2.1. Initial Assumptions
We will consider the area of small incidence angles, when, in the case of the sea waves, the quasi-specular backscattering mechanism is dominant, and the Kirchhoff method is used to find the reflected signal, for example, [
19,
20,
21,
22]. It is obvious that for the ice cover, the concept of the Doppler spectrum exists only when measured from a moving carrier. In previous studies, it was shown that in the case of a fast-moving carrier, the Doppler spectrum width depends on the parameters of sea waves only for a radar with a wide antenna beam [
23]; for a narrow antenna beam, the Doppler spectrum width depends only on velocity of movement. The calculations were performed for an orbital radar (velocity 7000 m/s), so we will repeat them for the aircraft version, when the flight speed is less than, for example, 200 m/s.
Figure 3 shows the dependence of the Doppler spectrum width on the antenna beam width for two wind speeds (5 and 10 m/s, fully developed wind waves) for a moving carrier (velocity 200 m/s along the axis
, probing direction 45°, radar wavelength 0.021 m, and an incidence angle 5°). The used probing scheme is shown in
Figure 4.
It can be seen from
Figure 3 that despite the fact that the wave parameters for different wind speeds are very different, in the case of a narrow antenna beam, this does not affect the width of the Doppler spectrum, and the curves coincide. Therefore, for a narrow antenna beam, the main factor affecting the width of the Doppler spectrum is the velocity of radar movement.
With an increase in the width of the antenna beam, the dependences of the Doppler spectrum width for different sea wave intensities (in this case, wind speeds) are separated. Thus, a radar with a wide antenna beam begins to “see” the reflecting surface, and the width of the Doppler spectrum depends not only on the velocity of movement but also on the parameters of sea waves.
From the theoretical model of the Doppler spectrum, it follows that for a radar with a wide antenna beam, with an increase in the movement velocity of radar, the key factor is not the orbital velocities (movement of the reflecting surface) but rather the mean square slopes (
) of large-scale waves compared to the radar wavelength [
23,
24].
If we make a number of simplifying assumptions about the direction of probing, the direction of carrier movement, and the direction of wave propagation, the formula for the width of the Doppler spectrum [
24] can be greatly simplified and written in the following form:
where
is the incidence angle;
is the mean square slopes (
) of large scale, in comparison with the radar wavelength, sea waves (large-scale waves) along axis
;
is the velocity of radar movement;
is the width of the antenna beam at the level 0.5 on power.
It can be seen from the formula that if it is to use a narrow antenna beam (), it is possible to neglect the of large-scale waves in the sum, and then the will be reduced. As a result, the width of the Doppler spectrum will be proportional to the width of the antenna beam.
Conversely, for a wide antenna beam, the fraction will be reduced in such a way that the Doppler spectrum width (see Formula (1)) will be proportional to the of large-scale waves. Therefore, a change in the of the reflecting surface leads to a change in the surface scattering diagram (dependence of the RCS on the angle of reflection), which ultimately affects the Doppler spectrum width when measured from a moving carrier.
The of the ice cover and sea waves are very different, so it was assumed that, when measuring from a moving carrier in terms of the width and shift of the Doppler spectrum, it would be easy to separate the ice cover and sea waves. This work is devoted to testing this assumption.
2.2. Semi-Empirical Model of the Doppler Spectrum for Ice Cover
For the sea surface, the description in terms of the wave spectrum is generally accepted, and many models of wave spectra are currently known, for example, [
25,
26,
27,
28,
29]. Due to this, it is possible to obtain analytical formulas for the Doppler spectrum at small incidence angles, for example, [
30,
31,
32,
33,
34,
35,
36].
To study the properties of the Doppler spectrum backscattered by the sea surface, numerical methods are used, for example, [
37,
38,
39,
40,
41]. A wave spectrum of sea waves is used to model a scattering surface, so a spectral description is also required when using the standard ice cover modeling approach. There is no spectral description for the ice cover, so it is necessary to use another approach to develop a semi-empirical model of the Doppler spectrum, which will be based on the available experimental data.
The measurement scheme is shown in
Figure 4. The radar is mounted on an aircraft that is moving at a velocity of
along the
axis at a height of
. The incidence angle is equal to
and the sounding is carried out at an angle of
in the
plane. The slant range to the reflection point is
. Then, the radial velocity component for the reflecting point is given by the following formula:
When measuring the Doppler spectrum from a moving carrier, the width of the antenna beam is important, determining the size of the reflecting area (footprint) and the spread of radial velocities in the reflected radar signal. In calculations, it was assumed that the antenna beam is Gaussian and is written in the following form:
where
and
are the antenna beam width at a 0.5 power level;
and
are the incidence angle and azimuth angle within the antenna beam, respectively, measured from the beam axis (
,
), i.e.,
and
.
A change in the azimuth angle leads to a change in the incidence angle; therefore, to correctly calculate the radial velocity, it is necessary to recalculate the incidence angle using the following formula:
To find the Doppler spectrum of the backscattered signal, it is necessary to integrate over the scattering area:
After integration, we obtained the spectrum of Doppler velocities (the distribution function of the radial velocity component). It is more common to represent the Doppler spectrum on the frequency axis; thus, to obtain the conventional Doppler spectrum, it is necessary to use the following formula:
where
is the radar wavelength.
Figure 5 shows examples of Doppler spectra for a moving carrier (
= 200 m/s), incidence angle
= 5°, azimuth angle
= 45°, and four values of the antenna beam: 2° × 2°, 2° × 20°, 20° × 2°, and 20° × 20°. For the convenience of comparison, we will always normalize each Doppler spectrum to its maximum. The first two spectra (2° × 2° and 2° × 20°) are shown in
Figure 5a, and the last two Doppler spectra are shown in
Figure 5b.
When transitioning from a narrow antenna beam (2° × 2°—black curve) to a knife-like beam (2° × 20°—blue curve), due to the wide antenna beam in the azimuthal plane, the range of incidence angles increases (see Formula (4)), which leads to a noticeable increase in the Doppler spectrum width. In this case, the width of the antenna beam in terms of the incidence angle is only 2°; therefore, a change in the azimuth angle (+/−10°) provides a noticeable increase in the range of incidence angles. It leads to increasing the Doppler spectrum width.
This effect practically does not manifest itself when transitioning from a knife-like antenna (20° × 2°—red curve) to a wide antenna (20° × 20°—green curve). This is because, in contrast to the first case, the change in the incidence angle due to a change in the azimuth angle (+/−10°) will be small compared to the width of the antenna beam along the incidence angle (20°).
In calculations, it was assumed that all surface points have the same reflection coefficient, which is not true. Thus, the next step in developing a semi-empirical model of the Doppler spectrum is related to taking into account the scattering diagram of the ice cover (or the dependence of the RCS on the incidence angle).
In our research, we use Ku-band (
= 0.021 m) precipitation radar data from the TRMM (Tropical Rain Measuring Mission) and GPM (Global Precipitation Measurement) satellites [
42,
43].
Precipitation measurement is an important task, and a joint project between Japan and the United States was implemented to solve it. The TRMM (Tropical Rainfall Measuring Mission) satellite was the first precipitation satellite and was launched on 28 November 1997 from the Tanegashima Space Center (TNSC) (JAXA-TRMM). Precipitation radar (PR—Ku-band) on board the TRMM satellite measured the spatial distribution of rain in the tropical area. The TRMM satellite made observations for 17 years.
The dual-frequency precipitation radar (DPR) is a successor to the PR (13.6 GHz) loaded onto the GPM’s (Global Precipitation Measurement) predecessor TRMM (JAXA-TRMM). The 35.5 GHz radar was additionally installed for high-accuracy observation of low-intensity rain. The launch of the core observatory for the GPM mission aboard was successfully performed on 28 February 2014. It can observe not only the tropical zone but also mid-to-high-latitude areas due to an orbit inclination of 65°.
DPR operates at wavelengths of 2.2 cm and 0.8 cm, and the probing scheme is shown in
Figure 6. DPR and PR are designed to measure the spatial distribution of precipitation as well as its vertical profile to determine the precipitation intensity. The last resolution element contains data on backscattering from water or land surface.
These data were used to determine the scattering diagram of the underlying surface.
In the Ku-band, measurements were taken for the range of incidence angles of 0°–19°. Precipitation radar data obtained over the Sea of Okhotsk were used to perform regression and derive formulas for an ice and sea surface backscatter diagram. An example of the dependence of the RCS on the incidence angle for a dry ice cover (negative air temperature, first-year ice) is shown in
Figure 7 [
44,
45]. In the figure, stars of different colors represent different days.
A more extensive analysis of the dependence of the RCS on the incidence angle for ice cover and sea waves is undertaken in a paper currently under review [
46] (private communication). However, using another dependency will not lead to fundamental changes in the results obtained, so we will use simpler formulas in the paper.
As a result of the regression analysis, the angular dependence of the backscatter diagram for the ice cover was approximated by the following formula:
where
= −3.1518,
= −0.008708,
= −0.016928,
= 26.013,
= 0.5288.
Thus, to calculate the Doppler spectrum of the backscattered radar signal, it is necessary to integrate over the scattering area:
It should be noted that for the ice cover, the azimuthal dependence of the RCS (from a probing direction) can be neglected, since, in contrast to sea waves, the ice surface can be considered isotropic.
2.5. Comparison of Analytical and Semi-Empirical Models of Doppler Spectrum
As noted earlier, for small incidence angles of probing radiation on the sea surface, there is a theoretical model of the Doppler spectrum, which was obtained in the Kirchhoff approximation. This will allow to evaluate the correctness of the method used to develop a semi-empirical model of the Doppler spectrum by comparing it with the theoretical model.
The theoretical model of the Doppler spectrum [
24,
31] includes the statistical characteristics of sea waves, which can be calculated from the wave spectrum model. The input parameters of the wave spectrum model are the wind speed and the nondimensional wind fetch [
28]. Formula (9) describes the averaged dependence of the RCS on the incidence angle obtained from the DPR data. In the general case, the dependence of the RCS on the incidence angle is not unambiguous (see
Figure 1), which complicates the problem of determining the kind of the scattering surface.
In the Kirchhoff approximation, the formula for the RCS for sea waves has the following form (Bass, Fuchs 1972) [
19]:
where
and
are the
of large-scale waves along axis
and axis
, respectively;
is the non-normalized correlation coefficient between the slopes along the axes
and
(hereinafter, the correlation coefficient);
is the effective reflection coefficient introduced to take into account the influence of a ripple on the power of the reflected signal.
Thus, the problem is reduced to determining the wind speed, which will give the best match between the model dependence from incidence angle (Formula (14)) and the experiment (Formula (9)). To achieve this, it is necessary to determine the wind speed that will provide the
of large-scale waves observed in the experiment. The
determines the form of the dependence of the RCS on the incidence angle (
Figure 8), which makes it easy to estimate the accuracy of selecting the wind speed.
The performed analysis showed that if we consider sea waves propagating at an azimuthal angle of 45°, then for a fully developed wind wave, it is necessary to set the wind speed equal to 9.7 m/s. Based on the wave spectrum, the
of large-scale waves were calculated and the result is shown in
Figure 8: asterisks are obtained by Formula (9) and the red curve is plotted by Formula (14). That is, the form of the dependence of the RCS on the incidence angle is important to us; therefore, for the convenience of comparison, the theoretical and experimental dependences were equated at a zero incidence angle when plotting the graph.
It can be seen from the
Figure 9 that a good agreement between the theoretical dependence and the experiment was obtained; therefore, for further estimates, we will also assume that there is a fully developed wind wave on the surface, which was formed at a wind speed of 9.7 m/s and propagates at an azimuth angle of 45°.
Wave spectrum [
28] was used to calculate all statistical moments of the second order for a wind speed of 9.7 m/s and substituted into the theoretical formula for the Doppler spectrum [
24]. Calculations were made for the following parameters: radar velocity of 200 m/s, incidence angle of 5°, and sounding direction of 45°.
For calculations, a radar with a knife-like antenna beam (14° × 2°) was chosen. In
Figure 10, the Doppler spectrum calculated from the semi-empirical model is shown with a green curve, and the theoretical model is shown with a dotted line. For ease of comparison, the spectra are normalized at their own maximum. It can be seen from the figure that for a fast-moving radar, both models of the Doppler spectrum show close results, i.e., the proposed approach to developing a semi-empirical model of the Doppler spectrum is effective for a fast-moving radar.
Discrepancies between the theoretical model and the semi-empirical model may appear at low velocity because the velocity of the radar and the orbital velocities of the sea surface become comparable. This must be taken into account when making measurements and analyzing data. For example,
Figure 11 shows the results of numerical simulation of the Doppler spectrum for a carrier velocity of 4 m/s (
Figure 11a) and 20 m/s (
Figure 11b). The blue curve in
Figure 11a is built according to the theoretical model of the Doppler spectrum for a motionless radar (
= 0). The direction of wave propagation is from the radar, so the Doppler spectrum shift is negative.
The black curve is derived from a theoretical model of the Doppler spectrum for a radar velocity of 4 m/s and a sounding direction of 45°. A radar movement occurs along the
axis, which results in a positive Doppler shift and, as seen in
Figure 11a, this almost cancels out the negative Doppler shift caused by sea waves.
The red curve was obtained for a radar moving over a stationary “sea” surface (Formula (8)). The surface is not moving, so the Doppler spectrum has narrowed considerably. In this case, the width is determined only by the movement of the radar, and the Doppler spectrum shift has a positive sign.
In
Figure 11b, the calculations were made for a radar velocity of 20 m/s, the black curve was built using the theoretical model of the Doppler spectrum for sea waves, and the red curve was built using the semi-empirical model of the Doppler spectrum for a stationary “sea” surface. It can be seen that, even at such a velocity, the widths of the Doppler spectra become close and are determined by the
and not by the orbital velocities. The difference in the shift of the Doppler spectra still remains. Thus, the comparison showed that the proposed approach to developing a semi-empirical model of the Doppler spectrum is effective, provided that the statistical characteristics of the scattering surface are reliably described.