Achievable parameters of the X-band synthetic aperture radar based on small satellite

. Space-based synthetic aperture radar systems are actively used for remote sensing of the Earth and have significant capabilities. Advances in microelectronics have made it possible to create and launch into orbit compact radars based on small satellites. Analysis of the capabilities of such radars is an actual problem. In this paper the issues of simulation and estimation of the antenna area, noise equivalent sigma zero, spatial and radiometric resolution of space-based synthetic aperture radars (SAR) operating on the basis of small spacecraft are considered. The necessary mathematical apparatus is given and the above values are calculated for the X-band SAR operating in strip mode at maximum resolution. The influence of the main technical parameters of space SAR on dimensions of antenna, on sensitivity, spatial and radiometric resolution is analyzed.


Introduction
Space-based synthetic aperture radars (SAR) are efficient, multifunctional and promising means of remote sensing with a wide range of capabilities [1][2][3][4].At the same time, in recent years, there has been a transition to a wider use of relatively compact radars based on small satellite and light-class launch vehicles [5][6][7][8][9][10][11].The area associated with the development, production and operation of small satellites is known as "New Space".There are several factors that influence the trend of this transition [5,6]: -improvement of technologies in the fields of microelectronics and surveillance equipment; -possibility of photo and video filming, radar surveillance using high-quality miniature equipment; -growth of proposals for means of launching small satellites, including group launches, launches with the help of ultra-light rockets, launches from the International Space Station; -development of unified small-sized space platforms, which makes it possible to reduce the cost, increase reliability and reduce the manufacturing time of small spacecraft; -increase in the number of tasks solved with the help of small satellites: geoinformation and communication services, meteorological tasks, monitoring of the environment and natural resources, defense tasks, interplanetary astronautics.
This paper simulates and estimates the achievable values of antenna area, radiometric sensitivity, spatial and radiometric resolution of X-band space-based synthetic aperture radars when they operate on the basis of small satellites in side-looking mode.

SAR system geometry
In most cases, SARs use a pulsed mode which allows you to simplify the antenna system and gives more possibilities for variations in the technical characteristics and parameters of synthetic aperture radar.When calculating the survey parameters in the time domain, the basic tactical and technical characteristics of the SAR are the initial ones: look angle , transmitter wavelength , range resolution у, azimuth resolution х, length of synthetic aperture Lsa, aperture synthesis time Tsa, slant range R0, as well as some parameters of the spacecraft orientation at orbit.
Consider the scene containing the SAR platform moving at a constant orbital velocity 0 V at an altitude of H (Fig. 1).The survey of the observed area of the terrain is carried out at a look angle  .The grazing angle  and slant range 0 R depend on the values of H and  [1,4].We will consider the case of side-looking mode when the viewing angle v = 90° respect to the SAR platform velocity vector.The wavelength of radiated signal is In Fig. 1 points 1 and 2 indicate the beginning and end of the synthesis of the antenna aperture; X  and Y  are the dimensions of the subject in azimuthal and range coordinates, respectively; R  is the size of the shooting area along the slant range; 1 R and 2 R are the distances to the near and far boundaries of the subject; y  is the viewing angle of the scene in the angular (vertical) plane; point O is the center point of the scene being shot.

Antenna area
The main survey parameter is the probing pulse repetition frequency PRF which is inversely proportional to the probing pulse repetition period The use of periodic probing pulses in SAR leads to the appearance of ambiguity in the radar image.Therefore, when calculating the pulse repetition frequency, it is necessary to ensure the unambiguity of measurements in the frequency and time domains.
In space-based SAR, where the maximum slant range max 0

R
reaches hundreds of kilometres, unambiguous range measurement can only be achieved by taking into account the delays in the swath Y and not the maximum range (Fig. 1a).In this case, the maximum repetition frequency max PRF is determined by the path difference R  from the radar to the near end of the swath 1 R and from the radar to the far end of the swath 2 R in the elevation plane [12]: where c is the speed of light.Thus, the smaller the distance R  , the higher the value of the maximum repetition frequency.The value of max PRF is determined by the satellite altitude, the width of the SAR antenna pattern in the elevation plane, and the look angle.
On the other hand, the minimum value of the pulse repetition frequency min PRF is selected from the condition for disambiguating in azimuth (i.e., from the condition for receiving the full Doppler spectrum of the return signal) and is determined by the maximum Doppler frequency max D f of the returned signal at the ends of the synthesized aperture (or by the linear dimension of the SAR antenna in the azimuth plane x l ) [1,4]: Expressions ( 1) and ( 2) imply a double inequality which can be rewritten as The radar swath can be described by the expression [1,4]   where y l is the size of the antenna in the elevation plane.Substituting the last expression and expression (2) into inequality (4), we obtain The orbital velocity of the spacecraft where  is the gravitational constant, E R is the Earth radius.Based on the expressions obtained above, we can write an expression for the SAR antenna area Parameters 0 R and  are functions of the satellite altitude and look angle, therefore, from (5) one can obtain the formula for calculating the minimum area of SAR antenna Figure 2a shows graphs obtained for three values of the viewing angle: 35, 45 and 55 degrees.From these graphs, it follows that with increasing satellite altitude and look angle, the minimum required antenna area increases.Based on the data in Fig. 2a, it can be concluded that at a satellite altitude of 520-580 km, the minimum values of the antenna area are in the range of 3-4 m 2 , which is quite achievable in modern small satellites.Figure 2b shows a comparative analysis of the area of the antennas for some X-band SAR in the   S H, plane.We add that expressions ( 5)-( 6) were obtained from the condition of the maximum ratio of the radar swath to the azimuth resolution of the SAR.This means that when designing a SAR system that does not have to simultaneously provide the best resolution and the best swath, the antenna area can be smaller [13].In any case, the final calculation of the SAR antenna parameters must take into account the antenna gain, the antenna efficiency, the antenna pattern and other SAR parameters such as slant range, pulse repetition frequency and processing bandwidth.

Spatial resolution
Expressions ( 1)-( 3) can also be used to estimate the achievable value of SAR spatial resolution  for the side-looking mode.Let the following parameters remain unchanged during this estimation: satellite altitude H , look angle  , grazing angle  , wavelength  , coordinates of the center of the observed terrain (Fig. 1).This means that the slant range will remain unchanged and equal to Let's perform the calculation under the following conditions: equality of the azimuthal resolution and ground range resolution , equality of the antenna azimuth and elevation beamwidths y x    .In this case, the slant range resolution will be determined by the expression Calculations according to formulas (1)-(3) will be carried out iteratively by changing the parameters of the probing pulses and recalculating the antenna pattern and the length of synthetic aperture until the minimum and maximum pulse repetition frequencies are equal Figure 3 shows the graphs of the achievable SAR spatial resolution obtained for the side-looking mode when equality (7) is satisfied.
At small look angles, when the distance R  is relatively small, there is a greater possibility of varying the value of the pulse repetition frequency.This, in turn, makes it possible to achieve lower  values, which means an improvement in SAR resolution.As follows from the above graphs, with a decrease in the satellite altitude and a corresponding increase in the orbital velocity, there is an increase in the achievable spatial resolution.At the same time, at altitudes typical for small satellites (Fig. 2b), the limiting value of the spatial resolution in the side-looking mode is about 1 m.
These results do not contradict the well-known statement that at small look angles, the ground range resolution deteriorates.This conclusion is valid when  changes and other SAR parameters remain unchanged.The graphs     shown in Fig. 3 were obtained by calculating the allowable values of the SAR parameters for each value of the look angle.

Noise equivalent sigma zero
The expression for calculating the noise equivalent sigma zero can be obtained from the equation for the maximum target detection range [1,2]   where peak P is a peak transmitting RF power,  is duration of probing pulses, is a number of coherent pulses emitted during aperture synthesis sа T , is an antenna gain,  is an aperture efficiency of the antenna,  is a radar cross section of a target, NF is a noise figure of receiver system, Q is signal- to-noise ratio, L is a system loss.
For side-looking mode The radar cross-section of a target is determined by the expression [1,2] where 0  is a radar cross-section per unit observed area.
The noise equivalent sigma zero (NESZ) ne 0  is a radar cross-section per unit observed area for which signal-to-noise ratio is unity.NESZ characterizes the effect of receiver noise [1,2,5].This parameter determines the sensitivity of SAR and is widely used as an index of SAR image quality.
Let us substitute formulas ( 9)-( 10) into equation (8).Let us take into account that at the value and in the absence of incoherent accumulation in the resolution element, the signal-to-noise ratio at the SAR output is 0 dB.Finally, we can write the expression for the noise equivalent sigma zero [14]: where is an average transmitting RF power.
Let us calculate the noise equivalent sigma zero (NESZ) of the X-band SAR operating in the strip mode with parameters typical for small satellites [5][6][7][8][9][10][11] (Table 1).The graphs of the noise equivalent sigma zero ne 0  obtained by changing the spatial resolution  for various values of the satellite altitude H are shown in Fig. 4.
From the graphs obtained, as well as from expression (11), it is obvious that an increase in the resolution of the SAR and a decrease in the transmitting RF power lead to a proportional degradation in the NESZ of the system.An increase in the satellite altitude also leads to degradation in value of ne 0  due to a decrease in the orbital velocity of SAR platform but this response is not linear.As follows from the graphs in Fig. 4, when using the strip mode in SAR based on small satellites with the currently implemented values of  , H , peak P [5][6][7][8][9] and the parameters given in Table 1, the achievable values of the noise equivalent sigma zero are within The desire of space-borne SAR developers to minimize the size of antennas, while providing the required technical parameters and performance of the radar, is in conflict, among other things, with the fact that the SAR noise equivalent sigma zero decreases inversely with the antenna area (Fig. 5).The graphs shown in Fig. 5    value of the  , an increase in the look angle leads to a decrease in the NESZ.This is due to a decrease in the level of return signal.The graphs obtained in Fig. 6 are non-linear.At large values of the look angle, the level of the returned signal from the most of spatially distributed objects becomes close to the clutter level.Therefore, in small SAR satellites, the use of look angles greater than (55...60)° is inappropriate [14].We add that at a fixed look angle  , a decrease in the transmitting RF power and an increase in the spatial resolution lead to a proportional decrease in the SAR sensitivity.

Radiometric resolution
The radar radiometric resolution characterizes the possibility of distinguishing objects that differ in radar cross-section per unit observed area [1,[3][4][5].The radiometric resolution can be calculated in decibels using the formula [14]: where N is the number of independent observations accumulated in the radar image resolution element.SAR radiometric resolution graphs plotted using expressions (11)-( 12) at km 500  H and W 800  peak P as functions of the number of independent observations N for different values of spatial resolution and radar cross-section per unit observed area are shown in Fig. 7.As follows from these graphs, a decrease in the value of 0  and an increase in  lead to deterioration in the radiometric resolution of SAR.These plots allow determining the required number of incoherent accumulations to provide the required radiometric resolution value.In this case, the spatial resolution is exchanged for the radiometric one.In most SARs with high-performance processing systems, such an exchange is usually performed by using inter-element incoherent accumulation over the area of the observed object [1].

Conclusion
The graphs obtained in the paper make it possible to perform analysis of the influence of the main technical characteristics of space-borne SAR operating on the basis of small satellites in the strip mode on its parameters: the minimum value of antenna area, noise equivalent sigma zero, spatial and radiometric resolution.
The resulting formulas make it possible to calculate the achievable values of the SAR parameters, take into account the influence of the SAR design on the main technical characteristics, and also perform a preliminary design of the X-band SAR.

Fig. 1 .
Fig. 1.The geometry of the survey when using side-looking mode: (a) elevation plane and (b) azimuth plane.

.
As the satellite altitude increases by 100 km, the SAR sensitivity decreases by about 3 dB.Increasing the SAR resolution without changing the antenna area will lead to a proportional decrease in the radiometric sensitivity.

Fig. 5 .R
Fig. 5. SAR NESZ versus antenna area S at different values of satellite altitude: (a) δ = 1 m and (b) δ = 3 m.The SAR noise equivalent sigma zero is also significantly affected by the look angle  which, for a constant satellite altitude H , determines the slant range

E3SFig. 7 .
Fig. 7. Radiometric resolution of SAR versus number of independent observations N and different values of radar cross-section per unit observed area: (a) δ = 1 m and (b) δ = 3 m.

Table 1
correspond to the parameters presented in