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.     

Semi-empirical correction algorithm for AC-9 measurements in a coccolithophore bloom

Values for the coefficients of absorption (a) and attenuation (c) obtained from AC-9 measurements in coccolithophore blooms do not provide satisfactory inputs for radiance transfer models. We have therefore modified the standard AC-9 scattering correction algorithm by including an extra term, F(lambda, lambda(r)), which allows for possible wavelength dependence in the scattering phase function. We estimated the magnitude of F(lambda, lambda(r)), which is unity in the standard algorithm, by adjusting the absorption and scattering values in Hydrolight radiance transfer calculations until the depth profiles of downward irradiance (E(d)) and upward radiance (L(u)) matched those measured in situ. The modified algorithm was tested with data from a phytoplankton bloom dominated by the coccolithophore Emiliania huxleyi, which occurred in the western English Channel in May 2001. In this paper, we only have sufficient data to adequately constrain the radiance transfer model in one wave band centered on 488 ma. A single value of F(lambda, lambda(r)) = 1.4 was found to produce satisfactory agreement between modeled and observed profiles at four widely spaced stations within the bloom. Measurements of the ratio of backscattering (b(b)) to total scattering (b) showed significant wavelength dependence at these stations.     

Measured and modeled radiometric quantities in coastal waters: toward a closure

Accurate radiative transfer modeling in the coupled atmosphere-sea system is increasing in importance for the development of advanced remote-sensing applications. Aiming to quantify the uncertainties in the modeling of coastal water radiometric quantities, we performed a closure experiment to intercompare theoretical and experimental data as a function of wavelength lambda and water depth z. Specifically, the study focused on above-water downward irradiance E(d)(lambda, 0+) and in-water spectral profiles of upward nadir radiance L(u)(lambda, z), upward irradiance E(u)(lambda, z), downward irradiance E(d)(lambda, z), the E(u)(lambda, z)/L(u)(lambda, z) ratio (the nadir Q factor), and the E(u)(lambda, z)/E(d)(lambda, z) ratio (the irradiance reflectance). The theoretical data were produced with the finite-element method radiative transfer code ingesting in situ atmospheric and marine inherent optical properties. The experimental data were taken from a comprehensive coastal shallow-water data set collected in the northern Adriatic Sea. Under various measurement conditions, differences between theoretical and experimental data for the above-water E(d)(lambda, 0+) and subsurface E(d)(lambda, 0-) as well as for the in-water profiles of the nadir Q factor were generally less than 15%. In contrast, the in-water profiles of L(u)(lambda, z), E(d)(lambda, z), E(u)(lambda, z) and of the irradiance reflectance exhibited larger differences [to approximately 60% for L(u)(lambda, z) and E(u)(lambda, z), 30% for E(d)(lambda, z), and 50% for the irradiance reflectance]. These differences showed a high sensitivity to experimental uncertainties in a few input quantities used for the simulations: the seawater absorption coefficient; the hydrosol phase function backscattering probability; and, mainly for clear water, the bottom reflectance.     

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.     

Ocean inherent optical property determination from in-water light field measurements

An algorithm is described and evaluated for determining the absorption and backscattering coefficients a(z) and bb(z) from measurements of the nadir-viewing radiance Lu(z) and downward irradiance Ed(z). The method, derived from radiative transfer theory, is similar to a previously proposed one for Eu(z) and Ed(z)and both methods are demonstrated with numerical simulations and field data. Numerical simulations and a sensitivity analysis show that good estimates of a(z) and bb(z) can be obtained if the assumed scattering phase function is approximately correct. In an experiment in Long Island Sound, estimates of a(z) derived with these methods agreed well with those obtained from an in situ reflecting tube instrument.     

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.     

Estimation of the inherent optical properties of natural waters from the irradiance attenuation coefficient and reflectance in the presence of Raman scattering

By means of radiative transfer simulations we developed a model for estimating the absorption a, the scattering b, and the backscattering b(b) coefficients in the upper ocean from irradiance reflectance just beneath the sea surface, R(0-), and the average attenuation coefficient for downwelling irradiance, 1, between the surface and the first attenuation depth. The model accounts for Raman scattering by water, and it does not require any assumption about the spectral shapes of a, b, and b(b). The best estimations are obtained for a and b(b) in the blue and green spectral regions, where errors of a few percent to <10% are expected over a broad range of chlorophyll concentration in water. The model is useful for satellite ocean color applications because the model input, R(0-) and 1, can be retrieved from remote sensing and the model output, a and b(b), is the major determinant of remote-sensing reflectance.     

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.     

Toward closure of upwelling radiance in coastal waters

We present three methods for deriving water-leaving radiance L(w)(lambda) and remote-sensing reflectance using a hyperspectral tethered spectral radiometer buoy (HyperTSRB), profiled spectroradiometers, and Hydrolight simulations. Average agreement for 53 comparisons between HyperTSRB and spectroradiometric determinations of L(w)(lambda) was 26%, 13%, and 17% at blue, green, and red wavelengths, respectively. Comparisons of HyperTSRB (and spectroradiometric) L(w)(lambda) with Hydrolight simulations yielded percent differences of 17% (18%), 17% (18%), and 13% (20%) for blue, green, and red wavelengths, respectively. The differences can be accounted for by uncertainties in model assumptions and model input data (chlorophyll fluorescence quantum efficiency and the spectral chlorophyll-specific absorption coefficient for the red wavelengths, and scattering corrections for input ac-9 absorption data and volume scattering function measurements for blue wavelengths) as well as radiance measurement inaccuracies [largely differences in the depth of the L(u)(lambda, z) sensor on the HyperTSRB].     

Mueller matrices and information derived from linear polarization lidar measurements: theory

A Mueller matrix M is developed for a single-scattering process such that G(theta, phi) = T (phi(a))M T (phi(p))u, where u is the incident irradiance Stokes vector transmitted through a linear polarizer at azimuthal angle phi(p), with transmission Mueller matrix T (phi(p)), and G(theta, phi) is the polarized irradiance Stokes vector measured by a detector with a field of view F, placed after an analyzer with transmission Mueller matrix T (phi(a)) at angle phi(a). The Mueller matrix M is a function of the Mueller matrix S (theta) of the scattering medium, the scattering angle (theta, phi), and the detector field of view F. The Mueller matrixM is derived for backscattering and forward scattering, along with equations for the detector polarized irradiance measurements (e.g., cross polarization and copolarization) and the depolarization ratio. The information that can be derived from the Mueller matrix M on the scattering Mueller matrixS (theta) is limited because the detector integrates the cone of incoming radiance over a range of azimuths of 2pi for forward scattering and backscattering. However, all nine Mueller matrix elements that affect linearly polarized radiation can be derived if a spatial filter in the form of a pie-slice slit is placed in the focal plane of the detector and azimuthally dependent polarized measurements and azimuthally integrated polarized measurements are combined.     

Algorithms for ocean-bottom albedo determination from in-water natural-light measurements

A method for determining the ocean-bottom optical albedo R(b) from in-water upward and downward irradiance measurements at a shallow site is presented, tested, and compared with a more familiar approach that requires additional measurements at a nearby deep-water site. Also presented are two new algorithms for estimating R(b) from measurements of the downward irradiance and vertically upward radiance. All methods performed well in numerical situations at depths at which the influence of the bottom on the light field was significant.     

Absorption and backscattering coefficients and their relations to water constituents of Poyang Lake, China

The measurement and analysis of inherent optical properties (IOPs) of the main water constituents are necessary for remote-sensing-based water quality estimation and other ecological studies of lakes. This study aimed to measure and analyze the absorption and backscattering coefficients of the main water constituents and, further, to analyze their relations to the water constituent concentrations in Poyang Lake, China. The concentrations and the absorption and backscattering coefficients of the main water constituents at 47 sampling sites were measured and analyzed as follows. (1) The concentrations of chlorophyll a (C(CHL)), dissolved organic carbon (C(DOC)), suspended particulate matter (C(SPM)), including suspended particulate inorganic matter (C(SPIM)) and suspended particulate organic matter (C(SPOM)), and the absorption coefficients of total particulate (a(p)), phytoplankton (a(ph)), nonpigment particulate (a(d)), and colored/chromophoric dissolved organic matter (a(g)) were measured in the laboratory. (2) The total backscattering coefficients, including the contribution of pure water at six wavelengths of 420, 442, 470, 510, 590, and 700 nm, were measured in the field with a HydroScat-6 backscattering sensor. (3) The backscattering coefficients without the contribution of pure water (b(b)) were then derived by subtracting the backscattering coefficients of pure water from the total backscattering coefficients. (4) The C(CHL), C(SPM), C(SPIM), C(SPOM), and C(DOC) of the 41 remaining water samples were statistically described and their correlations were analyzed. (5) The a(ph), a(d), a(p), a(g), and b(b) were visualized and analyzed, and their relations to C(CHL), C(SPM), C(SPIM), C(SPOM), or C(DOC) were studied. Results showed the following. (1) Poyang Lake was a suspended particulate inorganic matter dominant lake with low phytoplankton concentration. (2) One salient a(ph) absorption peak was found at 678 nm, and it explained 72% of the variation of C(CHL). (3) The a(d) and a(p) exponentially decreased with increasing wavelength, and they explained 74% of the variation of C(SPIM) and 71% variation of C(SPM), respectively, at a wavelength of 440 nm. (4) The a(g) also exponentially decreased with increasing wavelength, and it had no significant correlation to C(DOC) at a significance level of 0.05. (5) The b(b) decreased with increasing wavelength, and it had strong and positive correlations to C(SPM), C(SPIM) and C(SPOM), a strong and negative correlation to C(CHL), and no correlation to C(DOC) at a significance level of 0.05. Such results will be helpful for the understanding of the IOPs of Poyang Lake. They, however, only represented the IOPs during the sampling time period, and more measurements and analyses in different seasons need to be carried out in the future to ensure a comprehensive understanding of the IOPs of Poyang Lake.     

Comparison of the ocean inherent optical properties obtained from measurements and inverse modeling

A model developed recently by Loisel and Stramski [Appl. Opt. 39, 3001-3011 (2000)] for estimating the spectral absorption a(lambda), scattering b(lambda), and backscattering b(b)(lambda) coefficients in the upper ocean from the irradiance reflectance just beneath the sea surface R(lambda, z = 0(-)) and the diffuse attenuation of downwelling irradiance within the surface layer ?K(d)(lambda)?(1) is compared with measurements. Field data for this comparison were collected in different areas including off-shore and near-shore waters off southern California and around Europe. The a(lambda) and b(b)(lambda) values predicted by the model in the blue-green spectral region show generally good agreement with measurements that covered a broad range of conditions from clear oligotrophic waters to turbid coastal waters affected by river discharge. The agreement is still good if the model estimates of a(lambda) and b(b)(lambda) are based on R(lambda, z = 0(-)) used as the only input to the model available from measurements [as opposed to both R(lambda, z = 0(-)) and ?K(d)(lambda)?(1) being measured]. This particular mode of operation of the model is relevant to ocean-color remote-sensing applications. In contrast to a(lambda) and b(b)(lambda) the comparison between the modeled and the measured b(lambda) shows large discrepancies. These discrepancies are most likely attributable to significant variations in the scattering phase function of suspended particulate matter, which were not included in the development of the model.     

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.     

Vertical attenuation coefficient of photosynthetically active radiation as a function of chlorophyll concentration and depth in case 1 waters

Based on empirical relations found in the literature, relatively simple mathematical models of the average of the total absorption of seawater [a(T(z,Chl))], the chlorophyll-specific absorption coefficient of phytoplankton [a*(ph(z,Chl))], and the backscattering coefficient [b(b(Chl))], weighted by the in situ spectral distribution of photosynthetically active scalar irradiance (PAR), as functions of chlorophyll concentration and depth, were developed. The models for a(T(z,Chl)) and b(b(Chl)) can be used to calculate the coefficient of vertical attenuation of PAR [K(o(z,Chl))] and therefore to estimate the vertical profile of PAR as an input to algorithms for primary production. One application of a*(ph(z,Chl)) may be in the correction of the initial slope of the photosynthesis-irradiance curve [alpha*((z))] for the in situ spectral distribution of PAR and the package effect. Also, a*(ph(z,Chl)) may be used to calculate phi((z)), the in situ quantum yield of photosynthesis, from phi(max) and irradiance.     

Spectral parameters of inherent optical property models: method for satellite retrieval by matrix inversion of an oceanic radiance model

Inherent optical property (IOP) spectral models for the phytoplankton absorption coefficient, chromophoric dissolved organic matter (CDOM) absorption coefficient, and total constituent backscattering (TCB) coefficient are linear in the reference wavelength IOP and nonlinear in the spectral parameters. For example, the CDOM absorption coefficient IOP a(CDOM)(lambda(i)) = a(CDOM)(lambda(ref))exp[-S(lambda(i)- lambda(ref))] is linear in a(CDOM)(lambda(ref)) and nonlinear in S. Upon linearization by Taylor's series expansion, it is shown that spectral model parameters, such as S, can be concurrently accommodated within the same conventional linear matrix formalism used to retrieve the reference wavelength IOP's. Iteration is used to adjust for errors caused by truncation of the Taylor's series expansion. Employing an iterative linear matrix inversion of a water-leaving radiance model, computer simulations using synthetic data suggest that (a) no instabilities or singularities are introduced by the linearization and subsequent matrix inversion procedures, (b) convergence to the correct value can be expected only if starting values for a model parameter are within certain specific ranges, (c) accurate retrievals of the CDOM slope S (or the phytoplankton Gaussian width g) are generally reached in 3-20 iterations, (d) iterative retrieval of the exponent n of the TCB wavelength ratio spectral model is not recommended because the starting values must be within approximately +/-5% of the correct value to achieve accurate convergence, and (e) concurrent retrieval of S and g (simultaneously with the phytoplankton, CDOM, and TCB coefficient IOP's) can be accomplished in a 5 x 5 iterative matrix inversion if the starting values for S and g are carefully chosen to be slightly higher than the expected final retrieved values.     

Evidence for wavelength dependence of the scattering phase function and its implication for modeling radiance transfer in shelf seas

More than 90% of stations from the Irish and Celtic Seas are found to have significantly higher back-scattering ratios in the blue (470 nm) than in the red (676 nm) wave band. Attempts to obtain optical closure by use of radiance transfer modeling were least successful for stations at which backscattering ratios are most strongly wavelength dependent. Significantly improved radiance transfer simulation results were obtained with a modified scattering correction algorithm for AC-9 absorption measurements that took wavelength dependency in the scattering phase function into account.     

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.     

An inherent-optical-property-centered approach to correct the angular effects in water-leaving radiance

Remote-sensing reflectance (R(rs)), which is defined as the ratio of water-leaving radiance (L(w)) to downwelling irradiance just above the surface (E(d)(0?)), varies with both water constituents (including bottom properties of optically-shallow waters) and angular geometry. L(w) is commonly measured in the field or by satellite sensors at convenient angles, while E(d)(0?) can be measured in the field or estimated based on atmospheric properties. To isolate the variations of R(rs) (or L(w)) resulting from a change of water constituents, the angular effects of R(rs) (or L(w)) need to be removed. This is also a necessity for the calibration and validation of satellite ocean color measurements. To reach this objective, for optically-deep waters where bottom contribution is negligible, we present a system centered on water's inherent optical properties (IOPs). It can be used to derive IOPs from angular R_rs and offers an alternative to the system centered on the concentration of chlorophyll. This system is applicable to oceanic and coastal waters as well as to multiband and hyperspectral sensors. This IOP-centered system is applied to both numerically simulated data and in situ measurements to test and evaluate its performance. The good results obtained suggest that the system can be applied to angular R(rs) to retrieve IOPs and to remove the angular variation of R(rs).