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Hyperspectral remote sensing for shallow waters. I. A semianalytical model

Zhongping Lee, Kendall L. Carder, Curtis D. Mobley, Robert G. Steward, and Jennifer S. Patch

For analytical or semianalytical retrieval of shallow-water bathymetry andyor optical properties of the water column from remote sensing, the contribution to the remotely sensed signal from the water column has to be separated from that of the bottom. The mathematical separation involves three diffuse attenuation coefficients: one for the downwelling irradiance ~Kd!, one for the upwelling radiance of the water column ~Ku

C!, and one for the upwelling radiance from bottom reflection ~Ku B!. Because of the

differences in photon origination and path lengths, these three coefficients in general are not equal, although their equality has been assumed in many previous studies. By use of the Hydrolight radiative- transfer numerical model with a particle phase function typical of coastal waters, the remote-sensing reflectance above ~Rrs! and below ~rrs! the surface is calculated for various combinations of optical properties, bottom albedos, bottom depths, and solar zenith angles. A semianalytical ~SA! model for rrs of shallow waters is then developed, in which the diffuse attenuation coefficients are explicitly expressed as functions of in-water absorption ~a! and backscattering ~bb!. For remote-sensing inversion, param- eters connecting Rrs and rrs are also derived. It is found that rrs values determined by the SA model agree well with the exact values computed by Hydrolight ~;3% error!, even for Hydrolight rrs values calculated with different particle phase functions. The Hydrolight calculations included bbya values as high as 1.5 to simulate high-turbidity situations that are occasionally found in coastal regions. © 1998 Optical Society of America

OCIS codes: 010.0010, 280.0280.

b u

b s c e w f o

1. Introduction

Single- or quasi-single-scattering theory1 or numeri- cal simulations suggest that subsurface upwelling signals ~SUS2, see Section 8 for symbols used in this text! can be approximated as a sum of contributions rom the water column and from the bottom:2–6

SUS < SUSdp@1 2 exp~ 2 2KH!# 1 SUSB exp~ 2 2KH!.

(1)

Here SUS can designate either the subsurface up- welling radiance ~Lu2!, the subsurface irradiance re- flectance ~R!, or the subsurface remote-sensing

Z. Lee, K. L. Carder, R. G. Steward, and J. S. Patch are with the Department of Marine Science, University of South Florida, St. Petersburg, Florida 33701. C. D. Mobley is with Sequoia Scien- tific, Incorporated, 9725 S. E. 36th Street, Suite 308, Mercer Is- land, Washington 98040.

Received 3 December 1997; revised manuscript received 18 May 1998.

reflectance ~rrs!; we use subsurface to mean just be- low the air–water surface. SUS

dp is for optically deep waters, whereas SUS

B is for the upwelling sig- nals just above the bottom. H is the depth of the ottom. K is usually described as an effective atten- ation coefficient.2–4,6,7 The first term on the right

side of approximation ~1! is the water-column contri- bution, although in some studies this term was re- placed by a nearby deep-water signal.2,4,7,8 The econd term is the bottom contribution. Because the effective attenuation coefficient is am-

iguous and hard to determine from remotely mea- ured data, and because the diffuse attenuation oefficient for downwelling light is not generally qual to the diffuse attenuation coefficients for up- elling light, a more general expression for subsur-

ace remote-sensing reflectance, defined as the ratio f upwelling radiance ~Lu2! to downwelling irradi-

ance ~Ed2! evaluated just below the surface, is

rrs < rrsdp$1 2 A0 exp@2~Kd 1 KuC!H#% 1 A1r

3 exp@2~Kd 1 Ku B!H#, (2)

0 September 1998 y Vol. 37, No. 27 y APPLIED OPTICS 6329

c t t

where rrs is the rrs value for optically deep waters, Kd is the vertically averaged diffuse attenuation co- efficient for downwelling irradiance, Ku

C is the ver- tically averaged diffuse attenuation coefficient for upwelling radiance from water-column scattering, and Ku

B is the vertically averaged diffuse attenuation oefficient for upwelling radiance from bottom reflec- ance. r is the bottom irradiance reflectance, and he bottom is assumed to be a Lambertian reflector.

We introduce two additional factors into approxi- ation ~2!: A0 and A1. According to quasi-single-

scattering theory1, A0 would be 1.0 and A1 would be 1yp for a Lambertian bottom. However, we want to determine values for A0 and A1 from the exact nu- merical simulations to make the proposed approxi- mation ~2! as accurate as possible.

If we define

a ; a 1 bb, (3)

then any diffuse attenuation coefficient K can be re- lated to a through an appropriate distribution func- tion ~D! ~Ref. 9!:

K 5 Da. (4)

In particular,

Kd 5 Dda, Ku C 5 Du

Ca, Ku B 5 Du

Ba, (5)

which allows us to rewrite approximation ~2! as

rrs < rrsdp$1 2 A0 exp@2~Dd 1 DuC!aH#%

1 A1r exp@2~Dd 1 Du B!aH#. (6)

ote that a is an inherent optical property.9 Thus all the variability in the K’s associated with the di- rectional structure of the light field is now trans- ferred to the distribution functions ~D!.

In many previous remote-sensing applications it was assumed that Dd 5 Du

C 5 Du B 5 a constant,2,4,7

which is equivalent to using the same attenuation coefficient for downwelling and upwelling light. However, studies of the subsurface light field1,10 sug- gest that in general Dd Þ Du

C Þ Du B Þ constant, and

that the D’s vary with the inherent optical properties of the water column. Incorrect assumptions for the values of the D’s will cause errors in water-depth derivations. To improve remote sensing of shallow- water bathymetry, for example, we need to more ac- curately determine how D varies with in-water optical properties.

Our goal in this study is to develop a simple, ex- plicitly invertable, analytical formula for remote- sensing reflectance for homogeneous waters. To achieve this, the Hydrolight numerical model11,12 was used to compute remote-sensing reflectance above the surface @Rrs, the ratio of water-leaving radiance Lw! to downwelling irradiance ~Ed1! just above the

surface# and remote-sensing reflectance below the surface ~rrs! for a series of water, bottom type, and depth parameters. From these calculated rrs values, parameterizations of D were empirically derived, and a semianalytical ~SA! model was developed for rrs of

330 APPLIED OPTICS y Vol. 37, No. 27 y 20 September 1998

deep and shallow waters. This SA model is explic- itly invertable for the calculation of depth and the optical properties of the water column, whereas Hy- drolight is strictly a numerical, forward model and is therefore not explicitly invertable.

Although it would be preferable to develop our SA model using an extensive data set of actual observa- tions, data sets covering the needed range of environ- mental conditions do not exist. Hydrolight gives us a controlled environment in which pseudodata can be generated for any desired environmental conditions. Any SA model based on numerically generated data must of course be tested with actual data whenever the opportunity arises.

2. Semianalytical Model for rrs

A. Deep-Water Formulation

Defining

u ; bby~a 1 bb!, (7)

rrs dp can be expressed as in Gordon et al.13 and Morel

and Gentili14 as

rrs dp < gu, (8)

with g itself expressed as a function of u:

g < g0 1 g1 ug2. (9)

B. Shallow-Water Formulation

To further simplify the SA model, we assume that15

Dd < 1ycos~uw!,

where uw is the subsurface solar zenith angle, and we keep Dd as a spectral constant. We thus attribute all the variations of the D’s in approximation ~6! to Du

C and Du B only. However, we must keep in mind

that both Dd and Du do vary with bb and a, which eans that part of the variations of Du

C and Du B

versus bb and a are a result of that for Dd. For remote-sensing applications, it is not necessary to separate Du

C and Du B from Dd, because approxima-

tion ~6! depends only on the sum of Dd and Du C and

the sum of Dd and Du B. For in-water Kd studies,

however, it is necessary to know the relationship be- tween Dd versus bb and a, as in Kirk.10

A Kirk10 K type of formula,

Du 5 D0~1 1 D1 u! 0.5 (10)

is applied to describe the relationships between Du and bby~a 1 bb!, where D0 and D1 are constant coef- ficients to be determined.

b t o b m

g o h A c

Table 1. Environmental Input used in the Hydrolight Simulations Table 2. Parameters for the Empirical a ~l! Simulation by Eq. ~12!a

Our proposed SA formula for deep and shallow water rrs is then

rrs 5 ~g0 1 g1 u g2!uS1 2 A0 expH2F 1cos~uw!

1 D0~1 1 D1 u! 0.5GaHJD 1 A1r expH2F 1cos~uw!

1 D09~1 1 D19u! 0.5GaHJ . (11)

The values of g0,1,2, A0,1, D0,1, and D0,19 are derived rom Hydrolight-generated rrs values.

3. Computer Simulations of Rrs and rrs Hydrolight version 3.0 ~Ref. 16! was employed for the eneration of nadir-viewing Rrs and rrs values. We

chose the Hydrolight radiative transfer numerical model11,12 for our simulations for several reasons:

~1! Hydrolight has been used by many researchers and for many applications ~e.g., Refs. 17–19! and con- sequently has been extensively debugged and vali- dated.

~2! Hydrolight is much faster than Monte Carlo simulation for problems involving the upwelling ra- diance,11 as in the present study. Moreover, Hy- drolight computes all radiances with the same accuracy, whereas Monte Carlo simulations contain much more statistical noise in upwelling radiances than in downwelling11,12 because of the fewer number of backscattered solar photons. Hydrolight does not have any statistical noise at all arising from the so- lution of the radiative transfer equation.

~3! Hydrolight contains the needed physics ~wind- lown surfaces, finite or infinite bottom depth, spec- ral dependence of inherent optical properties, all rders of multiple scattering within the water and etween the water surface and bottom, etc.!. Not all odels now in use contain all the needed features.

Table 1 summarizes the inputs used in the Hydroli- ht simulations. For all calculations, a wind speed f 5 mys was used, and the water body was assumed omogeneous with a refractive-index value of 1.34. bove-surface downwelling spectral irradiances were

omputed by the model of Gregg and Carder,20 using

Variable Inputs

Solar zenith angle 0°, 30°, 60° Particle phase function Petzold average particle Chlorophyl a @chl-a#

~mg m23! 0.4, 1.0, 2.0, 5.0

ag~440! ~m 21! 0.05, 0.1, 0.3

B 0.3, 1.0, 5.0 r 0, 0.1, 0.3, 1.0 H ~m! 0.5, 1, 3, 8, 16, 32, infinite l ~nm! 400–700, every 20 nm

solar zenith angles of 0°, 30°, and 60°. Both clear and cloudy skys were simulated.

The total absorption coefficient is expressed as a sum of the absorption coefficients for pure water, phy- toplankton pigments, and gelbstoff. Absorption val- ues for pure water are taken from Pope and Fry,21 whereas absorption for phytoplankton pigments and gelbstoff were modeled with simple bio-optical mod- els, which give a realistic simulation of a variety of waters.

The phytoplankton pigment absorption coefficient is simulated using the empirical model in Lee,22 ex- pressed as

af~l! 5 $~a0~l! 1 a1~l!ln@af~440!#%af~440!. (12)

Values for a0~l! and a1~l! have been already empiri- cally determined and are shown in Table 2. Thus, given an af~440! value, an af~l! spectrum is con- structed with Eq. ~12!. As can be seen in Fig. 1, this approach allows the af~l! curvature to change with af~440!, which is consistent with field observations, t least to the first order. In the Hydrolight calculations, af~440! can be an

nput by itself or can be linked to another parameter

Wavelength a0~l! a1~l!

390 0.5813 0.0235 400 0.6843 0.0205 410 0.7782 0.0129 420 0.8637 0.006 430 0.9603 0.002 440 1.0 0 450 0.9634 0.006 460 0.9311 0.0109 470 0.8697 0.0157 480 0.789 0.0152 490 0.7558 0.0256 500 0.7333 0.0559 510 0.6911 0.0865 520 0.6327 0.0981 530 0.5681 0.0969 540 0.5046 0.09 550 0.4262 0.0781 560 0.3433 0.0659 570 0.295 0.06 580 0.2784 0.0581 590 0.2595 0.054 600 0.2389 0.0495 610 0.2745 0.0578 620 0.3197 0.0674 630 0.3421 0.0718 640 0.3331 0.0685 650 0.3502 0.0713 660 0.561 0.1128 670 0.8435 0.1595 680 0.7485 0.1388 690 0.389 0.0812 700 0.136 0.0317 710 0.0545 0.0128 720 0.025 0.005

aTaken from Lee.22

September 1998 y Vol. 37, No. 27 y APPLIED OPTICS 6331

c a t a

c a

T b F

p b s t t

t u fl

t w

w i

b c

such as chlorophyll a concentration ~@chl-a#!. To be onsistent with the calculations of other researchers nd to keep the input values for absorption and scat- ering within the appropriate ranges, we used @chl-a# s input to determine the af~440! and bp~550! values

~see below!. af~440! is determined by23

af~440! 5 0.06@chl-a# 0.65. (13)

Gelbstoff absorption is expressed as in Bricaud et al.24:

ag~l! 5 [email protected]~l 2 440!#, (14)

with ag~440! an independent variable ~see Table 1!. The total scattering coefficient is expressed as a

sum of the scattering coefficients for pure seawater and particles. Scattering coefficients for pure sea- water come from Morel,25 whereas scattering coeffi- ients for particles come from the model of Gordon nd Morel26:

bp~l! 5 B@chl-a# 0.62550yl. (15)

Here B is an empirical value, which was 0.3 in Gor- don and Morel.26 We used B values of 0.3, 1.0, and 5.0 to simulate a range from normal to highly turbid coastal waters.

Most simulations used the Petzold-type average par- ticle phase function described in Mobley et al.11 and in

able 3.10 of Ref. 12. This phase function has a bpybp ratio of 1.9%. This phase function is shown in ig. 2 along with a more highly backscattering ~bbpybp

ratio of 3.7%! Kopelevich phase function12 for sus- ended sediments, which are discussed in Section 5 elow. The pure seawater scattering was treated as a eparate component with a Rayleigh-like phase func- ion.11,12 The ratio of pure seawater to particle scat- ering varied with the particle scattering coefficient.

A spectrally constant bottom albedo r was used in he calculations; r values of 0, 0.1, 0.3, and 1.0 were sed in the simulations. Although the bottom re- ectance was taken to be independent of wavelength,

Fig. 1. Examples of simulated pigment absorption coefficients as computed by Eqs. ~12! and ~13! for three different chlorophyll a concentrations; curves are normalized to 1.0 at 440 nm to show the curvature dependence on @chl-a#.

332 APPLIED OPTICS y Vol. 37, No. 27 y 20 September 1998

he water-column inherent optical properties were avelength dependent.

4. Results and Discussion

A. Deep Waters

As discovered in Gordon et al.13 and Morel and Gen- tili14 through Monte Carlo simulations, nadir- viewing rrs is almost independent of solar zenith angle @see Fig. 3 for rrs values versus bby~a 1 bb! for the three Sun angles#. This result is different from hat for subsurface irradiance reflectance ~R, ratio of

upwelling irradiance to downwelling irradiance!, which is a stronger function of solar zenith an- gle.10,27,28 Detailed discussions regarding this dif- ference can be found in Morel and Gentili.14

Figure 4 demonstrates the changes in rrs values ith bbya. Similar to earlier Monte Carlo stud-

es,13,14 it was found that rrs increases with ratios bby~a 1 bb! or bbya, but at a different rate. rrs quickly increases with bby~a 1 bb!, especially when

Fig. 2. Particle phase functions used in the Hydrolight calcula- tions. The solid curve represents the calculations used in the parameter determination; the dashed curve represents the calcu- lations used in the test.

Fig. 3. Dependence of deep water rrs on bby~a 1 bb!, as simulated y Hydrolight. The simulations include various environmental onditions as shown in Table 1.

w a

c i F

m ~ i t

f s t T s t u

d n

the ratio is relatively large. However, rrs increases more slowly with bbya, especially for bbya . 0.3, in a manner similar to that found by Jerome et al.29 and Kirk10 for subsurface irradiance reflectance.

The values of the empirical parameters g0,1,2 in Eq. ~11! were determined first, by use of Hydrolight sim- ulations for infinitely deep water. rrs values com- puted by Hydrolight were compared with the values given by approximations ~8! and ~9! as g0,1,2 were systematically varied. The best-fit set of g values

as determined by minimizing the following quantity s in Gordon and Boynton30:

d 5 meanUlnSrrssrrsHDU , where rrs

s is a SA model simulated value and rrs H is

a Hydrolight-calculated value. The mean here is based on all Hydrolight calculations, i.e., over all sets of inherent optical properties and sky conditions and over all wavelengths. The resulting best-fit set of g parameters are g0 ' 0.070, g1 ' 0.155, and g2 ' 0.752. Therefore

rrs dp < ~0.070 1 0.155u0.752!u. (16)

Figure 5 presents the deep-water rrs values as cal- ulated by Hydrolight and as obtained from approx- mation ~16!; the error ~ed 2 1! is approximately 1%. igure 6 shows examples of rrs spectra as computed

by Hydrolight and by approximation ~16!; the differ- ent curves are for different optical property values. As expected, the rrs values from both calculation

ethods agree very well. Note that approximation 16! includes data with both low and high u values, .e., data for a wide range of inherent optical proper- ies for waters.

We could also try to express rrs dp as a simple func-

ion of bbya, instead of bby~a 1 bb!, as has been done in many studies of subsurface irradiance reflec- tance.10,22,28,29,31 For coastal waters, however, one expression cannot adequately cover both small and large bbya situations.10,29 Two or more expressions with different g parameters for different bbya ranges

ave to be applied.29

Fig. 4. Dependence of deep water rrs on bbya.

Gordon et al.13 found through Monte Carlo simu- lation that, for most oceanic waters, rrs

dp can be ex- pressed as

rrs dp < ~0.0949 1 0.0794u!u. (17)

orel and Gentili14 found that for case 1 waters,32

rrs dp < 0.0922bbya,

which is equivalent to

rrs dp <

0.0922 1 2 u

u. (18)

Figure 7 compares approximations ~16!, ~17!, and ~18! or deep-water remote-sensing reflectance. We can ee that for u less than 0.3 ~bbya less than ;0.4!, the hree expressions provide very close rrs

dp values. his agreement demonstrates that the three expres- ions are all consistent with each other, even though hey were developed through different computer sim- lations. The slightly smaller rrs

dp values found us- ing approximation ~16! rather than using approximation ~17! for u values less than 0.05 ~see inset in Fig. 7! may be due to technical differences

Fig. 5. Comparison of deep water rrs values as computed by Hy- rolight and by the SA model. Values for the different solar ze- ith angles are identified.

Fig. 6. Examples of rrs spectra as computed by Hydrolight ~filled symbols! and by the SA model ~open symbols!.

September 1998 y Vol. 37, No. 27 y APPLIED OPTICS 6333

E r

o u o m

t b r

between the Hydrolight and Monte Carlo simulations as pointed out in Mobley et al.,11 especially for the calculation of upwelling radiance.11 This difference of roughly 10% is also within the error bounds of Gordon et al.13 Because the expressions of Gordon et al.13 and Morel and Gentili14 were developed for open-ocean applications, extrapolation of approxima- tions ~17! and ~18! to waters with u values larger than 0.3 may not be appropriate. Therefore we suggest that approximations ~17! and ~18! are consistent for open-ocean waters, and that approximation ~16! ex- tends those expressions to the realm of turbid coastal waters. A test using a more mineral-rich expression for the particle phase function shows only a small effect on the SA model performance ~see Section 5!.

B. Shallow Waters

The remaining parameters A0,1, D0,1, and D0,19 were simultaneously determined by comparing Hydrolight-calculated rrs values with those given by

q. ~11! and minimizing the d value as above. The esulting best-fit parameter values are

A0 < 1.03, A1 < 0.31,

Du C < 1.2~1 1 2.0u!0.5, (19)

Du B < 1.1~1 1 4.9u!0.5. (20)

Therefore Eq. ~11! takes the specific form

rrs < ~0.070 1 0.155u0.752!uS1 2 1.03 expH2F 1cos~uw! 1 1.2~1 1 2.0u!0.5GaHJD 1 0.31r expH2F 1cos~uw! 1 1.1 3 ~1 1 4.9u!0.5GaHJ . (21)

Figure 8 compares rrs values calculated by Hydro- light versus those determined by approximation ~21!

Fig. 7. Comparison of rrs dp values as determined by three differ-

nt expressions.

334 APPLIED OPTICS y Vol. 37, No. 27 y 20 September 1998

for shallow waters; the error is approximately 3%. Although reasonably close agreement is to be antici- pated because the SA model parameters were deter- mined with Hydrolight data, 63% agreement nevertheless indicates the excellent predictive power of the SA model over a wide range of optical environ- ments. Note that approximation ~21! predicts rrs given only the absorption and backscattering proper- ties of the water column, the solar zenith angle, the water depth, and the bottom reflectance. This is the desired SA model for rrs.

We did not try to fit the Hydrolight rrs values of hallow, black-bottom environments to derive A0 and

D0,1 only, because such rrs values lack the interaction between photons scattered by the water column and by the bottom.

The values for A0 and A1 are almost identical ~;3% difference! to their values suggested by use of single- or quasi-single-scattering theory2–6,8,33 ~recall Sec- tion 1!, which suggests that the SA model and the empirically derived values are consistent with the- ory. The small difference found may be due to the fact that approximation ~6! is a simplified expression based on use of quasi-single-scattering theory. It therefore cannot be expected to perfectly describe a complicated shallow-water environment. We must keep in mind that the simple truncation of approxi- mation ~1! @the term in the bracket on the right side; r approximation ~2! with A0 5 1.0# for water col- mns with black bottoms only accounts to the first rder for the reduction of the number of particles aking contributions to rrs. A black-bottom-

truncated water column, however, reduces not only the number of scattering particles, but also the multiple-scattering probabilities, especially for cases with higher bbya. Also, as Gordon and Brown34 in- dicated, it is not true that part of the photons will just hit bottom while the other part will not. Thus the slight departure of A0 from 1.0 and of A1 from 1yp for he current calculation reflects in part the difference etween approximation ~6! and the exact Hydrolight rs values. Simply setting A0 to 1.0 and A1 to 1yp

would require the values of D1 in Eq. ~11! be negative

Fig. 8. Comparison of shallow water rrs values as computed by Hydrolight and by the SA model for different solar zenith angles.

F t s

to fit all the Hydrolight-generated rrs values. This suggests that Du

C will be smaller for high bbya wa- ters, which is contrary to the findings of Kirk.10 To force A0 to 1.0, A1 to 1yp, and allow appropriate val- ues for D0,1 and D0,19, a more complicated expression than approximation ~6! @or Eq. ~11!# is required. We herefore choose to retain the best-fit values of A0 and

A1 in approximation ~21!, so as to provide the SA model greater possible applicability.

Figure 9 shows the behavior of Du C and Du

B for u ess than 0.6 ~bbya less than ;1.5!. The ranges for

Du C and Du

B are 1.2–1.7 and 1.1–2.2, respectively. This indicates that the paths of photons reflected by the bottom are longer than those of photons scattered by the water column, especially for waters of high bbya. Earlier, Lee et al.5 empirically suggested an average value of 1.5 for Du

B; this average is consis- tent with the finding here.

Figure 10 shows the ratios of rrs at two wave- lengths as a function of H; note that the absorption oefficient is much smaller at l1 than at l2. It

should be noted that for some combinations of water- column optical properties and bottom albedo, there are two depths that correspond to the same rrs ratio.

nly when the water is very clear and shallow, i.e.,

Fig. 9. Dependence of distribution functions on bby~a 1 bb!.

Fig. 10. Ratios of rrs at two wavelengths as a function of bottom epth and reflectance.

when the water-column contributions are negligible, do spectral ratios of rrs become almost independent of the bottom albedo values. These observations sug- gest that use of just two wavelength channels may be insufficient to retrieve the bottom depth even when the optical properties of the water column and the bottom albedo are known. For shallow-water re- mote sensing, additional channels are almost cer- tainly required for the retrieval of both depth and optical properties of the water column.

5. Test with a Different Particle Phase Function

To evaluate the validity of approximations ~16! and ~21! for a variety of water environments, another rrs data set was generated by use of a different particle phase function as input to Hydrolight. This particle phase function is given by the formula of Kopelevich12 and has a bbpybp ratio of 3.7%, which is approxi- mately a factor of 2 larger than the particle phase function used in the parameter development ~recall

ig. 2!. The particle phase function is assumed to be he same for all wavelengths. Figures 11 and 12 how rrs values computed by Hydrolight with the

Kopelevich phase function versus those determined

Fig. 11. Comparison of deep water rrs values computed by Hy- drolight with the Kopelevich phase function of Fig. 2 with rrs values given by the SA model, which is based on the coastal par- ticle phase function of Fig. 2.

Fig. 12. Comparison of shallow water rrs values computed by ydrolight with the Kopelevich phase function of Fig. 2 with rrs

values given by the SA model, which is based on the coastal par- ticle phase function of Fig. 2.

September 1998 y Vol. 37, No. 27 y APPLIED OPTICS 6335

i i f s

1 i a

b f

a t

c r i f

a t

with approximations ~16! and ~21! for deep and shal- low waters, respectively. As can be seen, the rrs val- ues calculated by both methods still agree quite well with each other ~approximately 3% error for both deep and shallow situations!. These results suggest that there is a potential for wide application of ap- proximations ~16! and ~21! to water environments with considerably different scattering properties.

6. Across the Water–Air Interface

Approximation ~21! provides only the rrs values below the surface, but a remote sensor measures above- surface values. Therefore the relation between Rrs and rrs needs to be known. From Mobley12 and Gor- don et al.,13 there is

Rrs 5 Lw Ed1

5 t2t1 n2

R Q~1 2 gR!

, (22)

where t2 is the radiance transmittance from below to above the surface, t1 is the irradiance transmittance from above to below the surface, n is the refractive index of water, R is the subsurface irradiance reflec- tance, g is the water-to-air internal reflection coeffi- cient, and Q is the ratio of Eu2 to Lu2.

Noting that

rrs 5 Lu2 Ed2

5 R Q

, (23)

and defining

z 5 t2t1 n2

as the water-to-air radiance-divergence factor, we can write

Rrs 5 zrrs

1 2 Grrs , (24)

where G 5 Q g. Note that Eq. ~24! is exact. The denominator 1 2 G rrs accounts for the effects of nternal reflection from water to air. Although this nternal-reflection term is often ignored, especially or open-ocean waters, we retain it in the present tudy because rrs can be quite large for turbid or

shallow waters. For remote-sensing applications, the values of z and G in Eq. ~24! need to be deter-

ined. By comparing the Hydrolight-generated Rrs and rrs values, it is found that z ' 0.518 and G ' .562. This value of z is consistent with the analysis n Gordon et al.13 Recall that G is the product of g nd Q. If g is approximately 0.48 ~Lyzenga35 and

Ref. 13!, then Q has a value of 3.25, which is consis- tent with the nadir values seen in Morel and Gen- tili.14 For other view angles, values of z and G may e slightly different. Thus Eq. ~24! takes the specific orm

Rrs < 0.518rrs

1 2 1.562rrs . (25)

336 APPLIED OPTICS y Vol. 37, No. 27 y 20 September 1998

Figure 13 shows Rrs as calculated by approximation ~25! and by Hydrolight.

If we consider the idealized case of perfectly clear water ~u 5 0!, a perfectly reflecting bottom ~r 5 1.0! nd infinitesimally deep water ~H 5 0!, approxima- ion ~21! provides an rrs value of ;0.31 sr

21, whereas approximation ~25! also results in Rrs ' 0.31 sr

21. These results are consistent with Hydrolight- calculated values. However, if the internal reflec- tion term @the denominator of approximation ~25!#

ad been omitted, Rrs ' 0.16 sr 21 from approxima-

tion ~25!, which is in serious error. These results learly show that it is necessary to retain the internal eflection term in approximation ~25! when working n shallow or turbid waters, and that the above values or z and G are appropriate.

7. Summary

By use of the Hydrolight radiative transfer numerical model with inherent optical properties typical of coastal waters, Rrs and rrs values were calculated for both deep and shallow waters. The calculated rrs values were used to determine the empirical param- eters for a SA model for rrs. All model parameters are explicitly determined as functions of a and bb.

he resulting deep-water rrs expression is consistent with earlier Monte Carlo studies for deep, low- scattering waters and has potential for application to turbid waters with high bbya. Because the SA model explicitly includes bottom effects, it provides a basis for analytical inversion of rrs spectra to obtain shallow-water properties such as depth and bottom albedo, as well as water-column absorption and back- scatter. The SA model predictions were tested against rrs values calculated by Hydrolight using two different particle phase functions, and it was found that the rrs values from both Hydrolight and the SA model agree with each other quite well ~approximate- ly 3% error for both deep- and shallow-water rrs val- ues!. This indicates that predictions made by the SA model are not strongly dependent on the ~gener- lly unknown! scattering phase function. Therefore he SA model can be used to accurately and quickly

Fig. 13. Comparison of Rrs values computed by Hydrolight and by approximation ~25!.

v s t t

predict the rrs spectrum of various shallow waters, in contrast to time-consuming Monte Carlo or Hydro- light calculations. Also, parameters connecting Rrs and rrs are also derived from Hydrolight-generated values. This bodes well for rapid model inversions of high-resolution, measured, spectral Rrs data sets to obtain values of H, r, a~l!, and bb~l!. Knowing these alues may in turn enable the recovery of quantities uch as chlorophyll a, gelbstoff, and suspended par- iculate matter concentrations from empirical rela- ionships with a~l! and bb~l!.

8. Symbols Used in Text

aw absorption coefficient of pure seawater, m 21;

af absorption coefficient of phytoplankton pigments, m21;

ag absorption coefficient of gelbstoff, m 21;

a absorption coefficient of the total, m21; B an empirical value to describe particle scattering

coefficient; b scattering coefficient, m21;

bb backscattering coefficient, m 21;

bp scattering coefficient of suspended particles, m 21;

bbp backscattering coefficient of suspended particles, m21;

D distribution function; Ed downwelling irradiance, Wynmym

2; Ed1,d2 downwelling irradiance just above ~1! or below ~2!

the surface, Wynmym2; Eu upwelling irradiance, Wynmym

2; f water-to-air divergence factor;

H bottom depth, m; K diffuse attenuation coefficients, m21;

Lu2 subsurface upwelling radiance, Wym 2ynmysr;

Lw water-leaving radiance, Wym 2ynmysr;

n refractive index of water; Q ratio of subsurface upwelling irradiance to up-

welling radiance, sr; rrs subsurface remote-sensing reflectance, sr

21; rrs

dp subsurface remote-sensing reflectance for opti- cally deep waters, sr21;

R subsurface irradiance reflectance; Rrs above-surface remote-sensing reflectance, sr

21; SUS subsurface upwelling signal; t1,2 diffuse transmittance from above ~below! to below

~above! the surface; @chl-a# chlorophyll a concentration, mgym3;

a attenuation coefficient ~5 a 1 bb!, m 21;

r bottom albedo; uw subsurface solar zenith angle, rad; g water-to-air internal reflection.

Financial support was provided by NASA through grant NAS5-31716 and by the U.S. Office of Naval Research through grant N00014-96-I-5013. The de- velopment of Hydrolight was supported in part by the U.S. Office of Naval Research Environmental Optics Program under various contracts to C. D. Mobley, who was supported for the present research through contract N00014-97-C-0024. The authors thank an

anonymous reviewer for helpful comments on the paper.

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