Radiance-irradiance inversion algorithm for estimating the absorption and backscattering coefficients of natural waters: homogeneous waters

A full multiple-scattering algorithm for inverting upwelling radiance (L(u)) or irradiance (E(u)) and downwelling irradiance (E(d)) profiles in homogeneous natural waters to obtain the absorption (a) and backscattering (b(b)) coefficients is described and tested with simulated data. An attractive feature of the algorithm is that it does not require precise knowledge of the scattering phase function of the medium. For the E(u)-E(d) algorithm, tests suggest that the error in the retrieved a should usually be ?1%, and the error in b(b)?10-20%. The performance of the L(u)-E(d) algorithm is not as good because it is more sensitive to the scattering phase function employed in the inversions; however, the error in a is usually still small, i.e., ?3%. When the algorithm is extended to accommodate the presence of a Lambertian-reflecting bottom, the retrievals of a are still excellent, even when the presence of the bottom significantly influences the upwelling light field; however, the error in b(b) can be large.     

Irradiance inversion algorithm for estimating the absorption and backscattering coefficients of natural waters: Raman-scattering effects

We modify an algorithm for retrieving the absorption (a) and backscattering (b(b)) coefficient profiles in natural waters by inverting profiles of downwelling and upwelling irradiance so as to include the presence of Raman scattering. For a given wavelength of interest, lambda, the light field at the appropriate Raman excitation wavelength lambda(e) is first inverted to obtain the Raman source function at lambda. Starting from estimates of the inherent optical properties at lambda, the contribution to the irradiances at lambda from Raman scattering is then estimated and subtracted from the total irradiances to obtain the elastically scattered irradiances. We then inverted the elastically scattered irradiances to find new estimates of a and b(b) using our original method [Appl. Opt. 37, 3886 (1998)]. The algorithm then operates iteratively: The new estimates are used with the Raman source function to derive a new estimate of the Raman contribution, etc. Sample results are provided that demonstrate the working of the algorithm and show that the absorption and scattering coefficients can be retrieved with accuracies similar to those in the absence of Raman scattering down to depths at which the light field is significantly perturbed by it, e.g., with ~90% of the upwelling light field originating from Raman scattering.     

Irradiance inversion algorithm for absorption and backscattering profiles in natural waters: improvement for clear waters

Our iterative inversion algorithm for retrieving absorption a(z) and backscattering b(b)(z) from profiles of upwelling and downwelling irradiance, on the basis of assuming a depth-independent phase function for the medium, was found to have unsatisfactory accuracy for b(b)(z) in clear waters. We modified the algorithm here by assuming a depth-independent phase function for the particles and then performing an additional iteration over the fraction of total scattering that is due to the water itself. The modified algorithm's accuracy is considerably improved over the original in clear waters and reduces to the original in waters for which the particle contribution to b(b)(z) is dominant.     

Analytic model of ocean color

Ocean color is determined by spectral variations in reflectance at the sea surface. In the analytic model presented here, reflectance at the sea surface is estimated with the quasi-single-scattering approximation that ignores transspectral processes. The analytic solutions we obtained are valid for a vertically homogeneous water column. The solution provides a theoretical expression for the dimensionless, quasi-stable parameter (r), with a value of ~0.33, that appears in many models in which reflectance at the sea surface is expressed as a function of absorption coefficient (a) and backscattering coefficient (b(b)). In the solution this parameter is represented as a function of the mean cosines for downwelling and upwelling irradiances and as the ratio of the upward-scattering coefficient to the backscattering coefficient. Implementation of the model is discussed for two cases: (1) that in which molecular scattering is the main source of upwelling light, and (2) that in which particle scattering is responsible for all the upwelled light. Computations for the two cases are compared with Monte Carlo simulations, which accounts for processes not considered in the analytic model (multiple scattering, and consequent depth-dependent changes in apparent optical properties). The Monte Carlo models show variations in reflectance with the zenith angle of the incident light. The analytic model can be used to reproduce these variations fairly well for the case of molecular scattering. For the particle-scattering case also, the analytic and Monte Carlo models show similar variations in r with zenith angle. However, the analytic model (as implemented here) appears to underestimate r when the value of the backscattering coefficient b(b) increases relative to the absorption coefficient a. The errors also vary with the zenith angle of the incident light field, with the maximum underestimate being approximately 0.06 (equivalent to relative errors from 12 to 17%) for the range of b(b)/a studied here. One implication of this result is that the model could also be used to obtain approximate solutions for the Q factor, defined for a given look angle as the ratio of the upwelling irradiance at the surface to the upwelling radiance at the surface at that angle. This is a quantity that is important in remote-sensing applications of ocean-color models. An advantage of the model discussed here is that its implementation requires inputs that are in principle accessible only in a remote-sensing context.     

Irradiance inversion theory to retrieve volume scattering function of seawater

An attempt to retrieve the volume scattering function (VSF) of source-free and no-inelastic-scattering ocean water is made from the upwelling irradiance Eu and downwelling irradiance Ed. It will be shown, from the radiative transfer equation, that the VSF of seawater can be calculated by the planar irradiances when the scattering phase function of the suspended particles in the backward direction and the molecular VSF are known. On the derivation of the hydrosol VSF, several optical properties such as the absorption coefficient a; the scattering coefficients of hydrosol, b, b(f), b(b) and those of the suspended particles, b(p), b(fp), b(bp); the beam attenuation coefficient c; the average cosines mu, mu(d), and mu(u); and the backscattering shape factor for the downwelling light stream, r(du), will also be obtained. On the derivation of those optical parameters, classical knowledge related to interrelationships between inherent optical properties and apparent optical properties and obtained with Monte Carlo numerical simulations is analytically verified. The present theory can be applied to surface waters and any wavelengths, except for waters and wavelengths with an extremely low b(b)/a ratio.     

Ocean inherent optical property estimation from irradiances

A method is evaluated for estimating the absorption coefficient a and the backscattering coefficient b(b) from measurements of the upward and downward irradiances E(u)(z) and E(d)(z). With this method, the reflectance ratio R(z) and the downward diffuse attenuation coefficient K(d)(z) obtained from E(u)(z) and E(d)(z) are used to estimate the inherent optical properties R(infinity) and K(infinity) that are the asymptotic values of R(z) and K(d)(z), respectively. For an assumed scattering phase function beta , there are unique correlations between the values of R(infinity) and K(infinity) and those of a and b(b) that can be derived from the radiative transfer equation. Good estimates of a and the Gordon parameter G = b(b)/(a + b(b)) can be obtained from R(infinity) and K(infinity) if the true scattering phase function is not greatly different from the assumed function. The method works best in deep, homogeneous waters, but can be applied to some cases of stratified waters. To improve performance in shallow waters where bottom effects are important, the deep- and shallow-measurement reflectance models also are developed.     

Satellite retrieval of the absorption coefficient of phytoplankton phycoerythrin pigment: theory and feasibility status

Oceanic radiance model inversion methods are used to develop a comprehensive algorithm for retrieval of the absorption coefficients of phycourobilin (PUB) pigment, type I phycoerythrobilin (PEB) pigment rich in PUB, and type II PEB deficient in PUB pigment (together with the usual "big three" inherent optical properties: the total backscattering coefficient and the absorption coefficients of chromophoric dissolved organic matter (CDOM)-detritus and phytoplankton). This fully modeled inversion algorithm is then simplified to yield a hybrid modeled-unmodeled inversion algorithm in which the phycoerythrin (PE) absorption coefficient is retrieved as unmodeled 488-nm absorption (which exceeds the modeled phytoplankton and the CDOM-detritus absorption coefficients). Each algorithm was applied to water-leaving radiances, but only hybrid modeled-unmodeled inversions yielded viable retrievals of the PE absorption coefficient. Validation of the PE absorption coefficient retrieval was achieved by relative comparison with airborne laser-induced PEB fluorescence. The modeled-unmodeled retrieval of four inherent optical properties by direct matrix inversion is rapid and well conditioned, but the accuracy is strongly limited by the accuracy of the three principal inherent optical property models across all four spectral bands. Several research areas are identified to enhance the radiance-model-based retrievals: (a) improved PEB and PUB absorption coefficient models, (b) PE spectral shifts induced by PUB chromophore substitution at chromophore binding sites, (c) specific absorption-sensitive phytoplankton absorption modeling, (d) total constituent backscattering modeling, (e) unmodeled carotinoid and phycocyanin absorption that are not now accounted for in the chlorophyll-dominated phytoplankton absorption coefficient model, and (f) iterative inversion techniques to solve for six constituents with only five radiances. Although considerable progress has been made toward the satellite recovery of PE absorption, the maturity of the retrieval is presently insufficient for routine global application. Instead it must currently be used on a regional basis where localized ship and aircraft validation can be made available. The algorithm was developed for the MODIS (Moderate-Resolution Imaging Spectroradiometer) sensor but is applicable to any sensor having comparable band locations.     

Analytic algorithms for determining radiative transfer optical properties of ocean waters

A synthetic model for the scattering phase function is used to develop simple algebraic equations, valid for any water type, for evaluating the ratio of the backscattering to absorption coefficients of spatially uniform, very deep waters with data from upward and downward planar irradiances and the remotely sensed reflectance. The phase function is a variable combination of a forward-directed Dirac delta function plus isotropic scattering, which is an elementary model for strongly forward scattering such as that encountered in oceanic optics applications. The incident illumination at the surface is taken to be diffuse plus a collimated beam. The algorithms are compared with other analytic correlations that were previously derived from extensive numerical simulations, and they are also numerically tested with forward problem results computed with a modified FN method.     

Regularized algorithm for Raman lidar data processing

A regularized algorithm that has the potential to improve the quality of Raman lidar data processing is presented. Compared to the conventional scheme, the proposed algorithm has the advantage, which results from the fact that it is based on a well-posed procedure. That is, the profile of the aerosol backscatter coefficient is computed directly, using the explicit relationships, without numerical differentiation. Thereafter, the profile of the lidar ratio is retrieved as a regularized solution of a first-kind Volterra integral equation. Once these two steps have been completed, the profile of the aerosol extinction coefficient is computed by a straightforward multiplication. The numerical simulations demonstrated that the proposed algorithm provides good accuracy and resolution of aerosol profile retrievals. The error analysis showed that the retrieved profiles are continuous functions of the measurement errors and of the a priori information uncertainties.     

Diffuse reflection coefficient of a stratified sea

A differential equation of a Riccati type for the diffuse reflection coefficient of a stratified sea is proposed. For a homogeneous sea with arbitrary inherent optical properties this equation is solved analytically. For an inhomogeneous sea it is solved approximately for any arbitrary stratification. The resulting equation expresses the diffuse reflection coefficient of the sea through vertical profiles of absorption and backscattering coefficients, bottom albedo, and sea depth. The results of calculations with this equation are compared with Monte Carlo computations. It was found that the precision of this approach is in the range of 15%.     

Analytic remote-sensing optical algorithms requiring simple and practical field parameter inputs

Spectral in-water measurements of downward irradiance (E(d)), upward irradiance (E(u)), and nadir radiance (L(u)) are sufficient to calculate the scalar irradiances E(0), E(0d), and E(0u), the average cosines mu, mu(d), and mu(u), the light absorption coefficient a, the backscattering coefficient b(b), and the so-called f factor that relates to R, a, and b(b). The solar elevation of 42 degrees is a special case in which mu(d) is independent of all variables except solar elevation. The algorithms are valid for solar elevations between 12 degrees and 81 degrees for horizontally stratified clear and turbid deep waters.     

Instrument self-shading in underwater optical measurements: experimental data

Self-shading error of in-water optical measurements has been experimentally estimated for upwelling radiance and irradiance measurements taken just below the water surface. Radiance and irradiance data have been collected with fiber optics that terminated with 1°, 18°, and 2π optics housed in the center of a disk that simulated the size of the instrument. Analysis of measurements taken at 500, 600, and 640 nm in lake waters have shown errors ranging from a few percent up to several tens of percent as a function of the size of the radiometer, the absorption coefficient of the medium, the Sun zenith, and the atmospheric turbidity. Comparisons between experimental and theoretical errors, the latter computed according to a scheme suggested by other authors, have shown absolute differences generally lower than 5% for radiances and lower than 3% for irradiances. Analysis of radiance measurements taken with 1° and 18° fields of view have not shown appreciable differences in the self-shading error. This finding suggests that correction schemes for self-shading error developed for narrow-field-of-view radiance measurements could also be applied to measurements taken with relatively larger fields of view.     

Cloud Detection and Clearing for the Earth Observing System Terra Satellite Measurements of Pollution in the Troposphere (MOPITT) Experiment

The Measurements of Pollution in the Troposphere (MOPITT) instrument, which was launched aboard the Earth Observing System (EOS) Terra spacecraft on 18 December 1999, is designed to measure tropospheric CO and CH(4) by use of a nadir-viewing geometry. The measurements are taken at 4.7 mum in the thermal emission and absorption for the CO mixing ratio profile retrieval and at 2.3 and 2.2 mum in the reflected solar region for the total CO column amount and CH(4) column amount retrieval, respectively. To achieve the required measurement accuracy, it is critical to identify and remove cloud contamination in the radiometric signals. We describe an algorithm to detect cloudy pixels, to reconstruct clear column radiance for pixels with partial cloud covers, and to estimate equivalent cloud top height for overcast conditions to allow CO profile retrievals above clouds. The MOPITT channel radiances, as well as the first-guess calculations, are simulated with a fast forward model with input atmospheric profiles from ancillary data sets. The precision of the retrieved CO profiles and total column amounts in cloudy atmospheres is within the expected ?10% range. Validations of the cloud-detecting thresholds with the moderate-resolution imaging spectroradiometer airborne simulator data and MOPITT airborne test radiometer measurements were performed. The validation results showed that the MOPITT cloud detection thresholds work well for scenes covered with more than 5-10% cloud cover if the uncertainties in the model input profiles are less than 2 K for temperature, 10% for water vapor, and 5% for CO and CH(4).     

Retrieval of the Columnar Aerosol Phase Function and Single-Scattering Albedo from Sky Radiance over Land: Simulations

We present a retrieval scheme that can be used to derive the aerosol phase function and single-scattering albedo from the sky radiance over land. The retrieval algorithm iteratively corrects the aerosol volume scattering function, the product of the single-scattering albedo and the phase function, based on the difference between the measured sky radiance and the radiance calculated by solving the radiative transfer equation. It is tested first under ideal conditions, i.e., the approximations made in the retrieval algorithm totally agree with actual conditions assumed in creating the pseudodata for sky radiance. It is then tested under more realistic conditions to assess its susceptibility to measurement errors and effects of conditions not recognized in the retrieval algorithm, e.g., surface horizontal inhomogeneity, departures of the surface from Lambertian, and aerosol horizontal inhomogeneity. These simulations show that, in most cases, this scheme can retrieve the aerosol single-scattering albedo with high accuracy (within 1%) and can therefore be used to identify strongly absorbing aerosols. It can also produce meaningful retrievals of most aerosol phase functions: less than 5% error at 865 nm and less than 10% at 443 nm in most cases. Typically, the error in the volume scattering function is small for scattering angles ?90 degrees , then increases for larger angles. Disappointing results in both the single-scattering albedo and the scattering phase function occur at 443 nm, either when there are large calibration errors in the radiometer used to measure the sky radiance or when the land reflection properties are significantly inhomogeneous.     

Chlorophyll biomass in the global oceans: satellite retrieval using inherent optical properties

In the upper layer of the global ocean, 2082 in situ chlorophyll biomass values (Chl) are retrieved by concurrent satellite-derived inherent optical properties (IOP). It is found that (1) the phytoplankton absorption coefficient IOP alone does not provide satisfactory (Chl) retrieval; (2) the chromophoric dissolved organic matter (CDOM) absorption coefficient IOP must also be used to obtain satisfactory retrieval through (Chl) alpha a ph + pa CDOM where p is a constant and a ph and aCDOM are, respectively, the phytoplankton and CDOM absorption coefficients; (3) the IOP-based (Chl) retrieval performance is comparable to standard satellite reflectance ratio retrievals (that have CDOM absorption intrinsically embedded within them); (4) inclusion of the total backscattering coefficient IOP does not contribute significantly to (Chl) retrieval; and (5) the new IOP-based algorithm may provide the possibility for future research to establish the actual role of extracellular CDOM from all sources in the intracellular production of chlorophyll biomass.     

Vertical versus scanning lidar measurements in a horizontally homogeneous atmosphere

This study compares the aerosol backscatter and extinction coefficients retrieved from vertical elastic and Raman channels with those derived from measurements with multiangle elastic channels. Retrievals from simulated vertical signals at 355 nm, 387 nm, 532 nm, and 607 nm are compared with those from multiangle measurements (at 15 elevation angles) at 355 nm and 532 nm. The atmosphere is considered horizontally homogeneously stratified. For the backscatter coefficient, the Raman backscatter solution and the multiangle solution are considered. For the extinction coefficient, retrievals from the Raman channel and multiangle measurements are compared. The comparison shows that in the presence of horizontal homogeneity, multiangle measurements provide more reliable results, especially for the aerosol extinction coefficient. The uncertainty in the measured signals is considered in an alternative approach to quantify the relative error of the retrieved profiles with respect to the models (linear regression between retrieval and model).     

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

For analytical or semianalytical retrieval of shallow-water bathymetry and/or optical properties of the water column from remote sensing, the contribution to the remotely sensed signal from the water column has to be separated from that of the bottom. The mathematical separation involves three diffuse attenuation coefficients: one for the downwelling irradiance (K(d)), one for the upwelling radiance of the water column (K(u)(C)), and one for the upwelling radiance from bottom reflection (K(u)(B)). Because of the differences in photon origination and path lengths, these three coefficients in general are not equal, although their equality has been assumed in many previous studies. By use of the Hydrolight radiative-transfer numerical model with a particle phase function typical of coastal waters, the remote-sensing reflectance above (R(rs)) and below (r(rs)) the surface is calculated for various combinations of optical properties, bottom albedos, bottom depths, and solar zenith angles. A semianalytical (SA) model for r(rs) of shallow waters is then developed, in which the diffuse attenuation coefficients are explicitly expressed as functions of in-water absorption (a) and backscattering (b(b)). For remote-sensing inversion, parameters connecting R(rs) and r(rs) are also derived. It is found that r(rs) values determined by the SA model agree well with the exact values computed by Hydrolight (~3% error), even for Hydrolight r(rs) values calculated with different particle phase functions. The Hydrolight calculations included b(b)/a values as high as 1.5 to simulate high-turbidity situations that are occasionally found in coastal regions.     

Aerosol observations by lidar in the nocturnal boundary layer

Aerosol observations by lidar in the nocturnal boundary layer (NBL) were performed in Potenza, Southern Italy, from 20 January to 20 February 1997. Measurements during nine winter nights were considered, covering a variety of boundary-layer conditions. The vertical profiles of the aerosol backscattering coefficient at 355 and 723.37 nm were determined through a Klett-modified iterative procedure, assuming the extinction-to-backscattering ratio within the NBL has a constant value. Aerosol average size characteristics were retrieved from almost simultaneous profiles of the aerosol backscattering coefficient at 355 and 723.37 nm, the measurements being consistent with an accumulation mode radius not exceeding 0.4 mum. Similar results in terms of aerosol sizes were obtained from measurements of the extinction-to-backscattering ratio profile at 355 nm performed on six nights during the measurement campaign. Backscattering profiles at 723.37 nm were also converted into profiles of aerosol liquid water content.     

Effects of molecular and particle scatterings on the model parameter for remote-sensing reflectance

For optically deep waters, remote-sensing reflectance (r(rs)) is traditionally expressed as the ratio of the backscattering coefficient (b(b)) to the sum of absorption and backscattering coefficients (a + b(b)) that multiples a model parameter (g, or the so-called f'/Q). Parameter g is further expressed as a function of b(b)/(a + b(b)) (or b(b)/a) to account for its variation that is due to multiple scattering. With such an approach, the same g value will be derived for different a and b(b) values that provide the same ratio. Because g is partially a measure of the angular distribution of upwelling light, and the angular distribution from molecular scattering is quite different from that of particle scattering; g values are expected to vary with different scattering distributions even if the b(b)/a ratios are the same. In this study, after numerically demonstrating the effects of molecular and particle scatterings on the values of g, an innovative r(rs) model is developed. This new model expresses r(rs) in two separate terms: one governed by the phase function of molecular scattering and one governed by the phase function of particle scattering, with a model parameter introduced for each term. In this way the phase function effects from molecular and particle scatterings are explicitly separated and accounted for. This new model provides an analytical tool to understand and quantify the phase-function effects on r(rs), and a platform to calculate r(rs) spectrum quickly and accurately that is required for remote-sensing applications.     

Retrieval of atmospheric profiles from satellite sounder measurements by use of the discrepancy principle

It is known that an infrared or a microwave remote-sensing equation is an integral equation of the first kind. As a result, it is ill-posed, the solution is unstable, and difficulties arise in its retrieval. To make the solution stable, either an a priori error covariance matrix or a smoothing factor gamma is necessary as a constraint. However, if the error covariance matrix is not known or if it is estimated incorrectly, the solution will be suboptimal. The smoothing factor gamma depends greatly on the observations, the observation error, the spectral coverage of channels, and the initial state or the first guess of the atmospheric profile. It is difficult to determine this factor properly during the retrieval procedure, so the factor is usually chosen empirically. We have developed a discrepancy principle (DP) to determine the gamma in an objective way. An approach is formulated for achieving an optimal solution for the atmospheric profile together with the gamma from satellite sounder observations. The DP method was applied to actual Geostationary Operational Environment Satellite (GOES-8) sounder data at the Southern Great Plains Cloud and Radiation Testbed site. Results show that the DP method yields a 21.7% improvement for low-level temperature and a 23.9% improvement for total precipitable water (TPW) retrievals compared with the traditional minimum-information method. The DP method is also compared with the Marquardt-Levenberg algorithm used in current operational GOES data processing. Results of the comparison show significant improvement, 6.5% for TPW and 11% for low-level water-vapor retrievals, in results obtained with the DP method compared with the Marquardt-Levenberg approach.