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when the number of unknowns increases. For the surfaces with the same size, the CPU time increases with h because of the slow convergence of the CGM for rougher surfaces. For example, it only needs 2 iterations (n = 1) with 40 CG terms in the first iteration and 1 CG term in the second iteration for the 8 x 8>'~ surface with h = 0.05>'0' In contrast, it needs 3 iterations (n = 2) with 100 CG terms in the first iteration, 41 CG terms in the second iteration and 4 CG terms in the last iteration when h increases to 0.3>'0' The speedup factors become large when the CPU time increases. It is shown that the best speedup factor is around 13.7 for the PEC surface with 32,768 surface unknowns and h = 0.1>.0' and 13.4 for the lossy dielectric surface with 0.4 million surface unknowns for the same h. For the PEC surfaces, CPU times of the parallel code are obtained when the strong matrix is computed once

Total time Communication time Near interaction calculation time Far interaction calculation time Emissivity calculation time

0.05'\0'

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and stored When the number of unknowns exceeds 131,000, we recalculate the strong matrix in the sequential code when necessary In the last two rows of Table 661, we give the comparison of CPU time for one CG term when the surface sizes are 32 x 32 and 64 x 64>'~, respectively The strong matrix is recalculated in the sequential code One can see that the speedup factor can be up to 1156 for 1 CG term To complete the conjugate gradient solution of sequential code will be prohibitive long For dielectric surfaces, when the number of unknowns exceeds 15 millions, the code on a single processor cannot be run and therefore parallel computation is the only alternative In the 15 million unknown example, the computer code requires about 78% of the 40 GBytes of RAM.

4

We should emphasize that the CPU time comparisons shown in Table 661 and 662 are based on the collocation method For the lossy dielectric surface, it is found that near-field integration is necessary Although this near-field integration requires additional CPU time, the resulting matrix is better conditioned so that the number of CG terms may be reduced For examples, with near-field integration, the CPU time for the 8 x 8 and 16 x 16>'~ surface with h = 005>'0 reduces from 5,051 to 4,585 and 20,474 to 19,085 seconds, respectively It is expected that similar speed-up factors are obtained when replacing the collocation method with the nearfield integration Table 663 shows the CPU time distribution for a 8 x 8>'~ surface with near-field integration, where h = 005>'0' The communication time is only a small part of the total CPU time.

Most of the CPU time is spent on calculating the sparse matrix and sparse-matrix-vector multiplication Communication time only attributes to 26% of the total CPU time while that of the far interaction calculation based on FFT is 63% Table 664 shows the comparison of the computed brightness temperature versus surface size and rms height, and the comparison between the results obtained with and without the near-field integration The incident angle for the 8 x 8>'~ surfaces is 30 while that of the remaining ones is 50 The result of an infinite flat surface with the same permittivity is also given for reference It should be pointed out that the brightness temperatures of the rough surfaces are obtained from one realization Different realizations.

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Use regions to organize the structure of your code By de ning a standard set of regions, you can more easily work with code written by any member of your coding team And by expanding and collapsing regions, you can more easily focus on one set of code at a time

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