Synthetic aperture radar (SAR) is a coherent active microwave imaging method. In remote sensing it is used for mapping the scattering properties of the Earth's surface in the respective wavelength domain. The algorithms for the formation of 3D radar images in multiposition interferometric systems for remote sensing of the Earth are considered. Examples of reconstruction of the relief map for systems with one and two transmit antenna are presented.
Figure 1 shows the basic geometric equations for the 3D SAR system, for the case when the aircraft is equipped with two transmitter and two receivers, operating through its own spaced from each other antenna.
Using the basis of the Pythagorean theorem we can write:
(H1 h)2 r12 R12
2
(H2 h)2 (r1 d )2 R2
(
Values of the heights H1 and H 2 of the antennas and antenna spacing d in the horizontal plane
can be considered known while SAR is working. Also known and sloped range R1 and R2 . You need
to find the horizontal distance r1 and the height of the object h . Before solving the equation (
H1 R12 H1 R22 H 2 R12 H 2 R22 d (H12 2 H1 H 2 H 22 R12 2 R1 R2 R22 d 2 )1/ 2 (2 H1 H 2 H12 H 22 R12 2 R1 R2 R22 d 2 )1/ 2 ) r
1 2 (H12 2 H1 H 2 H 22 d 2 ) ; R12 R22 d 2 (H 2 h)2 (H1 h)2
2 d
On the practice ranges R1 and R2 are known with an item resolution accuracy, so the calculations
of value h and r1 use value R R2 R1 , calculated on the basis of the measurement result of the
phase difference
R (w) 2 1 , (
4 4 where 1 and 2 - phases of the signals received from the first and second antennas, respectively; wave`s length.
Because the value of the phase shift is in the range [0,2 ) , as a rule, there is ambiguity of the measurement values R . To fix it uses the "unwrap" phase. Consider possible approaches to its implementation [4].
The first approach is based on the preliminary construction of the dependence of phase shifts to the zero level, depending on slant range to the antenna A1 [5]:
0R1 4R0R1 / 4 ((H22 ((R12 H12 )1/ 2 d )2 )1/ 2 R1) /
Then the value of the phase shift uR1 used to calculate the difference between the sloping
distances, determined from the relationship:
uR1 w trunc 0R1 2 (
2 The value of the slant range R2 used for the calculation h and r1 is determined from the relation: u R2 R1
2m 4
To determine the value m , you can use the following approach. Typically, the height difference between adjacent pixels is relatively small and we can assume that the height of the object in some neighborhood is constant. Presented according to m estimates of the altitude differences in the neighboring pixels have a pronounced minimum, and calculations show that this minimum is achieved when the value of the parameter mˆ corresponding to the true values of the altitude and slant range. The algorithm of the calculation value mˆ is the following: 1. Around the current image point with coordinates x0 , y0 , set the gate; 2. Sets the range of values m : m mmin...mmax 3. For each point in the gate with coordinates x , y , a values rx, y and hx, y are calculated with current value m ; 4. Calculate total measurement error of the heights and horizontal distances. Terms y1 y0 and y2 y0 take into account the current offset of pixels in the horizontal range: hm rm x0dx y0dy x0dx y0dy
x1x0dx y1 y0dy x2x0dx y2 y0dy x0dx y0dy x0dx y0dy
yx1,x2,y1,y2 x1x0dx y1 y0dy x2x0dx y2 y0dy yx1,x2,y1,y2 rx1,y1 y1 y0 rx2,y2 y2 y0 hx1,y1 hx2,y2
hmˆ min , кmˆ min (9)
Because of spacing antennas, the resulting images have some mutual shift, and its magnitude will
depend on the range. In this regard, before calculating the elevation, you must perform the mutual
correction of the shifts of image elements. For this calculate values of distances R2 corresponding to
(
J Rк1о,рx J 2
R2 ,x R2 (H22 ((R12 H12 )1/ 2 d )2 )1/ 2 (10) where J 2
R2,x - image from the second antenna.
S2 S1 m, m ...,2,1,0,1,2,... (11)
2
Knowing the values of the parameters S1 , S2 , H1 , H 2 , d , you can find value and a horizontal range r1 :
D2 (d 2 (d 2 S12 2 H12 S 22 2 H 22 S 22 2 S1 d 2 S 2 S 24 H 22 S12 4 H13 H 2 2 d 2 S 22 H12 S12 d 4 2 H 22 d 2 H14 H 24 2 H12 d 2 6 H12 H 22 2 S1 S 2 H12 2 S1 S 23 4 H1 H 2 d 2 4 H1 H 23 4 H1 H 2 S12
4 H1 H 2 S1 S 2 2 H 22 S1 S 2 4 H1 H 2 S 22 S12 S 22 ))0.5 h H1 S1 S 2 H 2 d 2 H12 H 2 H13 H 23 H1 d 2 H 2 S 22
2 (H12 H 22 d 2 2 H1 H 2) H1 S 22 H 2 S1 S 2 H1 H 22 D2
2 (H12 H 22 d 2 2 H1 H 2) As in the first case, measuring the phase shift may be ambiguous. The dependence of phase shifts on the zero level from the slant range to the antenna A1 is described by the expression: 0R1 2 (H22 ((R12 H12 )1/ 2 d )2 )1/ 2 R1
Then the value of the phase shift uR1 is used to calculate the difference between the sloping
distances is determined from the equation (
S2 2R1 u 2m 2
Still from the spacing antennas, the elements of the resulting images can have some mutual shift. To compensate for this shift is necessary to perform the image correction according to the relation [7]: J Rк1о,рx J R21R2 ,x (15)
2 , where R2 is computed as in equation (10).
Figure 4 shows an example of the recovery bump maps for the considered case when the following system parameters H1 H2 10m , d 5m the maximum altitude of the relief 5m. (12) (13) (14)
Solution of a problem of restoration of a landscape`s in a synthetic aperture radar at various configuration of a reception-transmitting path is considered in this work. An original phase unwrapping algorithm based on joint minimization of the estimating error of object`s height and sloped range in neighboring pixels of the image is proposed. Examples of the restored 3D images are presented.
The work is executed at financial support of the Ministry of Education and Science of the Russian Federation, project No. 2.2226.2017/Project Part and project No. 2.9140.2017/Basic Part. The work is performed under financial support of Russian Found of Basic Research, research project No. 15-0799514.