This paper describes the radar image processing model for modern synthetic aperture radars and other airborne radar systems. The main attention in the paper is paid to elaboration of the method of combination of various well-known and enhanced mathematical models of forest surface types, radar signals, radar scene and scenario, etc. The method is applied to the e ective radar image calculation in the rangeDoppler coordinates. Also, bene ts of the radar echoes computation algorithm of the radar scene are discussed. In conclusion, an overview of the model features and some future investigations are presented.
Nowadays computer modeling technologies and digital signal processing are
ubiquitous and extremely useful in many various radar applications [
Solution of the autonomous navigation problem for di erent types of aircrafts
is very important. Especially it is so for aircraft that y at extremely low altitudes
above typical Earth surfaces, that include forest terrain. There are many known
works on the study of signals re ected from various natural surfaces [
So, a mathematical model of the re ected signal of a pulsed radio altimeter that uses a rectangular waveform is considered in the paper on the example of operating under various forest surfaces in the centimeter range of radio waves for a ight altitude of about 100 meters and below. The rst feature of the model is the attempt to solve the problem of image synthesizing for the vertical illumination of a surface. The second feature is the calculation of Earth radar images of forest vegetation in the range-Doppler coordinates (in the range-Doppler domain).
To create a mathematical model of the Earth surface radar response for the low altitude radar system with an active monostatic radiolocation method and pulsed signals, it is necessary to successively solve the following problems.
1. Construction of a mathematical surface model. For the mathematical
description of the surface with forest vegetation, the facet model is chosen. It is the
most universal for various types of concentrated, surface and volume-distributed
targets. It is based on a geometric model of propagation and re ection of the
radio waves (which here are similar to the light beams) from the whole set of
illuminated re ectors, which are presented in the form of conditional facets, i:e:,
simpli ed at areas with dimensions less than the radar resolution. So, this
provides the adequate and experimentally con rmed results under circumstances of
correct de nition of the modeled facet number and re ection parameters [
2. Development of a model for the re ected signal formation. The modeling
method used in the work assumes the joint application of a phenomenological
approach and sets of empirical data [
3. Calculation of the radar image of the previously simulated surface in the
range-Doppler coordinates. Here, our solution of this problem is based on an
algorithm similar to the one described in [
4. Calculation of the received radar signal. To calculate the signal that is the radar response for each emitted pulse, we nd the sum of low frequency replicas of the emitted pulse, then attenuate and shift them by the delay and frequency due to the corresponding radar image pixel value and their position inside the radar image. So, as it was mentioned above, the amplitude of each copy corresponds to the radio brightness of the pixel of the previously obtained radar image. The number of components is equal to the number of non-zero or the most "bright" elements of the radar image corresponding to the given radar antenna direction and pattern.
The above sequence of operations makes it possible to signi cantly reduce the amount of computation, especially, for the complex signals with the intrapulse modulation. Methods of the digital signal generation in the model are e ectively implemented by matrix calculations in the used MATLAB system. Let us now consider the solution of these problems using the example of forest terrain modeling.
Our choice of the forest terrain for research is based on a classi cation by the
type of prevailing vegetation. It is so due to the presence of appropriate tree
models from the standard set of primitives presented in the Autodesk 3ds Max.
Moreover, a su cient amount of empirical data is taken on a corresponding
vegetated ground echoes [
The block diagram of constructing the mathematical model of the forest
terrain and the results of modeling are shown in Figs. 1 and 2, respectively [
Simulation of the re ected signals for the canopy, branches, and trunks of the tree triangular facets is possible with the reduction (by random selection) of the number of elementary re ectors. It is made in order to leave about 100 facets per element of the modeled radar image (the resolution will be given below). Therefore, here, in order to save computational resources, a simpli ed discarding of a part of the facets can be additionally performed. For the remaining facets, a proportional scale conversion is performed to increase their relative power. Simulation of the re ected signal begins with a sequential calculation of the slant range, Doppler frequency shift, and power for each elementary re ector the facet. The power of a re ected signal from a facet is found from the radar equation and according to the expression
P =
P0 (4 )3 (1) where P0 is the transmitter power; is the wavelength of the emitted signal; Kap is the coe cient of antenna pattern; Kref is the coe cient of re ection to account the speci c surface RCS; Kbp is the coe cient of accounting for the backscattering pattern; 4S is the facet area; R is the slant range, that are demonstrated for the sample i-th facet in Fig. 3; is the propagation loss factor (including some tuneable shading because of propagation through the vegetated layers).
Values of Kref and Kbp take into account speci c types of surface material of the facets. In the general case, they are di erently set to represent the canopy leaves (needles), branches, tree trunks, and, also, the terrain ground layer model. They are commonly used by selecting corresponding features for desired carrier frequency in order to achieve the expected results that will correspond to the experimental studies. More accurate signal shadowing by the canopy can be resolved by implementing the backward raytracing method. But this is too di cult, since there are so many facets in our tree models.
Slant range R is the physical absolute distance from the onboard radar to the facet. It is obviously determined for each facet from the geometry of the model by the di erences in the relative coordinates (dimensions) where 4x, 4y, and 4z are the relative coordinates.
The calculation of the Doppler frequency shift of the facet re ected signal is carried out according to the expression 4f = 2 Vr = 2 V cos( ) cos( ) ; (2) (3) (4) where Vr is the radial velocity of the radar platform in the direction of the facet; and are the angles to account the individual direction of the facet (from the radar antenna relatively to the velocity vector V , which are also demonstrated for the sample facet in Fig. 3) in the horizontal and vertical planes. 3.3
For radars such as the synthetic aperture radar (SAR), the mean value of the resolution in the along-track dimension is determined by the synthesizing time according to the expression [1{3, etc.] 4 = 2 V Ts sin( view) ; where 4 is the azimuth resolution; V is the aircraft speed; Ts is the time of synthesizing (accumulation of re ected pulses); view is the mean viewing angle.
Under the vertical illumination (because of the small value of the
denominator in (4)), the resolution is relatively low for the qualitative (or the best) radar
application. Therefore, the vertical illumination mode is not traditionally used
for ground-mapping radars. However, take into account the fact that the width
of the antenna beam of a typical radio altimeter at half power is about 40-60
degrees. So, the re ected signal from each of the 4 quadrants of the illuminated zone
in spatial coordinates can be considered as a re ected signal in the
front/rearside-looking observation (the squint surveillance). Therefore, it is possible to use
a typical radio altimeter for synthesizing the aperture, to extract and analyze
additional radar information [
The radar images in the range-Doppler coordinates are useful from the point of view of the correspondence/equality of the discrete step of the radar image pixels to the radar resolution. So it becomes suitable for detecting the radio-contrast objects with dimensions near to the actual resolution of the radar system.
For constructing the radar images in the range-Doppler coordinates
(dimensions), the underlying surface is divided into separate areas called the resolution
elements or cells. These calls are bounded by the iso-range lines (lines of the
same range) and isodopes (lines of the same Doppler frequency) as it is
simpli ed shown in Fig. 4 for the horizontal motion of the radar platform under a
plane surface.
Each area cell (resolved by an imaging radar) has its own power proportional
to the area RCS. In such a grid, the pixel step of the radar corresponds to the
radar resolution in the frequency and range. Notice for further that when using
such a radar system, it is necessary to take into account the variable dimensions
of the cells, the nonlinearity of distances, and the mutual overlapping of the
facets located at the same range to the left and right of the ight axis (for
example, points A and A* in Fig. 4). Practically the eld of view with high resolution
is formed by the radar motion from one synthesis interval to another with the
single beam formation at each synthesis interval [
Algorithm for the formation of the radar image of the Earth surface in the range-Doppler coordinates consists of the following steps.
1) Determination of surface resolution elements in the range-Doppler coordinates. The resolution of the frequency and range is determined by the parameters of the emitted signal and the principles of the signal processing. For radars with the synthesis of the antenna aperture and the compression of the received chirp pulses, the following expressions can be used: 4fdop = 1 Ts ;
c 4r = 2 W ; (5) where 4fdop is the frequency resolution; 4r is the resolution in slant range; c is the speed of light; W is the bandwidth of the linear-FM pulse currier frequency.
2) Distribution-grouping the facets according to the cell midpoints that now correspond to the radar image pixels.
3) Calculation of the total power of each resolution element, i:e:; the image pixel.
4) Threshold digital image processing for the selection of some tall trees and analysis their properties.
In Fig. 5 an example of constructing the radar image for a test surface with a single tree is shown. The Doppler frequency is plotted along the abscissa axis and the slant range along the ordinate. The ight altitude is 50 m, the speed of the aircraft is 100 m/s, and the carrier frequency is 10 GHz. Here and further to increase the contrast, the zero values of the image pixels are replaced by a white (transparent) color.
The simulation results for models of the forest terrain depicted earlier in Fig. 2 are presented in Fig. 6. The simulation parameters are similar to the previous test.
Based on the results of modeling, it is possible to distinguish the following features of the radar images of the forest surfaces: { The prominent trace of the soil layer on the radar image is characteristic only of the deciduous forest model. In the coniferous forest, the attenuation created by the canopy (treetops) signi cantly weakens the relative level of the soil signal. { In the absence of the large-scale relief, the soil layer signal forms a gure bounded by two hyperbolas. The bottom hyperbola is the nadir track. { There is ambiguity in the azimuth that is resulted in the fact that the trees located symmetrically relatively to the velocity vector of the aircraft on the radar image are overlayed. { There is a relationship between the clarity, the scale of the image, and the viewing angle. This is resulted, for example, in the fact that the side-located trees (located at the angles about 90 degrees relatively to the velocity vector) have clear outlines of the treetops and trunks in contrast to trees located along the ight direction.
We note that the process of obtaining such radar image is, in fact, close to nding an impulse response of a surface. Then, it allows us to calculate the response of the surface by using one of the convolution methods with the emitted pulse. In our approach, all information about delays, amplitudes, and Doppler frequency shifts for all re ectors on the surface remains in the radar image. Therefore, using on this information, we can then calculate the re ected signal for the radar. 3.4
The process of re ected signal modeling assumes a consecutive calculation of the re ected signal for each emitted pulse. To do this, for each emitted pulse, the low frequency replicas of the emitted pulse (or, possibly, only its envelope) are summed by the delayed and frequency-shifted values at the corresponding position of the radar image pixel. The amplitude corresponds to the radio brightness (equal to the square root of the total facet backscattering) of the pixel as the radar image element. So, a complex form of the signal with some additive noise for practical using is calculated as
S(t) =
X i; where Ui>Umin
UiA t 2Ri c ej2 4fi t + N (t); (6) where Ui is the amplitude corresponding to the pixel, which must be greater than attunable level Umin; A(t) is the emitted waveform, which also should be de ned in a complex form; 2Ri=c = i is the delay for corresponding slant range Ri for the i-th pixel; 4fi is the Doppler frequency shift for the i-th pixel; N (t) is the white/band noise with the desired signal-to-noise ratio corresponding to the radar receiver.
The number of emitted pulse replicas A(t) is equal to the number of nonzero radar image elements. Otherwise, we can use only the \bright" (Ui > Umin) elements in the radar geometric region of our interest and assuming that when the signal is processed by the radar receiver, a part of the partial signals can be e ectively ltered, suppressed, or gated.
Also note here that one can see the saving of computing resources when calculating the re ected signal, since in (6) the signals are summed not by the number of facets in the surface model, but by the number of resolution elements in the radar image.
Further, for the radar functioning, a typical or special signal processing is
possible. For example, an altitude measurement of an aircraft over the forest
terrain is measured by catching the relative position of the rising slope of the
rst echo impulse [
The dense deciduous forest model is characterized by the separation of the re ected signal into two commensurable layer responses: the forest canopy, and the soil layer. This fact allows one to measure the ight altitude of aircraft not only over the treetops, but, also, above the soil layer. Here, with the ight altitude of 50 m and a tree height of 28 m, the distance to the tops of the trees is 22.5 m, and the height above the soil layer is 50 m.
For the model of dense coniferous forest, there is a clear predominance of re ection from the canopy, which is explained by the high closeness of the trees' canopy and the high level of the signal attenuation in it.
When constructing the radar image with one emitted pulse, information about the volume distribution of re ectors is lost. However, based on the received re ection signal sequence for the radar, this information can be recovered with some precision. Also, note here that the range migration information is contained in the sequence of the re ected pulses collected through the image synthesis interval. But this is a separate and complex goal, which concerns the signal processing by the SAR techniques. 4
As a result, the mathematical model of the layered surface was constructed, the algorithm for calculating the radar image and the re ected signal of a pulsed radio altimeter is built. The model allows taking into account the di cult structure of natural surfaces, where forests are both typical and complex Earth cover (other terrain models are under considering). Also, we can vary the parameters of the aircraft and emitted signal in order to perform the analysis of the radar operation and to improve the signal processing parameters including the synthesis of the antenna aperture. The radar image in the range-Doppler coordinates is visual and informative mean for detecting radio-contrast objects. Moreover, this method is useful and practical from the point of view of the correspondence of the discrete pixel size and the radar resolution.
One of our main goals is to save computing resources. When calculating the
response signal from a surface, we use properties of the partial grouped signals.
So, the total complex signal re ected from all the scene facets is calculated
economically, but in accordance with the desired radar resolution. Additional
computational resources can be saved by reducing the number of considered
model facets when calculating each pixel of a radar image. Such scalable methods
of the surface representation are also promotional for implementation in the
HILsimulators [
The general appearance and characteristics of the modeled images in the Xband, for example, the presence of graininess in the image, correspond to the expectations for the selected simulation conditions. This fact con rms the correctness of the proposed method and the implementation of the modeling algorithm.
The results can be used to investigate the possibilities of applying the radio altimeter to the re ected signal processing and radar image constructing, particularly, in the range-Doppler coordinates. In the future, in order to increase the compliance of the modeled radar images with the experimental ones, it is proposed to calculate the amplitude and phase of the total signal for each element of the radar image and, also, to elaborate the algorithm for accounting the shading depending on the modeled thick-ness and moisture content of the forest canopy. This will allow better simulating the presence of the speckle noise on the image. It will be useful in calculating the radar ground-returns with intrinsic amplitude uctuations, and fading e ects of the backscattered signal. Acknowledgments. This work was supported by the Grant of the Ministry of Education and Science of the Russian Federation, Project no. 8.2538.2017/4.6.