Errors in the remotely sensed ocean reflectance

Adv. Space Res. Vol. 7, No. 2, pp. (2)31—(2)35, 1987 Printed in Great Britain. All rights reserved.

0273—1177/87 $0.00 + .50 Copyright © COSPAR

ERRORS IN THE REMOTELY SENSED OCEAN REFLECTANCE S. Keevallik and A. Heinlo Institute ofAstrophysics and Atmospheric Physics, 202444, Tôravere, Estonia, U.S.S.R.

ABSTRACT An approximate method has been described to determine optical properties of the ocean surface from the reflected short—wave radiance at the top of the atmosphere. The non—scattered and singly scattered components of the radiation field have been taken into account exactly, the multiply scattered components approximately. The calculations for a real atmospheric model have been compared with the results of an exact method for solving the radiation transfer equation. —

On the basis of the elaborated algorithm the sensitivity of the value of the remotely sensed spectral albedo of the ocean to the variations in optical thickness, single scattering albedo and aerosol phase function has been described as well as the influence of vertical inhomogeneity of the atmosphere. INTRODUCTION In the visible and near infrared spectral regions optical thickness of the atmosphere is usually of the order of 0.2 to 0.8. In this diapason the asymptotic methods neither for a thick scattering medium nor for a thin layer can be applied. To avoid extensive calculations some of them are still used in the correction algorithms to remove atmospheric effects from remotely sensed radiative data. The other possibility to reduce the radiation transfer calculations is the simplification of the atmospheric model. Both procedures induce errors in the values of remotely sensed surface properties. ERRORS DUE TO A HOMOGENEOUS MODEL ATMOSPHERE The most ordinary simplification for an atmospheric model is the neglect of the vertical inhomogeneity of the atmosphere, so that to characterize it, only total optical thickness8 mean single scattering albedo and mean phase function are used. The validity and limits of this approximation have been estimated by the authors in an earlier paper /1/. It has been shown that the computations on the basis of a homogeneous atmospheric model erroneously ascribe too high values to the reflectance of the underlying surface. The errors are mainly induced by the neglect of stratification of the aerosol phase function. If the shape of it varies with the a].-. titude as shown in Figure 1, in the case of a hazy atmosphere at large solar and viewing angles (over 45°)the errors of the averaged model can amount to 14.0 % with respect to the stratified one. This estimation has been obtained for the conditions of remote sensing of ocean surface in the spectral region from 0.4 pm to 1.1 pm. Ocean albedo was considered to be 0.1. It is obvious that over brighter surfaces the error would be less drastic. The errors, due to the neglect of the stratification of the single scatter-. ing albedo /2/, do not exceed 5 % in the most unfavourable cases. Thus, the stratification of the single scattering albedo may be disregarded in the computations.

JASR 7/2

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(2)31

(2)32

S. Keevallik and A. Heinlo

p4

~

2

1~

\. \ —.————

I_i

~

~

.

-1

-~

60

120

180

Pig. 1. The aerosol phase function in the layers of 0—1 km 1—10 km

(— —

—),

and 10—50 km

(_

(

),

.

ERRORS DUE TO THE SINGLE SCATTERING APPROXIMATION The radiance at the top of the atmosphere in the case of a Lambert surface

can be presented by a formula

L~ ~

L~(~, ~, ~) g

(1)

•

Here Z90 is the solar zenith angle, ‘iY- is the viewing zenith angle, ~ the azimuth, L~(~~P0Zt~) is the path radiance, 9 is the ocean spectral al— bedo, £~(t1P0) is the flux at the lower boundary of the atmosphere of the radiation that has not interacted with the ocean surface, ~ is the spherical albedo of the atmosphere for the reflected radiation, ‘Z(’ZP-) is the total (direct plus diffuse) atmospheric transmittance factor. The terms in formula (1) consist of several components; ~

=

E~(~) E =

~(~~) ~

~ (~0) *

~(so)

~i

~

~

(~e) ~

(~R).

(2)

In these formulas the lower indices show the number of ecatteringe in the atmosphere and the single scattering approximation means that the last component is neglected in each term. This leads to systematic errors in the spectral albedo of the ocean that may be estimated by means of the follow1t 4Vi-w) ing formula: z~O. 1 (3) where is the optical thickness of the atmosphere and cJ value of the single scattering albedo.

is the mean

Estimation (3) has been obtained from a set of oases where S varied from 0.2 to 0.8 and &) from 0.5 to 1.0. The value of the ocean albedo was 0.1. Pormula (3) shows that if c5> 0.5, the relative error of the ocean albedo easily exceeds 100 %. To avoid this kind of errors, a method has been proposed to take multiple scattering into account with a reasonable accuracy. The method is based on the single scattering approximation and does not require additional computation time.

Errors in Ocean Reflectance

(2)33

DESCRIPTION OP THE METHOD The method starts from evaluating the flux Z~Ethat remains in the atmosphere after the second scattering and, thus, participates in multiple scattering: ~E Here S,, denotes the input solar radiance and E~,1~(4.90)isthe flux at the upper boundary of the atmosphere of the singly scattered radiation that has not interacted with the ocean surface. To obtain from the flux ~ the multiply scattered fluxes ~/nd~f(’tP0) at the boundaries of the atmosphere, one must consider that a part of the flux i.1E will be absorbed: ~ where C(4)~ -f • The rest of the flux ~E that is not absorbed will be divided between the upper and lower boundaries of the atmosphere: E~(’tzP0)=

)c(’P0)~E,

where o~(’D’0) depends mainly on the shape of the total (molecular plus aerosol) phase function. The coefficients c(’zP0) and”<7’t~)may be determined /3/, using an assumption that the fluxes, leaving the atmosphere after different orders of scattering, form a geometrical progression. Within the limits for the optical parameters that are characteristic of a real atmosphere (ó>’í and phase function asymmetry <0.14.) these coefficients can be expressed through the non—scattered and singly scattered fluxes at the boundaries of the atmosphere and by integration of the phase function over a hemisphere. The next step after the determination of multiply scattered fluxes is the evaluation of L~~(~tP0tZfi, ~) on the basis of E~1(~Z’.,). For this purpose the angular distribution of the leaving radiance may be given by means of an empirical function, so that ~

~

To calculate the components g~ and ‘Z~(rZ9), the same procedure should be repeated for reflected radiation. Now the atmosphere should be assumed to be illuminated from below by a unit diffuse flux. The described method for calculation of the multiply scattered components in equations (1) (2) allows to obtain the value of the radiance L at the top of the atmosphere with an accuracy better than 3 %. —

It must be repeated that the method must not be regarded as an universal one. Its application should be restricted by real conditions of remote sensing (real atmosphere,t~6O°, ~ ~6O°). ERRORS CAUSED BY THE VARIABILITY OP ATMOSPBERIC PARAMETERS The error estimated above appears only due to the simplification of radiative transfer calculations. In reality the parameters of the atmosphere are not exactly known and can only be supposed to be iithin some limits. To estimate the errors in the values of remotely sensed spectral albedo of the ocean, the sensitivity of the latter to variations in atmospheric parameters has been investigated. Each atmospheric parameter has been varied separately while keeping the others constant. The parameters of the atmosphere were choaen within the following boundaries: O.2~-6
(2)34

S. Keevallik and A. Heinlo

1. The optical thickness of the real atmosphere is ~ , but in calculations we have used a somewhat different value of ~ ~ • An example of the de-

pendence of the relative error in the ocean albedo on the values of real albedo g and real optical depth S is shown in Figure 2. If/ /5/o.’f, an approximation for the relative error in the ocean albedo may be obtained: -~ --~‘—O.289 ~ ~ (4) 2. The absorption number in the real atmosphere (‘t-&) differs from the absorption number (4-~-~&~) used in the calculations. The relative error in the ocean albedo at the values of~=O.75 and U—-cZl&)is shown in Figure 3. The dependence of the relative error in the ocean albedo on the atmospheric parameters, if/’~Y~/.~ 0.1 , can be approximated by the formula: ______

_2/

~ 2

/

~

UU

(5)

—

C0577)C0.Y?)

~

9

9

08 0.8

~3 ~ =0295

02

____________________

0.0

(.3=075

0.08

0.10

0.12

0.l’i

Fig. 2. Relative error in the spectral albedo of the ocean in the case of a 10 % error in the atmospheric optical thickness, The solar and viewing angles are 30° and 0°respectively.

~ =030

0

___________________

0.0

0.08

0.10

0.12

014

Fig. 3. Relative error in the spectral albedo of the ocean in the case of a 10 % error in the single scattering albedo. The solar and viewing angles are 30°and 0°respectively.

3. Uncertainties in the estimates of the atmospheric ‘noise’ may also be induced by a different shape of the phase function. An analysis shows that this kind of errors depend mainly on the values of the backscattering and are less sensitive to the exact shape of the phase function. The relative error in the ocean albedo due to uncertainties in phase function is shown in Figure 11. For the range of’ the investigated atmospheric parameters the approximation takes place: L~9 —~-O~8~ ~ — . L/ O~W~ ~ (6 As can be seen from these formulas, absorption in the atmosphere reduces the errors due to the variations in the atmospheric parameters. But from approximation (5) and Figure 3 it follows that alight uncertainties in c.~ (in cases (I—A.)>>O) may easily induce significant errors in the ocean al— bedo. Thus, in the spectral regions where absorption occurs, the value of &, should be known with a good accuracy. As to the errors induced by erroneous phase function, the values shown in Figure 4 are nearly the maximal possible ones, while a relative error of

Errors in Ocean Reflectance

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10 % in backsoattering is rarely exceeded. Thus, the poor knowledge of the exact shape of the phase function is somewhat compensated by a weak dependence of ocean albedo error on the phase function error.

9 • -0.08

\~

8:0.315

50,2

0.06

0.08

0.10

0.12

0.14

Fig. .1-. Relative error in the spectral albedo of the ocean in the case of a 8 % error in atmospheric backacat-tering. The solar and viewing angles are 30° and 0° respectively. CONCLUSIONS 1. A method has been proposed -to calculate multiply scattered components of the radiance at the top of the atmosphere above the ocean surface. The accuracy of’ the determination of the total radiance at the top of the atmosphere is better than 3 %, while the method does not require additional computation time in comparison with the single scattering approximation. 2. The vertical inhomogeneity of the single scattering albedo may be disregarded at remote sensing. But if the stratification of the aerosol phase function is neglected, errors in the remotely sensed values of’ the ocean reflectance can amount to 40 %.

3. At the wavelengths, where absorption is substantial, the level of’ the atmospheric ‘noise’ is lower, but at the same time the inaccuracy in the absorption number produces relatively great errors in the ocean albedo. The errors caused by uncertainties in the shape of the atmospheric phase function are relatively small. REFERENCES 1. S. Keevallik and A. Heinlo, On the Validity of Vertically

Homogeneous Atmospheric Models for Remote Sensing,Acaderny of Sd. ESSR, Tartu, 1983 2. R.A. McClatchey, R.W. Penn, J.E.A. Selby, F.E. Volz, and J.S. Garing, Optical Properties of the Atmosphere, Environm. Res. Papers, No. 354, AFCRL—71 —0279, 1971

3. S. Keevallik and A. Heinlo, Multiple scattering in the remote sensing of the earth, Investigation of the earth from the space, No. 2, 84 (1986) (in Russian)