Electromagnetic Scattering from Foliage and Vegetation:
Modelling and applications in Active Microwave Remote Sensing
4. Monte Carlo Modesl
The Monte Carlo technique is statistical method of solving mathematical and physical problems by simulation of random quantities. It is particularly suitable for application to a class of physical problems that requires precise calculation of energy transport, taking into account a detailed radiation model of the medium and the multiple scattering processes and energy transfer in the medium. In this method, the multiple scattering processes in the random medium are treated as chains of collisions between the photons in the incident beam and the discrete scatterers. At each scattering event, a probability function in assigned to the following possibilities:
-
The probability that a photon is scattered or absorbed,
- The probability that photon encounters a particular type of scatterer in the midst of various types of scatterers.
- The probability that the orientation of the scatter is described by the Eulerian angles,
- The probability that a photon incident from direction W' is scattered into a new direction W.
The process is described the following equation:
Where In is the Stockes' vector of a particular photon that has arrived at the receiver R in the direction W after undergoing n collisions. P, W are the phase matrix describing the scatterer and the combined biasing probability matrix of the collision.
By using sufficient number of photons, an ensemble average of In can be obtained. Effective Attenuation coefficient Kq in the random medium can be calculated using the definition,
where Ioq and Iq are the incident and the transmitted intensity with q polarization, respectively, and d the distance traveled.
Similarly the backscatter coefficient s pq can be obtained using the definition:
Figure 5 shows the comparison of vertically polarized normalized cross-sections for different cylinder radii at 9 GHz. The Monte Carlo model is able to give values that match well with experimental data, both in trend and in value. Figure 6 contains the comparison of backscatter coefficient from Monte Carlo model and measurement from Japanese Cypress. Close agreement is obtained for both cases: one with leaves and one where the leaves are removed.
Fig 5 : Comparison between theory and measurements of normalised cross-section from a layer of
vertical stems with different cylinder radii
Fig 6(a) : Modelling of a forest canopy as a multilayer multiconstituent random medium
Fig 6(b) : Comparison of Monte Carlo calculations with measurements from Japanese Cypress trees (with leaves) at 2,75 GHz
Fig 6(c) : Comparison of Monte Carlo calculations with measurements from Japanese Cypress trees (without leaves) at 2,75 GHz
5. Conclusions
This paper gives a review of some of the modeling work done in electromagnetic scattering and propagation in foliage and vegetation, as well as application examples in active remote sensing. The work is by no means complete. Many other issues should be researched into, such as a better formulation of phase matrix to describe vegetative component, the near-field effect of vegetative components which appear physically to be close to each other, and the overall structure and shape of a tree. These studies would shed light and to enhance our understanding of wave-mater interaction.
References
-
M.A. Karam, A.K. Fung, R.H. Lang and N.S. Chauhan, "A Microwave Scattering Model for Layered Vegetation", IEEE Trans. Geosci. Rem. Sens., vol. 30, no. 4, pp. 767-784, 1992.
- H.T. Chuah and H.S. Tan, "A High Order Renormalization method for Radar Backscatter from a Random Medium", IEEE Trans. Geosci, Rem. Sens., Vol. 27, no. 1, pp. 79-85, 1989.
- H.T. Chuah and H.S. Tan, "A Radiative Transfer Backscatter Model for a Vegetative layer of a Mixture of Discrete Scatterers", Journal of Electromagnetic Waves and Applications, vol. 4, no. 3, pp. 247-268, 19990.
- H.T. Chuah and H.s. Tan, "A Radar Backscatter Model for A Forest Stand", Waves in Random Media, vol. 2, pp. 7-28, 1992.
