjacency for the weights matrix.
Solution:
Step 1: Calculate the mean (μ) of the values: μ = (10 + 12 + 8 + 15 + 9 + 11 + 7 + 13 + 14)
/ 9 = 11
Step 2: Create the weights matrix (W) using rook’s case adjacency:
𝑊=
[
0 1 0 1 0 0 0 0 0
1 0 1 0 1 0 0 0 0
0 1 0 0 0 1 0 0 0
1 0 0 0 1 0 1 0 0
0 1 0 1 0 1 0 1 0
0 0 1 0 1 0 0 0 1
0 0 0 1 0 0 0 1 0
0 0 0 0 1 0 1 0 1
0 0 0 0 0 1 0 1 0
]
Step 3: Calculate the numerator of Moran’s I:
𝑁=∑∑𝑤𝑖𝑗
𝑛
𝑗=1
𝑛
𝑖=1 (𝑥𝑖−𝜇)(𝑥𝑗−𝜇)
Step 4: Calculate the denominator of Moran’s I:
𝐷=∑(𝑥𝑖−𝜇)2
𝑛
𝑖=1
Step 5: Calculate Moran’s I:
𝐼=𝑛
𝑊⋅𝑁
𝐷
where n is the number of observations (9) and W is the sum of all weights (24).
Step 6: Interpret the result:
• I > 0: Positive spatial autocorrelation
• I < 0: Negative spatial autocorrelation
• I ≈ 0: No spatial autocorrelation
76 PROBLEM 4: TREND SURFACE ANALYSIS
Question: Perform a first-order trend surface analysis on the following data:
X
Y
Z
1
1
10
1
2
12
2
1
15
2
2
18
Solution:
Step 1: Set up the equation for a first-order trend surface:
𝑍=𝑏0+𝑏1𝑋+𝑏2𝑌
Step 2: Create the design matrix:
𝑋=[1 1 1
1 1 2
1 2 1
1 2 2]
Step 3: Set up the normal equations: (𝑋𝑇𝑋)𝑏=𝑋𝑇𝑍
Step 4: Solve for the coefficients (b0, b1, b2):
𝑏=(𝑋𝑇𝑋)−1𝑋𝑇𝑍
Step 5: Interpret the results:
• b0: Intercept
• b1: Coefficient for X (east-west trend)
• b2: Coefficient for Y (north-south trend)
Step 6: Write the equation of the fitted trend surface:
𝑍=𝑏0+𝑏1𝑋+𝑏2𝑌
77 PROBLEM 5: SPATIAL INTERPOLATION
Question: Using inverse distance weighting (IDW) with a power of 2, estimate the value at
point (2.5, 2.5) given the following data points:
(1, 1): 10 (1, 4): 15 (4, 1): 12 (4, 4): 18
Solution:
Step 1: Calculate the distances from the estimation point to each data point:
• d1 = sqrt((2.5-1)^2 + (2.5-1)^2) = 2.12
• d2 = sqrt((2.5-1)^2 + (2.5-4)^2) = 2.69
• d3 = sqrt((2.5-4)^2 + (2.5-1)^2) = 2.69
• d4 = sqrt((2.5-4)^2 + (2.5-4)^2) = 2.12
Step 2: Calculate the weights using the inverse distance formula:
𝑤𝑖=1
𝑑𝑖2
• w1 = 1 / 2.12^2 = 0.222
• w2 = 1 / 2.69^2 = 0.138
• w3 = 1 / 2.69^2 = 0.138
• w4 = 1 / 2.12^2 = 0.222
Step 3: Normalize the weights: 𝑤𝑖′= 𝑤𝑖
∑𝑤𝑗
𝑛
𝑗=1
Step 4: Calculate the estimated value:
𝑍∗=∑𝑤𝑖
𝑛
𝑖=1 ′𝑍𝑖
where Z_i are the known values at the data points.
Step 5: Interpret the result and discuss the limitations of IDW interpolation.
78 PROBLEM 6: VARIOGRAM CLOUD ANALYSIS
Question: Given the following spatial data:
X
Y
Z
1
1
10
2
1
12
1
2
15
2
2
18
3
3
20
Create a variogram cloud and interpret the results.
Solution:
Step 1: Calculate the distances and squared differences for all pairs of points:
For each pair (i, j):
• Distance: h_ij = sqrt((x_i - x_j)^2 + (y_i - y_j)^2)
• Squared difference: γ_ij = (z_i - z_j)^2 / 2
Step 2: Create a scatter plot with distance (h) on the x-axis and semivariance (γ) on the y-axis.
Step 3: Analyze the variogram cloud:
• Look for trends or patterns in the distribution of points
• Identify any outliers or anomalies
• Observe the overall shape and spread of the cloud
Step 4: Interpret the results:
• Discuss the presence or absence of spatial correlation
• Comment on the variability of the data at different distances
• Identify any potential anisotropy or directional effects
Step 5: Consider next steps, such as binning the data to create an experimental variogram.
79 PROBLEM 7: CROSS-VALIDATION FOR KRIGING
Question: Perform leave-one-out cross-validation for ordinary kriging using the following data:
X
Y
Z
1
1
10
1
3
15
3
1
12
3
3
18
2
2
14
Assume an exponential variogram model with a sill of 20 and a range of 2.5.
Solution:
Step 1: For each data point, remove it from the dataset and estimate its value using ordinary
kriging with the remaining points.
Step 2: Calculate the kriging equations for each iteration:
[𝛾(ℎ11)⋯ 𝛾(ℎ1𝑛)1
⋮ ⋱ ⋮ ⋮
𝛾(ℎ𝑛1)⋯ 𝛾(ℎ𝑛𝑛)1
1 ⋯ 1 0][𝑤1
⋮
𝑤𝑛
𝜇]=[𝛾(ℎ10)
⋮
𝛾(ℎ𝑛0)
1]
where γ(h) is the exponential variogram model:
𝛾(ℎ)=20(1−𝑒−3ℎ
2.5)
Step 3: Solve the kriging equations for each iteration to obtain the weights and Lagrange
multiplier.
Step 4: Calculate the estimated value for each removed point:
𝑍𝑖∗=∑𝑤𝑗
𝑛−1
𝑗=1 𝑍𝑗
Step 5: Calculate the cross-validation statistics:
• Mean Error (ME): 𝑀𝐸=1
𝑛∑(𝑍𝑖∗−𝑍𝑖)
𝑛
𝑖=1
• Mean Squared Error (MSE): 𝑀𝑆𝐸=1
𝑛∑(𝑍𝑖∗−𝑍𝑖)2
𝑛
𝑖=1
• Root Mean Squared Error (RMSE): 𝑅𝑀𝑆𝐸=√𝑀𝑆𝐸
• Mean Standardized Error (MSE): 𝑀𝑆𝐸=1
𝑛∑𝑍𝑖∗−𝑍𝑖
𝜎𝑖
𝑛
𝑖=1
• Root Mean Squared Standardized Error (RMSSE): 𝑅𝑀𝑆𝑆𝐸=√1
𝑛∑(𝑍𝑖∗−𝑍𝑖
𝜎𝑖)2
𝑛
𝑖=1
Step 6: Interpret the results:
• ME should be close to 0 for unbiased estimation
• RMSE should be as small as possible
• MSE should be close to 0
• RMSSE should be close to 1
Step 7: Based on the cross-validation results, discuss the performance of the kriging model and
suggest potential improvements.
80 PROBLEM 8: INDICATOR KRIGING
Question: Perform indicator kriging to estimate the probability that the value at location (2.5,
2.5) exceeds a threshold of 15, given the following data:
X
Y
Z
1
1
10
1
4
18
4
1
12
4
4
20
Assume a spherical variogram model for the indicator variable with a sill of 0.25 and a range of
3.
Solution:
Step 1: Transform the data into indicator variables:
𝐼(𝑥)={1if 𝑍(𝑥)>15
0if 𝑍(𝑥)≤15
Step 2: Calculate the distances between the estimation point and the data points:
• d1 = sqrt((2.5-1)^2 + (2.5-1)^2) = 2.12
• d2 = sqrt((2.5-1)^2 + (2.5-4)^2) = 2.69
• d3 = sqrt((2.5-4)^2 + (2.5-1)^2) = 2.69
• d4 = sqrt((2.5-4)^2 + (2.5-4)^2) = 2.12
Step 3: Calculate the semivariances using the spherical model:
𝛾(ℎ)={0.25[3ℎ
2(3)−1
2(ℎ
3)3]for ℎ≤3
0.25 for ℎ>3
Step 4: Set up the kriging system:
[
𝛾(𝑑11)𝛾(𝑑12)𝛾(𝑑13)𝛾(𝑑14)1
𝛾(𝑑21)𝛾(𝑑22)𝛾(𝑑23)𝛾(𝑑24)1
𝛾(𝑑31)𝛾(𝑑32)𝛾(𝑑33)𝛾(𝑑34)1
𝛾(𝑑41)𝛾(𝑑42)𝛾(𝑑43)𝛾(𝑑44)1
1 1 1 1 0
]
[
𝑤1
𝑤2
𝑤3
𝑤4
𝜇
]
=
[
𝛾(𝑑1)
𝛾(𝑑2)
𝛾(𝑑3)
𝛾(𝑑4)
1
]
Step 5: Solve for the weights (w1, w2, w3, w4) and Lagrange multiplier (μ).
Step 6: Calculate the estimated probability:
𝑃∗=𝑤1𝐼(𝑥1)+𝑤2𝐼(𝑥2)+𝑤3𝐼(𝑥3)+𝑤4𝐼(𝑥4)
Step 7: Interpret the result as the probability that the value at (2.5, 2.5) exceeds the threshold
of 15.
81 PROBLEM 9: BAYESIAN GEOSTATISTICS
Question: Using a Bayesian approach, estimate the mean and variance of a spatial random
field given the following observations:
X
Y
Z
1
1
10
1
2
12
2
1
15
2
2
18
Assume a Gaussian prior with mean μ_0 = 13 and variance σ_0^2 = 16, and a measurement
error variance of σ_ε^2 = 1.
Solution:
Step 1: Define the prior distribution: 𝜇∼𝑁(𝜇0,𝜎02)
Step 2: Define the likelihood function: 𝑍𝑖|𝜇∼𝑁(𝜇,𝜎𝜀2)
Step 3: Calculate the sample mean and variance:
𝑍‾=1
𝑛∑𝑍𝑖
𝑛
𝑖=1
𝑠2=1
𝑛−1∑(𝑍𝑖−𝑍‾)2
𝑛
𝑖=1
Step 4: Calculate the posterior mean:
𝜇𝑛=𝑛
𝜎𝜀2𝑍‾+1
𝜎02𝜇0
𝑛
𝜎𝜀2+1
𝜎02
Step 5: Calculate the posterior variance:
𝜎𝑛2=1
𝑛
𝜎𝜀2+1
𝜎02
Step 6: Calculate the 95
𝐶𝐼95%=𝜇𝑛±1.96√𝜎𝑛2
Step 7: Interpret the results:
• Discuss the posterior mean and how it compares to the prior mean and sample mean
• Explain the meaning of the posterior variance and credible interval
• Comment on the influence of the prior and the data on the posterior distribution
82 PROBLEM 10: MULTIVARIATE GEOSTATISTICS
Question: Perform a linear model of coregionalization (LMC) for two variables, Z1 and Z2,
given the following cross-variogram and direct variogram models:
• γ11(h) = 10 Sph(h/100) + 5 Exp(h/50)
• γ22(h) = 15 Sph(h/100) + 8 Exp(h/50)
• γ12(h) = 8 Sph(h/100) + 4 Exp(h/50)
Where Sph and Exp denote spherical and exponential models, respectively.
Solution:
Step 1: Identify the basic structures and their ranges:
• Structure 1: Spherical model with range a1 = 100
• Structure 2: Exponential model with range a2 = 50
Step 2: Set up the coregionalization matrices for each structure:
For the spherical model:
𝐵1=[𝑏11
(1)𝑏12
(1)
𝑏21
(1)𝑏22
(1)]
For the exponential model:
𝐵2=[𝑏11
(2)𝑏12
(2)
𝑏21
(2)𝑏22
(2)]
Step 3: Solve for the elements of B1 and B2 using the sill values:
• b11^(1) = 10, b22^(1) = 15, b12^(1) = b21^(1) = 8
• b11^(2) = 5, b22^(2) = 8, b12^(2) = b21^(2) = 4
Step 4: Verify that B1 and B2 are positive semi-definite matrices:
• Calculate the determinants: det(B1) ≥ 0 and det(B2) ≥ 0
• Check that the diagonal elements are non-negative
Step 5: Write the final LMC model:
[𝛾11(ℎ)𝛾12(ℎ)
𝛾21(ℎ)𝛾22(ℎ)]=𝐵1Sph(ℎ/100)+𝐵2Exp(ℎ/50)
Step 6: Interpret the results:
• Discuss the spatial correlation structure for each variable
• Explain the cross-correlation between the variables
• Comment on the different ranges and their implications for spatial prediction
Step 7: Discuss potential applications of the LMC model, such as cokriging for improved spatial
prediction.
Given the following paired data points (distance, semivariance), calculate the experimental
variogram and fit a spherical model to it:
(10, 2.1), (20, 3.5), (30, 4.2), (40, 4.8), (50, 5.1), (60, 5.3)
Solution:
Step 1: Plot the experimental variogram using the given data points.
Step 2: Estimate the sill, range, and nugget effect from the plot:
• Sill (C): Approximately 5.3
• Range (a): Approximately 50
• Nugget effect (C0): Approximately 0
Step 3: Use the spherical model equation:
𝛾(ℎ)={𝐶0+𝐶[3ℎ
2𝑎−1
2(ℎ
𝑎)3]for ℎ≤𝑎
𝐶0+𝐶 for ℎ>𝑎
Step 4: Substitute the estimated values into the equation:
𝛾(ℎ)={5.3[3ℎ
100−1
2(ℎ
50)3]for ℎ≤50
5.3 for ℎ>50
Step 5: Plot the fitted model alongside the experimental variogram to assess the fit.
83 PROBLEM 2: KRIGING ESTIMATION
Question: Using simple kriging, estimate the value at an unsampled location (5, 5) given the
following data points and their values:
(0, 0): 10 (0, 10): 15 (10, 0): 12 (10, 10): 18
Assume an exponential variogram model with a sill of 20 and a range of 8.
Solution:
Step 1: Calculate the distances between the unsampled location and the data points:
• d1 = sqrt((5-0)^2 + (5-0)^2) = 7.07
• d2 = sqrt((5-0)^2 + (5-10)^2) = 7.07
• d3 = sqrt((5-10)^2 + (5-0)^2) = 7.07
• d4 = sqrt((5-10)^2 + (5-10)^2) = 7.07
Step 2: Calculate the semivariances using the exponential model:
𝛾(ℎ)=20(1−𝑒−3ℎ
8)
• γ1 = γ2 = γ3 = γ4 = 20 * (1 - exp(-3 * 7.07 / 8)) = 15.8
Step 3: Set up the kriging system:
[015.8 15.8 15.8
15.8015.8 15.8
15.8 15.8 0 15.8
15.8 15.8 15.8 0 ][𝑤1
𝑤2
𝑤3
𝑤4]=[15.8
15.8
15.8
15.8]
Step 4: Solve for the weights (w1, w2, w3, w4).
Step 5: Calculate the estimated value:
𝑍∗=𝑤1∗10+𝑤2∗15+𝑤3∗12+𝑤4∗18
84 PROBLEM 3: SPATIAL AUTOCORRELATION
Question: Calculate Moran’s I for the following spatial data:
10
12
8
15
9
11
7
13
14
Assume rook’s case adjacency for the weights matrix.
Solution:
Step 1: Calculate the mean (μ) of the values: μ = (10 + 12 + 8 + 15 + 9 + 11 + 7 + 13 + 14)
/ 9 = 11
Step 2: Create the weights matrix (W) using rook’s case adjacency:
𝑊=
[
0 1 0 1 0 0 0 0 0
1 0 1 0 1 0 0 0 0
0 1 0 0 0 1 0 0 0
1 0 0 0 1 0 1 0 0
0 1 0 1 0 1 0 1 0
0 0 1 0 1 0 0 0 1
0 0 0 1 0 0 0 1 0
0 0 0 0 1 0 1 0 1
0 0 0 0 0 1 0 1 0
]
Step 3: Calculate the numerator of Moran’s I:
𝑁=∑∑𝑤𝑖𝑗
𝑛
𝑗=1
𝑛
𝑖=1 (𝑥𝑖−𝜇)(𝑥𝑗−𝜇)
Step 4: Calculate the denominator of Moran’s I:
𝐷=∑(𝑥𝑖−𝜇)2
𝑛
𝑖=1
Step 5: Calculate Moran’s I:
𝐼=𝑛
𝑊⋅𝑁
𝐷
where n is the number of observations (9) and W is the sum of all weights (24).
Step 6: Interpret the result:
• I > 0: Positive spatial autocorrelation
• I < 0: Negative spatial autocorrelation
• I ≈ 0: No spatial autocorrelation
85 PROBLEM 4: TREND SURFACE ANALYSIS
Question: Perform a first-order trend surface analysis on the following data:
X
Y
Z
1
1
10
1
2
12
2
1
15
2
2
18
Solution:
Step 1: Set up the equation for a first-order trend surface:
𝑍=𝑏0+𝑏1𝑋+𝑏2𝑌
Step 2: Create the design matrix:
𝑋=[1 1 1
1 1 2
1 2 1
1 2 2]
Step 3: Set up the normal equations: (𝑋𝑇𝑋)𝑏=𝑋𝑇𝑍
Step 4: Solve for the coefficients (b0, b1, b2):
𝑏=(𝑋𝑇𝑋)−1𝑋𝑇𝑍
Step 5: Interpret the results:
• b0: Intercept
• b1: Coefficient for X (east-west trend)
• b2: Coefficient for Y (north-south trend)
Step 6: Write the equation of the fitted trend surface:
𝑍=𝑏0+𝑏1𝑋+𝑏2𝑌
86 PROBLEM 5: SPATIAL INTERPOLATION
Question: Using inverse distance weighting (IDW) with a power of 2, estimate the value at
point (2.5, 2.5) given the following data points:
(1, 1): 10 (1, 4): 15 (4, 1): 12 (4, 4): 18
Solution:
Step 1: Calculate the distances from the estimation point to each data point:
• d1 = sqrt((2.5-1)^2 + (2.5-1)^2) = 2.12
• d2 = sqrt((2.5-1)^2 + (2.5-4)^2) = 2.69
• d3 = sqrt((2.5-4)^2 + (2.5-1)^2) = 2.69
• d4 = sqrt((2.5-4)^2 + (2.5-4)^2) = 2.12
Step 2: Calculate the weights using the inverse distance formula:
𝑤𝑖=1
𝑑𝑖2
• w1 = 1 / 2.12^2 = 0.222
• w2 = 1 / 2.69^2 = 0.138
• w3 = 1 / 2.69^2 = 0.138
• w4 = 1 / 2.12^2 = 0.222
Step 3: Normalize the weights: 𝑤𝑖′= 𝑤𝑖
∑𝑤𝑗
𝑛
𝑗=1
Step 4: Calculate the estimated value:
𝑍∗=∑𝑤𝑖
𝑛
𝑖=1 ′𝑍𝑖
where Z_i are the known values at the data points.
Step 5: Interpret the result and discuss the limitations of IDW interpolation.
87 PROBLEM 6: VARIOGRAM CLOUD ANALYSIS
Question: Given the following spatial data:
X
Y
Z
1
1
10
2
1
12
1
2
15
2
2
18
3
3
20
Create a variogram cloud and interpret the results.
Solution:
Step 1: Calculate the distances and squared differences for all pairs of points:
For each pair (i, j):
• Distance: h_ij = sqrt((x_i - x_j)^2 + (y_i - y_j)^2)
• Squared difference: γ_ij = (z_i - z_j)^2 / 2
Step 2: Create a scatter plot with distance (h) on the x-axis and semivariance (γ) on the y-axis.
Step 3: Analyze the variogram cloud:
• Look for trends or patterns in the distribution of points
• Identify any outliers or anomalies
• Observe the overall shape and spread of the cloud
Step 4: Interpret the results:
• Discuss the presence or absence of spatial correlation
• Comment on the variability of the data at different distances
• Identify any potential anisotropy or directional effects
Step 5: Consider next steps, such as binning the data to create an experimental variogram.
88 PROBLEM 7: CROSS-VALIDATION FOR KRIGING
Question: Perform leave-one-out cross-validation for ordinary kriging using the following data:
X
Y
Z
1
1
10
1
3
15
3
1
12
3
3
18
2
2
14
Assume an exponential variogram model with a sill of 20 and a range of 2.5.
Solution:
Step 1: For each data point, remove it from the dataset and estimate its value using ordinary
kriging with the remaining points.
Step 2: Calculate the kriging equations for each iteration:
[𝛾(ℎ11)⋯ 𝛾(ℎ1𝑛)1
⋮ ⋱ ⋮ ⋮
𝛾(ℎ𝑛1)⋯ 𝛾(ℎ𝑛𝑛)1
1 ⋯ 1 0][𝑤1
⋮
𝑤𝑛
𝜇]=[𝛾(ℎ10)
⋮
𝛾(ℎ𝑛0)
1]
where γ(h) is the exponential variogram model:
𝛾(ℎ)=20(1−𝑒−3ℎ
2.5)
Step 3: Solve the kriging equations for each iteration to obtain the weights and Lagrange
multiplier.
Step 4: Calculate the estimated value for each removed point:
𝑍𝑖∗=∑𝑤𝑗
𝑛−1
𝑗=1 𝑍𝑗
Step 5: Calculate the cross-validation statistics:
• Mean Error (ME): 𝑀𝐸=1
𝑛∑(𝑍𝑖∗−𝑍𝑖)
𝑛
𝑖=1
• Mean Squared Error (MSE): 𝑀𝑆𝐸=1
𝑛∑(𝑍𝑖∗−𝑍𝑖)2
𝑛
𝑖=1
• Root Mean Squared Error (RMSE): 𝑅𝑀𝑆𝐸=√𝑀𝑆𝐸
• Mean Standardized Error (MSE): 𝑀𝑆𝐸=1
𝑛∑𝑍𝑖∗−𝑍𝑖
𝜎𝑖
𝑛
𝑖=1
• Root Mean Squared Standardized Error (RMSSE): 𝑅𝑀𝑆𝑆𝐸=√1
𝑛∑(𝑍𝑖∗−𝑍𝑖
𝜎𝑖)2
𝑛
𝑖=1
Step 6: Interpret the results:
• ME should be close to 0 for unbiased estimation
• RMSE should be as small as possible
• MSE should be close to 0
• RMSSE should be close to 1
Step 7: Based on the cross-validation results, discuss the performance of the kriging model and
suggest potential improvements.
89 PROBLEM 8: INDICATOR KRIGING
Question: Perform indicator kriging to estimate the probability that the value at location (2.5,
2.5) exceeds a threshold of 15, given the following data:
X
Y
Z
1
1
10
1
4
18
4
1
12
4
4
20
Assume a spherical variogram model for the indicator variable with a sill of 0.25 and a range of
3.
Solution:
Step 1: Transform the data into indicator variables:
𝐼(𝑥)={1if 𝑍(𝑥)>15
0if 𝑍(𝑥)≤15
Step 2: Calculate the distances between the estimation point and the data points:
• d1 = sqrt((2.5-1)^2 + (2.5-1)^2) = 2.12
• d2 = sqrt((2.5-1)^2 + (2.5-4)^2) = 2.69
• d3 = sqrt((2.5-4)^2 + (2.5-1)^2) = 2.69
• d4 = sqrt((2.5-4)^2 + (2.5-4)^2) = 2.12
Step 3: Calculate the semivariances using the spherical model:
𝛾(ℎ)={0.25[3ℎ
2(3)−1
2(ℎ
3)3]for ℎ≤3
0.25 for ℎ>3
Step 4: Set up the kriging system:
[
𝛾(𝑑11)𝛾(𝑑12)𝛾(𝑑13)𝛾(𝑑14)1
𝛾(𝑑21)𝛾(𝑑22)𝛾(𝑑23)𝛾(𝑑24)1
𝛾(𝑑31)𝛾(𝑑32)𝛾(𝑑33)𝛾(𝑑34)1
𝛾(𝑑41)𝛾(𝑑42)𝛾(𝑑43)𝛾(𝑑44)1
1 1 1 1 0
]
[
𝑤1
𝑤2
𝑤3
𝑤4
𝜇
]
=
[
𝛾(𝑑1)
𝛾(𝑑2)
𝛾(𝑑3)
𝛾(𝑑4)
1
]
Step 5: Solve for the weights (w1, w2, w3, w4) and Lagrange multiplier (μ).
Step 6: Calculate the estimated probability:
𝑃∗=𝑤1𝐼(𝑥1)+𝑤2𝐼(𝑥2)+𝑤3𝐼(𝑥3)+𝑤4𝐼(𝑥4)
Step 7: Interpret the result as the probability that the value at (2.5, 2.5) exceeds the threshold
of 15.
90 PROBLEM 9: BAYESIAN GEOSTATISTICS
Question: Using a Bayesian approach, estimate the mean and variance of a spatial random
field given the following observations:
X
Y
Z
1
1
10
1
2
12
2
1
15
2
2
18
Assume a Gaussian prior with mean μ_0 = 13 and variance σ_0^2 = 16, and a measurement
error variance of σ_ε^2 = 1.
Solution:
Step 1: Define the prior distribution: 𝜇∼𝑁(𝜇0,𝜎02)
Step 2: Define the likelihood function: 𝑍𝑖|𝜇∼𝑁(𝜇,𝜎𝜀2)
Step 3: Calculate the sample mean and variance:
𝑍‾=1
𝑛∑𝑍𝑖
𝑛
𝑖=1
𝑠2=1
𝑛−1∑(𝑍𝑖−𝑍‾)2
𝑛
𝑖=1
Step 4: Calculate the posterior mean:
𝜇𝑛=𝑛
𝜎𝜀2𝑍‾+1
𝜎02𝜇0
𝑛
𝜎𝜀2+1
𝜎02
Step 5: Calculate the posterior variance:
𝜎𝑛2=1
𝑛
𝜎𝜀2+1
𝜎02
Step 6: Calculate the 95
𝐶𝐼95%=𝜇𝑛±1.96√𝜎𝑛2
Step 7: Interpret the results:
• Discuss the posterior mean and how it compares to the prior mean and sample mean
• Explain the meaning of the posterior variance and credible interval
• Comment on the influence of the prior and the data on the posterior distribution
91 PROBLEM 10: MULTIVARIATE GEOSTATISTICS
Question: Perform a linear model of coregionalization (LMC) for two variables, Z1 and Z2,
given the following cross-variogram and direct variogram models:
• γ11(h) = 10 Sph(h/100) + 5 Exp(h/50)
• γ22(h) = 15 Sph(h/100) + 8 Exp(h/50)
• γ12(h) = 8 Sph(h/100) + 4 Exp(h/50)
Where Sph and Exp denote spherical and exponential models, respectively.
Solution:
Step 1: Identify the basic structures and their ranges:
• Structure 1: Spherical model with range a1 = 100
• Structure 2: Exponential model with range a2 = 50
Step 2: Set up the coregionalization matrices for each structure:
For the spherical model:
𝐵1=[𝑏11
(1)𝑏12
(1)
𝑏21
(1)𝑏22
(1)]
For the exponential model:
𝐵2=[𝑏11
(2)𝑏12
(2)
𝑏21
(2)𝑏22
(2)]
Step 3: Solve for the elements of B1 and B2 using the sill values:
• b11^(1) = 10, b22^(1) = 15, b12^(1) = b21^(1) = 8
• b11^(2) = 5, b22^(2) = 8, b12^(2) = b21^(2) = 4
Step 4: Verify that B1 and B2 are positive semi-definite matrices:
• Calculate the determinants: det(B1) ≥ 0 and det(B2) ≥ 0
• Check that the diagonal elements are non-negative
Step 5: Write the final LMC model:
[𝛾11(ℎ)𝛾12(ℎ)
𝛾21(ℎ)𝛾22(ℎ)]=𝐵1Sph(ℎ/100)+𝐵2Exp(ℎ/50)
Step 6: Interpret the results:
• Discuss the spatial correlation structure for each variable
• Explain the cross-correlation between the variables
• Comment on the different ranges and their implications for spatial prediction
Step 7: Discuss potential applications of the LMC model, such as cokriging for improved spatial
prediction.
Given the following paired data points (distance, semivariance), calculate the experimental
variogram and fit a spherical model to it:
(10, 2.1), (20, 3.5), (30, 4.2), (40, 4.8), (50, 5.1), (60, 5.3)
Solution:
Step 1: Plot the experimental variogram using the given data points.
Step 2: Estimate the sill, range, and nugget effect from the plot:
• Sill (C): Approximately 5.3
• Range (a): Approximately 50
• Nugget effect (C0): Approximately 0
Step 3: Use the spherical model equation:
𝛾(ℎ)={𝐶0+𝐶[3ℎ
2𝑎−1
2(ℎ
𝑎)3]for ℎ≤𝑎
𝐶0+𝐶 for ℎ>𝑎
Step 4: Substitute the estimated values into the equation:
𝛾(ℎ)={5.3[3ℎ
100−1
2(ℎ
50)3]for ℎ≤50
5.3 for ℎ>50
Step 5: Plot the fitted model alongside the experimental variogram to assess the fit.
92 PROBLEM 2: KRIGING ESTIMATION
Question: Using simple kriging, estimate the value at an unsampled location (5, 5) given the
following data points and their values:
(0, 0): 10 (0, 10): 15 (10, 0): 12 (10, 10): 18
Assume an exponential variogram model with a sill of 20 and a range of 8.
Solution:
Step 1: Calculate the distances between the unsampled location and the data points:
• d1 = sqrt((5-0)^2 + (5-0)^2) = 7.07
• d2 = sqrt((5-0)^2 + (5-10)^2) = 7.07
• d3 = sqrt((5-10)^2 + (5-0)^2) = 7.07
• d4 = sqrt((5-10)^2 + (5-10)^2) = 7.07
Step 2: Calculate the semivariances using the exponential model:
𝛾(ℎ)=20(1−𝑒−3ℎ
8)
• γ1 = γ2 = γ3 = γ4 = 20 * (1 - exp(-3 * 7.07 / 8)) = 15.8
Step 3: Set up the kriging system:
[015.8 15.8 15.8
15.8015.8 15.8
15.8 15.8 0 15.8
15.8 15.8 15.8 0 ][𝑤1
𝑤2
𝑤3
𝑤4]=[15.8
15.8
15.8
15.8]
Step 4: Solve for the weights (w1, w2, w3, w4).
Step 5: Calculate the estimated value:
𝑍∗=𝑤1∗10+𝑤2∗15+𝑤3∗12+𝑤4∗18
93 PROBLEM 3: SPATIAL AUTOCORRELATION
Question: Calculate Moran’s I for the following spatial data:
10
12
8
15
9
11
7
13
14
Assume rook’s case adjacency for the weights matrix.
Solution:
Step 1: Calculate the mean (μ) of the values: μ = (10 + 12 + 8 + 15 + 9 + 11 + 7 + 13 + 14)
/ 9 = 11
Step 2: Create the weights matrix (W) using rook’s case adjacency:
𝑊=
[
0 1 0 1 0 0 0 0 0
1 0 1 0 1 0 0 0 0
0 1 0 0 0 1 0 0 0
1 0 0 0 1 0 1 0 0
0 1 0 1 0 1 0 1 0
0 0 1 0 1 0 0 0 1
0 0 0 1 0 0 0 1 0
0 0 0 0 1 0 1 0 1
0 0 0 0 0 1 0 1 0
]
Step 3: Calculate the numerator of Moran’s I:
𝑁=∑∑𝑤𝑖𝑗
𝑛
𝑗=1
𝑛
𝑖=1 (𝑥𝑖−𝜇)(𝑥𝑗−𝜇)
Step 4: Calculate the denominator of Moran’s I:
𝐷=∑(𝑥𝑖−𝜇)2
𝑛
𝑖=1
Step 5: Calculate Moran’s I:
𝐼=𝑛
𝑊⋅𝑁
𝐷
where n is the number of observations (9) and W is the sum of all weights (24).
Step 6: Interpret the result:
• I > 0: Positive spatial autocorrelation
• I < 0: Negative spatial autocorrelation
• I ≈ 0: No spatial autocorrelation
94 PROBLEM 4: TREND SURFACE ANALYSIS
Question: Perform a first-order trend surface analysis on the following data:
X
Y
Z
1
1
10
1
2
12
2
1
15
2
2
18
Solution:
Step 1: Set up the equation for a first-order trend surface:
𝑍=𝑏0+𝑏1𝑋+𝑏2𝑌
Step 2: Create the design matrix:
𝑋=[1 1 1
1 1 2
1 2 1
1 2 2]
Step 3: Set up the normal equations: (𝑋𝑇𝑋)𝑏=𝑋𝑇𝑍
Step 4: Solve for the coefficients (b0, b1, b2):
𝑏=(𝑋𝑇𝑋)−1𝑋𝑇𝑍
Step 5: Interpret the results:
• b0: Intercept
• b1: Coefficient for X (east-west trend)
• b2: Coefficient for Y (north-south trend)
Step 6: Write the equation of the fitted trend surface:
𝑍=𝑏0+𝑏1𝑋+𝑏2𝑌
95 PROBLEM 5: SPATIAL INTERPOLATION
Question: Using inverse distance weighting (IDW) with a power of 2, estimate the value at
point (2.5, 2.5) given the following data points:
(1, 1): 10 (1, 4): 15 (4, 1): 12 (4, 4): 18
Solution:
Step 1: Calculate the distances from the estimation point to each data point:
• d1 = sqrt((2.5-1)^2 + (2.5-1)^2) = 2.12
• d2 = sqrt((2.5-1)^2 + (2.5-4)^2) = 2.69
• d3 = sqrt((2.5-4)^2 + (2.5-1)^2) = 2.69
• d4 = sqrt((2.5-4)^2 + (2.5-4)^2) = 2.12
Step 2: Calculate the weights using the inverse distance formula:
𝑤𝑖=1
𝑑𝑖2
• w1 = 1 / 2.12^2 = 0.222
• w2 = 1 / 2.69^2 = 0.138
• w3 = 1 / 2.69^2 = 0.138
• w4 = 1 / 2.12^2 = 0.222
Step 3: Normalize the weights: 𝑤𝑖′= 𝑤𝑖
∑𝑤𝑗
𝑛
𝑗=1
Step 4: Calculate the estimated value:
𝑍∗=∑𝑤𝑖
𝑛
𝑖=1 ′𝑍𝑖
where Z_i are the known values at the data points.
Step 5: Interpret the result and discuss the limitations of IDW interpolation.
96 PROBLEM 6: VARIOGRAM CLOUD ANALYSIS
Question: Given the following spatial data:
X
Y
Z
1
1
10
2
1
12
1
2
15
2
2
18
3
3
20
Create a variogram cloud and interpret the results.
Solution:
Step 1: Calculate the distances and squared differences for all pairs of points:
For each pair (i, j):
• Distance: h_ij = sqrt((x_i - x_j)^2 + (y_i - y_j)^2)
• Squared difference: γ_ij = (z_i - z_j)^2 / 2
Step 2: Create a scatter plot with distance (h) on the x-axis and semivariance (γ) on the y-axis.
Step 3: Analyze the variogram cloud:
• Look for trends or patterns in the distribution of points
• Identify any outliers or anomalies
• Observe the overall shape and spread of the cloud
Step 4: Interpret the results:
• Discuss the presence or absence of spatial correlation
• Comment on the variability of the data at different distances
• Identify any potential anisotropy or directional effects
Step 5: Consider next steps, such as binning the data to create an experimental variogram.
97 PROBLEM 7: CROSS-VALIDATION FOR KRIGING
Question: Perform leave-one-out cross-validation for ordinary kriging using the following data:
X
Y
Z
1
1
10
1
3
15
X
Y
Z
3
1
12
3
3
18
2
2
14
Assume an exponential variogram model with a sill of 20 and a range of 2.5.
Solution:
Step 1: For each data point, remove it from the dataset and estimate its value using ordinary
kriging with the remaining points.
Step 2: Calculate the kriging equations for each iteration:
[𝛾(ℎ11)⋯ 𝛾(ℎ1𝑛)1
⋮ ⋱ ⋮ ⋮
𝛾(ℎ𝑛1)⋯ 𝛾(ℎ𝑛𝑛)1
1 ⋯ 1 0][𝑤1
⋮
𝑤𝑛
𝜇]=[𝛾(ℎ10)
⋮
𝛾(ℎ𝑛0)
1]
where γ(h) is the exponential variogram model:
𝛾(ℎ)=20(1−𝑒−3ℎ
2.5)
Step 3: Solve the kriging equations for each iteration to obtain the weights and Lagrange
multiplier.
Step 4: Calculate the estimated value for each removed point:
𝑍𝑖∗=∑𝑤𝑗
𝑛−1
𝑗=1 𝑍𝑗
Step 5: Calculate the cross-validation statistics:
• Mean Error (ME): 𝑀𝐸=1
𝑛∑(𝑍𝑖∗−𝑍𝑖)
𝑛
𝑖=1
• Mean Squared Error (MSE): 𝑀𝑆𝐸=1
𝑛∑(𝑍𝑖∗−𝑍𝑖)2
𝑛
𝑖=1
• Root Mean Squared Error (RMSE): 𝑅𝑀𝑆𝐸=√𝑀𝑆𝐸
• Mean Standardized Error (MSE): 𝑀𝑆𝐸=1
𝑛∑𝑍𝑖∗−𝑍𝑖
𝜎𝑖
𝑛
𝑖=1
• Root Mean Squared Standardized Error (RMSSE): 𝑅𝑀𝑆𝑆𝐸=√1
𝑛∑(𝑍𝑖∗−𝑍𝑖
𝜎𝑖)2
𝑛
𝑖=1
Step 6: Interpret the results:
• ME should be close to 0 for unbiased estimation
• RMSE should be as small as possible
• MSE should be close to 0
• RMSSE should be close to 1
Step 7: Based on the cross-validation results, discuss the performance of the kriging model and
suggest potential improvements.
98 PROBLEM 8: INDICATOR KRIGING
Question: Perform indicator kriging to estimate the probability that the value at location (2.5,
2.5) exceeds a threshold of 15, given the following data:
X
Y
Z
1
1
10
1
4
18
4
1
12
4
4
20
Assume a spherical variogram model for the indicator variable with a sill of 0.25 and a range of
3.
Solution:
Step 1: Transform the data into indicator variables:
𝐼(𝑥)={1if 𝑍(𝑥)>15
0if 𝑍(𝑥)≤15
Step 2: Calculate the distances between the estimation point and the data points:
• d1 = sqrt((2.5-1)^2 + (2.5-1)^2) = 2.12
• d2 = sqrt((2.5-1)^2 + (2.5-4)^2) = 2.69
• d3 = sqrt((2.5-4)^2 + (2.5-1)^2) = 2.69
• d4 = sqrt((2.5-4)^2 + (2.5-4)^2) = 2.12
Step 3: Calculate the semivariances using the spherical model:
𝛾(ℎ)={0.25[3ℎ
2(3)−1
2(ℎ
3)3]for ℎ≤3
0.25 for ℎ>3
Step 4: Set up the kriging system:
[
𝛾(𝑑11)𝛾(𝑑12)𝛾(𝑑13)𝛾(𝑑14)1
𝛾(𝑑21)𝛾(𝑑22)𝛾(𝑑23)𝛾(𝑑24)1
𝛾(𝑑31)𝛾(𝑑32)𝛾(𝑑33)𝛾(𝑑34)1
𝛾(𝑑41)𝛾(𝑑42)𝛾(𝑑43)𝛾(𝑑44)1
1 1 1 1 0
]
[
𝑤1
𝑤2
𝑤3
𝑤4
𝜇
]
=
[
𝛾(𝑑1)
𝛾(𝑑2)
𝛾(𝑑3)
𝛾(𝑑4)
1
]
Step 5: Solve for the weights (w1, w2, w3, w4) and Lagrange multiplier (μ).
Step 6: Calculate the estimated probability:
𝑃∗=𝑤1𝐼(𝑥1)+𝑤2𝐼(𝑥2)+𝑤3𝐼(𝑥3)+𝑤4𝐼(𝑥4)
Step 7: Interpret the result as the probability that the value at (2.5, 2.5) exceeds the threshold
of 15.
99 PROBLEM 9: BAYESIAN GEOSTATISTICS
Question: Using a Bayesian approach, estimate the mean and variance of a spatial random
field given the following observations:
X
Y
Z
1
1
10
1
2
12
2
1
15
2
2
18
Assume a Gaussian prior with mean μ_0 = 13 and variance σ_0^2 = 16, and a measurement
error variance of σ_ε^2 = 1.
Solution:
Step 1: Define the prior distribution: 𝜇∼𝑁(𝜇0,𝜎02)
Step 2: Define the likelihood function: 𝑍𝑖|𝜇∼𝑁(𝜇,𝜎𝜀2)
Step 3: Calculate the sample mean and variance:
𝑍‾=1
𝑛∑𝑍𝑖
𝑛
𝑖=1
𝑠2=1
𝑛−1∑(𝑍𝑖−𝑍‾)2
𝑛
𝑖=1
Step 4: Calculate the posterior mean:
𝜇𝑛=𝑛
𝜎𝜀2𝑍‾+1
𝜎02𝜇0
𝑛
𝜎𝜀2+1
𝜎02
Step 5: Calculate the posterior variance:
𝜎𝑛2=1
𝑛
𝜎𝜀2+1
𝜎02
Step 6: Calculate the 95
𝐶𝐼95%=𝜇𝑛±1.96√𝜎𝑛2
Step 7: Interpret the results:
• Discuss the posterior mean and how it compares to the prior mean and sample mean
• Explain the meaning of the posterior variance and credible interval
• Comment on the influence of the prior and the data on the posterior distribution
100 PROBLEM 10: MULTIVARIATE GEOSTATISTICS
Question: Perform a linear model of coregionalization (LMC) for two variables, Z1 and Z2,
given the following cross-variogram and direct variogram models:
• γ11(h) = 10 Sph(h/100) + 5 Exp(h/50)
• γ22(h) = 15 Sph(h/100) + 8 Exp(h/50)
• γ12(h) = 8 Sph(h/100) + 4 Exp(h/50)
Where Sph and Exp denote spherical and exponential models, respectively.
Solution:
Step 1: Identify the basic structures and their ranges:
• Structure 1: Spherical model with range a1 = 100
• Structure 2: Exponential model with range a2 = 50
Step 2: Set up the coregionalization matrices for each structure:
For the spherical model:
𝐵1=[𝑏11
(1)𝑏12
(1)
𝑏21
(1)𝑏22
(1)]
For the exponential model:
𝐵2=[𝑏11
(2)𝑏12
(2)
𝑏21
(2)𝑏22
(2)]
Step 3: Solve for the elements of B1 and B2 using the sill values:
• b11^(1) = 10, b22^(1) = 15, b12^(1) = b21^(1) = 8
• b11^(2) = 5, b22^(2) = 8, b12^(2) = b21^(2) = 4
Step 4: Verify that B1 and B2 are positive semi-definite matrices:
• Calculate the determinants: det(B1) ≥ 0 and det(B2) ≥ 0
• Check that the diagonal elements are non-negative
Step 5: Write the final LMC model:
[𝛾11(ℎ)𝛾12(ℎ)
𝛾21(ℎ)𝛾22(ℎ)]=𝐵1Sph(ℎ/100)+𝐵2Exp(ℎ/50)
Step 6: Interpret the results:
• Discuss the spatial correlation structure for each variable
• Explain the cross-correlation between the variables
• Comment on the different ranges and their implications for spatial prediction
Step 7: Discuss potential applications of the LMC model, such as cokriging for improved spatial
prediction.
Given the following paired data points (distance, semivariance), calculate the experimental
variogram and fit a spherical model to it:
(10, 2.1), (20, 3.5), (30, 4.2), (40, 4.8), (50, 5.1), (60, 5.3)
Solution:
Step 1: Plot the experimental variogram using the given data points.
Step 2: Estimate the sill, range, and nugget effect from the plot:
• Sill (C): Approximately 5.3
• Range (a): Approximately 50
• Nugget effect (C0): Approximately 0
Step 3: Use the spherical model equation:
𝛾(ℎ)={𝐶0+𝐶[3ℎ
2𝑎−1
2(ℎ
𝑎)3]for ℎ≤𝑎
𝐶0+𝐶 for ℎ>𝑎
Step 4: Substitute the estimated values into the equation:
𝛾(ℎ)={5.3[3ℎ
100−1
2(ℎ
50)3]for ℎ≤50
5.3 for ℎ>50
Step 5: Plot the fitted model alongside the experimental variogram to assess the fit.
101 PROBLEM 2: KRIGING ESTIMATION
Question: Using simple kriging, estimate the value at an unsampled location (5, 5) given the
following data points and their values:
(0, 0): 10 (0, 10): 15 (10, 0): 12 (10, 10): 18
Assume an exponential variogram model with a sill of 20 and a range of 8.
Solution:
Step 1: Calculate the distances between the unsampled location and the data points:
• d1 = sqrt((5-0)^2 + (5-0)^2) = 7.07
• d2 = sqrt((5-0)^2 + (5-10)^2) = 7.07
• d3 = sqrt((5-10)^2 + (5-0)^2) = 7.07
• d4 = sqrt((5-10)^2 + (5-10)^2) = 7.07
Step 2: Calculate the semivariances using the exponential model:
𝛾(ℎ)=20(1−𝑒−3ℎ
8)
• γ1 = γ2 = γ3 = γ4 = 20 * (1 - exp(-3 * 7.07 / 8)) = 15.8
Step 3: Set up the kriging system:
[015.8 15.8 15.8
15.8015.8 15.8
15.8 15.8 0 15.8
15.8 15.8 15.8 0 ][𝑤1
𝑤2
𝑤3
𝑤4]=[15.8
15.8
15.8
15.8]
Step 4: Solve for the weights (w1, w2, w3, w4).
Step 5: Calculate the estimated value:
𝑍∗=𝑤1∗10+𝑤2∗15+𝑤3∗12+𝑤4∗18
102 PROBLEM 3: SPATIAL AUTOCORRELATION
Question: Calculate Moran’s I for the following spatial data:
10
12
8
15
9
11
7
13
14
Assume rook’s case adjacency for the weights matrix.
Solution:
Step 1: Calculate the mean (μ) of the values: μ = (10 + 12 + 8 + 15 + 9 + 11 + 7 + 13 + 14)
/ 9 = 11
Step 2: Create the weights matrix (W) using rook’s case adjacency:
𝑊=
[
0 1 0 1 0 0 0 0 0
1 0 1 0 1 0 0 0 0
0 1 0 0 0 1 0 0 0
1 0 0 0 1 0 1 0 0
0 1 0 1 0 1 0 1 0
0 0 1 0 1 0 0 0 1
0 0 0 1 0 0 0 1 0
0 0 0 0 1 0 1 0 1
0 0 0 0 0 1 0 1 0
]
Step 3: Calculate the numerator of Moran’s I:
𝑁=∑∑𝑤𝑖𝑗
𝑛
𝑗=1
𝑛
𝑖=1 (𝑥𝑖−𝜇)(𝑥𝑗−𝜇)
Step 4: Calculate the denominator of Moran’s I:
𝐷=∑(𝑥𝑖−𝜇)2
𝑛
𝑖=1
Step 5: Calculate Moran’s I:
𝐼=𝑛
𝑊⋅𝑁
𝐷
where n is the number of observations (9) and W is the sum of all weights (24).
Step 6: Interpret the result:
• I > 0: Positive spatial autocorrelation
• I < 0: Negative spatial autocorrelation
• I ≈ 0: No spatial autocorrelation
103 PROBLEM 4: TREND SURFACE ANALYSIS
Question: Perform a first-order trend surface analysis on the following data:
X
Y
Z
1
1
10
1
2
12
2
1
15
2
2
18
Solution:
Step 1: Set up the equation for a first-order trend surface:
𝑍=𝑏0+𝑏1𝑋+𝑏2𝑌
Step 2: Create the design matrix:
𝑋=[1 1 1
1 1 2
1 2 1
1 2 2]
Step 3: Set up the normal equations: (𝑋𝑇𝑋)𝑏=𝑋𝑇𝑍
Step 4: Solve for the coefficients (b0, b1, b2):
𝑏=(𝑋𝑇𝑋)−1𝑋𝑇𝑍
Step 5: Interpret the results:
• b0: Intercept
• b1: Coefficient for X (east-west trend)
• b2: Coefficient for Y (north-south trend)
Step 6: Write the equation of the fitted trend surface:
𝑍=𝑏0+𝑏1𝑋+𝑏2𝑌
104 PROBLEM 5: SPATIAL INTERPOLATION
Question: Using inverse distance weighting (IDW) with a power of 2, estimate the value at
point (2.5, 2.5) given the following data points:
(1, 1): 10 (1, 4): 15 (4, 1): 12 (4, 4): 18
Solution:
Step 1: Calculate the distances from the estimation point to each data point:
• d1 = sqrt((2.5-1)^2 + (2.5-1)^2) = 2.12
• d2 = sqrt((2.5-1)^2 + (2.5-4)^2) = 2.69
• d3 = sqrt((2.5-4)^2 + (2.5-1)^2) = 2.69
• d4 = sqrt((2.5-4)^2 + (2.5-4)^2) = 2.12
Step 2: Calculate the weights using the inverse distance formula:
𝑤𝑖=1
𝑑𝑖2
• w1 = 1 / 2.12^2 = 0.222
• w2 = 1 / 2.69^2 = 0.138
• w3 = 1 / 2.69^2 = 0.138
• w4 = 1 / 2.12^2 = 0.222
Step 3: Normalize the weights: 𝑤𝑖′= 𝑤𝑖
∑𝑤𝑗
𝑛
𝑗=1
Step 4: Calculate the estimated value:
𝑍∗=∑𝑤𝑖
𝑛
𝑖=1 ′𝑍𝑖
where Z_i are the known values at the data points.
Step 5: Interpret the result and discuss the limitations of IDW interpolation.
105 PROBLEM 6: VARIOGRAM CLOUD ANALYSIS
Question: Given the following spatial data:
X
Y
Z
1
1
10
2
1
12
1
2
15
2
2
18
3
3
20
Create a variogram cloud and interpret the results.
Solution:
Step 1: Calculate the distances and squared differences for all pairs of points:
For each pair (i, j):
• Distance: h_ij = sqrt((x_i - x_j)^2 + (y_i - y_j)^2)
• Squared difference: γ_ij = (z_i - z_j)^2 / 2
Step 2: Create a scatter plot with distance (h) on the x-axis and semivariance (γ) on the y-axis.
Step 3: Analyze the variogram cloud:
• Look for trends or patterns in the distribution of points
• Identify any outliers or anomalies
• Observe the overall shape and spread of the cloud
Step 4: Interpret the results:
• Discuss the presence or absence of spatial correlation
• Comment on the variability of the data at different distances
• Identify any potential anisotropy or directional effects
Step 5: Consider next steps, such as binning the data to create an experimental variogram.
106 PROBLEM 7: CROSS-VALIDATION FOR KRIGING
Question: Perform leave-one-out cross-validation for ordinary kriging using the following data:
X
Y
Z
1
1
10
1
3
15
3
1
12
3
3
18
2
2
14
Assume an exponential variogram model with a sill of 20 and a range of 2.5.
Solution:
Step 1: For each data point, remove it from the dataset and estimate its value using ordinary
kriging with the remaining points.
Step 2: Calculate the kriging equations for each iteration:
[𝛾(ℎ11)⋯ 𝛾(ℎ1𝑛)1
⋮ ⋱ ⋮ ⋮
𝛾(ℎ𝑛1)⋯ 𝛾(ℎ𝑛𝑛)1
1 ⋯ 1 0][𝑤1
⋮
𝑤𝑛
𝜇]=[𝛾(ℎ10)
⋮
𝛾(ℎ𝑛0)
1]
where γ(h) is the exponential variogram model:
𝛾(ℎ)=20(1−𝑒−3ℎ
2.5)
Step 3: Solve the kriging equations for each iteration to obtain the weights and Lagrange
multiplier.
Step 4: Calculate the estimated value for each removed point:
𝑍𝑖∗=∑𝑤𝑗
𝑛−1
𝑗=1 𝑍𝑗
Step 5: Calculate the cross-validation statistics:
• Mean Error (ME): 𝑀𝐸=1
𝑛∑(𝑍𝑖∗−𝑍𝑖)
𝑛
𝑖=1
• Mean Squared Error (MSE): 𝑀𝑆𝐸=1
𝑛∑(𝑍𝑖∗−𝑍𝑖)2
𝑛
𝑖=1
• Root Mean Squared Error (RMSE): 𝑅𝑀𝑆𝐸=√𝑀𝑆𝐸
• Mean Standardized Error (MSE): 𝑀𝑆𝐸=1
𝑛∑𝑍𝑖∗−𝑍𝑖
𝜎𝑖
𝑛
𝑖=1
• Root Mean Squared Standardized Error (RMSSE): 𝑅𝑀𝑆𝑆𝐸=√1
𝑛∑(𝑍𝑖∗−𝑍𝑖
𝜎𝑖)2
𝑛
𝑖=1
Step 6: Interpret the results:
• ME should be close to 0 for unbiased estimation
• RMSE should be as small as possible
• MSE should be close to 0
• RMSSE should be close to 1
Step 7: Based on the cross-validation results, discuss the performance of the kriging model and
suggest potential improvements.
107 PROBLEM 8: INDICATOR KRIGING
Question: Perform indicator kriging to estimate the probability that the value at location (2.5,
2.5) exceeds a threshold of 15, given the following data:
X
Y
Z
1
1
10
1
4
18
4
1
12
4
4
20
Assume a spherical variogram model for the indicator variable with a sill of 0.25 and a range of
3.
Solution:
Step 1: Transform the data into indicator variables:
𝐼(𝑥)={1if 𝑍(𝑥)>15
0if 𝑍(𝑥)≤15
Step 2: Calculate the distances between the estimation point and the data points:
• d1 = sqrt((2.5-1)^2 + (2.5-1)^2) = 2.12
• d2 = sqrt((2.5-1)^2 + (2.5-4)^2) = 2.69
• d3 = sqrt((2.5-4)^2 + (2.5-1)^2) = 2.69
• d4 = sqrt((2.5-4)^2 + (2.5-4)^2) = 2.12
Step 3: Calculate the semivariances using the spherical model:
𝛾(ℎ)={0.25[3ℎ
2(3)−1
2(ℎ
3)3]for ℎ≤3
0.25 for ℎ>3
Step 4: Set up the kriging system:
[
𝛾(𝑑11)𝛾(𝑑12)𝛾(𝑑13)𝛾(𝑑14)1
𝛾(𝑑21)𝛾(𝑑22)𝛾(𝑑23)𝛾(𝑑24)1
𝛾(𝑑31)𝛾(𝑑32)𝛾(𝑑33)𝛾(𝑑34)1
𝛾(𝑑41)𝛾(𝑑42)𝛾(𝑑43)𝛾(𝑑44)1
1 1 1 1 0
]
[
𝑤1
𝑤2
𝑤3
𝑤4
𝜇
]
=
[
𝛾(𝑑1)
𝛾(𝑑2)
𝛾(𝑑3)
𝛾(𝑑4)
1
]
Step 5: Solve for the weights (w1, w2, w3, w4) and Lagrange multiplier (μ).
Step 6: Calculate the estimated probability:
𝑃∗=𝑤1𝐼(𝑥1)+𝑤2𝐼(𝑥2)+𝑤3𝐼(𝑥3)+𝑤4𝐼(𝑥4)
Step 7: Interpret the result as the probability that the value at (2.5, 2.5) exceeds the threshold
of 15.
108 PROBLEM 9: BAYESIAN GEOSTATISTICS
Question: Using a Bayesian approach, estimate the mean and variance of a spatial random
field given the following observations:
X
Y
Z
1
1
10
1
2
12
2
1
15
2
2
18
Assume a Gaussian prior with mean μ_0 = 13 and variance σ_0^2 = 16, and a measurement
error variance of σ_ε^2 = 1.
Solution:
Step 1: Define the prior distribution: 𝜇∼𝑁(𝜇0,𝜎02)
Step 2: Define the likelihood function: 𝑍𝑖|𝜇∼𝑁(𝜇,𝜎𝜀2)
Step 3: Calculate the sample mean and variance:
𝑍‾=1
𝑛∑𝑍𝑖
𝑛
𝑖=1
𝑠2=1
𝑛−1∑(𝑍𝑖−𝑍‾)2
𝑛
𝑖=1
Step 4: Calculate the posterior mean:
𝜇𝑛=𝑛
𝜎𝜀2𝑍‾+1
𝜎02𝜇0
𝑛
𝜎𝜀2+1
𝜎02
Step 5: Calculate the posterior variance:
𝜎𝑛2=1
𝑛
𝜎𝜀2+1
𝜎02
Step 6: Calculate the 95
𝐶𝐼95%=𝜇𝑛±1.96√𝜎𝑛2
Step 7: Interpret the results:
• Discuss the posterior mean and how it compares to the prior mean and sample mean
• Explain the meaning of the posterior variance and credible interval
• Comment on the influence of the prior and the data on the posterior distribution
109 PROBLEM 10: MULTIVARIATE GEOSTATISTICS
Question: Perform a linear model of coregionalization (LMC) for two variables, Z1 and Z2,
given the following cross-variogram and direct variogram models:
• γ11(h) = 10 Sph(h/100) + 5 Exp(h/50)
• γ22(h) = 15 Sph(h/100) + 8 Exp(h/50)
• γ12(h) = 8 Sph(h/100) + 4 Exp(h/50)
Where Sph and Exp denote spherical and exponential models, respectively.
Solution:
Step 1: Identify the basic structures and their ranges:
• Structure 1: Spherical model with range a1 = 100
• Structure 2: Exponential model with range a2 = 50
Step 2: Set up the coregionalization matrices for each structure:
For the spherical model:
𝐵1=[𝑏11
(1)𝑏12
(1)
𝑏21
(1)𝑏22
(1)]
For the exponential model:
𝐵2=[𝑏11
(2)𝑏12
(2)
𝑏21
(2)𝑏22
(2)]
Step 3: Solve for the elements of B1 and B2 using the sill values:
• b11^(1) = 10, b22^(1) = 15, b12^(1) = b21^(1) = 8
• b11^(2) = 5, b22^(2) = 8, b12^(2) = b21^(2) = 4
Step 4: Verify that B1 and B2 are positive semi-definite matrices:
• Calculate the determinants: det(B1) ≥ 0 and det(B2) ≥ 0
• Check that the diagonal elements are non-negative
Step 5: Write the final LMC model:
[𝛾11(ℎ)𝛾12(ℎ)
𝛾21(ℎ)𝛾22(ℎ)]=𝐵1Sph(ℎ/100)+𝐵2Exp(ℎ/50)
Step 6: Interpret the results:
• Discuss the spatial correlation structure for each variable
• Explain the cross-correlation between the variables
• Comment on the different ranges and their implications for spatial prediction
Step 7: Discuss potential applications of the LMC model, such as cokriging for improved spatial
prediction.
Given the following paired data points (distance, semivariance), calculate the experimental
variogram and fit a spherical model to it:
(10, 2.1), (20, 3.5), (30, 4.2), (40, 4.8), (50, 5.1), (60, 5.3)
Solution:
Step 1: Plot the experimental variogram using the given data points.
Step 2: Estimate the sill, range, and nugget effect from the plot:
• Sill (C): Approximately 5.3
• Range (a): Approximately 50
• Nugget effect (C0): Approximately 0
Step 3: Use the spherical model equation:
𝛾(ℎ)={𝐶0+𝐶[3ℎ
2𝑎−1
2(ℎ
𝑎)3]for ℎ≤𝑎
𝐶0+𝐶 for ℎ>𝑎
Step 4: Substitute the estimated values into the equation:
𝛾(ℎ)={5.3[3ℎ
100−1
2(ℎ
50)3]for ℎ≤50
5.3 for ℎ>50
Step 5: Plot the fitted model alongside the experimental variogram to assess the fit.
110 PROBLEM 2: KRIGING ESTIMATION
Question: Using simple kriging, estimate the value at an unsampled location (5, 5) given the
following data points and their values:
(0, 0): 10 (0, 10): 15 (10, 0): 12 (10, 10): 18
Assume an exponential variogram model with a sill of 20 and a range of 8.
Solution:
Step 1: Calculate the distances between the unsampled location and the data points:
• d1 = sqrt((5-0)^2 + (5-0)^2) = 7.07
• d2 = sqrt((5-0)^2 + (5-10)^2) = 7.07
• d3 = sqrt((5-10)^2 + (5-0)^2) = 7.07
• d4 = sqrt((5-10)^2 + (5-10)^2) = 7.07
Step 2: Calculate the semivariances using the exponential model:
𝛾(ℎ)=20(1−𝑒−3ℎ
8)
• γ1 = γ2 = γ3 = γ4 = 20 * (1 - exp(-3 * 7.07 / 8)) = 15.8
Step 3: Set up the kriging system:
[015.8 15.8 15.8
15.8015.8 15.8
15.8 15.8 0 15.8
15.8 15.8 15.8 0 ][𝑤1
𝑤2
𝑤3
𝑤4]=[15.8
15.8
15.8
15.8]
Step 4: Solve for the weights (w1, w2, w3, w4).
Step 5: Calculate the estimated value:
𝑍∗=𝑤1∗10+𝑤2∗15+𝑤3∗12+𝑤4∗18
111 PROBLEM 3: SPATIAL AUTOCORRELATION
Question: Calculate Moran’s I for the following spatial data:
10
12
8
15
9
11
7
13
14
Assume rook’s case adjacency for the weights matrix.
Solution:
Step 1: Calculate the mean (μ) of the values: μ = (10 + 12 + 8 + 15 + 9 + 11 + 7 + 13 + 14)
/ 9 = 11
Step 2: Create the weights matrix (W) using rook’s case adjacency:
𝑊=
[
0 1 0 1 0 0 0 0 0
1 0 1 0 1 0 0 0 0
0 1 0 0 0 1 0 0 0
1 0 0 0 1 0 1 0 0
0 1 0 1 0 1 0 1 0
0 0 1 0 1 0 0 0 1
0 0 0 1 0 0 0 1 0
0 0 0 0 1 0 1 0 1
0 0 0 0 0 1 0 1 0
]
Step 3: Calculate the numerator of Moran’s I:
𝑁=∑∑𝑤𝑖𝑗
𝑛
𝑗=1
𝑛
𝑖=1 (𝑥𝑖−𝜇)(𝑥𝑗−𝜇)
Step 4: Calculate the denominator of Moran’s I:
𝐷=∑(𝑥𝑖−𝜇)2
𝑛
𝑖=1
Step 5: Calculate Moran’s I:
𝐼=𝑛
𝑊⋅𝑁
𝐷
where n is the number of observations (9) and W is the sum of all weights (24).
Step 6: Interpret the result:
• I > 0: Positive spatial autocorrelation
• I < 0: Negative spatial autocorrelation
• I ≈ 0: No spatial autocorrelation
112 PROBLEM 4: TREND SURFACE ANALYSIS
Question: Perform a first-order trend surface analysis on the following data:
X
Y
Z
1
1
10
1
2
12
2
1
15
2
2
18
Solution:
Step 1: Set up the equation for a first-order trend surface:
𝑍=𝑏0+𝑏1𝑋+𝑏2𝑌
Step 2: Create the design matrix:
𝑋=[1 1 1
1 1 2
1 2 1
1 2 2]
Step 3: Set up the normal equations: (𝑋𝑇𝑋)𝑏=𝑋𝑇𝑍
Step 4: Solve for the coefficients (b0, b1, b2):
𝑏=(𝑋𝑇𝑋)−1𝑋𝑇𝑍
Step 5: Interpret the results:
• b0: Intercept
• b1: Coefficient for X (east-west trend)
• b2: Coefficient for Y (north-south trend)
Step 6: Write the equation of the fitted trend surface:
𝑍=𝑏0+𝑏1𝑋+𝑏2𝑌
113 PROBLEM 5: SPATIAL INTERPOLATION
Question: Using inverse distance weighting (IDW) with a power of 2, estimate the value at
point (2.5, 2.5) given the following data points:
(1, 1): 10 (1, 4): 15 (4, 1): 12 (4, 4): 18
Solution:
Step 1: Calculate the distances from the estimation point to each data point:
• d1 = sqrt((2.5-1)^2 + (2.5-1)^2) = 2.12
• d2 = sqrt((2.5-1)^2 + (2.5-4)^2) = 2.69
• d3 = sqrt((2.5-4)^2 + (2.5-1)^2) = 2.69
• d4 = sqrt((2.5-4)^2 + (2.5-4)^2) = 2.12
Step 2: Calculate the weights using the inverse distance formula:
𝑤𝑖=1
𝑑𝑖2
• w1 = 1 / 2.12^2 = 0.222
• w2 = 1 / 2.69^2 = 0.138
• w3 = 1 / 2.69^2 = 0.138
• w4 = 1 / 2.12^2 = 0.222
Step 3: Normalize the weights: 𝑤𝑖′= 𝑤𝑖
∑𝑤𝑗
𝑛
𝑗=1
Step 4: Calculate the estimated value:
𝑍∗=∑𝑤𝑖
𝑛
𝑖=1 ′𝑍𝑖
where Z_i are the known values at the data points.
Step 5: Interpret the result and discuss the limitations of IDW interpolation.
114 PROBLEM 6: VARIOGRAM CLOUD ANALYSIS
Question: Given the following spatial data:
X
Y
Z
1
1
10
2
1
12
1
2
15
2
2
18
3
3
20
Create a variogram cloud and interpret the results.
Solution:
Step 1: Calculate the distances and squared differences for all pairs of points:
For each pair (i, j):
• Distance: h_ij = sqrt((x_i - x_j)^2 + (y_i - y_j)^2)
• Squared difference: γ_ij = (z_i - z_j)^2 / 2
Step 2: Create a scatter plot with distance (h) on the x-axis and semivariance (γ) on the y-axis.
Step 3: Analyze the variogram cloud:
• Look for trends or patterns in the distribution of points
• Identify any outliers or anomalies
• Observe the overall shape and spread of the cloud
Step 4: Interpret the results:
• Discuss the presence or absence of spatial correlation
• Comment on the variability of the data at different distances
• Identify any potential anisotropy or directional effects
Step 5: Consider next steps, such as binning the data to create an experimental variogram.
115 PROBLEM 7: CROSS-VALIDATION FOR KRIGING
Question: Perform leave-one-out cross-validation for ordinary kriging using the following data:
X
Y
Z
1
1
10
1
3
15
3
1
12
3
3
18
2
2
14
Assume an exponential variogram model with a sill of 20 and a range of 2.5.
Solution:
Step 1: For each data point, remove it from the dataset and estimate its value using ordinary
kriging with the remaining points.
Step 2: Calculate the kriging equations for each iteration:
[𝛾(ℎ11)⋯ 𝛾(ℎ1𝑛)1
⋮ ⋱ ⋮ ⋮
𝛾(ℎ𝑛1)⋯ 𝛾(ℎ𝑛𝑛)1
1 ⋯ 1 0][𝑤1
⋮
𝑤𝑛
𝜇]=[𝛾(ℎ10)
⋮
𝛾(ℎ𝑛0)
1]
where γ(h) is the exponential variogram model:
𝛾(ℎ)=20(1−𝑒−3ℎ
2.5)
Step 3: Solve the kriging equations for each iteration to obtain the weights and Lagrange
multiplier.
Step 4: Calculate the estimated value for each removed point:
𝑍𝑖∗=∑𝑤𝑗
𝑛−1
𝑗=1 𝑍𝑗
Step 5: Calculate the cross-validation statistics:
• Mean Error (ME): 𝑀𝐸=1
𝑛∑(𝑍𝑖∗−𝑍𝑖)
𝑛
𝑖=1
• Mean Squared Error (MSE): 𝑀𝑆𝐸=1
𝑛∑(𝑍𝑖∗−𝑍𝑖)2
𝑛
𝑖=1
• Root Mean Squared Error (RMSE): 𝑅𝑀𝑆𝐸=√𝑀𝑆𝐸
• Mean Standardized Error (MSE): 𝑀𝑆𝐸=1
𝑛∑𝑍𝑖∗−𝑍𝑖
𝜎𝑖
𝑛
𝑖=1
• Root Mean Squared Standardized Error (RMSSE): 𝑅𝑀𝑆𝑆𝐸=√1
𝑛∑(𝑍𝑖∗−𝑍𝑖
𝜎𝑖)2
𝑛
𝑖=1
Step 6: Interpret the results:
• ME should be close to 0 for unbiased estimation
• RMSE should be as small as possible
• MSE should be close to 0
• RMSSE should be close to 1
Step 7: Based on the cross-validation results, discuss the performance of the kriging model and
suggest potential improvements.
116 PROBLEM 8: INDICATOR KRIGING
Question: Perform indicator kriging to estimate the probability that the value at location (2.5,
2.5) exceeds a threshold of 15, given the following data:
X
Y
Z
1
1
10
1
4
18
4
1
12
4
4
20
Assume a spherical variogram model for the indicator variable with a sill of 0.25 and a range of
3.
Solution:
Step 1: Transform the data into indicator variables:
𝐼(𝑥)={1if 𝑍(𝑥)>15
0if 𝑍(𝑥)≤15
Step 2: Calculate the distances between the estimation point and the data points:
• d1 = sqrt((2.5-1)^2 + (2.5-1)^2) = 2.12
• d2 = sqrt((2.5-1)^2 + (2.5-4)^2) = 2.69
• d3 = sqrt((2.5-4)^2 + (2.5-1)^2) = 2.69
• d4 = sqrt((2.5-4)^2 + (2.5-4)^2) = 2.12
Step 3: Calculate the semivariances using the spherical model:
𝛾(ℎ)={0.25[3ℎ
2(3)−1
2(ℎ
3)3]for ℎ≤3
0.25 for ℎ>3
Step 4: Set up the kriging system:
[
𝛾(𝑑11)𝛾(𝑑12)𝛾(𝑑13)𝛾(𝑑14)1
𝛾(𝑑21)𝛾(𝑑22)𝛾(𝑑23)𝛾(𝑑24)1
𝛾(𝑑31)𝛾(𝑑32)𝛾(𝑑33)𝛾(𝑑34)1
𝛾(𝑑41)𝛾(𝑑42)𝛾(𝑑43)𝛾(𝑑44)1
1 1 1 1 0
]
[
𝑤1
𝑤2
𝑤3
𝑤4
𝜇
]
=
[
𝛾(𝑑1)
𝛾(𝑑2)
𝛾(𝑑3)
𝛾(𝑑4)
1
]
Step 5: Solve for the weights (w1, w2, w3, w4) and Lagrange multiplier (μ).
Step 6: Calculate the estimated probability:
𝑃∗=𝑤1𝐼(𝑥1)+𝑤2𝐼(𝑥2)+𝑤3𝐼(𝑥3)+𝑤4𝐼(𝑥4)
Step 7: Interpret the result as the probability that the value at (2.5, 2.5) exceeds the threshold
of 15.
117 PROBLEM 9: BAYESIAN GEOSTATISTICS
Question: Using a Bayesian approach, estimate the mean and variance of a spatial random
field given the following observations:
X
Y
Z
1
1
10
1
2
12
2
1
15
2
2
18
Assume a Gaussian prior with mean μ_0 = 13 and variance σ_0^2 = 16, and a measurement
error variance of σ_ε^2 = 1.
Solution:
Step 1: Define the prior distribution: 𝜇∼𝑁(𝜇0,𝜎02)
Step 2: Define the likelihood function: 𝑍𝑖|𝜇∼𝑁(𝜇,𝜎𝜀2)
Step 3: Calculate the sample mean and variance:
𝑍‾=1
𝑛∑𝑍𝑖
𝑛
𝑖=1
𝑠2=1
𝑛−1∑(𝑍𝑖−𝑍‾)2
𝑛
𝑖=1
Step 4: Calculate the posterior mean:
𝜇𝑛=𝑛
𝜎𝜀2𝑍‾+1
𝜎02𝜇0
𝑛
𝜎𝜀2+1
𝜎02
Step 5: Calculate the posterior variance:
𝜎𝑛2=1
𝑛
𝜎𝜀2+1
𝜎02
Step 6: Calculate the 95
𝐶𝐼95%=𝜇𝑛±1.96√𝜎𝑛2
Step 7: Interpret the results:
• Discuss the posterior mean and how it compares to the prior mean and sample mean
• Explain the meaning of the posterior variance and credible interval
• Comment on the influence of the prior and the data on the posterior distribution
118 PROBLEM 10: MULTIVARIATE GEOSTATISTICS
Question: Perform a linear model of coregionalization (LMC) for two variables, Z1 and Z2,
given the following cross-variogram and direct variogram models:
• γ11(h) = 10 Sph(h/100) + 5 Exp(h/50)
• γ22(h) = 15 Sph(h/100) + 8 Exp(h/50)
• γ12(h) = 8 Sph(h/100) + 4 Exp(h/50)
Where Sph and Exp denote spherical and exponential models, respectively.
Solution:
Step 1: Identify the basic structures and their ranges:
• Structure 1: Spherical model with range a1 = 100
• Structure 2: Exponential model with range a2 = 50
Step 2: Set up the coregionalization matrices for each structure:
For the spherical model:
𝐵1=[𝑏11
(1)𝑏12
(1)
𝑏21
(1)𝑏22
(1)]
For the exponential model:
𝐵2=[𝑏11
(2)𝑏12
(2)
𝑏21
(2)𝑏22
(2)]
Step 3: Solve for the elements of B1 and B2 using the sill values:
• b11^(1) = 10, b22^(1) = 15, b12^(1) = b21^(1) = 8
• b11^(2) = 5, b22^(2) = 8, b12^(2) = b21^(2) = 4
Step 4: Verify that B1 and B2 are positive semi-definite matrices:
• Calculate the determinants: det(B1) ≥ 0 and det(B2) ≥ 0
• Check that the diagonal elements are non-negative
Step 5: Write the final LMC model:
[𝛾11(ℎ)𝛾12(ℎ)
𝛾21(ℎ)𝛾22(ℎ)]=𝐵1Sph(ℎ/100)+𝐵2Exp(ℎ/50)
Step 6: Interpret the results:
• Discuss the spatial correlation structure for each variable
• Explain the cross-correlation between the variables
• Comment on the different ranges and their implications for spatial prediction
Step 7: Discuss potential applications of the LMC model, such as cokriging for improved spatial
prediction.
Given the following paired data points (distance, semivariance), calculate the experimental
variogram and fit a spherical model to it:
(10, 2.1), (20, 3.5), (30, 4.2), (40, 4.8), (50, 5.1), (60, 5.3)
Solution:
Step 1: Plot the experimental variogram using the given data points.
Step 2: Estimate the sill, range, and nugget effect from the plot:
• Sill (C): Approximately 5.3
• Range (a): Approximately 50
• Nugget effect (C0): Approximately 0
Step 3: Use the spherical model equation:
𝛾(ℎ)={𝐶0+𝐶[3ℎ
2𝑎−1
2(ℎ
𝑎)3]for ℎ≤𝑎
𝐶0+𝐶 for ℎ>𝑎
Step 4: Substitute the estimated values into the equation:
𝛾(ℎ)={5.3[3ℎ
100−1
2(ℎ
50)3]for ℎ≤50
5.3 for ℎ>50
Step 5: Plot the fitted model alongside the experimental variogram to assess the fit.
119 PROBLEM 2: KRIGING ESTIMATION
Question: Using simple kriging, estimate the value at an unsampled location (5, 5) given the
following data points and their values:
(0, 0): 10 (0, 10): 15 (10, 0): 12 (10, 10): 18
Assume an exponential variogram model with a sill of 20 and a range of 8.
Solution:
Step 1: Calculate the distances between the unsampled location and the data points:
• d1 = sqrt((5-0)^2 + (5-0)^2) = 7.07
• d2 = sqrt((5-0)^2 + (5-10)^2) = 7.07
• d3 = sqrt((5-10)^2 + (5-0)^2) = 7.07
• d4 = sqrt((5-10)^2 + (5-10)^2) = 7.07
Step 2: Calculate the semivariances using the exponential model:
𝛾(ℎ)=20(1−𝑒−3ℎ
8)
• γ1 = γ2 = γ3 = γ4 = 20 * (1 - exp(-3 * 7.07 / 8)) = 15.8
Step 3: Set up the kriging system:
[015.8 15.8 15.8
15.8015.8 15.8
15.8 15.8 0 15.8
15.8 15.8 15.8 0 ][𝑤1
𝑤2
𝑤3
𝑤4]=[15.8
15.8
15.8
15.8]
Step 4: Solve for the weights (w1, w2, w3, w4).
Step 5: Calculate the estimated value:
𝑍∗=𝑤1∗10+𝑤2∗15+𝑤3∗12+𝑤4∗18
120 PROBLEM 3: SPATIAL AUTOCORRELATION
Question: Calculate Moran’s I for the following spatial data:
10
12
8
15
9
11
7
13
14
Assume rook’s case adjacency for the weights matrix.
Solution:
Step 1: Calculate the mean (μ) of the values: μ = (10 + 12 + 8 + 15 + 9 + 11 + 7 + 13 + 14)
/ 9 = 11
Step 2: Create the weights matrix (W) using rook’s case adjacency:
𝑊=
[
0 1 0 1 0 0 0 0 0
1 0 1 0 1 0 0 0 0
0 1 0 0 0 1 0 0 0
1 0 0 0 1 0 1 0 0
0 1 0 1 0 1 0 1 0
0 0 1 0 1 0 0 0 1
0 0 0 1 0 0 0 1 0
0 0 0 0 1 0 1 0 1
0 0 0 0 0 1 0 1 0
]
Step 3: Calculate the numerator of Moran’s I:
𝑁=∑∑𝑤𝑖𝑗
𝑛
𝑗=1
𝑛
𝑖=1 (𝑥𝑖−𝜇)(𝑥𝑗−𝜇)
Step 4: Calculate the denominator of Moran’s I:
𝐷=∑(𝑥𝑖−𝜇)2
𝑛
𝑖=1
Step 5: Calculate Moran’s I:
𝐼=𝑛
𝑊⋅𝑁
𝐷
where n is the number of observations (9) and W is the sum of all weights (24).
Step 6: Interpret the result:
• I > 0: Positive spatial autocorrelation
• I < 0: Negative spatial autocorrelation
• I ≈ 0: No spatial autocorrelation
121 PROBLEM 4: TREND SURFACE ANALYSIS
Question: Perform a first-order trend surface analysis on the following data:
X
Y
Z
1
1
10
1
2
12
2
1
15
2
2
18
Solution:
Step 1: Set up the equation for a first-order trend surface:
𝑍=𝑏0+𝑏1𝑋+𝑏2𝑌
Step 2: Create the design matrix:
𝑋=[1 1 1
1 1 2
1 2 1
1 2 2]
Step 3: Set up the normal equations: (𝑋𝑇𝑋)𝑏=𝑋𝑇𝑍
Step 4: Solve for the coefficients (b0, b1, b2):
𝑏=(𝑋𝑇𝑋)−1𝑋𝑇𝑍
Step 5: Interpret the results:
• b0: Intercept
• b1: Coefficient for X (east-west trend)
• b2: Coefficient for Y (north-south trend)
Step 6: Write the equation of the fitted trend surface:
𝑍=𝑏0+𝑏1𝑋+𝑏2𝑌
122 PROBLEM 5: SPATIAL INTERPOLATION
Question: Using inverse distance weighting (IDW) with a power of 2, estimate the value at
point (2.5, 2.5) given the following data points:
(1, 1): 10 (1, 4): 15 (4, 1): 12 (4, 4): 18
Solution:
Step 1: Calculate the distances from the estimation point to each data point:
• d1 = sqrt((2.5-1)^2 + (2.5-1)^2) = 2.12
• d2 = sqrt((2.5-1)^2 + (2.5-4)^2) = 2.69
• d3 = sqrt((2.5-4)^2 + (2.5-1)^2) = 2.69
• d4 = sqrt((2.5-4)^2 + (2.5-4)^2) = 2.12
Step 2: Calculate the weights using the inverse distance formula:
𝑤𝑖=1
𝑑𝑖2
• w1 = 1 / 2.12^2 = 0.222
• w2 = 1 / 2.69^2 = 0.138
• w3 = 1 / 2.69^2 = 0.138
• w4 = 1 / 2.12^2 = 0.222
Step 3: Normalize the weights: 𝑤𝑖′= 𝑤𝑖
∑𝑤𝑗
𝑛
𝑗=1
Step 4: Calculate the estimated value:
𝑍∗=∑𝑤𝑖
𝑛
𝑖=1 ′𝑍𝑖
where Z_i are the known values at the data points.
Step 5: Interpret the result and discuss the limitations of IDW interpolation.
123 PROBLEM 6: VARIOGRAM CLOUD ANALYSIS
Question: Given the following spatial data:
X
Y
Z
1
1
10
2
1
12
1
2
15
2
2
18
3
3
20
Create a variogram cloud and interpret the results.
Solution:
Step 1: Calculate the distances and squared differences for all pairs of points:
For each pair (i, j):
• Distance: h_ij = sqrt((x_i - x_j)^2 + (y_i - y_j)^2)
• Squared difference: γ_ij = (z_i - z_j)^2 / 2
Step 2: Create a scatter plot with distance (h) on the x-axis and semivariance (γ) on the y-axis.
Step 3: Analyze the variogram cloud:
• Look for trends or patterns in the distribution of points
• Identify any outliers or anomalies
• Observe the overall shape and spread of the cloud
Step 4: Interpret the results:
• Discuss the presence or absence of spatial correlation
• Comment on the variability of the data at different distances
• Identify any potential anisotropy or directional effects
Step 5: Consider next steps, such as binning the data to create an experimental variogram.
124 PROBLEM 7: CROSS-VALIDATION FOR KRIGING
Question: Perform leave-one-out cross-validation for ordinary kriging using the following data:
X
Y
Z
1
1
10
1
3
15
X
Y
Z
3
1
12
3
3
18
2
2
14
Assume an exponential variogram model with a sill of 20 and a range of 2.5.
Solution:
Step 1: For each data point, remove it from the dataset and estimate its value using ordinary
kriging with the remaining points.
Step 2: Calculate the kriging equations for each iteration:
[𝛾(ℎ11)⋯ 𝛾(ℎ1𝑛)1
⋮ ⋱ ⋮ ⋮
𝛾(ℎ𝑛1)⋯ 𝛾(ℎ𝑛𝑛)1
1 ⋯ 1 0][𝑤1
⋮
𝑤𝑛
𝜇]=[𝛾(ℎ10)
⋮
𝛾(ℎ𝑛0)
1]
where γ(h) is the exponential variogram model:
𝛾(ℎ)=20(1−𝑒−3ℎ
2.5)
Step 3: Solve the kriging equations for each iteration to obtain the weights and Lagrange
multiplier.
Step 4: Calculate the estimated value for each removed point:
𝑍𝑖∗=∑𝑤𝑗
𝑛−1
𝑗=1 𝑍𝑗
Step 5: Calculate the cross-validation statistics:
• Mean Error (ME): 𝑀𝐸=1
𝑛∑(𝑍𝑖∗−𝑍𝑖)
𝑛
𝑖=1
• Mean Squared Error (MSE): 𝑀𝑆𝐸=1
𝑛∑(𝑍𝑖∗−𝑍𝑖)2
𝑛
𝑖=1
• Root Mean Squared Error (RMSE): 𝑅𝑀𝑆𝐸=√𝑀𝑆𝐸
• Mean Standardized Error (MSE): 𝑀𝑆𝐸=1
𝑛∑𝑍𝑖∗−𝑍𝑖
𝜎𝑖
𝑛
𝑖=1
• Root Mean Squared Standardized Error (RMSSE): 𝑅𝑀𝑆𝑆𝐸=√1
𝑛∑(𝑍𝑖∗−𝑍𝑖
𝜎𝑖)2
𝑛
𝑖=1
Step 6: Interpret the results:
• ME should be close to 0 for unbiased estimation
• RMSE should be as small as possible
• MSE should be close to 0
• RMSSE should be close to 1
Step 7: Based on the cross-validation results, discuss the performance of the kriging model and
suggest potential improvements.
125 PROBLEM 8: INDICATOR KRIGING
Question: Perform indicator kriging to estimate the probability that the value at location (2.5,
2.5) exceeds a threshold of 15, given the following data:
X
Y
Z
1
1
10
1
4
18
4
1
12
4
4
20
Assume a spherical variogram model for the indicator variable with a sill of 0.25 and a range of
3.
Solution:
Step 1: Transform the data into indicator variables:
𝐼(𝑥)={1if 𝑍(𝑥)>15
0if 𝑍(𝑥)≤15
Step 2: Calculate the distances between the estimation point and the data points:
• d1 = sqrt((2.5-1)^2 + (2.5-1)^2) = 2.12
• d2 = sqrt((2.5-1)^2 + (2.5-4)^2) = 2.69
• d3 = sqrt((2.5-4)^2 + (2.5-1)^2) = 2.69
• d4 = sqrt((2.5-4)^2 + (2.5-4)^2) = 2.12
Step 3: Calculate the semivariances using the spherical model:
𝛾(ℎ)={0.25[3ℎ
2(3)−1
2(ℎ
3)3]for ℎ≤3
0.25 for ℎ>3
Step 4: Set up the kriging system:
[
𝛾(𝑑11)𝛾(𝑑12)𝛾(𝑑13)𝛾(𝑑14)1
𝛾(𝑑21)𝛾(𝑑22)𝛾(𝑑23)𝛾(𝑑24)1
𝛾(𝑑31)𝛾(𝑑32)𝛾(𝑑33)𝛾(𝑑34)1
𝛾(𝑑41)𝛾(𝑑42)𝛾(𝑑43)𝛾(𝑑44)1
1 1 1 1 0
]
[
𝑤1
𝑤2
𝑤3
𝑤4
𝜇
]
=
[
𝛾(𝑑1)
𝛾(𝑑2)
𝛾(𝑑3)
𝛾(𝑑4)
1
]
Step 5: Solve for the weights (w1, w2, w3, w4) and Lagrange multiplier (μ).
Step 6: Calculate the estimated probability:
𝑃∗=𝑤1𝐼(𝑥1)+𝑤2𝐼(𝑥2)+𝑤3𝐼(𝑥3)+𝑤4𝐼(𝑥4)
Step 7: Interpret the result as the probability that the value at (2.5, 2.5) exceeds the threshold
of 15.
126 PROBLEM 9: BAYESIAN GEOSTATISTICS
Question: Using a Bayesian approach, estimate the mean and variance of a spatial random
field given the following observations:
X
Y
Z
1
1
10
1
2
12
2
1
15
2
2
18
Assume a Gaussian prior with mean μ_0 = 13 and variance σ_0^2 = 16, and a measurement
error variance of σ_ε^2 = 1.
Solution:
Step 1: Define the prior distribution: 𝜇∼𝑁(𝜇0,𝜎02)
Step 2: Define the likelihood function: 𝑍𝑖|𝜇∼𝑁(𝜇,𝜎𝜀2)
Step 3: Calculate the sample mean and variance:
𝑍‾=1
𝑛∑𝑍𝑖
𝑛
𝑖=1
𝑠2=1
𝑛−1∑(𝑍𝑖−𝑍‾)2
𝑛
𝑖=1
Step 4: Calculate the posterior mean:
𝜇𝑛=𝑛
𝜎𝜀2𝑍‾+1
𝜎02𝜇0
𝑛
𝜎𝜀2+1
𝜎02
Step 5: Calculate the posterior variance:
𝜎𝑛2=1
𝑛
𝜎𝜀2+1
𝜎02
Step 6: Calculate the 95
𝐶𝐼95%=𝜇𝑛±1.96√𝜎𝑛2
Step 7: Interpret the results:
• Discuss the posterior mean and how it compares to the prior mean and sample mean
• Explain the meaning of the posterior variance and credible interval
• Comment on the influence of the prior and the data on the posterior distribution
127 PROBLEM 10: MULTIVARIATE GEOSTATISTICS
Question: Perform a linear model of coregionalization (LMC) for two variables, Z1 and Z2,
given the following cross-variogram and direct variogram models:
• γ11(h) = 10 Sph(h/100) + 5 Exp(h/50)
• γ22(h) = 15 Sph(h/100) + 8 Exp(h/50)
• γ12(h) = 8 Sph(h/100) + 4 Exp(h/50)
Where Sph and Exp denote spherical and exponential models, respectively.
Solution:
Step 1: Identify the basic structures and their ranges:
• Structure 1: Spherical model with range a1 = 100
• Structure 2: Exponential model with range a2 = 50
Step 2: Set up the coregionalization matrices for each structure:
For the spherical model:
𝐵1=[𝑏11
(1)𝑏12
(1)
𝑏21
(1)𝑏22
(1)]
For the exponential model:
𝐵2=[𝑏11
(2)𝑏12
(2)
𝑏21
(2)𝑏22
(2)]
Step 3: Solve for the elements of B1 and B2 using the sill values:
• b11^(1) = 10, b22^(1) = 15, b12^(1) = b21^(1) = 8
• b11^(2) = 5, b22^(2) = 8, b12^(2) = b21^(2) = 4
Step 4: Verify that B1 and B2 are positive semi-definite matrices:
• Calculate the determinants: det(B1) ≥ 0 and det(B2) ≥ 0
• Check that the diagonal elements are non-negative
Step 5: Write the final LMC model:
[𝛾11(ℎ)𝛾12(ℎ)
𝛾21(ℎ)𝛾22(ℎ)]=𝐵1Sph(ℎ/100)+𝐵2Exp(ℎ/50)
Step 6: Interpret the results:
• Discuss the spatial correlation structure for each variable
• Explain the cross-correlation between the variables
• Comment on the different ranges and their implications for spatial prediction
Step 7: Discuss potential applications of the LMC model, such as cokriging for improved spatial
prediction.
Given the following paired data points (distance, semivariance), calculate the experimental
variogram and fit a spherical model to it:
(10, 2.1), (20, 3.5), (30, 4.2), (40, 4.8), (50, 5.1), (60, 5.3)
Solution:
Step 1: Plot the experimental variogram using the given data points.
Step 2: Estimate the sill, range, and nugget effect from the plot:
• Sill (C): Approximately 5.3
• Range (a): Approximately 50
• Nugget effect (C0): Approximately 0
Step 3: Use the spherical model equation:
𝛾(ℎ)={𝐶0+𝐶[3ℎ
2𝑎−1
2(ℎ
𝑎)3]for ℎ≤𝑎
𝐶0+𝐶 for ℎ>𝑎
Step 4: Substitute the estimated values into the equation:
𝛾(ℎ)={5.3[3ℎ
100−1
2(ℎ
50)3]for ℎ≤50
5.3 for ℎ>50
Step 5: Plot the fitted model alongside the experimental variogram to assess the fit.
128 PROBLEM 2: KRIGING ESTIMATION
Question: Using simple kriging, estimate the value at an unsampled location (5, 5) given the
following data points and their values:
(0, 0): 10 (0, 10): 15 (10, 0): 12 (10, 10): 18
Assume an exponential variogram model with a sill of 20 and a range of 8.
Solution:
Step 1: Calculate the distances between the unsampled location and the data points:
• d1 = sqrt((5-0)^2 + (5-0)^2) = 7.07
• d2 = sqrt((5-0)^2 + (5-10)^2) = 7.07
• d3 = sqrt((5-10)^2 + (5-0)^2) = 7.07
• d4 = sqrt((5-10)^2 + (5-10)^2) = 7.07
Step 2: Calculate the semivariances using the exponential model:
𝛾(ℎ)=20(1−𝑒−3ℎ
8)
• γ1 = γ2 = γ3 = γ4 = 20 * (1 - exp(-3 * 7.07 / 8)) = 15.8
Step 3: Set up the kriging system:
[015.8 15.8 15.8
15.8015.8 15.8
15.8 15.8 0 15.8
15.8 15.8 15.8 0 ][𝑤1
𝑤2
𝑤3
𝑤4]=[15.8
15.8
15.8
15.8]
Step 4: Solve for the weights (w1, w2, w3, w4).
Step 5: Calculate the estimated value:
𝑍∗=𝑤1∗10+𝑤2∗15+𝑤3∗12+𝑤4∗18
129 PROBLEM 3: SPATIAL AUTOCORRELATION
Question: Calculate Moran’s I for the following spatial data:
10
12
8
15
9
11
7
13
14
Assume rook’s case adjacency for the weights matrix.
Solution:
Step 1: Calculate the mean (μ) of the values: μ = (10 + 12 + 8 + 15 + 9 + 11 + 7 + 13 + 14)
/ 9 = 11
Step 2: Create the weights matrix (W) using rook’s case adjacency:
𝑊=
[
0 1 0 1 0 0 0 0 0
1 0 1 0 1 0 0 0 0
0 1 0 0 0 1 0 0 0
1 0 0 0 1 0 1 0 0
0 1 0 1 0 1 0 1 0
0 0 1 0 1 0 0 0 1
0 0 0 1 0 0 0 1 0
0 0 0 0 1 0 1 0 1
0 0 0 0 0 1 0 1 0
]
Step 3: Calculate the numerator of Moran’s I:
𝑁=∑∑𝑤𝑖𝑗
𝑛
𝑗=1
𝑛
𝑖=1 (𝑥𝑖−𝜇)(𝑥𝑗−𝜇)
Step 4: Calculate the denominator of Moran’s I:
𝐷=∑(𝑥𝑖−𝜇)2
𝑛
𝑖=1
Step 5: Calculate Moran’s I:
𝐼=𝑛
𝑊⋅𝑁
𝐷
where n is the number of observations (9) and W is the sum of all weights (24).
Step 6: Interpret the result:
• I > 0: Positive spatial autocorrelation
• I < 0: Negative spatial autocorrelation
• I ≈ 0: No spatial autocorrelation
130 PROBLEM 4: TREND SURFACE ANALYSIS
Question: Perform a first-order trend surface analysis on the following data:
X
Y
Z
1
1
10
1
2
12
2
1
15
2
2
18
Solution:
Step 1: Set up the equation for a first-order trend surface:
𝑍=𝑏0+𝑏1𝑋+𝑏2𝑌
Step 2: Create the design matrix:
𝑋=[1 1 1
1 1 2
1 2 1
1 2 2]
Step 3: Set up the normal equations: (𝑋𝑇𝑋)𝑏=𝑋𝑇𝑍
Step 4: Solve for the coefficients (b0, b1, b2):
𝑏=(𝑋𝑇𝑋)−1𝑋𝑇𝑍
Step 5: Interpret the results:
• b0: Intercept
• b1: Coefficient for X (east-west trend)
• b2: Coefficient for Y (north-south trend)
Step 6: Write the equation of the fitted trend surface:
𝑍=𝑏0+𝑏1𝑋+𝑏2𝑌
131 PROBLEM 5: SPATIAL INTERPOLATION
Question: Using inverse distance weighting (IDW) with a power of 2, estimate the value at
point (2.5, 2.5) given the following data points:
(1, 1): 10 (1, 4): 15 (4, 1): 12 (4, 4): 18
Solution:
Step 1: Calculate the distances from the estimation point to each data point:
• d1 = sqrt((2.5-1)^2 + (2.5-1)^2) = 2.12
• d2 = sqrt((2.5-1)^2 + (2.5-4)^2) = 2.69
• d3 = sqrt((2.5-4)^2 + (2.5-1)^2) = 2.69
• d4 = sqrt((2.5-4)^2 + (2.5-4)^2) = 2.12
Step 2: Calculate the weights using the inverse distance formula:
𝑤𝑖=1
𝑑𝑖2
• w1 = 1 / 2.12^2 = 0.222
• w2 = 1 / 2.69^2 = 0.138
• w3 = 1 / 2.69^2 = 0.138
• w4 = 1 / 2.12^2 = 0.222
Step 3: Normalize the weights: 𝑤𝑖′= 𝑤𝑖
∑𝑤𝑗
𝑛
𝑗=1
Step 4: Calculate the estimated value:
𝑍∗=∑𝑤𝑖
𝑛
𝑖=1 ′𝑍𝑖
where Z_i are the known values at the data points.
Step 5: Interpret the result and discuss the limitations of IDW interpolation.
132 PROBLEM 6: VARIOGRAM CLOUD ANALYSIS
Question: Given the following spatial data:
X
Y
Z
1
1
10
2
1
12
1
2
15
2
2
18
3
3
20
Create a variogram cloud and interpret the results.
Solution:
Step 1: Calculate the distances and squared differences for all pairs of points:
For each pair (i, j):
• Distance: h_ij = sqrt((x_i - x_j)^2 + (y_i - y_j)^2)
• Squared difference: γ_ij = (z_i - z_j)^2 / 2
Step 2: Create a scatter plot with distance (h) on the x-axis and semivariance (γ) on the y-axis.
Step 3: Analyze the variogram cloud:
• Look for trends or patterns in the distribution of points
• Identify any outliers or anomalies
• Observe the overall shape and spread of the cloud
Step 4: Interpret the results:
• Discuss the presence or absence of spatial correlation
• Comment on the variability of the data at different distances
• Identify any potential anisotropy or directional effects
Step 5: Consider next steps, such as binning the data to create an experimental variogram.
133 PROBLEM 7: CROSS-VALIDATION FOR KRIGING
Question: Perform leave-one-out cross-validation for ordinary kriging using the following data:
X
Y
Z
1
1
10
1
3
15
3
1
12
3
3
18
2
2
14
Assume an exponential variogram model with a sill of 20 and a range of 2.5.
Solution:
Step 1: For each data point, remove it from the dataset and estimate its value using ordinary
kriging with the remaining points.
Step 2: Calculate the kriging equations for each iteration:
[𝛾(ℎ11)⋯ 𝛾(ℎ1𝑛)1
⋮ ⋱ ⋮ ⋮
𝛾(ℎ𝑛1)⋯ 𝛾(ℎ𝑛𝑛)1
1 ⋯ 1 0][𝑤1
⋮
𝑤𝑛
𝜇]=[𝛾(ℎ10)
⋮
𝛾(ℎ𝑛0)
1]
where γ(h) is the exponential variogram model:
𝛾(ℎ)=20(1−𝑒−3ℎ
2.5)
Step 3: Solve the kriging equations for each iteration to obtain the weights and Lagrange
multiplier.
Step 4: Calculate the estimated value for each removed point:
𝑍𝑖∗=∑𝑤𝑗
𝑛−1
𝑗=1 𝑍𝑗
Step 5: Calculate the cross-validation statistics:
• Mean Error (ME): 𝑀𝐸=1
𝑛∑(𝑍𝑖∗−𝑍𝑖)
𝑛
𝑖=1
• Mean Squared Error (MSE): 𝑀𝑆𝐸=1
𝑛∑(𝑍𝑖∗−𝑍𝑖)2
𝑛
𝑖=1
• Root Mean Squared Error (RMSE): 𝑅𝑀𝑆𝐸=√𝑀𝑆𝐸
• Mean Standardized Error (MSE): 𝑀𝑆𝐸=1
𝑛∑𝑍𝑖∗−𝑍𝑖
𝜎𝑖
𝑛
𝑖=1
• Root Mean Squared Standardized Error (RMSSE): 𝑅𝑀𝑆𝑆𝐸=√1
𝑛∑(𝑍𝑖∗−𝑍𝑖
𝜎𝑖)2
𝑛
𝑖=1
Step 6: Interpret the results:
• ME should be close to 0 for unbiased estimation
• RMSE should be as small as possible
• MSE should be close to 0
• RMSSE should be close to 1
Step 7: Based on the cross-validation results, discuss the performance of the kriging model and
suggest potential improvements.
134 PROBLEM 8: INDICATOR KRIGING
Question: Perform indicator kriging to estimate the probability that the value at location (2.5,
2.5) exceeds a threshold of 15, given the following data:
X
Y
Z
1
1
10
1
4
18
4
1
12
4
4
20
Assume a spherical variogram model for the indicator variable with a sill of 0.25 and a range of
3.
Solution:
Step 1: Transform the data into indicator variables:
𝐼(𝑥)={1if 𝑍(𝑥)>15
0if 𝑍(𝑥)≤15
Step 2: Calculate the distances between the estimation point and the data points:
• d1 = sqrt((2.5-1)^2 + (2.5-1)^2) = 2.12
• d2 = sqrt((2.5-1)^2 + (2.5-4)^2) = 2.69
• d3 = sqrt((2.5-4)^2 + (2.5-1)^2) = 2.69
• d4 = sqrt((2.5-4)^2 + (2.5-4)^2) = 2.12
Step 3: Calculate the semivariances using the spherical model:
𝛾(ℎ)={0.25[3ℎ
2(3)−1
2(ℎ
3)3]for ℎ≤3
0.25 for ℎ>3
Step 4: Set up the kriging system:
[
𝛾(𝑑11)𝛾(𝑑12)𝛾(𝑑13)𝛾(𝑑14)1
𝛾(𝑑21)𝛾(𝑑22)𝛾(𝑑23)𝛾(𝑑24)1
𝛾(𝑑31)𝛾(𝑑32)𝛾(𝑑33)𝛾(𝑑34)1
𝛾(𝑑41)𝛾(𝑑42)𝛾(𝑑43)𝛾(𝑑44)1
1 1 1 1 0
]
[
𝑤1
𝑤2
𝑤3
𝑤4
𝜇
]
=
[
𝛾(𝑑1)
𝛾(𝑑2)
𝛾(𝑑3)
𝛾(𝑑4)
1
]
Step 5: Solve for the weights (w1, w2, w3, w4) and Lagrange multiplier (μ).
Step 6: Calculate the estimated probability:
𝑃∗=𝑤1𝐼(𝑥1)+𝑤2𝐼(𝑥2)+𝑤3𝐼(𝑥3)+𝑤4𝐼(𝑥4)
Step 7: Interpret the result as the probability that the value at (2.5, 2.5) exceeds the threshold
of 15.
135 PROBLEM 9: BAYESIAN GEOSTATISTICS
Question: Using a Bayesian approach, estimate the mean and variance of a spatial random
field given the following observations:
X
Y
Z
1
1
10
1
2
12
2
1
15
2
2
18
Assume a Gaussian prior with mean μ_0 = 13 and variance σ_0^2 = 16, and a measurement
error variance of σ_ε^2 = 1.
Solution:
Step 1: Define the prior distribution: 𝜇∼𝑁(𝜇0,𝜎02)
Step 2: Define the likelihood function: 𝑍𝑖|𝜇∼𝑁(𝜇,𝜎𝜀2)
Step 3: Calculate the sample mean and variance:
𝑍‾=1
𝑛∑𝑍𝑖
𝑛
𝑖=1
𝑠2=1
𝑛−1∑(𝑍𝑖−𝑍‾)2
𝑛
𝑖=1
Step 4: Calculate the posterior mean:
𝜇𝑛=𝑛
𝜎𝜀2𝑍‾+1
𝜎02𝜇0
𝑛
𝜎𝜀2+1
𝜎02
Step 5: Calculate the posterior variance:
𝜎𝑛2=1
𝑛
𝜎𝜀2+1
𝜎02
Step 6: Calculate the 95
𝐶𝐼95%=𝜇𝑛±1.96√𝜎𝑛2
Step 7: Interpret the results:
• Discuss the posterior mean and how it compares to the prior mean and sample mean
• Explain the meaning of the posterior variance and credible interval
• Comment on the influence of the prior and the data on the posterior distribution
136 PROBLEM 10: MULTIVARIATE GEOSTATISTICS
Question: Perform a linear model of coregionalization (LMC) for two variables, Z1 and Z2,
given the following cross-variogram and direct variogram models:
• γ11(h) = 10 Sph(h/100) + 5 Exp(h/50)
• γ22(h) = 15 Sph(h/100) + 8 Exp(h/50)
• γ12(h) = 8 Sph(h/100) + 4 Exp(h/50)
Where Sph and Exp denote spherical and exponential models, respectively.
Solution:
Step 1: Identify the basic structures and their ranges:
• Structure 1: Spherical model with range a1 = 100
• Structure 2: Exponential model with range a2 = 50
Step 2: Set up the coregionalization matrices for each structure:
For the spherical model:
𝐵1=[𝑏11
(1)𝑏12
(1)
𝑏21
(1)𝑏22
(1)]
For the exponential model:
𝐵2=[𝑏11
(2)𝑏12
(2)
𝑏21
(2)𝑏22
(2)]
Step 3: Solve for the elements of B1 and B2 using the sill values:
• b11^(1) = 10, b22^(1) = 15, b12^(1) = b21^(1) = 8
• b11^(2) = 5, b22^(2) = 8, b12^(2) = b21^(2) = 4
Step 4: Verify that B1 and B2 are positive semi-definite matrices:
• Calculate the determinants: det(B1) ≥ 0 and det(B2) ≥ 0
• Check that the diagonal elements are non-negative
Step 5: Write the final LMC model:
[𝛾11(ℎ)𝛾12(ℎ)
𝛾21(ℎ)𝛾22(ℎ)]=𝐵1Sph(ℎ/100)+𝐵2Exp(ℎ/50)
Step 6: Interpret the results:
• Discuss the spatial correlation structure for each variable
• Explain the cross-correlation between the variables
• Comment on the different ranges and their implications for spatial prediction
Step 7: Discuss potential applications of the LMC model, such as cokriging for improved spatial
prediction.
