Atmospheric correction over case 2 waters with an ... - OSA Publishing

Atmospheric correction over case 2 waters with an iterative fitting algorithm Peter E. Land and Joanna D. Haigh

A modular atmospheric correction algorithm is proposed that uses atmospheric and water contents models to predict the visible and near-infrared reflectances observed by a satellite over water. These predicted values are compared with the satellite reflectances at each pixel, and the model parameters changed iteratively with an error minimization algorithm. The default atmospheric model uses singlescattering theory with a correction for multiple scattering based on lookup tables. With this model we used parameters of the proportions of three tropospheric aerosol types. For the default water content model we need the parameters of the concentrations of chlorophyll, inorganic sediment, and gelbstoff. The diffuse attenuation and backscatter coefficients attributed to these constituents are calculated and used to derive the water-leaving reflectance. Products include water-leaving reflectance, concentrations of water constituents, and aerosol optical depth and type. We demonstrate the application of the method to sea-viewing wide field-of-view sensor by using model data. © 1996 Optical Society of America Key words: Ocean color, SeaWiFS, optical properties, multiple scattering, error minimization.

1. Introduction

Since the launch of the Coastal Zone Color Scanner, atmospheric correction of ocean color data has relied mainly on an approach based on the assumption that the water-leaving reflectance Rw 5 0 in one or more red or near-infrared bands, and this approach has continued to be applied to the new generation of sensors such as sea-viewing wide field-of-view sensor ~SeaWiFS!.1 This assumption is valid over case 1 waters, particularly at the longer wavelengths available from the new sensors, but is invalid in turbid case 2 waters that contain inorganic suspended sediment that backscatters strongly at visible and nearinfrared wavelengths ~Fig. 1!.2 Various attempts have been made to measure sediment concentration with visible reflectances,2 but these attempts relied on the reflectances being dominated by the sediment ~the opposite of the Rw 5 0 assumption!, and the results were strongly dependent on sediment type.3 In this paper we consider an alternative approach: to model the optical properties of the water with respect to its constituents. Rw at each wavelength l can be related to the subsurface reflectance ratio

The authors are with Space & Atmospheric Physics, Imperial College, London SW7 2BZ, United Kingdom. Received 16 August 1995; revised manuscript received 1 April 1996. 0003-6935y96y275443-09$10.00y0 © 1996 Optical Society of America

Rss~l!, which in turn can be related to optical properties of the water.4 The contributions of the water constituents to inherent optical properties such as the backscatter coefficient are strictly additive and those to the diffuse attenuation coefficient are approximately additive,5 unlike contributions to Rw~l!. The approach we describe is to use models of the water and atmosphere to predict Rs~l!, the reflectance observed at the satellite in each of the eight SeaWiFS bands. We compared these predictions with the observed Rs~l!, and we minimized a measure of the difference by iteratively changing the water and atmospheric models. Similar approaches have been used by many authors: Jain and Miller ~1976!6 characterized the ocean by a backscattering coefficient with a fixed spectral dependence and an absorption coefficient proportional to the chlorophyll concentration; Morel and Prieur ~1977!7 used absorption proportional to chlorophyll and sediment concentrations and backscattering proportional to sediment; Bukata et al. ~1981!8 used backscatter and absorption proportional to chlorophyll, organic and inorganic sediments, and gelbstoff; Sathyendranath et al. ~1989!9 used absorption proportional to chlorophyll, sediment, and gelbstoff and backscatter that has a power-law dependence on chlorophyll10 and a component resulting from sediment with a fixed spectral dependence; Tassan ~1994!11 used locally determined correlations among chlorophyll, sediment, and gelbstoff; Doerffer and Fischer ~1994!12 used diffuse attenuation and absorption propor20 September 1996 y Vol. 35, No. 27 y APPLIED OPTICS

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Fig. 2. Algorithm flow chart. Fig. 1. Relationship between radiance and wavelength for ten levels of sediment in milligrams per liter.

2. Method A.

tional to chlorophyll, sediment, and gelbstoff and included an atmospheric model with a fixed aerosol Ångstro¨m exponent and type, but with a variable optical depth. We describe a modular algorithm that we used to model the optical properties of the water and atmospheric effects independently and to minimize the error iteratively. The models given in the default algorithm are drawn from the current literature, and the algorithm is designed such that substitution of more appropriate models is relatively straightforward. In the default water content model, the constituents considered are phytoplankton ~represented by chlorophyll concentration C!, sediment, and gelbstoff. The diffuse attenuation coefficient is assumed proportional to sediment and gelbstoff, with a chlorophyll dependence given by Baker and Smith ~1982!.13 The backscatter coefficient is assumed proportional to sediment and has a power-law dependence on chlorophyll.14 The method of Gordon et al. ~1988!5 is used to calculate Rss~l! and hence Rw~l!. The default atmospheric model allows for variation in the type and optical depth of tropospheric aerosol. The optical depth ta~l! of the aerosol is assumed to follow Ångstro¨m’s law, and the aerosol is assumed to consist of a mixture of three types ~maritime, continental, and urban!. Single-scattering theory is used to find the value of ta~l! consistent with the Rw~l! given by the water model. This is then fitted to Ångstro¨m’s law, and the aerosol path reflectance Ra is calculated. These calculations are corrected for multiple-scattering effects with precalculated lookup tables.15 5444

APPLIED OPTICS y Vol. 35, No. 27 y 20 September 1996

Overview

This algorithm is iterative and modular in that the atmosphere and water are modeled separately, the reflectance predicted from these models is compared with Rs, and the models are adjusted accordingly until the error converges to a minimum ~Fig. 2!. The algorithm is designed such that the atmospheric and water models can be modified or replaced independently, as can the initialization and minimization techniques. The atmospheric model must be able to estimate Rs~l! from Rw~l! and vice versa given the atmospheric parameters. The water model must be able to estimate Rw~l! given the water parameters. The parameters of the two models must first be initialized. The importance of this step depends on the robustness of convergence of the algorithm, which will itself depend on the choice of models, and it must be tested. Ideally the convergence should be independent of the initialization, but this depends on the error measure having a single minimum, since a local minimization algorithm is used, and on the ability of the minimization algorithm to converge from any point, which is not generally true of such algorithms. Doerffer and Fischer ~1994!12 found no problems of multiple minima with Coastal Zone Color Scanner data over the North Sea, although their algorithm used fewer model parameters. One can apply the algorithm sequentially to each pixel so most pixels can be initialized within the context of the image as a whole, for example, by using the best-fitting parameters from an adjacent pixel, but at least one pixel must be initialized independently. After initialization, we calculated Rc~l!, which is Rs~l! corrected for molecular scattering and absorption, using collocated measurements of ozone concentration, surface pressure, and water vapor column

amount, if available. One can deal with the effect of O2 absorption on SeaWiFS band 7 ~0.765 mm! by changing the path radiance in this band, as described by Ding and Gordon ~1995!.16 This method assumes Rw 5 0 in this band, and so the effect of O2 absorption on any nonzero Rw needs to be accounted for. Calculations of the straight-line beam absorption for various air masses M of O2 suggest17 that the O2 absorption is mostly saturated beyond approximately M 5 2, and the transmission tO2 can best be modeled over the region of interest ~typical combined path air mass M 5 2–5!16 with the power-law dependence, tO2 5 0.9248M20.03856, corresponding to an almost airmass independent O2 absorption of approximately 12% of the total band 7 signal. This statement needs some qualification: if the O2 absorption lines are nearly saturated when the light reaches the water, then this light will not be significantly attenuated on its return journey to space ~i.e., attenuation is independent of air mass!. However, the light spends some time in the water, and if inelastic scattering ~e.g., Raman scattering! occurs, the O2 lines will start to fill, resulting in further O2 attenuation on the return journey through the atmosphere. The two simplest regimes to model are one in which inelastic scattering is unimportant, in which case the downward and return journeys are treated as one, and one in which inelastic scattering in the water completely refills the O2 absorption lines, in which case the downward and return journeys may be treated independently. For this study we assumed the first of these regimes. An iterative process now begins. In each iteration, we estimated Rw~l! from the water model parameters using the water model. We then recalculated Rc~l! from Rw~l! and the atmospheric parameters using the atmospheric model; and we minimized x2, the sum of squared differences between the observed and estimated values of Rc~l!, by varying the water and atmospheric parameters until a convergence criterion was met. An alternative error measure is udRcumax, the maximum absolute difference

Rss~l! 5

Rw~m0, m, w, l!Q~m0, m, w, l!n2~l! p@1 2 r1~l!#@1 2 r2~m9, l!#t~m0, l! 1 r2~l! Rw~m0, m, w, l!Q~m0, m, w, l!n2

between the observed and estimated values. Both quantities may be compared with the measurement error. B.

and 1% urban!. This ta~l! is the optical depth of the assumed aerosol in each band that would give the observed Rs in that band over a dark surface. Each of these apparent optical depths can be considered as consisting of a component resulting from the neglected Rw and a component resulting from the presence of the aerosols. At surface albedos greater than approximately 15%, this approach becomes invalid because, if the surface reflectance is greater than the path reflectance of an optically thick atmosphere, then the aerosol component is negative. No matter how much aerosol is assumed, it alone cannot account for the observed reflectance. However, the albedo of the ocean is almost always less than this except in regions of Sun glint or foam, or possibly in coccolithophore blooms or at high concentrations of highly reflective sediment. Modern sensors tilt away from Sun glint, and foam occurs on a scale much smaller than a pixel, so the approach is valid—the values represent a robust upper limit on the aerosol optical depth in each band, with the possible exceptions of coccolithophore blooms and some strongly turbid waters such as the plumes of highly sediment-laden rivers. We found an optical depth with an Ångstro¨m-law wavelength dependence that accounts for as much of the apparent ta as possible, by finding two bands in which ta can be assumed to equal its apparent value ~i.e., Rw 5 0!, all other Rw~l! being nonnegative. Modeling Rw~l! over the likely range of water content reveals that the furthest infrared band ~SeaWiFS band 8 at 0.865 mm! always has the smallest Rw, so this is assumed to be one of the two bands. The Ångstro¨m t~l! found from these two bands can be used to find Rw~l! in the other bands. t~l! represents an upper limit for ta~l! with a plausibly smooth wavelength variation. In essence, the approach is to divide Rc~l! into a smoothly varying component that is at least partly attributable to aerosols with an Ångstro¨m-law wavelength dependence and a nonmonotonically varying residue attributable solely to Rw~l!. We now convert Rw~l! to Rss~l! using

Initialization

One or more pixels must be initialized independently. By default, we did this by using the default atmospheric model to calculate an apparent ta~l! on the initial ~fallacious! assumptions that Rw~l! 5 0 and that the aerosol composition has a suitably chosen representative value ~95% maritime, 4% continental,

,

(1)

where ~m0, m, w! are the solar and view zenith angle cosines and the relative azimuth, n is the refractive index of the water, r1~l! is the diffuse reflectance of the surface from above and r2~l! from below, r2~m9, l! is the reflectance from below at m9, the angle cosine that would refract into the view direction for a flat ocean, t~m0, l! is the diffuse transmittance of the atmosphere in the solar direction ~including O2 absorption in SeaWiFS band 7!; and Q~m0, m, w, l! is the ratio of upwelling irradiance to radiance just below the surface. 20 September 1996 y Vol. 35, No. 27 y APPLIED OPTICS

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Q~m0, m, w, l! is uniformly equal to p for a Lambertian surface, but is typically 4 –5 for near-nadir water viewing.18 For this study we used the value Q 5 4.55. Following Gordon et al. ~1988!,5 we used the quantity Rq~l! 5 RssyQ, so the value of Q appears only in the relatively unimportant r2~l! Rw~m0, m, w, l!Q~m0, m, w, l!n2 term. We used Rq~l! to estimate the concentrations of chlorophyll, gelbstoff, and sediment. First, NASA’s universal pigment algorithm was used to estimate the chlorophyll concentration.19 This is of dubious application to case 2 waters, but, as more robust algorithms for the new sensors emerge, they may be substituted trivially. The diffuse downward attenuation and backscattering coefficients k~l! and bb~l! are approximated5 by k~l! 5 kw~l! 1 kc~C, l! 1 k*s~l!S 1 kg~l!, where kc~C, l! 5 k0c~l!C exp$2@k1c~l!log10~Cy0.5!#2% 1 0.001C2, and bb~l! 5 bbw~l! 1 bbc~C, l! 1 b*bs~l!S, where bbc~C, l! 5 A~l!CB~l!. k*s~l! and b*bs~l! were derived from Tassan,11 kg~l! 5 kg~l0!exp@2kg1~l 2 l0!#,20 where l0 is 0.38 mm and kg1 5 0.014 mm21, and the chlorophyll parameters k0c~l!, k1c~l!, A~l!, and B~l! were taken from the SeaWiFS literature.21 Rq~l! can then be approximated by Rq~l! 5 0.11bb~l!yk~l! ' bb~l!y 9k~l!.5 To estimate the sediment concentration, we used red to near-infrared bands ~bands 6 and 8 in the case of SeaWiFS!. Gelbstoff absorption is neglected in these bands, giving Rq~l! 5 @bbw~l! 1 bbc~C, l! 1 b*bs~l!S#y9@kw~l! 1 kc~C, l! 1 k*s~l!S#, from which S may be calculated. The two bands give two estimates of sediment concentration. These were weighted by b*bsy9k*s, the reflectance attributable to sediment alone, and the weighted mean was found. bb~l! and k~l! attributable to water, chlorophyll, and sediment may now be calculated, leaving only kg~l!, the diffuse attenuation coefficient of the gelbstoff, unknown. Little is known about gelbstoff, and this study follows others11,20 in assuming gelbstoff to be a pure absorber with a known exponential wavelength dependence of the diffuse attenuation coefficient. kg~l0! is calculated for each band with nonzero Rw and the values weighted by exp@2kg1~l 2 l0!# to find the mean. This initialization and subsequent minimization are performed for the first pixel, then one can initialize subsequent pixels by the best-fitting parameters of their neighbors. This makes use of any continuity of conditions that may exist from one pixel to the next. If convergence is poor, the pixel is reinitialized independently, as was done for the first pixel. If time permits, the fit may be further improved by back initialization, in which one can reinitialize pixels with poor convergence using the parameters of the next pixel. The reason for this is that, if the parameters change suddenly, for example, at an edge feature in the image, the parameters taken from the previous pixel will be inaccurate. One or more pixels may follow in which convergence is poor because both the previous pixel’s parameters and the independent initialization fail. The first pixel to converge successfully may then be used to initialize the failed pixels by back initialization to the edge feature. 5446


Back initialization may be applied after the whole image has been processed. Similar refinements may be easily implemented. C.

Default Models

The default water model takes as its parameters the concentrations of chlorophyll and sediment and k0~l0! resulting from gelbstoff. It uses the semianalytic radiance model of Gordon et al. ~1988!5 to calculate Rq as described above. The sediment characteristics are highly variable, depending on the local sediment type. However, this should not vary too much temporally ~although in some places effects such as variable mixing of sediments from different sources, changes in currents, or variations in river sediment may give rise to different sediment types!, so the sediment values may be specified regionally on scales to be determined empirically. In the method described by Doerffer and Fischer ~1994!,12 it was found that two different sediment types were required to characterize the waters of the North Sea.22 The default model uses values of k*s~l! and b*bs~l! derived from Tassan.11 The default atmospheric model takes as its parameters the proportions of two tropospheric aerosol types associated with land and human activities ~continental and urban!, relative to that which is normally found over oceans ~maritime!.23 Since the proportions must add up to 1, the proportions of continental and urban aerosols determine the proportion of maritime aerosol. These proportions are in terms of the optical effects of the aerosol rather than the physical aerosol amount, and the single-scattering albedo Ãa~l!, phase function Pa~C, l! at scattering angle C, and forward-scattering fraction Fa~l! of the composite aerosol are assumed to be the sums of those aerosol types, weighted by their proportions. This allows single-scattering theory ~q.v.! to be used to calculate ta ~or Rw!, given Rw ~or ta! and Rc in each band. The model makes a correction for multiple scattering based on lookup tables of the difference between Ra~l! calculated by multiple and single scattering for various aerosol types and geometries.15 It then finds the ta~l! with an Ångstro¨m-law wavelength dependence that best fits the calculated values with a least-squares fit and recalculates either Rc~l! from Rw~l! or vice versa. Sometimes it is found, particularly in early iterations when the fit is poor, that finding the maximum Ångstro¨m varying ta~l! not greater than the calculated values ~compare initialization! aids the convergence of the next leastsquares iteration, although it slows or prevents convergence in later iterations. These models have five variable parameters ~concentrations of chlorophyll, sediment, and gelbstoff and relative proportions of continental and urban aerosols! and two derived quantities ~aerosol optical depth and Ångstro¨m exponent! between them, and there are eight SeaWiFS bands, so there is some redundancy of information, which makes it less likely that the solution found is simply one of a degenerate set of equally valid solutions. In principle the spec-

tral variation of optical depth ~modeled by the Ångstro¨m exponent! is a function of aerosol type, but it is also strongly affected by relative humidity ~RH!.1 It may be possible to include RH as an atmospheric model parameter that together with the aerosol type can be used to determine the aerosol properties, including spectral variation of optical depth. The model would then fit this variation to that derived from the observations to determine the aerosol amount. The effect of this would be to add an explicit degree of freedom ~the RH! but to lose an implicit one ~the Ångstro¨m exponent! and would, of course, allow estimation of RH. If independent satellite measurements of RH were available, this would save a degree of freedom. A principal components study by Sathyendranath et al.24 suggests that the first six SeaWiFS bands may only have 5 effective degrees of freedom that can be used to retrieve the contents of case 2 waters. If this is true, this method as it stands uses all available degrees of freedom, making it likely that some parameters will be found to be degenerate. More complex models with more parameters may be introduced, with the potential of residual errors being reduced at the cost of the incidence of degeneracy being increased. To avoid degeneracy, it is important that at least as many independent observations are used as there are degrees of freedom and that each parameter uses a distinct subset of the observations. For example, an algorithm that depends on a single-band ratio ~e.g., a pigment algorithm! would use only one observation ~the ratio! rather than two, but if the same two bands were used to model another effect ~e.g., aerosol!, the two effects would not be independent, even though two observations are used to fix 2 degrees of freedom. As the reflectances in the two bands changed, the derived pigment and aerosol parameters would track one another.

given1 by Rs 5 Rr 1 Ra 1 Rra 1 Rg 1 t~m! Rw, where Rr is the multiple Rayleigh-scattering path reflectance, Ra is the multiple aerosol scattering path reflectance, Rra is the reflectance attributable to both Rayleigh and aerosol scattering of the same photon, Rg is the reflectance attributable to Sun glint and surface foam, and t~m! is the diffuse transmittance of the atmosphere at m, given approximately27 by t~m! 5 tmol~m!tr~m!ta~m! 5 tmol~m! exp

where m is the view angle cosine, tmol~m! is the transmittance attributable to molecular absorption at m ~including O2 in band 7!, tr~m! is the diffuse transmittance attributable to Rayleigh scattering at m, ta~m! is the diffuse transmittance attributable to aerosols at m, tr is the Rayleigh optical depth, ta is the aerosol optical depth, Ãa is the aerosol single-scattering albedo, and Fa is the probability that a photon scattered by the aerosol will be forward scattered. In the aerosol single-scattering approximation, Rr is precalculated ~including the band 7 correction for O2!, Rra and Rg are neglected, and Ra is approximated by its single-scattering value Ras. The molecular corrected reflectance Rc is then given by Rc 5

Minimization

For each pixel the model parameters are initialized, molecular scattering and absorption effects calculated, and Rc~l! derived. Each iteration starts with a calculation of Rw~l! from the water model with the water model parameters. The atmospheric model is then used to calculate Rc~l!. The error is minimized with Powell’s method25 ~others such as the simplex algorithm of Nelder and Mead26 may be trivially substituted! to change the water and atmospheric model parameters and to recalculate Rc~l!. Finally, we checked the residual error for convergence before starting the next iteration. The default convergence criteria are that either the fractional decrease in the best x2 found at each iteration must be less than a specified threshold for a specified number of iterations, x2 falls below a threshold value, or the total number of iterations exceeds its limit. E.

Single-Scattering Theory

In this subsection we omitted wavelength dependence for clarity. The reflectance Rs at the sensor is

~Rs 2 Rr! Ras < 1 tr~m!ta~m! Rw. tmol~m! tmol~m!

(2)

Ras consists of terms representing paths scattered in the atmosphere only, scattered then Fresnel reflected, and Fresnel reflected then scattered: Ras 5

D.

2tr 2~1 2 ÃaFa!ta exp , 2m m

H F S DGJ S S D S D H F S DGJ H F S DGJDG F

1 t*mol~m0m!Ãa Pa~C1! 1 1 2 exp 2ta 1 4 m 1 m0 m m0 1

Pa~C2! 2ta r1~m!exp 2 m 2 m0 m

3 1 2 exp ta

1 1 2 m m0

3 1 2 exp 2ta

1 1 2 m m0

2 r1~m0!exp 2

2ta m0

,

where t*mol~m0m! 5 tmol~m0!tmol~m!y@tO2~m0!tO2~m!# is the molecular absorption path transmission for the combined incoming and outgoing paths ~assumed to be above the aerosol layer, so excluding O2!, Pa~C1! is the aerosol phase function at the Sun-sensor angle, Pa~C2! is the aerosol phase function at the ~Fresnelreflected path! sensor angle, and m0 is the solar angle cosine. In the case of O2 absorption ~and H2O absorption!, the assumption that the absorber lies above the aerosol layer is obviously inaccurate, which is why a separate correction is made for O2. Ras is calculated without O2 absorption, and an air-mass dependent 20 September 1996 y Vol. 35, No. 27 y APPLIED OPTICS

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correction factor is applied to band 7 ~this is also done when we precalculate Rr!: Ras ~with O2! 5 @1 1 10Pa~M!#Ras ~no O2!.16 If ta is known, evaluation of Rw from Eq. ~2! is trivial. If Rw is known, Eq. ~2! may be solved for small ta ~typical atmospheric ta , 0.1! if we expand the exponentials and neglect high powers of ta. Gordon27 and others have discussed the first-order case, but the resulting polynomial remains analytically soluble up to order ta3, giving the cubic

SD SD

ta ta a1b 1c m m

2

ta 1d m

3

5 0,

(3)

where a 5 tr~m! Rw 2 Rc, tmol~m0!Ãa $Pa~C1! 1 Pa~C2!@r1~m! 1 r1~m0!#% 4m0

b5

2 tr~m! Rw~1 2 ÃaFa!, c52

S

1 tmol~m0!Ãa $Pa~C1!~m 1 m0! 1 Pa~C2! 2 4m02

3 @r1~m! ~3m0 1 m! 1 r1~m0!~m0 1 3m!#%

D

1 tr~m! Rw~1 2 ÃaFa!2 ,

S

1 tmol~m0!Ãa $Pa~C1!~m 1 m0!2 1 Pa~C2!@r1~m!~7m02 6 4m03

d5

1 4m0m 1 m2! 1 r1~m0!~m02 1 4m0m 1 7m2!#%

D

2 tr~m! Rw~1 2 ÃaFa!3 , except in band 7, where tmol~m0! must be replaced by tmol~m0!ytO2~m0m!. This is analytically soluble given Rc, Rw, tr~m!, tmol~m0!, tO2~m0m!, r1~m! and r1~m0! ~aerosol independent!, Ãa, Fa, Pa~C1!, and Pa~C2!. If this proves computationally excessive, or is shown to give no significant improvement, it is possible to neglect d ~second-order approximation! or c and d ~first order!; and in the unlikely event that the expansion to order ta3 is insufficiently accurate, the solution may be further refined at little extra computational cost by one or two iterations of the Newton–Raphson root finding with the unexpanded Eq. ~2! and its differential with respect to ta. F.

Multiple-Scattering Correction

Multiple-scattering correction is accomplished with lookup tables. Each table entry applies to a given wavelength, geometry, and aerosol type and consists of a set of nine biquadratic coefficients relating DRa 5 Ra 2 Ras to Rw and ta. We precalculated the coefficients using a multiple-scattering model of the atmosphere ~e.g., discrete ordinates, successive orders of scattering, or Monte Carlo! to generate sample corrections and then fitting a polynomial to these by 5448


least squares.15 We now have Rc 5

@Ras 1 DRa~Rw, ta!# 1 tr~m!ta~m! Rw. tmol~m!

(4)

If ta is known, this is a quadratic in Rw. If Rw is known, Eq. ~4! may be expanded as described above for Eq. ~2!, resulting in a modified version of Eq. ~3!, with the appropriate powers of ta from DRa inserted in a, b, and c. This is soluble in the same way as Eq. ~3!, with an implicit multiple-scattering correction. 3. Results

We tested the algorithm for convergence and accuracy by generating a set of model parameters, calculating the resulting Rs~l! to simulate SeaWiFS observations, and then attempting to retrieve the model parameters by the above inversion technique. The convergence criteria used were that the algorithm terminated if x2 # x2n ~see error analysis below!, 20 iterations elapsed, or ten iterations elapsed with no further improvement of the fit. A pixel was considered not to have converged adequately if the bestfitting x2 . 10x2n. If degeneracy occurs, this is shown by a successful fit ~small x2! that retrieves incorrect model parameters. This shows that different model parameters can produce indistinguishable reflectances, and so the physical situations represented by those parameters are indistinguishable with the given models. Independent initialization was used, as no attempt was made to simulate the continuity of physical parameters found in real data. The geometry was held constant, with a solar zenith angle of 59.4° and nadir viewing ~air mass M ' 3!. Two model atmospheres were used: ~i! composed of 70% maritime, 30% continental aerosol with an optical depth of ta 5 0.075 at 0.55 mm, and an Ångstro¨m exponent of 0.25, representative of a relatively clean marine atmosphere ~WMO MARI!,23 and ~ii! composed of 70% maritime, 25% continental, and 5% urban aerosol with the same optical depth and Ångstro¨m exponent to simulate advected pollution. Water constituents were included singly and in combination. The water parameters used were clear water, pure chlorophyll concentrations from 0.1 to 12.8 mg m23, pure sediment concentrations from 0.1 to 12.8 g m23, and pure gelbstoff absorption coefficients at 0.38 mm from 0.01 to 1.28 m21. Combinations included 1 mg m23 of chlorophyll with 0.1 to 12.8 g m23 of sediment and 0.01 to 1.28 m21 of gelbstoff, and 1 g m23 of sediment with 0.1 to 12.8 mg m23 of chlorophyll. The results were as follows: With chlorophyll only ~i.e., case 1 waters! and atmosphere ~i!, the algorithm performed well for small and large concentrations, less well for intermediate ones. Errors in the derived chlorophyll amounts changed from 210% to 233% as chlorophyll in-

Fig. 3. Retrieval results for case 1 waters.

Fig. 4. Retrieval results for sediment-laden case 2 waters.

creased from 0.01 to 6.4 mg m23, 110% for 12.8 mg m23. At 0.1 mg m23 a trace ~0.01 g m23! of sediment was retrieved. The aerosol had a maritime proportion ranging from 0 to 61%, with more urban than continental. The optical depth was 3–32% too low and the Ångstro¨m exponent 10% too low to 6% too high. Rw was generally 0 –10% too high with larger errors at 0.8 –3.2 mg m23 and 1–7% too low at 12.8 mg m23. Results were slightly better with atmosphere ~ii!. An example of the results for case 1 waters @6.4 mg m23, atmosphere ~i!# is shown in Fig. 3. With sediment only and atmosphere ~ii!, the algorithm was successful in the retrieval of water parameters, with the retrieved amounts of sediment within 2% of the true value except at 0.8 g m23, where a small amount ~0.07 mg m23! of chlorophyll was retrieved, resulting in a 16% underestimate of the sediment concentration. The aerosol maritime proportion was 63–100% with urban occurring more frequently than continental. The optical depth was 9% too low to 28% too high, and the Ångstro¨m exponent varied by 612% about the true value. Rw was uniformly within 62% of the true value. Results were slightly poorer with atmosphere ~i!. In three cases a small amount of chlorophyll ~0.02– 0.07 mg m23! was retrieved, and in two of these this resulted in a 5–16% underestimate of the sediment concentration. Rw was still only 3% too high in one case. An example of sediment-laden waters @3.2 g m23, atmosphere ~ii!# is shown in Fig. 4. With gelbstoff only and atmosphere ~i!, results were often poor because of the low values of Rw. Gelbstoff amounts ranged from 31% too low to 59% too high. The aerosol maritime proportion was 0 –77%, with continental and urban occurring equally frequently. The optical depth was from 33% too low to 7% too high, and the Ångstro¨m exponent was 20% too low to 6% too high. Variations in Rw ranged from 4% at 0.01 and 0.02 m21 to 46% at 0.32 m21 as the magnitude of Rw decreased. Results were similar with atmosphere ~ii!. With the combination of 1 mg m23 of chlorophyll and varying sediment the sediment concentration

was 20% too low at 0.1 g m23, 10% too low at 0.2 and 0.4 g m23, and 1– 4% too low thereafter. Chlorophyll concentrations ranged from 35% too low to 5% too high. The aerosol composition varied erratically with an optical depth from 33% too low to 29% too high and the Ångstro¨m exponent from 28% too low to 12% too high. Rw was 2–5% too high except at 0.4 and 0.8 g m23 where it was 4 –9% too low and 2– 8% too high, respectively. Results were similar with atmosphere ~ii!. With 1 mg m23 of chlorophyll and varying gelbstoff, results were again poor because of the low Rw. Gelbstoff was detected only at 0.64 and 1.28 m21 ~22% too high and 91% too low, respectively!, chlorophyll amounts rose steadily from 18% too low at 0.01 m21 to 36% too high at 0.32 m21, then rose to approximately 22 mg m21 at 1.28 m21, with 0.08 g m23 of sediment retrieved. The aerosol to as high as 0.64 m21 had a maritime proportion of 26 –73% with more urban than continental, an optical depth 31% too low to 19% too high, and an Ångstro¨m exponent 40% too low to 12% too high. At 1.28 m21 the aerosol was pure maritime with an optical depth 19% too high and an Ångstro¨m exponent 12% too high. Variations in Rw were as much as 75%. With 1 g m23 of sediment and varying chlorophyll, sediment amounts were retrieved to within 2– 6% except at 12.8 mg m23 when they were 14% too low, and chlorophyll amounts were 3–30% too low. At 0.8 g m23 a trace ~0.01 m21! of gelbstoff was retrieved. The aerosol maritime proportion was 0 – 63% with continental and urban occurring equally frequently, an optical depth 0 –35% too low, and an Ångstro¨m exponent 30% too low to 18% too high. Rw varied by 67% about the true value. 4. Discussion

The quality of the results is highly variable. In general, the easiest parameter to retrieve is sediment concentration, followed by chlorophyll and gelbstoff. This is in agreement with previous findings.24 Nominal SeaWiFS retrieval accuracy requirements28 are for Rw to within 5% and for chlorophyll concentra20 September 1996 y Vol. 35, No. 27 y APPLIED OPTICS

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tions to within 35%. The Rw requirement is satisfied for pure chlorophyll to 0.2 mg m23 or to 6.4 mg m23 in the presence of 1 g m23 of sediment, and for any amount of pure sediment. The chlorophyll requirement is satisfied in all the cases studied, including the presence of sediment and gelbstoff. The algorithm distinguishes well between the water components, only occasionally detecting traces of chlorophyll, sediment, or gelbstoff where there are none. The atmospheric fit is problematic. The algorithm fails to distinguish between continental and urban aerosols, and the maritime component is highly variable, although less so than the others. This can be explained using the single-scattering theory. The effect of the aerosol can be approximated1 by the factor ε~li, lj! 5

ta~li!Ãa~li!pa~u, u0, li! , ta~lj!Ãa~lj!pa~u, u0, lj!

which in the context of this study may be further separated into components because of spectral variations of ta~l! ~determined by the Ångstro¨m exponent! and of Ãa~l!pa~u, u0, l! ~determined by the aerosol type!. Ãa~l!pa~u, u0, l! has a similar spectral variation in the three aerosol types, especially continental and urban, which accounts for the inability to distinguish between these types. The main practical distinction among aerosol types is then the different Ångstro¨m exponents. It seems that if we allow free variation of both ta~l! and Ãa~l!pa~u, u0, l! it allows the aerosol to duplicate effects because of variations in Rw, and hence degeneracy is created. If the spectral variations of ta are linked to the aerosol type by way of the RH, it is therefore likely to give a significant improvement ~see default models, above! by a reduction in the scope for unrealistic atmospheric configurations that may account for effects actually attributable to Rw. In particular, unrealistically extreme Ångstro¨m exponents are avoided. More research is necessary to determine the best minimization technique to ensure convergence. The algorithm as it stands exhibits three consistent biases: the aerosol optical depth is an average of 12% too low, the Ångstro¨m exponent is 8% too low, and Rw is 5% too high. 5. Error Analysis

In an iterative fitting algorithm, an understanding of the propagation of errors is crucial, especially when the measurements of interest have different orders of error. In this system, a small effect with high natural variability ~Rw! propagates through a medium ~the atmosphere! that both changes it and adds its own naturally varying effect ~Ra!, which often has a much larger amplitude. The contribution of Rw to Rs may be as little as 1%,27 and generally varies strongly and nonmonotonically with wavelength, whereas the contribution of Ra has a weaker, smoother wavelength dependence. For atmospheric correction purposes Rw is the measurement of inter5450


est, and we wish simply to minimize the effects of the atmosphere on it. SeaWiFS radiometer noise has been well documented,28 allowing calculation of the noiseequivalent reflectance dRs as a function of Rs. Neglecting errors in Rr and tmol~m!, Eq. ~2! gives us the noise-equivalent molecular corrected reflectance dRc. This allows one to calculate the noiseequivalent x2, x2n 5 •~dRc!2. If x2 # x2n, the errors are indistinguishable from sensor noise. Finally, if we know the atmospheric transmission in the view direction ~including aerosols!, we can calculate the change in Rw that would result in a change in Rs of dRs. This is dRw, the noise-equivalent Rw. Errors in Rw less than this are indistinguishable from radiometer noise, and if dRw is greater than 5% of Rw, the required 5% tolerance in Rw retrieval is unobtainable. As with any atmospheric correction algorithm, operational errors must be investigated with real data, and the results will depend on the details of the methods used. 6. Concluding Remarks

An algorithm has been presented to derive waterleaving reflectances from SeaWiFS and similar ocean color satellites, particularly over case 2 waters. It is unlikely to perform as well over case 1 waters as algorithms tailored to case 1,1 but it will provide consistent data in mixed case images. The current default algorithm also estimates concentrations of chlorophyll, gelbstoff, and sediment and aerosol type, optical depth, and Ångstro¨m exponent. However, it may be used in conjunction with user-specified water models or atmospheric models so that aspects of interest to the user can be modeled in detail, whereas other aspects use the default models supplied with the basic algorithm. This approach integrates the task of atmospheric correction with that of interpretation, which is normally performed separately after atmospheric correction. A possibility remains for the use of the traditional approach if the results of the basic algorithm are accepted and then if a pigment algorithm ~for example! is separately applied to any water-leaving reflectances that are found. However, this would be a poor utilization of the method. Tests with model data indicate that the default algorithm is capable of retrieving Rw accurately in sediment-laden waters and those waters with low concentrations of chlorophyll. It does not perform as well in the presence of gelbstoff. Aerosol optical depth and Ångstro¨m exponent are fairly well retrieved ~with some bias!, although the aerosol types are not adequately distinguished. Inclusion of RH as a model parameter should resolve many of these problems. The Rw requirement is satisfied for pure chlorophyll to as much as 0.2 mg m23 or 6.4 mg m23 in the presence of 1 g m23 of sediment, and for any amount of pure sediment. The chlorophyll requirement is satisfied in all the cases studied, including the presence of sediment and gelbstoff. The algorithm dis-

tinguishes well between the water components; it only occasionally detects traces of chlorophyll, sediment, or gelbstoff where there are none. Before the algorithm can be made operational, it must be validated. The modular approach lends itself to validation of each module separately, as well as the algorithm as a whole. Much of this will be performed with data collected and calibrated by Andrew Wilson of the British National Space Centre, Monks Wood, UK, with compact airborne spectrographic imager data supplemented by ground-based aerosol measurements. The authors thank R. Doerffer for his helpful suggestions and constructive criticisms. This study was supported by the United Kingdom Natural Environmental Research Council under its SeaWiFS Exploitation Initiative. References 1. H. R. Gordon and M. H. Wang, “Retrieval of water-leaving radiance and aerosol optical thickness over the oceans with SeaWiFS: a preliminary algorithm,” Appl. Opt. 33, 443– 452 ~1994!. 2. P. J. Curran and E. M. M. Novo, “The relationship between suspended sediment concentration and remotely sensed spectral radiance: a review,” J. Coastal Res. 4 ~3!, 351–368 ~1988!. 3. E. M. M. Novo, J. D. Hansom, and P. J. Curran, “The effect of sediment type on the relationship between reflectance and suspended sediment concentration,” Int. J. Remote Sensing 10, 1283–1289 ~1989!. 4. H. R. Gordon, O. B. Brown, and M. M. Jacobs, “Computed relationships between the inherent and apparent optical properties of a flat homogeneous ocean,” Appl. Opt. 14, 417– 427 ~1975!. 5. H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semianalytic radiance model of ocean colour,” J. Geophys. Res. 93, 10,909 –10,924 ~1988!. 6. S. C. Jain and J. R. Miller, “Subsurface water parameters: optimization approach to their determination from remotely sensed water color data,” Appl. Opt. 15, 886 – 890 ~1976!. 7. A. Morel and L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709 –722 ~1977!. 8. R. P. Bukata, J. E. Bruton, J. H. Jerome, S. C. Jain, and H. H. Zwick, “Optical water quality model of Lake Ontario. 2: Determination of chlorophyll a and suspended mineral concentrations of natural waters from submersible and low altitude optical sensors,” Appl. Opt. 20, 1704 –1714 ~1981!. 9. S. Sathyendranath, L. Prieur, and A. Morel, “A threecomponent model of ocean colour and its application to remote sensing of phytoplankton pigments in coastal waters,” Int. J. Remote Sensing 10, 1373–1394 ~1989!. 10. A. Morel, “In water and remote measurements of ocean color,” Boundary-Layer Meteorol. 18, 177–201 ~1980!. 11. S. Tassan, “Local algorithms using SeaWiFS data for the retrieval of phytoplankton, pigments, suspended sediment, and yellow substance in coastal waters,” Appl. Opt. 33, 2369 –2378 ~1994!. 12. R. Doerffer and J. Fischer, “Concentrations of chlorophyll, sus-

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