The Construction of a Three-Dimensional Antenna Gain Matrix and Its Impact on Retrieving Sea Surface Mean Square Slope Based on Aircraft Wave Spectrometer Data
1. Introduction
With the development of radar technology, radar has been applied in ocean wave directional spectrum measurement (Jackson 1981), that is, the ocean wave spectrometer. This type of radar is different from synthetic aperture radar (SAR) systems, which are not straightforward in their use due to their motion (Hasselmann et al. 1996; and many others) and their high cost of development. Spectrometers are real aperture radars and are unable to provide radar “images” of the sea surface. However, by using the frequency-modulated continuous wave (FMCW) technique, they can achieve the same level of range resolution as SAR. Currently, there is no ocean wave spectrometer payload on satellites, but real aperture radars on aircraft used for ocean wave measurement have been developed, such as the radar ocean wave spectrometer (ROWS), developed by the United States (Jackson et al. 1985a,b; Jackson 1987); Radar pour l’Etude du Spectre des Surfaces par Analyse Circulaire (RESSAC), developed by the European Space Agency (ESA) and the French Space Agency (Hauser et al. 1992); and the Système de Télédétection pour l’Observation par Radar de la Mer (STORM), adapted from RESSAC (Hauser et al. 2003; Mouche et al. 2005, 2006a,b).
Around 2018, the China–France Oceanography Satellite (CFOSAT) will carry the ocean wave spectrometer Surface Waves Investigation and Monitoring (SWIM), with the goal of measuring the directional spectra of waves from space (Le Traon et al. 2015). The SWIM (Hauser et al. 2001; Calvary et al. 2002; Lorenzo et al. 2010; Hauser et al. 2014) is a Ku-band real aperture radar, whose incidence angle ranges from 0° to 10°. To obtain the two-dimensional ocean wave spectrum, the radar is designed to measure the electromagnetic echoes at low incidence angles from different azimuths, by means of antenna rotating. From the wave spectrum, the peak wavelength, wave propagating direction, and significant wave height may be acquired. Before the satellite is launched, the manufacture of spectrometer hardware and the study of its work mechanism and ocean wave spectrum retrieval algorithms must be conducted. To confirm its validation, the radar is first mounted on an aircraft that flies over the Bohai Sea in China, and the data acquired from the flights are processed.
However, the aircraft radar backscatter cross section of the ocean surface is usually calculated inaccurately, due to the use of an incorrect antenna gain in the radar equation caused by the roll, pitch, and yaw of the aircraft. In this way, the normalized radar cross section (NRCS) will be disturbed and the resulting parameter, mean square slope (MSS), which is obtained based on the radar echo and is important for wave propagation direction retrieval (Chu et al. 2012a,b), will be affected. Therefore, the antenna gain must be corrected according to the records of the aircraft attitude during flight experiments. Hauser et al. (1992) explained that the influence of attitude angles can be excluded, but they did not describe the specific steps in detail or how greatly the retrieved parameter could be impacted.
In this paper, we aim to eliminate the effects of roll, pitch, and yaw of aircraft on the NRCS of the sea surface; to describe the concrete steps for constructing a three-dimensional antenna gain matrix used in the radar equation; to calculate the backscattering cross section; and to reduce CPU consumption. Furthermore, the results of the MSS after correcting the NRCS are also analyzed. Therefore, we first introduce the radar equation and quasi-specular scatter process. Then, the construction method of the three-dimensional antenna gain matrix is proposed, and the validation of the results is confirmed. Finally, we analyze the MSS variation from the echoes of 200 circles of the airborne radar after using attitude angle correction.
2. NRCS of ocean waves
The airborne spectrometer is a monopolarized radar and is compatible with installation on aircraft; that is, it transmits horizontal-polarized (H polarized) spherical electromagnetic (EM) waves downward toward the sea surface and receives H-polarized EM waves at different moments (corresponding to different slant ranges and different incidence angles, as shown in Fig. 1), which constitute an echo by the reflected electromagnetic power from the sea surface. It is an FMCW radar, and the modulation is sawtooth-shaped with a 13.575-GHz mean frequency, a 320-MHz bandwidth, and a 4-ms duration, and the corresponding slant range resolution is 0.47 m. The radar antenna is placed in a radome installed under the aircraft and can rotate over 360° in the horizontal plane (assuming that the roll and pitch of the aircraft are both zero). The 3-dB beamwidth of the antenna gain is 4° in the azimuth direction, as shown in Fig. 2a; that is, the semiminor axis of the ellipses corresponds to an angle of 4° on the plane perpendicular to the incidence plane. It transmits 2500 pulses during one circle, and records one echo for each pulse. To suppress the thermal noise and speckle noise, the adjacent echoes are averaged (Hauser et al. 2001). Theoretically, the more echoes averaged, the larger the signal-to-noise ratio (SNR). However, in view of the resolution of azimuth, there is an upper limit of the amount. In fact, according to Jackson et al. (1985a), the azimuth resolution of two-dimensional ocean wave spectra by this type of radar is about 15°. The amount used to average by 10 or 20 echoes has little effect on the MSS. Here, 10 echoes are averaged and 1.44° of azimuth resolution is obtained.
Geometry of observation of the spectrometer mounted on the aircraft. The reference frame is a local coordinate system that is independent of the aircraft. The aircraft is believed to be located at point O, with a heading along the x axis. Points t, r, yaw, and rot are the angles of the pitch, roll, and yaw of the aircraft and the rotation of the antenna varying clockwise during flights. The positive directions of the three angles describing the aircraft attitudes conform to the right-hand screw rule. The ellipse shown with a solid line is the projection of the half power of the antenna gain on the sea surface with the zero aircraft attitudes, while the ellipses shown with a broken line is that with nonzero attitudes. Beam center incidence 
Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-15-0095.1
Antenna gain pattern measured in the laboratory. (a) The gain dependence on two angles (dB). (b) The definition of the axis of (a). In fact, if the gain is shown in natural units, then only the gains at 5°–25° of 
Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-15-0095.1
In Fig. 1, the antenna pattern is assumed to be in line with the two-dimensional Gaussian distribution (its true pattern is shown in Fig. 2), the ellipse shown with a solid line is assumed as the projection of the half power of the antenna gain on the sea surface with the zero aircraft attitudes (i.e., the r, t, and yaw are all zero), while the ellipse shown with a dashed line is the nonzero attitudes. The area S1 denotes the sea surface containing the gain with zero aircraft attitude, and S2 denotes the surface containing the gain with nonzero aircraft attitudes. It is clear that, in order to obtain an accurate NRCS in Eq. (4), it is necessary to integrate the gain over S2 rather than S1.
For an echo, the trend of NRCS with incidence angles will vary strongly by substituting S1 with S2, while the final two-dimensional sea wave spectrum may not be affected so significantly according to the retrieval method (because in wave spectrum retrieval, there is a procedure in which the trend of the NRCS is subtracted in order to get the echo fluctuation to prepare for the spectrum estimate; Hauser et al. 1992). The MSS can be used in wind speed (Hwang et al. 1998; Chu et al. 2010; Karaev et al. 2013) and spectrum retrieval; thus, it is important to determine how much the MSS can be corrected.
3. Construction of the three-dimensional antenna gain matrix
When the aircraft carrying the radar is in flight, the servo system installed on the aircraft will simultaneously record three attitude angles: yaw, pitch (t), and roll (r), as shown in Fig. 1. If the rotation angle (rot) is considered as the fourth angle, then there are four parameters determining the pointing of the beam centerline O-Pb (point Pb is the center of the ellipse shown with a solid line, where the gain is maximum in theory). Finally, the incidence angles for one azimuth are taken into account, and the gain 
However, it would require a great amount of CPU time to calculate the NRCS according to the five parameters, because the process includes integrating the annular gain in area S2. In fact, through the yaw, t, r, and rot, we can obtain the beam center incidence angle 
In Fig. 1, the “H′” is the point (0, 0, H) in the aircraft reference frame, which is not consistent with the local reference frame XYZ, except that the three angles r, t, and yaw are all zero. Point “H” is (0, 0, H) in the local reference frame. From the figure, we can observe that once the angle 
To examine the stability of the value in 
An example of the stability of the three-dimensional 
Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-15-0095.1
Figure 3b shows the standard deviation (STD) of the 46 gains at each incidence angle bin. It demonstrates that if the gains fall into a single three-dimensional bin, then the variation of the gains is very small (no larger than 0.1 dB) after 4° incidence angles. Although the STD becomes larger with decreasing incidence, it is smaller than 0.5 dB at incidence angles larger than 2°. In terms of accuracy, incidence angles used in the fitting process by Eq. (7) (to calculate the MSS) should be larger than 4°~5°. Therefore, the two figures tell us that the antenna gain of each element in 
4. Influence of antenna gain correction on the NRCS and MSS
We calculated the NRCS with 
Results of aircraft attitude correction on the echoes. (a) The NRCS vs incidence angles from 2° to 16°. The black solid line (
Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-15-0095.1
Figure 4b shows the linear fit to the echoes from Fig. 4a according to Eq. (7) at incidence angles 5°–12°. Here, the incidence angle 5° is selected for the sake of gain accuracy described above. It should be noted that the interpretation of MSS determined by Eq. (7) rests on the assumption that the backscatter RCS is dominated by quasi-specular reflection, that is, the form of Eq. (5). In fact, Bragg scattering increases at large incidence angles and its magnitude depends on the EM wavelength (Majurec et al. 2014). According to Hauser et al. (1992), the quasi-specular scatter process has an equal contribution from the Bragg resonant scatter process at the 16° incidence angle at 4 m s−1 of sea surface wind speed. Freilich and Vanhoff (2003) assumed that the incidence angles 0°–18° of the PR can be dominated by the quasi-specular scatter process at wind speeds smaller than 20 m s−1, and at the outer portion of the PR swath the data extend into the transition region, where resonant scattering from tilted, rough facets becomes significant. Although more accurate NRCS modeling than the simple Eq. (6) has appeared (Bringer et al. 2012; Boisot et al. 2015), the method of MSS acquisition here is sufficient to evaluate the effects of the flight attitude angle correction. Therefore, here the upper limited incidence angle 12° is selected, and we believe that the quasi-specular scatter mechanism is appropriate.
With the gain correction, different rotation angles and beam center angles may lead to different echo corrections; thus, there will be different MSS corrections. When the NRCS decreases more quickly with increasing incidence angles, the MSS will be smaller. In Fig. 4b, at 
Figure 5 shows the dependence of the differences of MSS with and without aircraft attitude correction (
Impact of 
Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-15-0095.1
The sea state is very low (about 3–4 m s−1 of the sea wind speed, according to ECMWF) during the time of the experiment in June 2014 in the Bohai Sea, and the direction of sea wave propagation is not clear enough; thus, the MSS from the measured echoes is very irregular for each circle. To observe the effects of attitude correction on MSS more clearly, we obtain the average of MSS from every 50 cycles. Figure 6 shows the successive MSS from data block 320, which corresponds to about 2000 s of the flight. Furthermore, in order to show the accuracy of the LUT correction, the standard attitude correction is executed. Because the MSS with LUT correction and with standard correction are very close to each other, the percent of error (red pluses) is shown. The difference between MSS with standard correction and those with LUT correction are within 5%, showing that there is limited error due to the discretization (the elements of the LUT have a 0.2° × 0.2° × 0.2° bin widths) of the gain of the radar antenna.
MSS variation within 50 circles averaged as one circle. The four images are from adjacent 50 circles of data block 320 and the 200 circles correspond to an ocean surface distance within 34 km. 
Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-15-0095.1
The four images indicate that the MSS peak is at about 110° and 290°, which differs by about 180°. Although the skewness of the sea surface slope denoting upwind/downwind asymmetry exists according to Cox and Munk (1956), Bréon and Henriot (2006), and Munk (2009), in theory the MSS upwind should be strictly equal to MSS downwind. However, there are many other factors affecting the MSS detected by radar, such as the horizontal resolution of surface, radar noise, and so on; the MSS upwind and downwind from the radar should be approximately equal. From Fig. 6, it is a clear improvement that the MSS upwind and downwind with attitude angle correction are all corrected to be approximately equal in contrast to those without correction. The corrected MSS may become either smaller or larger than that without correction, depending on the attitude angles.
Figure 7 shows comparisons of 
Comparisons of 
Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-15-0095.1
There are many other factors affecting the MSS derived by radars besides wind speed, such as swell, surface oil, the submarine topography, and so on. Generally, the two peak MSS due to the rotation of radar antenna appear, so the flight attitude angle correction in Eq. (4) is worthwhile and essential in order to determine the right NRCS of the sea surface using airborne radar. Furthermore, according to the computer with “Intel(R) Core(TM) i7-2600 CPU @3.40GHz, RAM 8.00GB,” the time of standard correction for 50 circles is about 5.54 h, while that of LUT correction is 78.55 s. So, the construction of LUT is necessary in view of the accuracy and the reduction of the CPU time. To be noted, most of the rolls, pitches, and yaws are in the range of −5° to 5°, and our matrix is constructed limited on this attitude range. The bin size 0.1° is also tested and the results are nearly the same as 0.2°. However, the gain matrix becomes too large to be used conveniently.
5. Conclusions
In this paper we reviewed the basic radar equation in order to calculate the NRCS of the ocean surface from the received power of the sea wave spectrometer, and a construction method of the three-dimensional antenna gain matrix used in Eq. (4) is described. It is based on the coordinate system transformation from the aircraft reference frame to the local reference frame, in order to determine the accurate antenna gain. It can be shown in Fig. 3 that the gain is approximately invariant at fixed angle 
We then calculate the NRCS according to the matrix based on Eq. (4) and retrieve the MSS of the sea surface for every azimuth by linear fitting according to Eq. (7). The results show that the most influential factor is the beam center incidence angle 
Because the acquisition of the two-dimensional sea wave spectrum is a complicated process and because it can be affected by other factors besides the flight attitudes, the resulting impacts on the sea spectrum are not shown. The MSS acquired here cannot be as stable as that of BH2006, so the conclusions of MSS from Ku-band radar over a large range of sea states may be reached after the CFOSAT mission.
As a general rule, the method to exclude the influence of flight attitude angles and to reduce CPU time for accurate NRCS may be used in other airborne radar systems, such as SAR, scatterometers, and possibly others. The difference is that the resulting gain matrix may not be three-dimensional, instead it may be two-dimensional. But the precondition is that the three angles of the flight attitude angles and antenna gain must be correctly measured.
Acknowledgments
This study is supported by the Jiangsu Natural Science Foundation under Grant BK2011008, and by the CFOSAT Project, China. We thank three anonymous reviewers for their constructive comments, which helped to greatly improve the manuscript.
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