### I. Introduction

*U*> 3.46 m/s, which led us to develop a new model that works at wind speeds of

*U*< 3.46 m/s as well as for other ranges of wind speed.

*n̂*. We focused on the X-band frequency, which is used by Korea Multi-Purpose Satellite 5 (KOMPSAT-5) and KOMPSAT-6.

### II. Extraction of Small-Roughness Parameters

*e*

^{−1}), and a type of surface correlation function. First, ocean spectra are generated, and then cut-off frequency points can be found for the small-roughness surfaces (i.e

*.*, high-frequency spectrum regions).

*S*(

*ω*) =

*αg*

^{2}

*ω*

^{−5}exp[−

*β*(

*ω*

_{0}

*/ω*)

^{4}], where

*α*= 8.1 × 10

^{−3},

*β*= 0.74,

*g*is the gravitational constant,

*ω*

_{0}=

*g*/

*U*

_{19.5}, and

*U*

_{19.5}is the wind speed at a height of 19.5 m. Fig. 2 shows the wave spectral densities at wind speeds of 15, 10, and 5 m/s. The peak of the spectrum is given as

*ω*

_{peak}= 0.877/

*U*

_{19.5}(e.g

*.*,

*ω*

_{peak}= 0.8382 rad/s at

*U*= 10 m/s). The maximum angular frequency

*ω*

_{max}is set to one hundred times of

*ω*

_{peak}(100

*ω*

_{peak}) for convenience.

*ω*

*for generation of a small-roughness sea surface, corresponding to the range of the radian frequency of*

_{c}*ω*

*≤*

_{c}*ω*≤

*ω*

_{max}. The cut-off frequency can be found with a trial-and-error method for each wind speed, according to the following procedure. First, we generated a small-roughness surface with a picked

*ω*

*. Second, we retrieved small-roughness parameters (RMS height and correlation length) from a numerically generated small-roughness surface. Third, we compared the backscattering coefficients of the SVF model with the data in [7]. Finally, we iterated the above procedure to get an optimum*

_{c}*ω*

*for each wind speed. From this procedure, we found that the optimum cut-off radian frequency*

_{c}*ω*

*is about 0.135*

_{c}*ω*

_{max}, or 13.5

*ω*

_{peak}. For example, the optimum

*ω*

*was found to be about 11.3 rad/s at*

_{c}*U*= 10 m/s.

*U*≤ 10 m/s at X-band, small-roughness sea surfaces were numerically generated. Then, we developed the following empirical expressions for the RMS height and surface correlation length as functions of wind speed, as shown in the following equations from the small-roughness sea surfaces that were generated using the Pierson-Moskowitz ocean spectrum.

*h*

_{rms}is the RMS height,

*l*

_{c}is the correlation length,

*U*is the wind speed, and

*φ*is the azimuthal angle (e.g.,

*φ*= 0° corresponds to the up-wind direction).

### III. Development of the Scattering Model

*n*

*-power roughness spectrum (Fourier transform of the*

^{th}*n*

*-power of correlation function) is required for the IEM model [2]. Since sea surfaces usually show an exponential-like correlation function, we utilized the modified roughness spectrum for an exponential function in the following form [3]:*

^{th}*z*is a decay factor that can be chosen appropriately,

*L*is the correlation length,

*K*is 2

*k*sin

*θ*, and

*k*is the wave number. Fig. 3 shows a comparison between the SVF model and the existing SESS model in choosing the decaying factor

*z*. As an example, for a wind speed of 4 m/s, the optimum

*z*value is about 0.019, as shown in Fig. 3.

*σ*

^{0}

_{IEM-pp}(

*θ′*,

*φ*) represents the backscattering coefficients computed by the IEM model [2, 3] for small-roughness facets and

*P*

*(*

_{θ}*Zx′*,

*Zy′*) is the slope PDF in [1]. In order to use IEM with a small-roughness condition, the

*kh*

*value must be less than 3, where*

_{rms}*k*is the wave number and

*h*

*is the RMS height. The local incidence angle*

_{rms}*θ′*can be computed using the normal unit vector of the facets:

*Zx*and

*Zy*are the surface slopes in

*x*and

*y*directions (i.e.,

*Zx*=

*∂Z*(

*x, y*)/

*∂x*and

*Zy*=

*∂Z*(

*x, y*)/

*∂y*, where

*Z*(

*x, y*) is the sea surface height).

*θ*

*= 30° and (b)*

_{i}*θ*

*= 45°. Our primary goal is to extend the validity region of the SESS model to lower wind speeds. In Fig. 4, the lines of the SVF model and the SESS model intersect at about*

_{i}*U*= 3.5 m/s for VV and HH polarizations at

*θ*

*= 30° and*

_{i}*θ*

*= 45°, as shown in Fig. 4(a) and (b). Therefore, we propose a new semi-empirical model for X-band radar backscattering from the sea surface by combining the SVF model and the SESS model in the following form.*

_{i}### IV Verification of the New Model

*θ*

*= 0° and*

_{i}*θ*

*= 45°. The first dataset [7] has 26 data points for each VV-and HH-polarization, and the second dataset [8] has a total of 35 data points for various wind speeds.*

_{i}*θ*

*= 30° and*

_{i}*θ*

*= 45° at X-band. The root-mean-square errors (RMSEs) have been computed with (∑(*

_{i}*M*–

*d*)

^{2}/

*N*)

^{0.5}, where

*M*is the model values,

*d*is the data,

*N*is the amount of data. The RMSE of the first dataset [7] at

*θ*

*= 30° for VV- and HH-polarizations are 2.50 dB and 2.29 dB, those at*

_{i}*θ*

*= 45° for VV and HH polarizations are 5.77 dB and 3.70 dB, and the RMSEs of the second dataset [8] at*

_{i}*θ*

*= 30° and*

_{i}*θ*

*= 45° for VV- polarization are 2.93 dB and 4.53 dB, respectively.*

_{i}*z*in Eq. (3) can be extracted only once by comparing the model with the first measurement dataset to increase the accuracy of the model.