Chapter 4 Numerical models of radiative transfer Remote Sensing of Ocean Color Instructor: Dr. Cheng-Chien LiuCheng-Chien Liu Department of Earth Science National Cheng-Kung University Last updated: 13 March 2003

4.1 Monte Carlo method  Origin World War II Atomic bomb Neutron diffusion problems Fermi, von Neumann and Ulam  Fundamental concept if we know the probability of occurrence of each separate event in a sequence of events, then we can determine the probability that the entire sequence of events will occur

4.1 Monte Carlo method (cont.)  Applications Extensive application of this method can be found in many different areas  Ocean optics The earliest work: Plass and Kattawar (1972) The basic principle:  simulate a beam of light by a very large number of photons. Following the path of each photon, we can use a series of random numbers to determine the photon’s life history according to different probabilities for different phenomena. The final light field is the cumulative contribution of total photons.  Processes of radiative transfer (Fig 4.1.1)

Fig 4.1.1

4.1 Monte Carlo method (cont.)  Example 1 (model-to-model comparison): Source code:  Description of the problem (detailed description of this example can be referred to §11.1 (Mobley 1994))  Optically homogeneous water  High albedo:  0 = 0.9  Averaged VSF  The sea surface is level  One unit of solar irradiance at zenith angle  s =60 0, and the sky is otherwise black  Calculate the radiance distribution at three selected optical depths  = 0 +, 5 m and 20 m, in the plane of the Sun

4.1 Monte Carlo method (cont.)  Example 1 (cont.): Functions:  main(int argc,char **argv)  Setup(void)  New_Photon(void)  Air_Sea_Up(void)  Trace_Down(void)  Trace_Up(void)  Scattering(void)  Output(void)  Result(void)  Output_Rad(char fname[],int MM)  RecordRad(int II,double SS,int JJ)  RTF2XYZ(double RR, double TT, double FF, double PP[])  Absorb(void)

Z X vv Fig Schematic description of the physical problem Fig 4.1.2

4.1 Monte Carlo method (cont.)  Example 1 (cont.) Results:  No. of Photon = 600  No. of Photon =  No. of Photon =

Fig Fig Model-to-model comparison. Compare to Fig 11.1 in (Mobley 1994)

4.1 Monte Carlo method (cont.)  Example 2 (model-to-data comparison): Source code:  Description of the problem (detailed description of this example can be referred to §11.1 (Mobley 1994))  Optically homogeneous water  High albedo:  0 = 0.7 (a=0.012, b=0.028)  Averaged VSF  The sea surface is level  One unit of solar irradiance at zenith angle  s =38 0, and the sky irradiance is distributed by E=0.1  E sky +0.9  E Sun  Calculate the radiance distribution at three selected depths z = 4.2 m, 29.0 m and 66.1 m, in the plane of the Sun

Fig Fig Model-to-data comparison

4.1 Monte Carlo method (cont.)  Accelerating Monte-Carlo calculations Backward ray tracing  reciprocity relation Variance reduction techniques

Fig Illustration of the original (forward) and adjoint (time-reversed) problems used to develop backward Monte-Carlo methods, reprinted from (Mobley 1994).

4.1 Monte Carlo method (cont.)  Example 3: 3D Light Description of the problem (detailed description of this example can be referred to (Carder et al. 2003)  Optically homogeneous water  Bottom reflectance  Ramp angle and height  Water depth  GC+KC+HC sky radiance distribution

Fig Fig Representation of a shallow bottom with sand waves and the associated coordinates; (B) Illustration of optical pathways that contribute to the sensor-detected radiance and the geometrical specifications of bottom, reprinted from Carder et al. (2003)

Fig Fig

4.1 Monte Carlo method (cont.)  Pros and Cons of MC method Clear & Direct 3-D problem Simulation of R rs Remote sensing application Computer time consuming Noise

4.2 Invariant imbedding method  Origin Chandrasekhar (1943) Mobley & Preisendorfer (1989)  Hydrolight (

4.2 Invariant imbedding method (cont.)  History of Hydrolight Natural Hydrosol V1.0 (1979 – 1988) (Preisendorfer and Mobley) Natural Hydrosol V2.1 (1992) Hydrolight V3.0 (1995) (US Office of Naval Research funded) Hydrolight V4.0 (1998) (Sequoia, $10,000) Hydrolight V4.2 (2001)

4.2 Invariant imbedding method (cont.)  Using the invariant imbedding technique to solve the RTE Governing equation (RTE) Air-water surface boundary conditions Bottom boundary conditions Invariant Imbedding: solving the RTE

Fig Illustration of the classic radiative transfer equation

Application in marine optics

dimensionless form of RTE

quad-averaged form of RTE

spectral decomposition of RTE discrete orthogonality relations Fourier polynomial analysis

Matrix form of RTE

4.2 Invariant imbedding method (cont.)  Rerun the example 2 using Hydrolight  Applications of Hydrolight See the user guide of Hydrolight v4.2