Extracting Information on Surface Properties From ... - Caltech GPS

JOURNAL

OF GEOPHYSICAL

RESEARCH,

VOL. 96, NO. D2, PAGES 2865-2874, FEBRUARY

20, 1991

Extracting Information on Surface Properties From Bidirectional Reflectance

Measurements

BERNARD

PINTY

Laboratoire d'Etudes et de Recherches en Tflfdftection Spatiale, Toulouse, France MICHEL

M.

VERSTRAETE

Department of Atmospheric, Oceanic and Space Sciences, University of Michigan, Ann Arbor The retrieval of surfaceparametersfrom remotely senseddata is of prime interest for the estimation of surface properties of various planets in the solar system, including the Earth. Bidirectional reflectance measurements taken over natural surfaces in visible and near-infrared spectral bands represent one data set from which these surfaceproperties could be estimated. To achieve this goal, it is necessaryto have both physicalmodelspredictingthe bidirectionalreflectancefield as a function of the relevant surfaceparameters,and numerical proceduresallowing the inversion of these models using a limited sampling of the bidirectional reflectance field. Given that theoretical models of the bidirectional reflectance have been published (Hapke, 1981, 1984, 1986; Verstraete et al., 1990) and that numerical procedures are available (Pinty et al., 1990), this paper focuses on the errors and uncertainties in the retrieved parameters which may arise because of (1) the weaknessesin our theoretical understandingand representationof the surfaceradiation transfer and (2) the errors in the bidirectional reflectance data. For instance, it is shown that the addition a posteriori of an amplitude parameter in the function accountingfor the opposition effect can drastically modify the retrieved values of the optical and morphologicalparametersof the surface.The consequencesof uncertainties in the reflectancedata are also investigated,and the redistribution, by the inversion procedure, of such uncertainties on the retrieved parametersis discussed.Finally, synthetic reflectance data contaminated by a known random noise are used to examine the numerical stability of the retrieval and the compatibility between the models.

1.

INTRODUCTION

Radiometric measurementsconstitute the major sourceof data used to study the surface properties of various solar system bodies. Except for the Moon where in situ analysis and collection of surface samplesbecame feasible, theoretical studiesof the interaction between electromagneticradiation and the surface properties of the asteroidsremain the only possible research avenue. On Earth, major advances in the design, performance, accuracy and reliability of satellite platforms over the past three decades have also resulted in much expanded and enhancedcapabilities to observe the nature and variability of surface properties. In various fields of geophysics(for example, atmosphericsciences,hydrology, oceanography,geology and environment monitoring), the acquisition of satellite remote sensingdata supports an increasingnumber of scientific investigations and thereby contributes directly to our knowledge and understandingof the Earth as a global and integratedsystem. These developmentsare particularly timely in the context of the current climate changes predicted by climate models, since satellitesare ideally suited to monitor the entire planet with an adequate spatial and temporal resolution. Among the various suspectedcausesof climate change, land use modifications and environmental degradation constitute a major focus of interest urgently requiring the design of etficient quantitative methods for interpreting satellite data in terms of physical properties of the surface and the atmosphere.

Although the complexity of the problem is different for Earth than for other celestialbodies, the basic approachmay be the same and requires (1) modeling the field of radiance emerging from the surface as a function of the physically relevant parameters, and (2) inverting these models to retrieve the surface parameters of interest from a finite sampling of the measured radiance field. In this paper, we will focus on the case of radiances

Copyright 1991 by the American Geophysical Union. Paper number 90JD02239. 0148-0227/91/90JD-02239505.00 2865

reflected

in the visible

and

near-infraredwavelengthsby homogeneousand semi-infinite media.

The achievement of the first goal is limited by our theoretical understanding of the radiation transfer through particulate surfaces. During the past decade, a number of models have been developed to describe light scattering from a semi-infinite medium composed of closely packed particles. These models have been applied to specific problems in planetology [e.g., Lumme and Bowell, 1981; Goguen, 1981; Hapke, 1981, 1984, 1986]. In these models, the reflected radiance is related to a set of physically meaningful parameters of the surface through simple analytical derivations. Using a similar approach, Verstraete et al. [ 1990], and Dickinson et al. [ 1990] have extended the original model of Hapke in two specificrespects: (1) to account for the finite size and orientation distribution of the scatterers,

and (2) to describethe "hot spot" phenomenonanalytically. The hot spot effect, also known as the "opposition" effect, is the relative increase in reflectance in the backscattering region due to the absence of shadows observable from the direction of illumination. These new physically based models allow the calculation of both the hemispherical reflectance (see, for example, Dickinson, 1983; Sellers, 1985] and bidirectional reflectancesin particular conditions of illumi-

2866

PINTY AND VERSTRAETE:

BIDIRECTIONAL

REFLECTANCE

MEASUREMENTS

2. DESCRIPTION OF THEORETICAL MODELS nation and observation. These models share simple, physically based analytical solutions to the equations of radiative transfer, and they can easily be generalized and applied to 2.1. Hapke's Model arbitrary porous media. One of the most comprehensivemodels of bidirectional The simplicity of these solutions satisfiesa fundamental requirement when dealing with radiometric measurements reflectance was developed by Hapke [1981, 1984, 1986]. since one is generally interested in the inversion of the From the fundamentalprinciples of radiative transfer theory, Hapke [1981] derived an analytical equation for the bidirecmodels with numerous data sets in order to retrieve the model parametersfrom an angular samplingof the reflected tional reflectancefunction of a medium composedof dimensionlessparticles. The singly scatteredradiance is derived radiancefield. Although this point has receivedlessattention exactly, whereas the multiply scatteredradiance is evaluated in the past than the development of theoretical solutions to from a two-stream approximation, assumingthat the scatterthe radiative transfer equation, this step is necessaryin order ers making up the surface are isotropic. The bidirectional to extract the surfaceproperties from radiometric measurereflectance p of a surface illuminated by the Sun from a ments. Inversion methods based on nonlinear least squares direction(01, 4h), observedfrom a direction (02, 4•2), and fitting algorithms have been developed (see, for example, normalized with respect to the reflectance of a perfectly Helfenstein and Veverka [1987], Camillo [1987], and Pinty et reflectingLambertian surfaceunder the same conditionsof al. [1989, 1990]), but they need to be tested more thoroughly. illumination and observation, is given by

These methods are based on the minimization

of the normal-

ized squared differencesbetween the model predictions and the observational data over the entire sample. However, for a given function, the performances of these numerical algorithms are sensitive to various problems such as the initial guess, the data sampling mode, the level of noise contaminating the data, and so on. The formulations

of the bidirectional

reflectance

ro

{[1 + B(#)]P(#) + H(/a,1)H(/a,2)- 1}

(1)

where

functions

are currently slightly different between the various authors. Furthermore, no method of inversion has been universally adopted by the scientific community. These two aspectsare linked because a complete validation of the models necessitates the use of an inverse method in order to compare the values of the surface physical parameters retrieved by the inversion with those measured independently using in situ observational techniques. Indeed, a good fit of the reflectance measurements by the models does not necessarily imply that the functional dependenciesbetween the reflectance and the surface parameters are correctly established. This is partly due to the fact that, except at low phaseangles, the reflectancesexhibit rather smoothangularvariations and can be reasonably well fitted by functions using fewer parameters than the physical models mentioned above. An additional difficulty already noted by Domingue and Hapke [1989] is the lack of uniqueness of the set of physical parametersleadingto a successfulfit to the data. Becauseof such problems, further pilot experiments are urgently needed, both in the laboratory and in the field. Our goal in this paper is to supportthe above statementby showinghow slightdifferencesin model formulation can lead to nontrivial differencesin retrieved parameter values and by discussing how such differences affect the expected "errors" between model predictions and actual data. We concentrate here on the theoretical developments proposed by Hapke [1984] and Verstraete et al. [1990], with special attention to the hot spot phenomenon. Specifically, we will address the need for a separate parameter to describe the amplitude of the hot spot, and the consequencesof this for the retrieval procedure when inverting the model on reflectance data. With respect to the inverse method, we will test the behavior of a nonlinear least squares technique when artificial noise is added to synthetic data in order to examine the redistribution on the various model parameters of the original "noise" contaminatingthe reflectance data.

1

p(O1, if>i; 02, •b2)=--•

/-•l =COS 01 •2 -- COS0 2

COS9 = COS01 COS02 + sin 01 sin 02 cos (qb1 - qb2) B0

S(g) =

[1 + (l/h) tan (g/2)] su(o) ooP(O) l+2x

H(x) =

1 + 2(1- oo)1/2.,12

In these equations,9 is the phase angle between the incoming and the outgoingrays, rois the averagesinglescattering albedo of the particles making up the surface, P(g) is the averagephase function of the particles, B(g) is a backscattering function that accountsfor the hot spot effect, and the term H(/xl)H(/x2) - 1 approximatesthe contributionfrom multiple scattering within the medium. For practical purposes,P(g) can be approximatedby a Legendre polynomial expansion (see, for example, Helfenstein and Veverka [1987]) or by the Henyey-6reenstein formula [Henyey and Greenstein, 1941] (see, for example, Pinty et al. [1990]). Equation (1) includes the improved formulation for B(g) [Hapke, 1986] instead of the original expressiongiven by Hapke [1981], but it neglects the macroscopic roughness effect discussedby Hapke [1984].

B0 correspondsto the amplitude of the hot spot or the oppositioneffect, i.e., the preferentialescapeof radiationin the direction of illumination: It is controlled by the relative value of the parameter Su, relative to the magnitude of wHO). The parameter h controlsthe width of this opposition effect and may be related to the scatterer size distribution and the gradient of compactionwith depth [Hapke, 1986].

PINTY AND VERSTRAETE: BIDIRECTIONAL

2.2.

Verstraete

et al.'s Model

Using the basic framework previously suggested by Hapke, Verstraete et al. [1990] developed a model for predictingthe bidirectional reflectanceexiting from a simple vegetation canopy. In their paper, they concentrate on the case of a fully covering, homogeneous and semi-infinite canopy made of leaves only. With the same notations as for (1), the parametric version of the derived model [seePinty et al., 1990] is as follows: o•

P(01, (•1; 02, (•2)=-

K1

4 Ki].•2 + K2].•1

{[1 + Pv(G)]P(g)

+ H(tx 1/K1)H(tx 2/1(2)- 1}

(2)

where

Pv(G) =

4)G/.6 2

G = [tan2 01+tan2 02 -- 2 tan 01tan 02 COS ((•1-- (•2)]1/2 COSg = COS01 COS02 + sin 01 sin 02 cos ((•1 -- (•2) l+x

1 + (1 -

In these equations, r is the radius of the Sun flecks on the inclined scatterers, A is the scatterer area density of the canopy (expressed as the scatterer surface per unit bulk volume), K1 and K2 describe the orientation distribution of the scatterers for the illumination and viewing angles, respectively, # is the phase angle as before, G is a geometric factor that generally takes on large values, except when the direction

of observation

is close to the direction

MEASUREMENTS

2867

scattering contribution in Verstraete et al.'s model reduces to that of Hapke when the scatterers are uniformly distributed in all directions. This extension of Hapke's model derives naturally from the consideration of scatterers of finite size and given orientation. This improvement, however, modifies both the single and multiple scattering contributions to the bidirectional reflectance. Figures l a and lb illustrate this point by contrasting the bidirectional reflectance field in the principal plane (i.e., the plane defined by the direction of illumination and the normal to the surface) for mostly horizontally and mostly vertically oriented scatterers, respectively. These figures clearly show significant differencesbetween the two cases at all illumination angles considered here. In this computation, the single scattering albedo was set at 0.2 (that is, the bulk of the bidirectional reflectance comes from the single scattering contribution), the phase function of the scatterers P(#) was described by a quasi-isotropicfunction, and the contribution of the hot spot

was ignored (i.e., Pv(G) = 0), which implies the neglect of

1 + Vv(G)

Vv(G) =4 1- • 2• •

H(x) =

REFLECTANCE

of illumina-

tion, Pv(G) is the function that accounts for the joint transmission of the incoming and outgoing radiation, and thereby also for the hot spot phenomenon, and the term H(tZl/•Cl)H(l•2/•c2)- 1 approximatesthe contributionfrom multiple scattering [see Dickinson et al., 1990]. This formulation of Pv(G) represents a parameterization of the full analytical theory. The function gx representsthe averageof the cosineof the angle between the normals to the scatterers and the direction of illumination (x = 1) and observation (x = 2), a value that can be computed if the orientation distribution of the scat-

terers is known [e.g., Verstraete, 1987]. As for P(#), the functions can be estimated with simple empirical expressionsas suggestedby Goudriaan [1977, 1988] and Dickinson et al. [1990]. In the special simple case of uniformly distributed scatterers (i.e., all scatterer orientations are equiprobable), Kxis constantand equal to 1/2, and the only difference between the two models (1) and (2) lies in their formulation of the hot spot term.

both the mutual shadowing effect and any correlation between leaf orientation and the directions of light propagation.

A second major difference between (1) and (2) concerns the function that accounts for the joint transmission of the incoming and outgoingradiation, and thereby also for the hot spot effect arising at low phase angles. The function derived by Verstraete et al. [1990] exhibits a more complex angular dependencythan that given by Hapke and does not contain

an amplitudeparameteranalogousto the B0 parameterin (1). In both models, the denominators in the B(#) and Pv(G) functions control the angular width of the hot spot effect. In the model of Verstraete et al. [1990], the maximum increase

in reflectance resulting from the joint transmissionis given by 1 + Pv(G) = 2, when G = 0. Clearly, in Hapke's model, this maximum, given by 1 + B(#), can be greater or lower than 2, dependingon the value of B0, when # = 0. It is important to note here that the amplitude parameter B0 was not derived rigorously from the theory, but was introduced intuitively by Hapke to account for second-order effects in the physical properties of scatterers. The reader is referred to Hapke' s original paper for a fuller discussionand justification of an amplitude parameter B0 different from unity. Hapke's strongest statement in support of a value of B0 lessthan unity is to accountfor the fact that, in a compact medium where the scatterers are partly illuminated and partly hidden from the sensor, a fraction of the refracted light might travel through the scatterer and emerge some large distance away from the point of entry, where "large" here is with respect to the size of the scatterers. An analogousphysical argument would suggestthe introduction of an additionalparameterP0 in Verstraete et al.'s model, so that P•(G) becomes P0

P•,(G) •

1+Vv(G)

(3)

Sv(O) wP(O)

2.3.

Model Differences

As can be seen by comparing (1) and (2) above, Verstraete et al.'s model differs from Hapke's by accountingexplicitly for the effects of scattererorientation, althoughthe multiple

From the above discussion, it follows that at least three different

versions of a bidirectional

reflectance

model can be

compared: (1) model A' the original Verstraete et al.' s model as given by Equation 2; (2) model B' equation (2) where the

2868

PINTY

AND VERSTRAETE:

BIDIRECTIONAL

REFLECTANCE

MEASUREMENTS

ß30 ß28 ß26 .2q ß 22

.20 .18

.16

.!2

.O8

ß06

.O2 o

- l O0

-80

-60

-qO

-20

0

20

qO

Viewing •lngle in the principol

60

80

O0

plone

Fig. la. Bidirectional reflectances computed in the principal plane, for mostly horizontally oriented scatterers correspondingto a Xt value of 0.6, for various solarzenith anglesbetween0ø(A) and 80ø(I) by incrementsof 10ø. The other model parameters are •o = 0.2, and t9 • 0.

Pv(G) function is given by equation (3); and (3) model C: Hapke's model as given by equation (1). Model C could be considered a limit case of model B, but it will be studied independently for historical reasons. For practical purposes, it is necessary to examine the consequencesof the addition of an amplitude parameter to the function describingthe hot spot phenomenon, as well as the introduction of a supplementary function accounting for the average scatterer orientation. The next section examines the variability in the values of the retrieved model parameters when inverting each of the three models with the same bidirectional

reflec-

was selectedfrom the Numerical Algorithms Group (NAG) library. The routine E04JAF implements a quasi-Newtonian algorithm for finding a minimum of a function, subject to fixed upper and lower bounds on the independentvariables, using function values only. An initial guess for each of the desiredparametersmust be provided to the routine. The root

mean square (rms) values ofthefits,i.e.,(•2/nf)1/2 where nf is the number of degreesof freedom, were calculated to give an indication of the quality of the optimization. Following Pinty et al. [1990], the Henyey-Greenstein expressionis used to represent the scatterer phasefunction,

tance data set. Clearly, the level of noise which is acceptable depends on the further use of the retrieved parameters. 3.

MODEL

COMPARISONS

EXERCISES

3.1.

IN RETRIEVAL

ON ACTUAL

DATA

Summary of the Inversion Procedure

The inversion procedure described by Pinty et al. [1989, 1990] is used here. Its purpose is to seek the values of the

modelparameters thatminimize•2 definedas

1 -(9 2

P(#)(1+(92_ 2(9 cos •-•)3/2

(5)

where the scattering angle 11 - rr - #, and (9 is the asymmetry factor ranging from - 1 (backward scattering) to + 1 (forward scattering). The parameterization developed by Goudriaan [1977] has been chosenfor a practical estimate of the Kxfunctions,that is,

Kx(•x ) = xXt 1 q- xXt2•x n

a2-- Z [Pk--p(01,k, •l,k; 02,k, qb2,k)] 2

(4)

xXt 1-' 0.5-- 0o6333)(!0.33,,¾•

(6)

k=l

where Pk is the measured and p the modeled bidirectional reflectance of the surface, for the relative geometry of

•2 = 0.877(1 - 2•b1) where x stands for 1 or 2, the two directions of illumination

illuminationandobservation definedby 01,k,•bl,k,02,•,and and observation,and where -0.4 < X• < 0.6. The value of X• •b2, •, and where n is the number of observations.The is more negative (-0.4) for an erectophile canopy (mostly nonlinear, least squares fitting algorithm used to solve (4)

vertical scatterers), 0 for a canopy with a uniform distribu-

PINTY AND VERSTRAETE: BIDIRECTIONAL

REFLECTANCE

MEASUREMENTS

2869

ß3o ß28 ß26 .24 .22

ß20 .18

.16 -

It

-

.12

-

o10

-

'

_

_

.08

-

.06 .

ß04

-

ß02 0 -100

I

I

-80

I

I

-60

I

-•0

I

I

-20

I

I

I

0

I

I

I

20

Viewing Rnõle in the principol

Fig. lb.

l1 -- gl(Ol) sin OldOl •0re/2

(7)

where Ot is the scatterer zenith angle, and 9t(Ot) is the scatterer angle distribution.

The

Application to Laboratory Data

The data set were used in this application was obtained by Woessnerand Hapke [1987], who observed a clover patch in the laboratory. The data acquisition system is describedby Woessner[ 1985], and this data set was investigatedby Pinty et al. [1990] where additional information can be found.

Bidirectional reflectances were measured every 10ø in the principal plane from 0ø to 80øfor two relative azimuths of 0ø and 180ø.The illumination anglesare successively0øand 60ø. The inversion procedure was applied to the three physical models A, B, and C presented in the previous section. Four parameters are to be retrieved for model A: the single scattering albedo w, the asymmetry factor of the phase function ©, the parameterXl for the scattererangledistribution, and the parameter 2 rA controlling the width of the hot spot. Model B uses five parameters: the same as those of

I

80

00

parameters: roand ©, in addition to the two parameters h and S•(0) which describethe width and the amplitude of the hot

reflectances

measured

at 0 ø were

in-

verted first with the three models. The model parameters retrieved by this procedure were then used to generate the reflectances

as observed

under

illumination

at both 0 ø and

60ø. The values retrieved when inverting the models with the data at 0ø, as well as the rms, are summarized in Table 1. It can be seen that all three models yield single scattering albedo values very close to 0.1 and predict a backward scattering component (indicated by a negative ©) in the phase function. Compared with model A, model B predicts more isotropic scattering properties of the leaves, while values are retrieved

with model

C. Because

ro

keeps almost the same value for the three models we studied, the large variations occurring in © would correspond to large variations in the hemispherical reflectance and transmittance factors of the leaves, a higher reflectance factor being obtained with model A. The variations in © are

balancedby the two parameters$ v(0) and Xl. If the valuesof $v(O) are placed in (3), the resulting values of P0 become greater than 1. Furthermore, the values of Xl depart significantly from 0, the value implied by model C. The values retrieved for the parameter 2 rA from the inversion of models A and B also differ noticeably; and the larger this value, the

TABLE

1.

Values

of Model

Parameters

as Retrieved

From

Woessner and Hapke' s [1987] Observations of a Clover

model A plus the parameter $v(O). Model C requires four

spot.

i

plone

bidirectional

intermediate

3.2.

I

60

Same as Figure la, except for Xt = -0.4.

tion (equal probability for all scatterer orientations, also called spherical distribution), and more positive (0.6) for a planophile canopy (mostly horizontal scatterers). The parameter Xt is a function of the scattererangle distributionsin the canopy:

Xl-- +

I

•0

Model A ModelB ModelC

to

t9

0.099 0.116 0.101

-0.391 -0.168 -0.263

Xt

Patch

h

0.115 -0.395

2r A 0.277 0.688

0.046

Sv,H(O) rms 0.618 0.589

0.007 0.005 0.005

2870

PINTY AND VERSTRAETE: BIDIRECTIONAL

REFLECTANCE MEASUREMENTS

.11

.lO

ßo9

.o8

c 0

'

07

_..

'-

06

o

õ .o5 ._

u

.--

.

04

.--

.O3

. O2

.01

•

O0

I

-80

f

I

-60

i

I

-40

I

I

-20

I

I

I

0

I

20

I

I

i

40

I

i

60

I

80

Viewing Angle in lhe principal plane Fig. 2a. Comparison between the bidirectional reflectances measured over a clover patch at 0.448/am and those modeled with models A, B, and C, using optimally fitted parameters. Dots indicate data points for illumination at 0ø zenith angle.

broader the hot spot effect. Such variations indicate significant differences regarding the relative arrangements of the leaves making up the clover patch in terms of their compaction [see Verstraete et al., 1990]. By contrast, it is interesting to note that the estimated hemispherical reflectances, obtained using a numerical integration scheme, are 0.034, 0.033, and 0.033 for models A, B, and C, respectively. Since the rms values are less than 10% of the reflectance

values, Figure 2a shows that each of the three models studied provides a good fit of the reflectance pattern observed at 0ø, including the sharp hot spot phenomenon. The model predictions for an illuminating source at 60øare given in Figure 2b, together with the observations. Upon first inspection, this figure reveals that the modeledand observed reflectances agree fairly well over the whole angular domain. However, it is noticeable that model B leads to a better agreement than models A or C with the data taken close to the hot spot region. The reflectance patterns obtained from models A and C are very close to each other indicating that

reflectance and transmittance factors of the scatterers (see, for example, Pinty et al. [1990] and Dickinson et al. [1990]). These physical quantities are measurable in the laboratory. Although they might be more difficult to measure, the parameters describingthe orientation of the scatterers and the degree of compaction are also measurable in principle. Unfortunately, such measurements are not available from the experiment made by Woessner and Hapke [1987], and from this standpoint, further discussion of the inversion results would necessarily involve rather speculative arguments.

3.3. Application to Laboratory Data Contaminated by "Noise"

In the above comparison between model prediction and

scatterer orientations, as described by model A. Despite the slightly better performance of model B as compared to the other two models, it remains difficult to

data, there is no allowance for errors in the data or for noise coming from various sources, including the inversion procedure itself. We soughtto find out whether the differencesin the physical parameters retrieved from the three models, for instance in the single scattering albedo toand the asymmetry factor O, are significant when considering that the data may be contaminated by "noise." To addressthis question, we added a random noise of zero

decide which model is the correct

mean and known

the amplitudeparameterB0 of the hot spotfunctionin model C balancesthe effect due to a nonsphericaldistribution in the

one from the results of the

above experiment alone. As discussedin section 1, the only way to select the best model is to compare the parameter values retrieved from the inversion procedure to the values measured separately using appropriate instrumentation. For instance, the single scattering albedo to and the phase function P(#) can be combined to estimate the hemispherical

variance

to each value of the reflectance

data set described above and repeated this operation 500 times. The inversion procedure was applied each time using model A alone, and the four physical parameters, as well as the rms of the fits, were averaged over all the retrievals. The standard deviation of the noise was chosen to generate averaged values for each parameter, plus an rms close to the

PINTY AND VERSTRAETE: BIDIRECTIONAL REFLECTANCE MEASUREMENTS

2871

.11

.10

ß09

•

c

0

o7

ß

05

.--

u -.

.

O•

-.

.03 -

ß02

-

o•1

-

0 -100

I

I

-80

I

•

-60

I

,

-•0

I

•

-20

I

0

I

I

,

20

Viewing Rngle in the principol

I

,

•0

I

60

,

I

i

80

plone

Fig. 2b. Comparison between the bidirectional reflectancesmeasured over a clover patch at 60ø zenith angle and those modeled with models A, B, and C, using optimally fitted parameters with data taken at 0ø zenith angle.

one in the simple application described in the previous section.

standard

It was found

deviation

that the addition

of 0.001

to the

of a noise with

reflectance

data

a

was

enough to satisfy this requirement within 0.1% of the retrieved values. Proceeding that way, the derived standard

deviationsare 0.22 x 10-2, 0.70 x 10-2, 0.29 x 10-] and

contaminatedby noise, the additional question then arises as to whether the inversion procedure retrieves stationary averaged values for the model parameters when increasing the level of an artificial random noise in the synthetic data. Two basic data sets were built, denoted by A and B referring to the model version we used. For data set A, to =

0.97 x 10-2, for to, ©, Xt and2rA, respectively. These 0.4, © = -0.2, 2rA = 0.2 and X• = 0. The corresponding values are indicative

of the errors which exist in the retrieval

we made in Table 1. They show, for instance, that the differencesin the values of the parameterswhich result from the inversion of the three models are significantlylarger than the variations one would expect from the noise in the input data. In other words, the three models are significantly different from each other, even in the presence of noise in the input data set. 4.

MODEL

COMPARISONS

IN RETRIEVAL

ON SYNTHETIC

EXERCISES

DATA

In the previous section, we investigated the sensitivity of the values of the model parameters to small noise levels in the input data. We now study a different kind of sensitivity, namely, the sensitivity of our results to small changesin the formulation

of the model. In order to examine the misinter-

pretation which may occur when inverting a "wrong model" against a given data set, a few experiments were performed using synthetic bidirectional reflectances, i.e., bidirectional reflectancesgenerated by models A and B. The basic experiment was to invert model A (which standsin that case as the "wrong model") with a synthetic data set generated by model B and, conversely, to invert model B with synthetic data produced by model A. Because observational data are

values for data set B are the same as above with Sv(O) = 0.5625 so that P0 = 0.75. In both cases, bidirectional reflectance values were computed in the principal plane for an illumination source at 0ø, with the viewing angle varying from 10ø to 70ø in increments of 10ø. To get a good representation for very small phase angles, an additional viewing condition at 2ø was also considered, so that 16 bidirectional reflectance values were generated. Using the procedure described in section 3.3, some random noise was added to these synthetic data. The standard deviation values we considered are successively 0.0001, 0.001, 0.005 and 0.01, correspondingapproximately to 0.1%, 1%, 5% and 10% of the hemisphericalreflectance value of the data sets. Because the optimization algorithm can sometimesfind an ambiguous minimum (in which case the routine issues an explicit message), the number of cases considered when averaging the parameter values can be less than 500. Table 2 gives the values of the various physical parameters when model A is inverted against the bidirectional reflectance data generated by model B, for different values of the standard deviation of noise. First, as already shown by Pinty et al. [1989], the noise inherent to the optimization procedure is an order of magnitude lower than the external perturbation since the rms of the fit is very close to the

2872

PINTY AND VERSTRAETE: BIDIRECTIONAL

TABLE

Nominal

value

2.

Inversion

•o

•)

0.400

-0.200

REFLECTANCE

of Model

A With

Xl

2r A

0.000

0.200

Standard Deviation,* Retrieved value

(Standard deviation)

0.394

-0.216

(0.0009)

Set B

Sv(O)

N

rms of Fit

500

0.0004

500

0.001

499

0.005

500

0.01

0.5625

0.0001

0.028

(0.0013)

Data

MEASUREMENTS

0.092

(0.0032)

(0.0011)

Standard Deviation, 0.001 Retrieved value

(Standard deviation)

0.394

-0.216

(0.009)

-0.027

(0.013)

(0.033)

0.093

(0.011)

Standard Deviation, 0.005 Retrieved value

(Standard deviation)

0.406

-0.192

(0.051)

-0.003

(0.079)

(0.17)

0.171

(0.18)

Standard Deviation, 0.01 Retrieved value

(Standard deviation)

0.403

-0.196

(0.068)

(0.10)

0.019

0.184

(0.24)

(0.22)

N denotes number of successfulcases, out of 500. *Of the input data.

standarddeviation of the random noise added to the synthetic data. Secondly, we found only one situationof non-

hemisphericalreflectancevalue), the correspondinguncertaintiesare 17%, 51% and 120% for to, 19and 2rA, respecconvergence over all the conditions studied (2000 data sets). tively. These results indicate that the uncertainty in the As clearly seen when the standard deviation of the noise retrieval of the morphologicalparameters is significantly (s.d.) is at its lowest value (i.e., 0.1% of the hemispherical larger than the uncertainty on the optical parameters. reflectance),model A acts to balancethe specificfeatures Resultsof the inversionof model B with data generatedby imposedon the data through model B by overestimating19 model A are given in Table 3. Compared with Table 2, and ,¾land, more significantly,by lowering the values of to situations of nonconvergence(column N) occur more freand 2rA. In that case, the relative errors between the

nominal values and the averaged retrieved values are 2%,

8% and 50%, in to, t9 and 2rA, respectively.Table 2 shows that when the standard deviation increases,the values of the retrieved parameters get closer to the nominal values, for example, the relative error in 2rA is less than 10% for a

quently. The inversion procedure itself works reasonably well sincethe averagedretrieved values are very closeto the nominal values for the lowest standard deviation we consid-

eredin the experiment.Note that the nominalvaluefor Sv(0) is implicitlyequalto 0.75 sincethe valuefor P0 is equalto 1. The most interestingfeature revealed by Table 3 lies in the standarddeviationequalto 0.01. Another interestingfeature of the experimentconcernsthe redistributionof the input occurrence of a systematic trend on the averaged values noiseinto the model parameters.Clearly, this redistribution when increasingthe level of the input noise. Indeed, with is not equally partitioned between the model parameters respect to their nominal values, to, Sv(O), and 2rA are since,for instance,the remainingstandarddeviationscorre- increasingwith the standard deviation of the noise, while 19 spond to 0.2%, 0.6% and 1.2% of the averaged retrieved and Xl are decreasing. Such a divergence on the averaged values in to, 19and 2rA when the standarddeviation equals retrieved values may indicate a lack of stability in model B. 0.0001. If the latter is increased to 0.01 (i.e., 10% of the As notedin the previousexperiment, a significantpart of the

TABLE

to Nominal

value

3.

Inversion

19

0.400

-0.200

0.406

-0.190

of Model

Xt 0.000

Standard Deviation,* Retrieved value

(Standarddeviation)

(0.0014)

(0.0026)

-0.020

(0.0052)

B With Data Set A

2r A

S v(0)

N

rms of Fit

493

0.0002

469

0.001

394

0.005

396

0.01

0.200

0.0001 0.216

(0.0084)

0.747

(0.0070)

Standard Deviation, 0.001 Retrieved value

(Standarddeviation)

0.408

(0.015)

-0.185

(0.027)

-0.029

(0.054)

0.232

0.771

(0.085)

(0.097)

Standard Deviation, 0.005 Retrieved value

(Standard deviation)

0.423

(0.061)

-0.149

(0.11)

-0.071

(0.21)

0.345

3.59

(0.41)

(4.19)

Standard Deviation, 0.01 Retrieved value

(Standard deviation)

0.423

(0.085)

-0.139

(0.14)

-0.077

(0.28)

N denotes number of successfulcases, out of 500. *Of the input data.

0.395

5.34

(0.69)

(4.54)

PINTY

AND VERSTRAETE:

BIDIRECTIONAL

input noise is redistributed onto the morphologicalparameters.

In conclusion, it appears from these two numerical experiments that model A may provide better estimates than model B, since the results of the inversion of model A are

less sensitive to input noise. In this sense, even if some intuitive physical expectationssupportthe use of a parameter P0 (or B0) with a value differentfrom unity, model A acts to balance this sensitivity to input noise, and leads to more statistically acceptable results; that is, the retrieved values remain close to the nominal values. By contrast, results from the inversion

of model

MEASUREMENTS

2873

arrangements and orientations of these scatterers (see, for example, Ranson et al. [1985]). These experiments would allow a true validation

of the radiative

transfer

models which

are discussedby allowing direct comparisons between the values of the parameters retrieved from the inversion and those directly available from the laboratory measurements. The study of vegetation canopies and bare soil surfaces could provide interesting comparisons for these models, since they offer media with widely different values of compaction, and therefore help to better understand the hot spot phenomenon.

B are not as stable in the retrieval.

This is a very critical point for practical applicationsbecause it implies that, for a given medium, the values of the physical parameters retrieved from an inversion with model B are biased by the level of the noise present in the observations. 5.

REFLECTANCE

CONCLUSIONS

In this work, a series of tests regarding the inversion of physical models with actual and synthetic bidirectional reflectancedata setswere performed. We addressedproblems arisingfrom the lack of a complete theoretical understanding of the radiative transfer occurring through a particulate semi-infinite medium. For instance, we examine the consequences of (1) considering a nonspherical shape for the scatterers as adopted by Verstraete et al. [1990], and (2) adding a posteriori an amplitude parameter on the "hot spot" function as done by Hapke [1986]. Our main goal was to show the sensitivity of the results obtained from the inversion of models including these two contributions. Using actual data, it was first demonstrated that various models can lead to an acceptable fit of the observed reflectance patterns. It was also shown that the parameter describing the shapeof the scattererphasefunction and the parameters that control the width of the hot spot are the most sensitive to the mathematical

formulation

of the model.

As

already noted by Domingue and Hapke [ 1989]when addressing the nonuniquenessof the fit, a reasonablygood stability is observed in the retrieval of the single scattering albedo. Synthetic reflectance data contaminated by a random noise were generated by a given model and then inverted with a different model. This experiment reveals that addinga varying amplitude parameter to the hot spot function does not yield stable values in the retrieved model parameters. We also observed systematic positive or negative trends on the retrieved values when the standarddeviation of the input noise added to the reflectances

was increased from 0.1% to

roughly 10% of the correspondinghemisphericalreflectance values. This behavior suggeststhat allowing the amplitude parameter to vary could lead to a strongly biased estimation of the physical parameters characterizingthe surface. From this standpoint, we believe that, at present, it is more reasonableto avoid this problem by constrainingthe amplitude parameter to a constant value equal to unity. Clearly, more experimental and theoretical work is urgently needed to clear up this critical point. Two possible ways can be suggested: (1) the use of Monte-Carlo simulation approaches (see, for example, Ross and Marshak [1984]), and (2) the simultaneous collection of bidirectional reflectance data over a surface and separate laboratory measurements of the optical properties of the scatterers making up the surface, as well as observationsof the relative

Acknowledgments. It is a pleasure to acknowledge the leadership role of B. Hapke in pioneering this field. We are very grateful for his willingness to read a draft of this paper and for encouraging us in our research. In the course of this study, we benefited from the computing facilities at the National Center for Atmospheric Research (NCAR). This work would not have been possible without the financial support of the European Space Agency (ESA), the Centre National de la Recherche Scientifique (CNRS), and the National Center for Atmospheric Research (NCAR) for B.P.M.V. acknowledges the support of the University of Michigan. The National Center for Atmospheric Research is operated by the University Corporation for Atmospheric Research under the sponsorship of the National Science Foundation.

REFERENCES

Camillo, P., A canopy reflectance model based on an analytical solution to the multiple scattering equation, Remote Sens. Environ., 23, 453-477, 1987. Dickinson, R. E., Land surface processes and climate--Surface albedos and energy balance, Adv. Geophys., 25, 305-353, 1983. Dickinson, R. E., B. Pinty, and M. M. Verstraete, Relating surface albedos in GCMs to remotely senseddata, Agric. For. Meteorol., in press, 1990. Domingue, D., and B. Hapke, Fitting theoretical photometric functions to asteroid phase curves, Icarus, 78, 330-336, 1989. Goguen, J., A theoretical and experimental investigation of the photometric functions of particulate surfaces, Ph.D. thesis, Cornell Univ., Ithaca, N.Y., 1981. Goudriaan, J., Crop micrometeorology: A simulation study, report, Wageningen Cent. for Agric. Publ. and Doc., Wageningen, The Netherlands, 1977. Goudriaan, J., The bare bones of leaf-angle distribution in radiation models for canopy photosynthesis and energy exchange, Agric. For. Meteorol., 43, 155-169, 1988. Hapke, B. W., Bidirectional reflectance spectroscopy,1, Theory, J. Geophys. Res., 86, 3039-3054, 1981. Hapke, B. W., Bidirectional reflectance spectroscopy,3, Correction for macroscopic roughness,Icarus, 59, 41-59, 1984. Hapke, B. W., Bidirectional reflectance spectroscopy, 4, The extinction coefficient and the opposition effect, Icarus, 67, 264-280, 1986.

Helfenstein, P., and J. Veverka, Photometric properties of lunar terrains derived from Hapke's equation, Icarus, 72, 342-357, 1987.

Henyey, L. G., and T. L. Greenstein, Diffuse radiation in the galaxy, Astrophys. J., 93, 70-83, 1941. Lumme, K., and E. Bowell, Radiative transfer in the surfaces of atmospherelessbodies, II, Interpretation of phase curves, Astron. J., 86, 1705-1721, 1981. Pinty, B., M. M. Verstraete, and R. E. Dickinson, A physical model for predicting bidirectional reflectances over bare soil, Remote Sens. Environ., 27, 273-288, 1989. Pinty, B., M. M. Verstraete, and R. E. Dickinson, A physical model of the bidirectional reflectance over vegetative canopies, 2, Inversion and validation, J. Geophys. Res., 95, 11,767-11,775, 1990. Ranson, K. J., L. L. Biehl, and M. E. Bauer, Variation in spectral response of soybeans with respect to illumination, view and canopy geometry, Int. J. Remote Sens., 6, 1827-1842, 1985. Ross, J., and A. L. Marshak, Calculation of the canopy bidirectional

2874

PINTY AND VERSTRAETE: BIDIRECTIONAL REFLECTANCE MEASUREMENTS

reflectance using the Monte-Carlo method, Remote Sens. Environ., 24, 213-225, 1984. Sellers, P. J., Canopy reflectance, photosynthesisand transpiration, Int. J. Remote $ens., 6, 1335-1372, 1985. Verstraete, M. M., Radiation transfer in plant canopies: Transmission of direct solar radiation and the role of leaf orientation, J. Geophys. Res., 92, 10,985-10,995, 1987. Verstraete, M. M., B. Pinty, and R. E. Dickinson, A physical model of the bidirectional reflectance of vegetation canopies, 1, Theory, J. Geophys. Res., 95, 11,755-11,765, 1990. Woessner, P., A study of the polarization of light scattered by vegetation, M.S. thesis, 91 pp., University of Pittsburgh, Pittsburgh, Pa., 1985.

Woessner, P., and B. Hapke, Polarization of light scattered by clover, Remote $ens. Environ., 21, 243-261, 1987.

B. Pinty, Laboratoire d'Etudes et de Recherches en T616d6tection Spatiale, 18, Avenue Edouard-Belin, 31055 Toulouse cedex, France.

M. M. Verstraete, Department of Atmospheric, Oceanic and Space Sciences, University of Michigan, Ann Arbor, MI 48109.

(Received April 27, 1990; revised August 22, 1990; accepted October 12, 1990.)