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Solving the radiative transfer equation (RTE) for the ocean or atmosphere is a linear two-point boundary value problem. That is to say, there are boundary conditions describing the radiance at the top and bottom of the atmosphere or ocean, and the propagation of radiance within the medium — between the boundaries — is governed by the integro-differential RTE, which is linear in the radiance. The essence of the invariant imbedding solution technique is that it converts the linear, integro-differential, two-point boundary value problem into a pair of non-linear, initial value problems formulated in terms of ordinary differential equations (ODEs).
As applied in HydroLight, for example, the upper boundary condition specifies the sky radiance incident onto the sea surface. The lower boundary condition specifies how the sea bottom (or infinitely deep layer of water below the deepest depth of interest) reflects the downwelling radiance.
To develop the invariant imbedding equations, the radiance and scattering phase function are azimuthally decomposed into Fourier amplitudes, and the RTE is split into separate equations for upwelling and downwelling radiances. One can then obtain a set of non-linear (Riccati) ODEs (for example, http://mathworld.wolfram.com/RiccatiDifferentialEquation.html) for transmittance and reflectance operators, which themselves depend on the absorption and scattering properties of the medium. The Riccati ODEs for the transmittance and reflectance operators are then solved as a pair of initial value problems. One set of ODEs is integrated downward from the surface, using a unit-input surface boundary condition to initialize the solution, and the other set is integrated upward from the bottom using the bottom boundary condition. The desired radiances are finally obtained from the transmittance and reflectance operators (which are functions of depth, direction, and wavelength), the actual boundary conditions, and a Fourier synthesis of the radiance amplitudes. The complicated mathematics of this entire process as applied in HydroLight is described by Mobley CD 1994 (Chapter 7). Preisendorfer 1976 (especially Vol. IV) discusses the general mathematics of invariant imbedding theory and the formulation of the RTE in terms of local and global transmittance and reflectance operators.
One virtue of invariant imbedding is that the numerical solution of the Riccati ODEs is extremely fast compared to the direct solution of the RTE by methods such as Monte Carlo simulation. The Ricatti ODEs also account for all orders of multiple scattering within the medium, and the solution time depends linearly on the optical depth of the medium. The optical properties of the medium can vary arbitrarily with depth. The price paid for these numerical advantages is that invariant imbedding is restricted to problems with only one spatial dimension. In the ocean this dimension is the depth in the water column. For a planetary atmosphere, the single spatial dimension could be the radial distance from the planet’s surface. Thus invariant imbedding (e.g., HydroLight) cannot solve the RTE for environments in which the optical properties of the medium vary spatially in three dimensions, or for which the boundary conditions depend on location (e.g., a sloping ocean bottom). Inherently 3D RTE problems are usually solved by Monte Carlo techniques, which are completely general but computationally slow.
See also RTE solution, methods, numerical, invariant imbedding and RTE solution for additional references.
| CITATION: Mobley C. D. 2008. Invariant imbedding method (www.tpdsci.com/Tpc/RTESolNumIi.php). In: Radiative transfer equation (RTE): Numerical solution methods. Kokhanovsky A. A. (ed.) (www.tpdsci.com/Tpc/RTESolNumIntro.php). Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 2008 Modified: 25-Aug-2008 Peer-reviewed: 06-Aug-2008 |
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