Statistic
Giego
all_earthsc_5642_ex2-3_09feb15-2_all.docEARTHSC 5642 – GEOMATHEMATICAL ANALYSIS
Exercise 2.1
Referring to Fig. 2.1 on page 24/100 of the notes=> 5642Lectures_2_4.pdf, consider a set of 5 horizontally infinite cylinders with the following parameters (
|
Cylinder # |
1 |
2 |
3 |
4 |
5 |
|
d (km) |
-34 |
-20 |
0 |
20 |
35 |
|
z (km) |
2 |
3 |
5 |
10 |
10 |
|
R (km) |
2 |
1 |
3 |
4 |
3.5 |
|
Δρ (gm/cm3) |
3.0 |
5.0 |
1.0 |
2.0 |
1.5 |
A) Along a bisecting profile extending from d = -64 km through 0 km to +64 km at 1-km intervals, compute the 5 gravity profiles in mgal by
gz = [41.93 Δρ(R2/z)]/[(d2/z2) + 1],
and plot them superimposed on a single graph using different colors or symbols. Computer software (e.g., IMSL, LINPACK, Matlab, Mathematica, MathCad, Maple, etc.) may be helpful here.
B) Compute and 1) plot the total gravity effect of the 5 cylinders by summing their effects at each observation point on the profile. 2) What is the mean value and standard deviation of the total gravity effect? 3) What is the utility of these statistics for graphing the profile?
C) Suppose you want to estimate the 5 densities (Δρi) from the total gravity observations in B-above. Determine 1) the [ATA]-matrix and 2) least-squares estimates of Δρi, and 3) compare the estimated densities with those in the above table.
D) Determine 1) the Choleski factorization of [ATA] – i.e., determine a lower triangular matrix L such that [LLT = ATA]. 2) Find the coefficients of [P] such that [LP = ATB], and 3) solve the system for the least-squares estimates of Δρi. 4) Compare your density estimates with those you obtained in C-above.
EARTHSC 5642 – GEOMATHEMATICAL ANALYSIS
Exercise 2.3
A) For the data set in Exercise 2.1.B, evaluate (XTATB), (BTB), and (BTB – XTATB).
For the solution that you obtained in Exercise 2.1.C (
B) Use the results from A-above to evaluate the correlation coefficient (r).
C) Evaluate (XT(ATA) X) and the unbiased
m
n
X
A
A
X
B
B
S
B
T
T
T
-
-
=
@
)
(
)
var(
2
.D) Calculate the variance/covariance matrix of AX = B.
E) Calculate the 95% confidence intervals for the xj ( X.
F) Calculate the eight values of var(
b
ˆ
i) at i = 1, 16, 32, …, 128.G) Calculate 95% confidence intervals on the eight
b
ˆ
i at i = 1, 16, 32, …, 128.H) Graph the model
B
ˆ
and its 95% confidence envelop on a scatter plot of the data.I) Make a plot of the residuals and analyze its characteristics.
J) Fill in the ANOVA table below and test the hypothesis Ho :
B
ˆ
( B with (-risk = 0.05ANOVA
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Source of Variation |
( |
CSS |
MS |
F |
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Total |
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Regression |
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Residual |
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EARTHSC 5642 – GEOMATHEMATICAL ANALYSIS
Exercise 5.1
Suppose the function g(t) has the Fourier Transform (FT) given by G(f).
A) Prove the FT-pair
∂g (t )
∂t
⇔ (j2πf) G(f).
B) Prove the FT-pair
∂ p g (t )
∂t p
⇔ (j2πf) p G(f).
C) Prove the FT-pair (-j2πt) g(t) ⇔
would be useful?
∂G( f ) . Give a practical example where this result
∂f
t1 ∞
D) Prove the FT-pair ∫−∞ g (t )dt ⇔ (1/ j2πf) G(f) iff ∫−∞ g (t1 )dt = G(0) = 0 .
Examples of the use of some of these operational properties of the FT are given in the attached APPENDIX_A2.1 from Jenkins and Watts (1968) .
Suppose function g(x, y) with Fourier Transform G(kx, ky) [ie., g(x, y) ⇔ G(kx, ky)] satisfies Laplace’s Equation => ∂2g/∂x2 + ∂2g/∂y2 + ∂2g/∂z2 = 0 for the z-dimension directed normal to the (x,y)-plane. Find the Fourier Transform pair for its
E) first derivative with respect to x. F) first derivative with respect to y. G) first derivative with respect to z.
H) second derivative with respect to x2.
I) second derivative with respect to y2.
J) second derivative with respect to z2.
K) second derivative with respect to x and y. L) second derivative with respect to x and z. M) second derivative with respect to y and z.
EARTHSC 5642 – GEOMATHEMATICAL ANALYSIS
Exercise 5.3
A) Show that the gravity effect of the horizontal cylinder=>
gz = [41.93 Δρ(R2/z)]/[(d2/z2) + 1]
satisfies Laplace’s equation=> ∂2gz/∂d2 + ∂2gz/∂z2 = 0.
For the gravity effects of the 5 buried horizontal cylinders obtained in Exercise 2.1
B) Compute, list, and plot the related amplitude and phase spectra.
C) Inverse transform the Fourier coefficients and compare the synthesized signal with the original. What are the sources of the mismatches?
D) Compute, list and plot the first horizontal derivative ∂gz/∂d from the FFT of (gz).
E) How do the results in D-above compare with the analytical horizontal first derivative gravity effects of the buried horizontal cylinders?
F) Compute, list, and plot the second horizontal derivative ∂2gz/∂d2 from the FFT of (gz).
G) How do the results in F-above compare with the analytical horizontal second derivative gravity effects of the buried horizontal cylinders?
H) Compute, list and plot the first vertical derivative ∂gz/∂z from the FFT of (gz).
I) How do the results in H-above compare with the analytical vertical first derivative gravity effects of the buried horizontal cylinders?
J) Compute, list, and plot the second vertical derivative ∂2gz/∂z2 from the FFT of (gz).
K) How do the results in J-above compare with the analytical vertical second derivative gravity effects of the buried horizontal cylinders?
L) Compute, list, and plot (gz) from the FFT of the analytical (∂2gz/∂z2) in K-above.
M) How do the FFT estimates from L-above compare with the gravity effects of the 5 buried horizontal cylinders obtained in Exercise 2.1?
