The small slope approximation (SSA) is an emission theory based
on an expansion in surface slope. It turns out that this theory
results in expressions identical to those of the small perturbation
method. A second order SSA method is therefore identical to
the two scale theory when no "tilting" is applied, except that
the "cutoff" wavenumber can be extended to include contributions
from long scale waves. Because the theory results in an integral
over the surface directional spectrum, it is easy to compute
emission zeroth and second azimuthal harmonics directly without
performing an azimuth sweep, as described in the second reference
listed below.

Note no first azimuthal harmonics can be captured by the second
order theory; a third order theory is necessary for this purpose.
Third order theories have been described in the two most recent
references listed below. No third order contributions are included
in the result files.

Input parameters to the code at 19.35 GHz were as follows:

Frequency                             19.35 GHz
Sea Water Dielectric Constant         29.04+i35.55 
Surface temperature                   285 K
Friction Velocity u                   0.312547 m/s
Resulting U_19.5                      9 m/s
Incidence angle Theta                 55 degrees
Longest sea wave in integration       10,000 m
Shortest sea wave in integration      0.1 mm

Value of spectral parameter          0.503723923 
D obtained by code

At 37 GHz parameters were the same as above, except

Frequency                             37 GHz
Sea Water Dielectric Constant         14.34+i24.17

Results are obtained in terms of zeroth and second
azimuthal harmonics of the four polarimetric quantities.
The files contain these data in the format:

u19.5 (m/s)|Theta (deg)|Th0(K)|Th2(K)|Tv(K)|Tv2(K)|U0(K)|U2(K)|V0(K)|V2(K)|

Integrations were performed using 16000 points in the
length direction, 64 points in the azimuth direction, except
in the "critical phenomenon" region where 16384 azimuth
points were employed. 

Useful references on the SSA emission theory are:

Irisov, V. G., ``Small-slope expansion for thermal
and reflected radiation from a rough surface,'' 
Waves in Random Media, vol. 7., pp. 1-10, 1997.

Johnson, J. T., and M. Zhang, ``Theoretical study of the 
small slope approximation for ocean polarimetric thermal emission,''
IEEE Trans. Geosc. Remote Sens., vol. 37, pp. 2305--2316, 1999.

Zhang, M. and J. T. Johnson, ``Comparison of modeled and measured 
second azimuthal harmonics of ocean surface brightness temperatures,'' 
IEEE Trans. Geosc. Remote Sens., vol. 39, pp. 448-452, 2001.

Irisov, V. G., ``Azimuthal variations of the microwave
radiation from a slightly non-Gaussian sea surface,''
Radio Science, vol. 53, pp. 65--82, 2000.

Johnson, J. T. and Y. Cai, ``A theoretical study of sea surface
up/down wind brightness temperature differences,"
submitted to IEEE Trans. Geosc. Remote Sens., 2001.
(available from author)


Please contact me to discuss how our codes compare.
I expect there may be some internal parameters not listed
above that may cause some differences between codes.
We can develop more detailed comparisons of internal
code variables to assist in locating the sources of
any differences observed. Other cases can also be tried as 
the study progresses.

--------------------------------------------------------------------------
Joel T. Johnson, Associate Prof. ------------- The Ohio State University
E-mail: johnson@ee.eng.ohio-state.edu -------- Dept. of Electrical Eng.
Voice: (614) 292-1593 or 1606    ------------- 205 Dreese Laboratories
FAX:   (614) 292-7297            ------------- 2015 Neil Ave
URL:http://eewww.eng.ohio-state.edu/~johnson - Columbus, OH 43210