## Abstract

The processing algorithms used for relating the apparent color of the ocean observed with the Coastal-Zone Color Scanner on Nimbus-7 to the concentration of phytoplankton pigments (principally the pigment responsible for photosynthesis, chlorophyll *a*) are developed and discussed in detail. These algorithms are applied to the shelf and slope waters of the Middle Atlantic Bight and also to Sargasso Sea waters. In all, four images are examined, and the resulting pigment concentrations are compared to continuous measurements made along ship tracks. The results suggest that over the 0.08–1.5-mg/m^{3} range the error in the retrieved pigment concentration is of the order of 30–40% for a variety of atmospheric turbidities. In three direct comparisons between ship-measured and satellite-retrieved values of the water-leaving radiance the atmospheric correction algorithm retrieved the water-leaving radiance with an average error of ~10%. This atmospheric correction algorithm does not require any surface measurements for its application.

© 1983 Optical Society of America

## I. Introduction

The Coastal-Zone Color Scanner (CZCS) on Nimbus-7 is a scanning radiometer which views the ocean in six coregistered spectral bands, five in the visible and near IR (443, 520, 550, 670, and 750 nm) and the sixth, a thermal IR band (10.5–12.5 *μ*m). The sensor has an active scan of 78° centered on nadir and a field of view of 0.0495°, which from a nominal height of 955 km produces a ground resolution of 825 m at nadir. The satellite is in a sun-synchronous orbit with the ascending node near local noon. The sensor is equipped with provision for tilting the scan plane ±20° from nadir in 2° increments along the satellite track to minimize the influence of direct sun glint (the contribution to the sensor radiance from photons which were specularly reflected from the sea surface without interacting with the atmosphere).

The purpose of the CZCS experiment is to provide estimates of the near-surface concentration of phytoplankton pigments and total seston by measuring the spectral radiance backscattered out of the ocean.[1] The radiance backscattered from the atmosphere and/or sea surface (specular reflection) is typically at least an order of magnitude larger than the desired radiance scattered out of the water *L** _{w}*. The process of retrieving

*L*

*from the total radiance measured at the sensor*

_{w}*L*

*is usually referred to as atmospheric correction, even though all surface reflection effects other than direct sun glint are removed with a single algorithm. The phytoplankton pigment concentration is estimated from the retrieved spectral radiances by the application of an in-water or bio-optical algorithm.*

_{t}In the first comparison between ship-measured and CZCS-derived pigment concentrations,[2] the atmospheric correction was effected by utilizing ship measurements of *L** _{w}* at a single location in the image. Also the prelaunch in-water algorithms[3] were used to derive the pigment concentration along the ship’s track. The two data sets tracked one another well, although their rms difference was a factor of about 2. Following this initial comparison, additional cruises in support of the CZCS experiment have been completed, resulting in a considerably larger data base on which to fine tune the in-water algorithms.[4] Also the recent quantitative development of the concept of clear water radiance[5] suggests the possibility that atmospheric corrections can be made without recourse to any surface measurements, removing the principal blemish on the initial comparisons. In this paper we apply these new developments to CZCS imagery acquired off the East Coast of the U.S.A. and compare the results with simultaneously collected ship data.

## II. Atmospheric Correction

As mentioned above, the first step in CZCS processing is the derivation of the water-leaving radiance *L** _{w}*(λ) at a wavelength λ by removing the effects of the intervening atmosphere. The severity of these effects can be appreciated by considering that at the ship station for Orbit 3240 described below the sensor radiance

*L*

*in the blue was about ten times the water-leaving radiance*

_{t}*L*

*. These effects are principally due to scattering by the air (Rayleigh scattering) and by microscopic particles suspended in the air (aerosol scattering), both of which increase the radiance detected at the sensor. (Note: direct sun glint, i.e., the contribution to*

_{w}*L*

*from photons specularly reflected from the sea surface without interaction with the atmosphere, is ignored since its contribution is decreased significantly because of the tilting capability of the CZCS.) In principle this added radiance could be removed if the concentration and optical properties of the aerosol were known throughout an image. The aerosol, however, is highly variable, and, unlike the Rayleigh scattering component, its effect on the imagery cannot be predicted*

_{t}*a priori*. The basis for the correction of these effects was developed by Gordon,[6] specifically applied to the problem of phytoplankton pigment retrieval by Gordon and Clark[3] and first applied to CZCS imagery by Gordon

*et al*.[7] The correction algorithm is most easily understood by considering first only single scattering. In this approximation, ignoring direct sun glint and assuming that the sea surface is flat, the sensor radiance

*L*

*(λ) can be divided into its components:*

_{t}*L*

*(λ) the contribution arising from Rayleigh scattering;*

_{r}*L*

*(λ) the contribution arising from aerosol scattering; and*

_{a}*t*(λ)-

*L*

*(λ) the water-leaving radiance diffusely transmitted[8] to the top of the atmosphere; i.e.,*

_{w}*L** _{r}* and

*L*

*are given by*

_{a}where

Ө_{0} and *ϕ*_{0} are, respectively, the solar zenith and azimuth angles, Ө and *ϕ* are the zenith and azimuth angles of a vector from the point on the sea surface under examination (pixel) to the sensor. *ρ*(Ө) is the Fresnel reflectance of the interface for an incident angle Ө, *P** _{x}*(Ө,λ) is the scattering phase function of component

*x*(

*x*=

*r*or

*a*) at λ,

*ω*

*(λ) is the single-scattering albedo of*

_{x}*x*(

*ω*

*= 1), and*

_{r}*τ*

*(λ) is the optical thickness of*

_{x}*x*. ${{F}^{\prime}}_{0}\left(\lambda \right)$ is the instantaneous extraterrestrial solar irradiance

*F*

_{0}(λ) reduced by two trips through the ozone layer, i.e., ${{F}^{\prime}}_{0}={F}_{0}\phantom{\rule{0.2em}{0ex}}\text{exp}\left[-{\tau}_{Oz}\left(1/\text{cos}\u04e8+1/\text{cos}{\u04e8}_{0}\right)\right]$, where

*τ*

*is the ozone optical thickness. The term involving Ө*

_{Oz}_{−}in Eq. (2) provides the contribution due to photons which are backscattered from the atmosphere without interacting with the sea surface. The terms involving Ө

_{+}account for those photons which are scattered in the atmosphere toward the sea surface (sky radiance) and then specularly reflected from the surface into the field of view of the sensor [

*ρ*(Ө) term] as well as photons which are first specularly reflected from the sea surface and then scattered by the atmosphere into the field of view of the sensor [

*ρ*(Ө

_{0}) term]. If the assumption of a flat surface is relaxed, these terms involving

*ρ*become integrals over solid angle of essentially the product of the reflectance, the phase function, and the Cox and Munk[9] surface slope probability density function.

*t*(λ) is the diffuse transmittance of the atmosphere between the sea surface and the sensor. It is given by

where *t** _{a}*(λ) = exp{−[l −

*ω*

*(λ)*

_{a}*F*(λ)]

*τ*

*(λ)/cosӨ}, and*

_{a}*F*is the probability that a photon scattered by the aerosol will be scattered through an angle of <90°. The upper limit to the factor [1 −

*ω*

*(λ)*

_{a}*F*(λ)] is ~⅙, so

*t*

*depends only weakly on the aerosol optical thickness. The rationale for using the diffuse transmittance rather than the direct transmittance is to account for the fact that when the sensor is viewing a given pixel some of the radiance it receives originates from neighboring pixels. The only unknowns in these equations are*

_{a}*ω*

*,*

_{a}*τ*

*, and the aerosol scattering phase function.*

_{a}From Eq. (2) it is seen that

where

Equations (1) and (2) are rigorously correct in the limit that the slant paths *τ** _{r}*/cosӨ,

*τ*

*/cosӨ*

_{r}_{0},

*τ*

*/cosӨ, and*

_{a}*τ*

*/cosӨ*

_{a}_{0}all approach zero. Although Eq. (2) provides a poor estimate of

*L*

*(λ) for values of*

_{a}*τ*

*(λ) where multiple scattering becomes important [i.e.,*

_{a}*τ*

*(λ) >0.1], it is shown in the Appendix that Eq. (1) is still approximately valid even for large optical thicknesses as long as the radiometer is viewing the ocean sufficiently far from the center of the glitter pattern, a condition normally insured by the glint avoidance procedure on the CZCS. It is important to note that for a given aerosol type, defined here to be a given normalized particle size distribution and refractive index,*

_{a}*∊*(λ

_{2},λ

_{1}) is independent of the aerosol concentration. Furthermore, since over the spectral region of interest here (443–670 nm) the aerosol scattering phase function usually depends only weakly on wavelength,

*∊*(λ

_{2},λ

_{1}) will be nearly independent of the aerosol phase function, hence independent of the viewing angle. For example, consider geometry for which the sun is directly behind the spacecraft (

*ϕ*−

*ϕ*

_{0}= 90°) at a zenith angle of 45° and the CZCS scans the ocean in an untilted mode; then over the full CZCS scan 45° < Ө

_{+}< 57.2° and 122° < Ө

_{−}< 135°. Using aerosol phase functions computed by Quenzel and Kastner[10] for marine aerosol models corresponding to 70 and 90% relative humidity it is found that the phase function-induced variation in

*∊*(443,670) from the center to the edge of the CZCS scan is 6.9 and 0.5%, respectively. Since

*ω*

*(λ) is also usually a weak function of wavelength,*

_{a}*∊*(λ

_{2},λ

_{2}) is principally determined by

*τ*

*(λ*

_{a}_{2})/

*τ*

*(λ*

_{a}_{1}). In what follows, it is assumed that

*∊*(λ

_{2},λ

_{1}) is constant even in the presence of a horizontally inhomogeneous aerosol. This is equivalent to assuming a constant aerosol type. The proportionality between

*L*

*(λ*

_{a}_{1}) and

*L*

*(λ*

_{a}_{2}) in this case [Eq. (4)] is still approximately valid even in the presence of multiple scattering. Combining Eqs. (1) and (5) we have

for *i* = 1, 2, and 3, where the indices *i* = 1, 2, 3, and 4 refer to the four visible CZCS bands in order of increasing wavelength, and

Equations (6) are three in number, but there appear to be eleven unknowns, *t*(λ* _{i}*) for

*i*= 1–4,

*S*(λ

*,λ*

_{i}_{4}) for

*i*= 1–3, and

*L*

*(λ*

_{w}*) for*

_{i}*i*= 1–4. However, for a given aerosol type the three

*S*(λ

*,λ*

_{i}_{4}) values can be determined everywhere once they are determined at one position in the image, reducing the number of unknowns to eight. Also

*t*(λ

*) is unknown only because the aerosol optical thickness is required for the computation of*

_{i}*t*

*(λ*

_{a}*). In most cases of practical interest,*

_{i}*t*

*(λ*

_{a}*) can be set to unity because the entire algorithm will break down before*

_{i}*τ*

*becomes large enough to significantly influence the results through its effect on*

_{a}*t*

*(λ*

_{a}*). Thus*

_{i}*t*(λ

*) can be taken to be known, and there are in fact only four unknowns in the three parts of Eq. (6). To close the system and enable a solution, an additional equation must be included. Smith and Wilson[11] use an empirical equation of the form*

_{i}while in the present work the equation used in Gordon *et al*.,[2]

is shown below to be satisfactory.

The key to effecting a solution to Eqs. (6) and (7) is the determination of *∊*(λ* _{i}*,λ

_{4}), which provides

*S*(λ

*,λ*

_{i}_{4}). This is accomplished here using the concept of clear water radiances. Gordon and Clark[5] have shown that for phytoplankton pigment concentrations

*C*less than ~0.25 mg/m

^{3}the water-leaving radiance in the green, yellow, and red CZCS bands can be written

where [*L** _{w}*]

*, the normalized water-leaving radiance, is 0.498, 0.30, and <0.015 mW/cm*

_{N}^{2}

*μ*m sr for 520, 550, and 670 nm, respectively. Thus, if a region of image for which

*C*< 0.25 mg/m

^{3}can be located, Eqs. (6) and (8) can be used to determine

*∊*(520,670),

*∊*(550,670), and

*∊*(670,670).

*∊*(443,670) can then be estimated by extrapolation since model calculations show that

*∊*(λ

*,λ*

_{i}_{4}) is a smooth function of λ

*. To automate this procedure, it is assumed for convenience that*

_{i}and then *∊*(λ_{1},λ_{4}) is determined from

An important aspect of this algorithm is that no surface measurements of either *L** _{w}*(λ) or any properties of the aerosol are required to effect the atmospheric correction with this scheme.

Application of the atmospheric correction algorithm requires a high degree of consistency between the calibration of the sensor and the instantaneous extraterrestrial solar irradiance *F*_{0}(λ).[12] This problem is most easily described with the aid of a numerical example. Consider the following problem. A location (I) is provided in an image for which all the water radiances *tL** _{w}* are given, and it is desired to find the

*tL*

*at a second position in the image (II), for which the aerosol radiance (concentration) is double that at position I. Assuming that the sensor calibration and*

_{w}*F*

_{0}are exact,

*L*

*(670) ≅ 0, and*

_{w}*S*(443,670) is position independent, this situation is described in Table I, wherein the boxed quantities are derived using the correction algorithm with

*S*determined at position I. Now assume that the sensor is calibrated to yield radiances that are 5% too high in the blue. This situation is described in Table II. Note that the 5% error in

*L*

*(443) has been magnified to the point of even yielding a negative value for the retrieved*

_{t}*t*(443)

*L*

*(443). (Note also that if the atmosphere were horizontally homogeneous, no error would be induced in*

_{w}*tL*

*in this example.) The sensor calibration and*

_{w}*F*

_{0}errors are completely independent, so trying to reduce this effect in future systems through careful calibration and careful measurements of

*F*

_{0}(λ) will be very difficult. One way of circumventing this problem directly is the addition of the capability of the sensor to view the sun in diffuse reflection as suggested by Yates.[12] The addition of such a capability should be considered for future ocean color sensors. In the present case, the sensor calibration was adjusted by trial and error to yield values of

*∊*(λ

*,λ*

_{i}_{4}) over clear water which were reasonably stable from day to day and were not unreasonable in terms of their dependence on λ

*. The fact that (see below) this calibration enables horizontally inhomogeneous atmospheres to be corrected with good accuracy attests to its reasonableness.*

_{i}If the radiance at the sensor is written *L** _{t}*(λ) = [

*A*(λ)

*N*(λ) +

*B*(λ)]

*C*(λ), where

*A*(λ) and

*B*(λ) are the calibration slope and intercept,

*N*(λ) is the digital count from the sensor, and

*C*(λ) is the calibration correction resulting from the adjustment mentioned above. These are provided in Table III along with the value of the mean extraterrestrial solar irradiance computed by weighing the Labs and Neckel[13] ${\overline{F}}_{0}\left(\lambda \right)$ with the spectral response of the CZCS (W. A. Hovis, NOAA/NESS; personal communication). These ${\overline{F}}_{0}$ values and the calibration slopes and intercepts are those that were actually used in producing the imagery presented in this paper. This imagery was all acquired at sensor gain 1(1–4), and the calibration corrections

*C*(λ) apply only to gain 1. R. W. Austin (Scripps Visibility Lab.; personal communication) has examined imagery for which the sensor gain was varied during acquisition and suggests use of

*A*(λ) and

*B*(λ) values which differ slightly from those in Table III. He also has provided a new set of ${\overline{F}}_{0}\left(\lambda \right)$ values obtained from the most recent determinations of Neckel and Labs.[14] These new calibration and solar irradiance values require reevaluation of the calibration adjustment

*C*(λ) to retain the

*F*

_{0}/calibration consistency. These new values of

*A*,

*B*,

*C*, and ${\overline{F}}_{0}$ are presented in Table IV. We prefer Table IV because the corrections

*C*(λ) are independent of the sensor gain (Gordon

*et al*., in preparation). It should be pointed out that the results presented below in no way depend on whether Table III or IV is used. They produce almost identical values of

*L*

*(λ) upon application of the atmospheric correction algorithm.*

_{w}## III. Bio-optical Algorithms

Most early studies concerning the remote sensing of ocean color[15],[16] were directed toward the extraction of the surface chlorophyll concentration from the spectral radiance upwelling above the sea surface. Chlorophyll *a* is the pigment present in living plants responsible for photosynthesis. In the ocean, this pigment is present in microscopic organisms called phytoplankton, which form the first link in the marine food chain. In productivity studies chlorophyll *a* is usually taken as a measure of phytoplankton biomass.[17]

In the ocean, degradation products of chlorophyll *a*, the phaeopigments, are also present. These are produced upon acidification of chlorophyll *a* as would happen, for example, in the gut of zooplankton feeding on phytoplankton. The phaeopigments have absorption characteristics which are so similar to chlorophyll *a* in the blue that separation of these pigments with an instrument with as few spectral bands as the CZCS is impossible. Because of this, for CZCS studies it is necessary to consider chlorophyll *a* and the phaeopigments together. The sum of the concentrations of chlorophyll *a* and the phaeopigments will henceforth be called the phytoplankton pigment concentration or just the pigment concentration and will be denoted by *C*.

To establish algorithms for retrieval of the pigment concentration from CZCS imagery a field program was initiated by NOAA NESS in 1975. This program consisted of measurements of vertical profiles of upwelled (traveling toward the zenith) spectral radiance [*L** _{u}*] and pigment concentration along with other optical, physical, and biological parameters of importance for interpretation of ocean color remote sensing data.[3],[4],[18] These measurements were made at the over sixty locations shown in Fig. 1. The measurements of chlorophyll

*a*and its associated phaeopigments were made fluorometrically using the technique described by Yentsch and Menzel[19] with the modifications given by Holm-Hansen

*et al*.[20] The upwelled spectral radiance measurements were made at 5-nm increments with a submergible radiometer covering a 400–700-nm spectral range. The spectral resolution of the instrument was 4 nm.

From the profiles of *L** _{u}*(λ,

*z*), where

*z*is depth, the attenuation coefficient of upwelled spectral radiance

*K*

*(λ) defined by*

_{u}was computed. This allowed determination of the upwelled spectral radiance just beneath the surface *L** _{u}*(λ,0) from

*L** _{u}*(λ,0) was then transmitted through the interface as described by Austin,[21] yielding the water-leaving spectral radiance

*L*

*(λ). Finally,*

_{w}*L*

*(λ) was weighted by the spectral responses of the CZCS (spectral resolution of ~20 nm) to provide 〈*

_{w}*L*

*(λ)〉, the CZCS weighted water-leaving radiance. This is the component of the upwelling radiance just above the sea surface which carries information concerning subsurface constituents. As discussed in the previous section, the goal of the atmospheric correction algorithm is the retrieval of 〈*

_{w}*L*

*(λ)〉 from the CZCS measured radiance*

_{w}*L*

*(λ).*

_{t}Since the solar irradiance backscattered out of the ocean may have actually penetrated to significant depths in the ocean, the relationship between *C* and the water-leaving radiance should depend on the vertical distribution of the phytoplankton. Through a series of Monte Carlo simulations of radiative transfer in a continuously stratified ocean, Gordon and Clark[22] have shown that the pigment concentration 〈*C*〉 for an optically homogeneous ocean which would produce the same water-leaving radiance as an optically stratified ocean with pigment concentration *C*(*z*) is

where

*K** _{d}*(

*z*′) is the attenuation coefficient for downwelling irradiance [given by Eq. (11) with

*L*

*replaced by*

_{u}*E*

*], and*

_{d}*z*

_{90}is the penetration depth defined to be the depth at which

*E*

*falls to 1/*

_{d}*e*of its value just beneath the surface. It is the depth above which 90% of the radiance contributing to

*L*

*originates in a homogeneous ocean.[23]*

_{w}*z*

_{90}is ~22% of the depth of the euphotic zone at the given wavelength. Since

*K*

*depends on wavelength, 〈*

_{d}*C*〉 should as well; however, direct computation[4] of 〈

*C*〉 from the data acquired at the locations in Fig. 1 shows that 〈

*C*〉 is the same in all the visible CZCS spectral bands. Furthermore, these computations show that there is no statistical difference between 〈

*C*〉 and the surface phytoplankton pigment concentration. This indicates no significant variation of

*C*within

*z*

_{90}. In all the bio-optical algorithms discussed here, 〈

*C*〉 was evaluated at 520 nm.

Most of the algorithms developed thus far to extract the pigment concentration from *L** _{w}*(λ) relate the surface

*C*or 〈

*C*〉 to the ratio of the water-leaving radiance at two different wavelengths.[3],[4],[24],[25] The basis for this lies in the fact that in the first approximation

*L*

*is proportional to the ratio of the backscattering coefficient*

_{w}*b*

*and the absorption coefficient*

_{b}*a*of the water plus its constituents,[24]–[26] i.e.,

*L*

*~*

_{w}*b*

*/*

_{b}*a*. Both

*a*and

*b*

*are linearly summable over the constituents, and the portion of*

_{b}*a*and

*b*

*arising from phytoplankton pigments is proportional to*

_{b}*C*. These pigments strongly influence

*a*but have little effect on backscattering, which primarily results from interactions with phytoplankton detrital material and inorganic suspended material of nonbiogenic origin. Thus the ratio of the water-leaving radiance at two wavelengths will be approximately inversely proportional to the ratio of the associated absorption coefficients. For a system composed only of water and phytoplankton, both of these absorption coefficients consist of the sum of a part due to the water itself and a part proportional to

*C*; hence the relationship between

*C*and

*L*

*(λ*

_{w}_{2})/

*L*

*(λ*

_{w}_{1}) is nonlinear.

Morel and Prieur[24] have optically classified seawater according to the constituents chiefly responsible for determining *a* and *b** _{b}*. Those waters for which phytoplankton and their covarying detrital material play the dominant role in determining the optical properties are called Case I waters; those for which inorganic suspended material (such as that which might be resuspended from the bottom in shallow areas), which does not covary with phytoplankton, plays an important role are referred to as Case II waters. Most of the open ocean waters are near Case I. These waters are the easiest to treat from a remote sensing point of view. The circled locations in Fig. 1 represent those stations felt to contain waters very close to Case I. Figure 2 shows the relationship between

*R*(13) ≡ 〈

*L*

*(443)〉/〈*

_{w}*L*

*(550)〉 and the pigment concentration 〈*

_{w}*C*〉 for all the stations in Fig. 1, while Fig. 3 gives the same quantities using the data from only those locations thought to be Case I. The lines in these figures are linear regressions on the log-transformed data. The reader should note the significantly tighter fit when the Case II waters are excluded.

At high pigment concentration 〈*L** _{w}*(443)〉 usually becomes too small to be retrieved from

*L*

*(443) with sufficient accuracy to be useful. In this case it is necessary to employ the ratio*

_{t}*R*(23) ≡ 〈

*L*

*(520)〉/〈*

_{w}*L*

*(550)〉 to extract the pigment concentration. This ratio and the associated regression for all the data are shown in Fig. 4. Since*

_{w}*R*(23) is usually to be used only in the case of large

*C*, a separate regression on the data for

*C*>1.5 mg/m

^{3}has been performed and is shown in Fig. 5. For

*R*(23) no attempt has been made to separate Case I waters from the data base, because most of the situations for which

*C*>1.5 mg/m

^{3}involve waters which are a mixture of Case I and Case II.

*R*(23) is less sensitive than

*R*(13) to variations in 〈

*C*〉.

All of the linear regressions shown in these figures are of the form

The values of *A*, *B*, *r*^{2}, the standard error of estimate *s*, and the number of samples in the regression *N* for the various algorithms are presented in Table V. The relative error in 〈*C*〉 is ~10* ^{s}* − 1. Both the ratios

*R*(13) and

*R*(23) (but derived from a significantly smaller data base) were used in processing the imagery presented by Gordon

*et al*.,[2] yielding two pigment displays for each scene: one for 〈

*C*〉 < 1 mg/m

^{3}[

*R*(13)] and one for 〈

*C*〉 > 1 mg/m

^{3}[

*R*(23)]. The specific algorithm which we now recommend and which has been adopted by NASA to compute 〈

*C*〉 is

where 〈*C*〉* _{i}* is in mg/m

^{3}, and 〈

*C*〉

_{2}and 〈

*C*〉

_{4}refer to algorithms 2 and 4 in Table V.

## IV. Application to CZCS Imagery and Comparison with Ship Measurements

Application of these algorithms to the processing of CZCS imagery requires two steps: (1) the removal of the atmospheric effects and the compution of *L** _{w}* via Eq. (6); and (2) the subsequent computation of 〈

*C*〉 using Eqs. (15) and (16). The first step begins with the computation of

*L*

*using Eq. (2) with*

_{r}*x*=

*r*. This requires determination of Ө,

*ϕ*, Ө

_{0}, and

*ϕ*

_{0}from the satellite ephemeris and the latitude and longitude of the pixel under examination. We obtain this ephemeris using orbital elements provided by NORAD and the acquisition time provided by the spacecraft clock. [Data from which these angles can be computed are also provided directly on the CZCS Calibrated Radiance and Temperature Tape (CRTT).] To save computation time

*L*

*is computed only every 1.25° in scan angle along a given scan line, and this computation is performed only every sixteenth scan line.*

_{r}*L*

*is then determined at each pixel by 2-D interpolation. The instantaneous extraterrestrial solar irradiance required in Eq. (2) is computed from ${F}_{0}={\overline{F}}_{0}{\left[1+e\phantom{\rule{0.2em}{0ex}}\text{cos}\left[2\pi \left(D-3\right)/365\right]\right]}^{2}$, where*

_{r}*D*is the Julian day (

*D*= 1 on 1 Jan. and

*D*= 365 on 31 Dec.), and

*e*, the eccentricity of the earth’s orbit, is 0.016. Equation (6) is then solved for

*L*

*(λ). If*

_{w}*L*

*(λ*

_{w}_{4}) ≠ 0, Eqs. (6) and (7a) must be solved iteratively[11] for

*L*

*(λ). The resulting*

_{w}*L*

*(λ) values are then used in Eq. (16) to determine 〈*

_{w}*C*〉.

This procedure assumes that *S*(λ* _{i}*,λ

_{4}), or equivalently

*∊*(λ

*,λ*

_{i}_{4}), is known, and thus these factors must be determined before the computation can begin.

*∊*(λ

*,λ*

_{i}_{4}) is determined using the clear water radiance concept. This is effected by, first, carrying out the above process with

*∊*(λ

*,λ*

_{i}_{4}) = 1 for

*i*= 1–3; next, locating the regions of lowest 〈

*C*〉; and, finally, solving Eqs. (6)–(10) for the unknown

*∊*values. To minimize the effect of noise on this

*∊*estimation,

*L*

*values averaged over a 5 × 5 pixel box, rather than single-pixel values, are used in Eq. (6). Often, especially on very clear atmospheres, the*

_{t}*∊*values will vary from one clear water area to another. In most cases this results from the fact that

*C*is not sufficiently small in the clear water regions. The criteria to determine the best clear water regions are as follows: (1) we select only regions for which the aerosol radiances (

*L*

*≡*

_{a}*L*

*−*

_{t}*L*

*−*

_{r}*tL*

*)*

_{w}*L*

*(520),*

_{a}*L*

*(550), and*

_{a}*L*

*(670) form a monatonic (usually decreasing) sequence; (2) of these, we select the region with the highest value for*

_{a}*L*

*(670). The location finally chosen is referred to below as the clear water calibration area. Since we can process a 512 × 512 pixel scene from*

_{a}*N*(λ) to 〈

*C*〉 in ~4 min (real time), and an individual determination of

*∊*given a trial clear water area (located using a data tablet and an interactive image display system) requires 5–10 sec, the entire procedure of determining and testing the atmospheric correction factors for a typical image can be effected in <10 min.

Comparing ship data and satellite imagery presents difficult problems. Ideally, in the case of the CZCS, it is desirable to compare ship and satellite measurements of both 〈*L** _{w}*(λ)〉 and 〈

*C*〉 for waters with a wide range of pigment concentrations. Effecting this for the low pigment concentrations and their associated weak horizontal gradients in both 〈

*C*〉 and the optical properties poses no problem, since the precise position of the ship is unnecessary. However, for high pigment concentrations and their associated strong horizontal gradients, it is necessary to know the exact pixel at which the ship is located. We cannot navigate CZCS imagery to within better than ~3 pixels; i.e., we can only locate the pixel in question to within a 3 × 3 box, so at present this is impossible. Even with precise navigation, gradients on the subpixel scale (825 × 825 m) would induce differences between the ship-measured and pixel-averaged values of 〈

*C*〉 and 〈

*L*

*(λ)〉.*

_{w}In this study we shall compare ship station data for regions displaying relatively weak horizontal gradients (i.e., horizontal scale of variability significantly greater than the CZCS pixel size) with simultaneously acquired CZCS imagery. This will provide an indication of the accuracy with which the atmospheric effects can be removed, and 〈*L** _{w}*(λ)〉 can be extracted from the CZCS imagery. Since the number of such simultaneous acquisitions is very small (here only three), we will also compare the continuous measurements of the surface pigments made along the ship track with 〈

*C*〉 extracted from the imagery. This will provide a statistical indication of the accuracy with which pigments can be extracted to the extent that the pigment concentration can be considered stable over time scales of the order of a few hours. For this purpose, the ship measurements and CZCS imagery will be considered coincident if they are separated in time by <12 h.

It is important to note that in both types of comparison the algorithms are applied as set forth above. No surface measurements of either *L** _{w}* or

*C*are used to bootstrap the derivation of 〈

*L*

*(λ)〉 or 〈*

_{w}*C*〉. The data from the three stations, however, have been included in the regressions leading to the bio-optical algorithms. Exclusion of these three data points produces negligible changes in the regressions, e.g., for the

*C*(13) Case I algorithm

*A*= 1.160 and

*B*= −1.711, compared to the 1.172 mg/m

^{3}and −1.705, respectively, from Table V.

The ship track and station locations of the RV Athena II are presented in Fig. 6. Those portions of the track centered on stations 4, 5, 8, and 11 will be compared with the CZCS imagery. Figure 7 provides an overview of the northern portion of the ship track (dotted line) superimposed on CZCS imagery of the pigment concentration [derived from the *C*(13) Case I algorithm] from Orbit 3226 (14 June 1979). In this presentation, land is a uniform gray with the coasts outlined in white. Clouds are black. The pigment concentration ranges from 0 (white) to 1 (black) mg/m^{3}. Areas with *C* >1 mg/m^{3}, such as Georges Bank, have been treated in a manner similar to clouds. The locations of the ship stations are indicated by the crosses on the ship track. The track is seen to cross the north wall of the Gulf Stream and pass through a warm core ring (confirmed by the CZCS thermal band) located near 39.4° N and 68.9°W.[27]

Figures 8 and 9, respectively, show *L** _{t}*(443) and 〈

*L*

*(443)〉 for the subscene of Orbit 3171 (10 June 1979) which contains ship station 5. Figures 10 and 11 provide the same information for λ = 550 mn. Note the very bright haze layer covering nearly half of this subscene. This image was atmospherically corrected to obtain*

_{w}*L*

*(λ) from*

_{w}*L*

*(λ) using the clear water radiance technique described above with the Gulf Stream serving as the clear water calibration area. It is fortuitous that this provides a reasonably good correction even in the region of the bright haze, indicating that the values of*

_{t}*∊*(λ

*,λ*

_{i}_{4}) for the aerosol over the Gulf Stream and for the haze layer are similar. The pigment concentration 〈

*C*〉 derived from

*R*(13) (Case 1) is given in Fig. 12, in a format similar to that for Fig. 7. [The CZCS-derived pigment concentration for this image is better quantified in the color-encoded format presented as Plate (b) on the cover of this issue.] In this display the data have been remapped to a standard grid with all the pixels having the same area on the earth’s surface. The pigment concentration computed along the portion of the ship track coincident with the imagery (in the sense of leading or lagging the satellite by <12 h) is compared with the surface pigments measured continuously along the ship track in Fig. 13. The pigment concentration was measured while the ship was underway using a Turner Designs flow-through fluorometer (model 10) to monitor the

*in vivo*fluorescence of water drawn continuously from ~1.5 m beneath the surface. Periodically, discrete samples were drawn from the same source and extractions performed to determine fluorometrically the concentrations of chlorophyll

*a*and the associated phaeopigments and hence calibrate the in vivo fluorescence-pigment concentration relationship. Zero on the distance axis in Fig. 13 is in the warm core ring (Fig. 12). Absent from the satellite determination are the regions covered by clouds. The agreement between the two data sets is excellent in the region where the atmosphere is clear, but large departures are seen in the region covered by haze, which begins near 240 km on the distance scale. The CZCS tracks the variations in surface pigments very well in the clear region with the exception of portions of the track which run parallel to and close to a strong color front [such as along the track just south of the cross on Fig. 12 or between the two crosses on Plate (b), corresponding to the region between 180 and 230 km on the distance scale in Fig. 13]. A small displacement of the color front perpendicular to the ship track during the time between the ship’s passage and the satellite overpass could account for this discrepancy. Imagery of 〈

*C*〉 from the previous day (Orbit 3157) coincident with the track through the warm core ring and Gulf Stream is presented in a similar format in Fig. 14 [color-encoded in Plate (a) on the cover of this issue], and Fig. 15 provides comparison between the ship- and satellite-determined pigments along the ship track. Figures 16 [color-encoded in Plate (c) on the cover of this issue] and 17, respectively, give a subscene of 〈

*C*〉 derived from Orbit 3240 (15 June 1979) and the associated ship- and satellite-determined pigments along the ship track. In this case the image did not extend as far south as the Gulf Stream, and the warm core ring was used as the clear water calibration area for atmospheric correction. Georges Bank is near zero on the distance scale in Fig. 17. Again the two data sets track one another very well. The overall agreement among the derived pigment concentrations for the three images can be assessed by comparing the color-encoded displays on the cover of this issue. Such comparison suggests that the pigment retrievals over the common areas of the images are within a ±1 color chip. Since one would not expect significant changes in 〈

*C*〉 in this area over a one-week time period in summer with calm weather, we find this overall agreement gratifying.

To provide an example of the degree to which surface pigments can be estimated in the open ocean, we have also examined Orbit 3351 (23 June 1979), which contains station 11 in the western Sargasso Sea occupied by the ship on 19 June 1979 (Fig. 6). Since most of this track is east of the Gulf Stream and near the Sargasso Sea, it was felt that the surface pigment concentration would be sufficiently stable that a comparison between the ship-measured and satellite-determined surface pigments would be meaningful even though they were acquired four days apart. Figure 18 shows the results of such a comparison. Zero on the distance scale is at the northern end of the track. Missing satellite estimates are due to clouds. The agreement between the two data sets is excellent and, we believe, demonstrates that CZCS-type sensors can be used to study surface pigment concentrations in the open ocean.

Comparison between the ship-measured and CZCS-estimated pigments for all of these orbits is summarized in Fig. 19. The data from Orbit 3171 (circles) which fall far from the line of equality are from those portions of the track either near clouds or in the haze. Table VI gives a statistical summary of the ship-satellite pigment comparisons for the four orbits mentioned above. The rms difference (in percent) between the ship and satellite estimates is provided. Also given in Table VI is the average aerosol radiance at 670 nm [〈*L** _{a}*(670)〉] for the cloud-free regions of the ship track. This provides a relative measure of the turbidity of the atmosphere present when the pigment concentrations were estimated. The fact that 〈

*L*

*(670)〉 varied by a factor of ~4 over the four images discussed here underscores the wide variety of atmospheric conditions that were encountered in this study. The relative standard deviation in 〈*

_{a}*L*

*(670)〉,*

_{a}*σ*

*L*

*/*

_{a}*L*

*indicates the degree of horizontal uniformity in the aerosol.*

_{a}*N*is the number of pixels from which the rms differences were determined. It is seen that within the range considered here (0.08–1.5 mg/m

^{3}) the pigment concentration can be extracted from the CZCS imagery to within ±30–40% with either of the

*C*(13) algorithms. It is important to notice that these results have been obtained without recourse to any surface measurements at the time of the satellite overpass.

The comparison between the shipboard station measurements and satellite estimates of 〈*L** _{w}*(λ)〉 and 〈

*C*〉, determined at the time of the satellite overpass, is presented in Tables VII–IX. 〈

*L*

*(670)〉 is again the computed aerosol radiance in band 4 at the station location and is intended to provide a measure with which to compare the severity of the atmospheric effects on the imagery for these three days. The stations for Orbits 3157 and 3171, respectively, were ~150 and 400 km from the position in the Gulf Stream used as the clear water calibration area, while for Orbit 3240, the station was ~200 km from the warm core ring, which was used for the clear water calibration area. The 〈*

_{a}*L*

*(λ)〉 agreement shown in these three determinations suggests that the clear water radiance technique has provided excellent atmospheric corrections in each case. Furthermore, comparison between 〈*

_{w}*C*〉 and the pigment concentrations determined from the various pigment algorithms suggests that 〈

*C*〉 can be determined from CZCS imagery to considerably better than the factor of ±2 initially found by Gordon

*et al*.[2] Although

*C*(13) provides a better estimate of 〈

*C*〉 than

*C*(13) Case I, we do not feel justified in suggesting this algorithm over that for Case I waters in this region. In any event, the retrieval of nearly correct values for 〈

*L*

*(λ)〉 is the more impressive part of these station comparisons, since it clearly demonstrates the viability of the atmospheric correction algorithms.*

_{w}As mentioned in Sec. II, atmospheric correction of the imagery presented here has been effected through employing the assumption that 〈*L** _{w}*(670)〉 = 0. This assumption is valid in these cases because the pigment concentration does not exceed 1.5 mg/m

^{3}, and for most of the track lines it is <0.8 mg/m

^{3}. In Fig. 20 we present the normalized water-leaving radiance at 670 nm [

*L*

*(670)]*

_{w}*as a function of 〈*

_{N}*C*〉 for the 0–1.5-mg/m

^{3}concentration range. [

*L*

*(670]*

_{w}*is given in units of CZCS Gain 1 (1–4) digital counts. The trend of the data suggests that over the range of 0 to ~0.8 mg/m*

_{N}^{3}[

*L*

*(670)]*

_{w}*will rarely reach the 2-count threshold. Since 〈*

_{N}*L*

*(670)〉 is always less than [*

_{w}*L*

*(670)]*

_{w}*[see above, Eq. (8)], and*

_{N}*t*(λ)〈

*L*

*(λ)〉 is even smaller, it is relatively safe to conclude that 〈*

_{w}*L*

*(670)〉 will rarely exceed the noise level of the sensor in this band (1–2 digital counts) for the pigment levels encountered in this work. At higher pigment concentrations, however, [*

_{w}*L*

*(670)]*

_{w}*can become significant,[5],[11] and Eq. (7) is no longer valid. In this case the Smith and Wilson[11] scheme of closing Eq. (6) must be used. The system is then nonlinear and is solved by an iterative procedure at each pixel.*

_{N}## V. Some Questions

We feel that the results presented in the previous section indicate that in a variety of atmospheric conditions pigment concentrations in the 0 < 〈*C*〉 < 1.5-mg/m^{3} range can be extracted from CZCS imagery with an accuracy of about ±30–40%. However, there are two obvious questions which can be raised considering the generality of the procedures which have been used.

- (1) The atmospheric corrections for Orbits 3157, 3151, and 3240 were effected by using either the Gulf Stream or the warm core Gulf Stream ring as the clear water calibration area. How strongly does the derived pigment concentration depend on the actual pigment concentration in the clear water area?
- (2) In general, how can atmospheric correction be effected in an image for which the aerosol type varies over the image?

We deal with these two questions now.

#### A. Choice of the Clear Water Calibration Area

The choice of the best clear water calibration area in a given image is relatively straightforward. As described above, our procedure is first to perform a preliminary atmospheric correction by choosing *∊*(λ* _{i}*,λ

_{4}) = 1. Then the pigment concentration is estimated using the

*C*(13) Case I algorithm. The obvious choice for the clear water calibration area is the one which yields the lowest value of 〈

*C*〉 < 0.25 mg/m

^{3}near the region under investigation. The rationale behind the choice of the Case I algorithm is that suitable clear water areas must of necessity belong to Case I waters.

To try to estimate the effect of the actual pigment concentration at the clear water calibration area, we compare 〈*C*〉 computed along the ship track on Fig. 16 (Orbit 3240) using three different clear water calibration points:

The resulting computations are shown in Fig. 21 along with the associated ship data. Note that the only significant effect of the choice of the calibration area seems to be an enhancement of the computed concentration in the higher pigment areas. This is due to the fact that as 〈*C*〉 increases for clear water 〈*L** _{w}*(550)〉 increases somewhat, while 〈

*L*

*(520)〉 remains essentially constant. The increased water radiance at 550 nm over the calibration area will be incorrectly ascribed to*

_{w}*L*

*(550) in Eq. (1), which will increase*

_{a}*n*(550) [Eq. (9)] and hence

*n*(443) in Eq. (10). [This is the rationale for requiring above that

*L*

*(λ) be monotonic with λ.] This in turn causes the retrieved values of 〈*

_{a}*L*

*(443)〉 to be everywhere too small, which will be interpreted, using the pigment algorithms involving*

_{w}*R*(13), as an increased pigment concentration. If the clear water calibration area also contains substantial quantities of suspended sediments, both 〈

*L*

*(520)〉 and 〈*

_{w}*L*

*(550)〉 will increase, and this effect will be further magnified. It should be noted that in general the resulting retrievals of 〈*

_{w}*C*〉 will be better the lower the pigment concentration in the clear water calibration area and the closer the region of interest is to the clear water calibration area.

#### B. Images with Variable Aerosol Type

In cases with variable aerosol type, the set of Eqs. (6) cannot be solved in general because the *S*(λ* _{i}*,λ

_{4}) (

*i*= 1–3) are unknown. There are, however, some particular cases which can be dealt with in a systematic manner. These include images which display

- (1) a variable aerosol type over water for which the pigment concentration is sufficiently low that each pixel could serve as the clear water calibration area;
- (2) two distinct aerosol types having very different concentrations with the more turbid atmosphere over some likely candidates for clear water calibration areas; and
- (3) two distinct aerosol types having very different concentrations with the more turbid atmosphere over a region containing no candidates for clear water calibration.

The first case poses no particular difficulty. Each pixel can be used as a clear water calibration area, and *n*(443) is determined there. This provides 〈*L** _{w}*(443)〉 and hence 〈

*C*〉. Figure 22 shows the result of applying this procedure to the ship track on Orbit 3351. Comparison between Figs. 19 and 22 reveals very little difference between the retrieved pigments over the first 200 km on the distance scale; however, as the clouds are approached the constant

*∊*(λ

*,λ*

_{i}_{4}) seems to provide a somewhat better fit to the ship track data.

The second case is also straightforward to handle. The correction factors *∊*(λ* _{i}*,λ

_{4}) can be readily determined in the turbid areas using the technique discussed in the previous section. These factors will be incorrect for the second aerosol type; however, because of its assumed low concentration, this error may be insignificant. For example, if the clearer atmosphere were similar to that over the Orbit 3171 ship station (Table VIII), and an error of ±0.5 (very large) was made in

*∊*(443,670), the error in the retrieved 〈

*L*

*(443)〉 would be about ±0.12 mW/cm*

_{w}^{2}

*μ*m sr or ±20%. Somewhat smaller errors with the same sign would be present at the other bands, which would tend to cancel the effect in the computation of 〈

*C*〉.

For the third case (the reverse of the second) the clear water radiance technique fails, since a small error in *∊*(λ* _{i}*,λ

_{4}) in the more turbid atmosphere will cause a very large error in 〈

*L*

*(λ)〉. Again, in the case of Orbit 3171, had the ship station been under the more turbid atmosphere for which 〈*

_{w}*L*

*(670)〉 reached as much as 1.5 mW/cm*

_{a}^{2}

*μ*m sr, a ±0.5 error in

*∊*(443,670) could yield a value for 〈

*L*

*(443)〉 between −0.33 and +1.51 mW/cm*

_{w}^{2}

*μ*m sr, which would be meaningless. Since adequate pigment concentrations can be derived in the clearer atmosphere even in the presence of significant errors in

*∊*(λ

*,λ*

_{i}_{4}), it is only necessary to find the correction factors for the more turbid atmospheric region. One systematic method of effecting this is simply to assume that all

*n*(λ

*) values in Eq. (9) are the same and vary this*

_{i}*n*until the boundary in the corrected image is no longer evident in any of the bands. A second method is to vary the

*n*(λ

*) or*

_{i}*∊*(λ

*,λ*

_{i}_{4}) values individually until all trace of the boundary vanishes from each 〈

*L*

*(λ)〉 image. Note, however, that when either of these procedures is used, neither area will be perfectly corrected, since it is really not possible to remove all traces of the aerosol over both regions with a single set of*

_{w}*∊*(λ

*,λ*

_{i}_{4}) values.

At present, only the three cases of variable aerosol type described above are amenable to atmospheric correction. This limitation is due to the small number of spectral bands available on the sensor. In Morel and Gordon[28] a set of spectral bands is discussed (about half of which are proposed solely for effecting atmospheric correction) with which atmospheres of variable aerosol type could be corrected automatically. The difficulty is with the present instrument and not due to a fundamental limitation of nature.

## VI. Summary and Conclusions

The processing algorithms for the Nimbus-7 CZCS have been described in detail. These algorithms are applied to the shelf and slope waters of the Middle Atlantic Bight and also to the Sargasso Sea waters. In all, four images have been processed and the resulting pigment concentrations compared with continuous measurements made along ship tracks. The results suggest that over the 0.08–1.5-mg/m^{3} range the error in the retrieved pigment concentration is of the order of 30–40% (Tables VI). Three direct comparisons between ship-measured and satellite-retrieved values of 〈*L** _{w}*(λ)〉 suggest that the atmospheric correction algorithm is capable of retrieving the water-leaving radiance to within ~10–15% (Tables VII–IX). It is important to note that these results have been obtained using an atmospheric correction algorithm which does not require any surface measurements at the time of the overpass for its application.

## Appendix: Influence of Multiple Scattering

Some of the effects of multiple scattering (MS) on the atmosphere correction algorithm will now be investigated. This process can influence the two central aspects of the single-scattering development given in the text: the separability of the Rayleigh and aerosol contributions to the total radiance at the sensor; and the constancy of *∊*(λ_{1},λ_{2}) in the presence of variations in aerosol concentrations. To assess the MS effects several exact solutions of the scalar radiative transfer equations have been generated using a doubling[29] code with Wiscombe’s[30] diamond initiation. The aerosol is assumed to be nonabsorbing, and *L** _{w}* is taken to be zero, since only the atmosphere is of interest here. (The lower boundary of the atmosphere, however, does have Fresnel reflectance characteristic of the sea–air interface.) For brevity only the nadir radiance will be discussed. In this Appendix the term radiance referring to
$L/{{F}^{\prime}}_{0}$ is designated by

*I*.

The influence of MS on the separability of the Rayleigh and aerosol contributions is studied by solving the radiative transfer equation for

- (a) ${I}_{r}^{\text{MS}}\left({\tau}_{r},0\right)$, the Rayleigh radiance at the sensor for an atmosphere with Rayleigh optical thickness
*τ*and aerosol optical thickness_{r}*τ*= 0 (aerosol free);_{a} - (b) ${I}_{a}^{\text{MS}}\left(0,{\tau}_{a}\right)$, the aerosol radiance at the sensor for an atmosphere with aerosol optical thickness
*τ*and Rayleigh optical thickness_{a}*τ*= 0 (no Rayleigh scattering); and_{r} - (c) ${I}_{t}^{\text{MS}}\left({\tau}_{r},{\tau}_{a}\right)$, the sensor radiance for an atmosphere of Rayleigh optical thickness
*τ*and aerosol optical thickness_{r}*τ*(the physical situation encountered in practice)._{a}

The two quantities ${I}_{a}^{\text{MS}}\left(0,{\tau}_{a}\right)$ and ${I}_{t}^{\text{MS}}\left({\tau}_{r},{\tau}_{a}\right)-{I}_{r}^{\text{MS}}\left({\tau}_{r},0\right)$, which are identical in the single-scattering approximation, are then compared.

Figure 23 shows the percent difference between these two quantities for the CZCS blue band (*τ** _{r}* = 0.232), the green band (

*τ*

*= 0.123), and the red band (*

_{r}*τ*

*= 0.044) for a*

_{r}*τ*

*of 0.6. Note that Eq. (1) is seen to be valid to within ~10% of the aerosol contribution over a range of solar angles from 20 to over 60°, and this error decreases somewhat with increasing wavelength. Also the errors in Eq. (1) are seen to have the tendency to be in the same direction for all the CZCS bands and hence to have the tendency to cancel in the computation of*

_{a}*S*(λ

*,λ*

_{i}_{4}). For smaller values of

*τ*

*similar results are obtained. It should be noted that for solar zenith angles of <20°, the sensor would be tilted to avoid direct sun glint, so this region of Fig. 23 would never be reached in practice. Thus we conclude that in typical conditions, Eq. (1) will probably be in error by no more than ~10% of*

_{a}*L*

*(λ).*

_{a}The effect of MS on *L** _{a}*(λ) itself will now be investigated. Figure 24 shows the aerosol radiance at nadir as a function of

*τ*

*using the phase functions shown in Fig. 25. These phase functions are two-term Henyey-Greenstein functions*

_{a}*P*

*(γ) =*

_{a}*αf*(γ,

*g*

_{1}) + (1 −

*α*)

*f*(γ,

*g*

_{2}), where

*f*(γ,

*g*) = (1 −

*g*

^{2})/(1 +

*g*

^{2}− 2

*g*cosγ)

^{3/2}, with

*g*

_{1}= 0.7130 and

*g*

_{2}= −0.7596. The values of

*α*are 0.9618 and 0.985, respectively, for the curves labeled

*P*1 and

*P*2 in Fig. 24.

*P*1 provides an analytic approximation[31] to the Deirmendjian[32] haze

*L*distribution.

*P*2 provides an alternative to

*P*1 and is used to examine the influence of spectral variations in

*P*

*(γ) on Eq. (4). The only justification for their choice is that they have shapes similar to those predicted from Mie theory. The dashed lines in Fig. 25 are the single-scattering predictions [Eq. (2)]. Notice that the nearly linear relationship between*

_{a}*I*

*and*

_{a}*τ*

*is approximately valid for small variations in*

_{a}*τ*

*, but Eq. (2) does not provide the correct relationship. This is displayed in a different and more useful manner in Fig. 26, in which*

_{a}*∊*(λ

_{2},λ

_{1}) is plotted against

*τ*

*(λ*

_{a}_{1}) for various aerosol types [combinations of

*τ*

*(λ*

_{a}_{2})/

*τ*

*(λ*

_{a}_{1}) and aerosol phase functions]. The solid curves are for the case where the phase function at both λ

_{1}and λ

_{2}is given by

*P*1 in Fig. 25, while the dashed curves correspond to phase function

*P*1 for λ

_{1}and

*P*2 for λ

_{2}. Equation (4) implies that

*∊*(λ

_{2},λ

_{1}) should depend only on aerosol type and not on aerosol concentration. The near independence of

*∊*(λ

_{2},λ

_{1}) with

*τ*

*(λ*

_{a}_{1}) for

*τ*

*(λ*

_{a}_{1}) > 0.1 shows that the algorithm as developed in the text should result in only small errors even in an inhomogeneous multiple-scattering atmosphere as long as the aerosol type is independent of position. For a quantitative estimate of the degradation caused by these nonlinearities and hence the usefulness of the algorithm, the magnitude of the error in the determination of

*I*

*(λ) should be compared with the magnitude of the desired water radiance. Table X gives typical values of*

_{a}*t*(λ)

*I*

*(λ) for representative low and high pigment concentrations derived from measurements made in the Gulf of Mexico at solar zenith angles near that used in the computations for Figs. 24 and 26.*

_{w}Consider an aerosol type represented by the solid *τ** _{a}*(λ

_{2})/

*τ*

*(λ*

_{a}_{1}) = 2 curve in Fig. 26. Assume that

*τ*

*(λ*

_{a}_{1}) = 0.1, where

*I*

*(λ*

_{a}_{2})/

*I*

*(λ*

_{a}_{1}) is determined (the clear water calibration point) and increases to 0.2, where

*I*

*(λ*

_{a}_{2}) is desired. According to Fig. 26 a value of 2.13 would be used for

*∊*(λ

_{2},λ

_{1}) instead of the correct 2.16 in the estimation of

*I*

*(λ*

_{a}_{2}). This results in underestimating

*I*

*(λ*

_{a}_{2}) by 0.126 × 10

^{−3}(see Fig. 24) or 3%. Comparing the magnitude of this error with the values of

*t*(λ)

*I*

*(λ) for low and high pigment concentration given in Table X shows that the error is at most 15% of the residual water radiance in those bands to which the algorithm would be applied (443, 520, and 550 nm for low*

_{w}*C*), and hence

*t*(λ

_{2})

*L*

*(λ*

_{w}_{2}) could be extracted from

*L*

*(λ*

_{a}_{2}) in this particular case to within 15%. [Retrieval of

*I*

*(λ) for the high*

_{w}*C*case would require application of the Smith and Wilson[11] iterative algorithm since

*I*

*(670) ≠ 0; however, assuming that their procedure is exact and not the result of a statistical analysis, the absolute error in*

_{w}*I*

*(λ*

_{a}_{2}) would be the same as that found for the low

*C*examples above and below.] If

*τ*

*(λ*

_{a}_{1}) was 0.4 at the point for which

*t*(λ

_{2})

*I*

*(λ*

_{w}_{2}) was desired, i.e., increased by a factor of 4 over that at the clear water calibration point, the error in

*I*

*(λ*

_{a}_{2}) would be 4.2%, which amounts to 0.46 × 10

^{−3}. This error is comparable with

*t*(λ

_{2})-

*I*

*(λ*

_{w}_{2}) and hence would be unacceptable. Note that we have assumed that the aerosol type [i.e.,

*τ*

*(λ*

_{a}_{2})/

*τ*

*(λ*

_{a}_{1})] is independent of position in the examples above. If

*τ*

*(λ*

_{a}_{2})/

*τ*

*(λ*

_{a}_{1}) varied by 10% in the first example the error in

*I*

*(λ*

_{a}_{2}) would be roughly 0.88 × 10

^{−3}or 20%, which would be totally unacceptable. This underscores the severity of the requirement that the aerosol type be position independent for successful application of this algorithm to a horizontally inhomogeneous atmosphere.

We acknowledge the advice and help of D. A. Kiefer and C. S. Yentsch in the proper application of fluorometry to the measurement of the pigment concentration and A. E. Strong for performing the pigment extractions. This research received support from NASA (NAS 5-22963, H.R.G., J.W.B.), NOAA (NA79SA00741, H.R.G.; 04-8-M01-129, W.W.B.), ONR (N00014-80-C-0042, O.B.B., R.H.E.), and NSF (OCE-80–16991, O.B.B., R.H.E.).

## Figures and Tables

## References

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