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The mean upward-scattering coefficient of the downward-traveling photons and the mean downward-scattering coefficient of the upward-traveling photons are two factors needed for the two-stream approximation to the radiative-transfer equation. Numerical values of each shape factor just beneath the surface and at asymptotic depths give an indication of the range of values at intermediate depths in spatially uniform waters with no sources and are used to obtain an approximate depth-dependent model for each shape factor. The shape factors are computed for different surface-illumination conditions, wavelengths, and chlorophyll concentrations.

The importance of multiple scattering for lidar detection of a spherical object obscured by an aerosol is assessed using Monte Carlo radiative transfer calculations. Multiple scattering correction factors are significant and depend on the location and size of the object, and the field of view and time resolution of the detector.

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.

The design of integrating cavity absorption meters of general geometry is analyzed for cases in which the incident illumination of the cavity is spatially uniform and isotropic, such as the meter of Fry et al. [Appl. Opt. 31, 2055 (1992)]. The analysis by Kirk [Appl. Opt. 34, 4397 (1995)] for the probability of photon survival in a spherical meter is extended to general geometries. An estimate of the effect of the shape of the cavity on the estimated absorption coefficient is given.

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.

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.

The approach of ocean optical radiance to an approximate asymptotic dependence with increasing depth in spatially uniform waters is numerically examined for a variety of sea surface illumination conditions.

The depth dependence for which the downward diffuse attenuation coefficient, the upward-to-downward plane irradiance ratio, the vertically upward radiance-to-downward plane irradiance ratio, and the mean cosine of the radiance depend negligibly on the surface incident illumination have been examined. The depths at which these coefficients approach to within a specified percent of their asymptotic values depends significantly on the characteristics of the incident illumination and on the inherent optical properties of the water. This information is useful when solving inverse ocean optics problems with a method for which the radiance is assumed to be approximately in the asymptotic regime.

A discrete ordinates code is developed with which to compute the beam spread function (BSF) without invoking the small-angle scattering approximation or performing Monte Carlo calculations. The computed BSF is used to predict the response of a detector versus its distance to the origin of a highly collimated beam, its angle with respect to the beam, and the two local angles that specify the detector orientation. Numerical results have been obtained for water models that simulate a clear ocean, a coastal ocean, and a turbid harbor. Six orders of magnitude or more change in the detector response caused by scattered photons can be predicted for different detector locations while simultaneously obtaining small changes for different detector orientations. This capability is useful for assessment of the sensitivity of the detector response to the interpretation of time-independent underwater imaging systems or visibility models.

Equations are obtained for determining the Legendre polynomial expansion coefficients of the phase function from angular or angular-spatial integrals of the radiance measured either in water or at the surface.

We propose and test an inverse ocean optics procedure with numerically simulated data for the determination of inherent optical properties using in-water radiance measurements. If data are available at only one depth within a deep homogeneous water layer, then the single-scattering albedo and the single parameter that characterizes the Henyey-Greenstein phase function can be estimated. If data are available at two depths, then these two parameters can be determined along with the optical thickness so that the absorption and scattering coefficients, and also the backscattering coefficient, can be estimated. With a knowledge of these parameters, the albedo and Lambertian fraction of reflected radiance of the bottom can be determined if measurements are made close to the bottom. A simplified method for determining the optical properties of the water also is developed for only three irradiance-type measurements if the radiance is approximately in the asymptotic regime.

Plane-to-point transformations are used to develop a version of the Hydrolight computer program with which to compute the spatial dependence of the irradiance and the scalar irradiance of the light field away from an isotropic point source deep within a spatially uniform ocean. The transformations are also used to derive analytic approximations for determining the diffuse attenuation coefficient and the mean cosine of the radiance far from an isotropic point source. Approximations for determining the asymptotic diffuse attenuation coefficient from measurements at only two distances far from the source are derived and numerically tested with the modified version of the Hydrolight computer program. New spatial integrals of the outward irradiance are also derived that provide a different way for correlating the inherent optical properties of seawater.

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.

A method was developed for determining the spatial distribution of a source from downward and upward irradiance measurements at a single wavelength in seawater of known optical properties. The algorithm uses measurements at two depths located an arbitrary distance apart and solves two nonlinear equations for two parameters that fit a globally exponential or linear source shape. Complex spatially dependent source shapes can be estimated from an irradiance profile by piecing together estimates from neighboring measurement pairs. Numerical tests illustrate the sensitivity of the algorithm to depth, measurement spacing, chlorophyll concentration, sensor noise, and uncertainty in the a priori assumed inherent optical properties. The algorithm works well with widely spaced measurements, moderate sensor noise, and uncertainties in the optical properties regardless of whether the assumed and true profiles are the same shape.

The Zaneveld-Wells algorithm for calculating N inherent optical expansion coefficients from N + 1 measured angle-integrated moments of the radiant light field is investigated. Because the algorithm is well conditioned but sensitive to errors in the spatial derivatives, different approximations for the spatial derivatives are considered. The effects of noise and sensor error on the performance of the algorithm have been evaluated analytically, and testing with randomly sampled simulated noise was performed to assess the stability and sensitivity of the algorithm. Results show that the algorithm is fairly insensitive to sensor noise, but neither using a higher-order finite-difference approximation for the derivatives nor reformulating the algorithm into an integral form was successful in overcoming the large errors observed.

The local diffuse attenuation coefficients for monochromatic net irradiance and monochromatic scalar irradiance differ from the corresponding spatially averaged coefficients except when both equal an eigenvalue of the radiative transfer equation. Explicit equations to relate spatially averaged to local coefficients that may be useful when inferring self-consistency (i.e., closure) of optical properties are given.

We solve an inverse problem of ocean optics for estimating spatially dependent absorption and scattering coefficients and for determining sources such as fluorescence, bioluminescence, or Raman scattering. The solution requires in situ measurement of the downward and upward plane irradiances and scalar irradiances and a priori estimation of the angular shape of the volume scattering function. Both an explicit algorithm and an implicit one are developed from new two-stream radiative-transfer equations that utilize an asymptotic radiance approximation to close the set of equations. A comparison of numerical tests for the two algorithms is given.

An algorithm is developed for estimating the spatial location and magnitude of a bioluminescent radiation source from measurements of the in situ irradiance and scalar irradiance at two depths. The algorithm is based on the principle of photon conservation. The most direct application of the algorithm requires that the absorption coefficient be known, but the algorithm is useful even if that coefficient is unknown. Numerical tests and an error analysis have been done to test the algorithm numerically. In addition we show that if the estimated source magnitude is nearly constant, that value can be used to estimate the vertical attenuation coefficient of the radiation field.

Methods of using cw and pulsed electromagnetic sources are compared for determination of the coefficients of the Legendre polynomial expansion of the single-scattering phase function from measurements of the multiply scattered radiance emerging from the surfaces of the atmosphere. Quadrature formulas are presented with which to calculate the angular moments of the radiance needed for the methods.

Numerical calculations for estimating the albedo of a surface and its distance behind an obscuring multiple-atmosphere are performed for the idealized case of an instantaneous pulse uniformly illuminating a scattering plane-geometry atmosphere of known properties. This inverse problem is complicated by the fact that the backscattered irradiance is broadened in time because of purely geometric effects arising from the geometry considered; the broadening increases with an increasing thickness of the obscuring medium and if the medium scatters in a more isotropic manner, such as with Rayleigh scattering. Isocline maps of different observables suggest that graphical inversion maps might be useful for some applications of this type.

A simple algorithm that solves the inverse problem is numerically tested for estimating the ratio of the absorption to the transport coefficients from measurements of the irradiance and scalar irradiance at depths sufficiently beneath the surface of seawater. Results of radiative transfer calculations with a phase function for San Diego harbor seawater were used to test the algorithm and show that the ratio is quite insensitive to the depth of the measurements and to the direction of incident solar radiation unless the absorption is very weak. A second algorithm, based on the assumption that the radiance in the diffusion regime is proportional to a (one-parameter) Henyey-Greenstein shape, is examined for estimating the ratio of the backward scattering to absorption coefficients from only irradiance measurements. The algorithm has the advantage that no scalar irradiance measurements are required, but it provides significantly poorer estimates because the radiance is not sufficiently close to the Henyey-Greenst in shape.