Restoring the height of the terrain taking into
account the statistical relationship of the
interferometric pair of radar images
Oleg Goryachkin Ivan Maslov
Image Processing Systems Institute of RAS - Branch of the FSRC Image Processing Systems Institute of RAS - Branch of the FSRC
"Crystallography and Photonics" RAS "Crystallography and Photonics" RAS
Samara, Russia Samara, Russia
oleg.goryachkin@gmail.com macloff@mail.ru
Abstract—An algorithm for reconstructing the height is Using the technology of multi-pass interferometric imaging,
proposed, which allows, based on the statistical relationship of it is possible to restore the height of the terrain in the vicinity
the interferometric pair of radar images arising from the of the ground receiving point, and further control its change.
influence of the Earth’s atmosphere, to clarify the height of the The necessary interferometric base can be formed due to the
terrain. The results of numerical simulation are presented with special ballistic construction of the orbit of the spacecraft.
the initial data corresponding to the parameters of the on-
board equipment of the P-band bistatic radar system installed II. ALTITUDE RECOVERY ALGORITHM BASED ON
on the Aist-2D small spacecraft. The results obtained confirm ATMOSPHERIC STATISTICS
the advisability of considering the statistical data on the state
of the ionosphere in the algorithm of radar interferometry. Consider the main stages of processing and obtaining a
digital elevation model for the interferometric survey mode
Keywords—height measurement error, ionosphere, P-band, in a synthetic aperture radar (SAR):
radar imaging, radar interferometry, synthetic aperture radar.
1. The exact combination of two images (interferometric
I. INTRODUCTION pair) obtained under the same conditions, but with a "small"
diversity in space.
Currently, spacecraft equipped with synthetic aperture
radar (SAR) allow you to receive radar (amplitude) images 2. Finding the interferometric phase difference of the two
with high spatial resolution. However, SARs also make it images.
possible to obtain phase information from reflecting objects 3. Filtering the resulting interferogram to reduce the
and use it to reconstruct the third dimension, i.e. topographic influence of speckle noise.
elevation. The most developed frequency ranges are X-, C-,
S- and L-bands. The launch of the next spacecraft with the P- 4. Elimination of linear phase incursion in range.
band SAR of the Biomass of the European Space Agency is
5. The elimination of the ambiguity of the interferometric
scheduled for 2021. The main difference between the P-
phase difference, which is due to the influence of the terrain.
range and the others used is high penetration and reflection
stability. There are two main schemes for shooting images 6. Recalculation of the interferometric phase in the height
using SAR: monostatic when the transmitter and receiver are of the terrain.
combined in space, and bistatic when the transmitter and
receiver are separated in space. The placement of P-band 7. The procedure for geocoding.
monostatic SARs is complicated by well-known technical Two images can be represented as:
problems [1-4]: the destructive effect of the ionosphere,
restrictions on the radio communication regulations, the need
I 1 f 1 ( h ) I 1 0 n1 and I 2 f 2 ( h ) I 2 0 n 2 , (1)
to use large antennas with a wide aperture, and a significant
pulse power of the transmitter. So, for example, the basic
design parameters of a BIOMASS spacecraft with a P-band where f 1 ( h ) e x p j 0 12 ( 0 , x 0 , y 0 , 0 ) h and
monostatic SAR, suggest that the spatial resolution is not
better than 50 m when using a 12-meter diameter antenna f 2 ( h ) e x p j 0 2 2 ( 0 , x 0 , y 0 , 0 ) h functions describing the
[5]. In [6–9], it was shown that multistatic (in particular dependence of the height of the target, 12 ( 0 , x 0 , y 0 , 0 ) and
bistatic, when the transmitter is placed on board the 2 2 ( 0 , x 0 , y 0 , 0 ) - regular component signal delay, h -
spacecraft and the receiving part on the Earth) radar
observations open up the possibility of creating space-based height, I 1 0 e x p j 0 1 ( t k ) and
radar sounding equipment in the P-bands of high-resolution. k
The need for a land-based stationary or mobile receiving I 2 0 e x p j 0 2 ( t k ) , 1 (t k ) and 2 ( t k ) - a random
station at a relatively short distance from the observed object k
limits the scope of application of such remote sensing component of the signal delay that occurs in the process of
systems. Nevertheless, it is possible to indicate some areas of signal propagation in the Earth’s atmosphere, n1 and n 2 -
application in which the proposed technologies have independent additive complex noises in SAR channels.
advantages: control of landscape changes; control of the ice
situation around offshore oil and gas production platforms; An estimate of the maximum likelihood of the desired
precision farming; tactical intelligence; monitoring of forest height under the conditions of known statistics of
resources, etc. The first in the history of remote sensing fluctuations in the time of arrival of a signal in the Earth's
spaceborne radar system operating in the P-frequency range atmosphere can be written:
is a bistatic SAR installed on a small spacecraft Aist-2D.
Copyright © 2020 for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0)
Image Processing and Earth Remote Sensing
h m ax p I1 , I 2 | h 1
h p I10 , I 20
4 R e 1 0 Im 1 0 R e 2 0 Im 2 0
2
D et
m a x p I 1 , I 2 | I 1 0 , I 2 0 , h p I 1 0 , I 2 0 d I 1 0 d I 2 0 (2)
h
x1 0 M 0
2
x 20 M 0
2
G
1
where G is the region of integration on the complex plane, exp D11 D 33
R e10 R e 20
2 2
2 D et
2
R e I 1 R e f 1 h I 1 0
p I1 , I 2 | I10 , I 20 , h
1
y1 0
2
x1 0 M 0 x 2 0 M 0
e x p
D 22 D13 D 31
2 n1 2 n1
2 2
Im 1 0 R e10 R e 20
2
y 1 0
2
Im I Im f h I
2
y 20 y 20
1 1 10 D 44 D 24 D 42 ,
Im 2 0 Im 2 0 Im 1 0
2
2 n1
2
where R e 1 0 , Im 1 0 , R e 2 0 , Im 2 0 - are the standard
2
1 R e I 2 R e f 2 h I 2 0 deviations of the real and imaginary parts of the first and
e x p
2 n 2
2
2 n 2
2 second images, respectively, and algebraic additions.
We find the height estimate by integrating analytically.
2
Im I 2 Im f 2 h I 2 0
,
h m ax p I1 , I 2 | h
2 n 2
2
h
R e I1 and Im I 1 - the real and imaginary part of the m a x p I 1 , I 2 | I 1 0 , I 2 0 , h p I 1 0 , I 2 0 d I 1 0 d I 2 0
h
image I 1 , R e I 2 and Im I 2 - the real and imaginary part m a x p x1 , y 1 , x 2 , y 2 | h
h
of the image I 2 , n21 and n2 2 - the noise variance of the first
m a x p x 1 , y 1 , x 2 , y 2 | x 1 0 , y 1 0 , x 2 0 , y 2 0 , h
and second image. h
p x1 0 , y 1 0 , x 2 0 , y 2 0 d x1 0 d y 1 0 d x 2 0 d y 2 0
We introduce the following notation: R e I 1 x1 ,
1. Simplify p x1 , y 1 , x 2 , y 2 | x1 0 , y 1 0 , x 2 0 , y 2 0 , h .
Im I 1 y 1 ; R e I 2 x 2 , Im I 2 y 2 ; R e I 1 0 x1 0 ,
Im I 1 0 y 1 0 ; R e I 20 x 20 , Im I 2 0 y 2 0 ; 1 1 x 2 y 2 x 2 y 2
p I1 , I 2 | I10 , I 20 , h exp
1 1 2 2
4 D n
2 2
R e f1 ( h ) k 1 x , Im f 1 ( h ) k 1 y ; R e f2 (h) k2x , 2 D n
Im f 2 ( h ) k 2 y . 1 1 1 1 1
exp x1 0 y10 x 20
2 2 2 2
y 20
2 D n Dn Dn Dn
Then
p I 1 0 , I 2 0 p x1 0 , y 1 0 , x 2 0 , y 2 0
2k x 2k
1x 1 1y
y1 2k y 2k x
1x 1 1y 1
x1 0 y1 0
Dn Dn
2
1 1 x1 0
exp D11 2 2k x2 2 k 2 y y2 2k y2 2 k 2 y x2
y 2 0 ,
2x 2x
2 D e t R e10 x 20
2
4
R e 1 0 Im 1 0 R e 2 0 Im 2 0 D et
Dn Dn
x1 0 y 1 0 x1 0 x 20 x1 0 y 20
D12 D13 D14 where D n n 1 n 2 . 2 2
R e 1 0 Im 1 0 R e10 R e 20 R e 1 0 Im 2 0
2 2. Simplify p x 1 0 , y 1 0 , x 2 0 , y 2 0 .
y 1 0 x1 0 y10 y10 x 20
D 21 D 22 D 23
R e 1 0 Im 1 0 Im 1 0 Im 1 0 R e 2 0
2
1
p x1 0 , y 1 0 , x 2 0 , y 2 0
R e 1 0 Im 1 0 R e 2 0 Im 2 0 4
2
y10 y 20 x 20 x1 0 x 20 y10 D et
D 24 D 31 D 32
Im 1 0 Im 2 0 R e 20 R e10 R e 2 0 Im 1 0 1 D11 M 0
2
D 33 M 0
2
D 1 3 D 3 1 2
2
exp M 0
2 D e t R e 1 0 R e 20 R e 10 R e 20
2 2
x 20 x 20 y 20 y 20 x1 0
D 33 D 34 D 41
R e 20 R e 2 0 Im 2 0 Im 2 0 R e 1 0
2
1 D11 D 22 D 33 D 44
exp x1 0 2 y1 0 2 x 20 2
2 2 2 2
2 y 20
y 20
2
y 20 y1 0 y 20 x 20
2 D e t
,
D 42 D 43 D 44 2 R e10 Im 1 0 R e 20 Im 2 0
Im 2 0 Im 1 0 Im 2 0 R e 2 0 R e 2 0
D13 D 31 2 M 0 D11 D13 D 31 M 0
where D e t - is the determinant of the correlation matrix x1 0 x 2 0 x1 0
R e 10 R e 20 R e10 R e 20
2
p R e I 1 0 , Im I 1 0 , R e I 2 0 , Im I 2 0 ,
R e10
D ij - is the
2 M 0 D 33 D13 D 31 M 0 D 24 D 42
algebraic complement of the element R i j in the determinant x 20 y 1 0 y 2 0
R e 20 R e 10 R e 20 Im 1 0 Im 2 0
2
D et .
3. We write p x 1 , y 1 , x 2 , y 2 | h down considering the above
After simplification, we write the multidimensional transformations.
probability density for the quantities x 1 0 , y 1 0 , x 2 0 , y 2 0 :
VI International Conference on "Information Technology and Nanotechnology" (ITNT-2020) 127
Image Processing and Earth Remote Sensing
1 1
p x1 , y 1 , x 2 , y 2 | h 2k x 2k y 2 M 0 D11 D13 D 31 M 0
1x 1 1y 1
1 6 D n R e 1 0 Im 1 0 R e 2 0 Im 2 0
4 2
D et 2
Dn R e10 D e t R e10 R e 20 D e t
1 x 2 y 2 x 2 y 2 D11 M 0
2
D 33 M 0
2
exp 2 2
2k y 2k x
1 1 2 2
2 Dn R e10 D e t R e 20 D e t 1x 1 1y 1
D13 D 31
Dn
1 1 D11 2 C
M 0 e x p 2 2k x 2k y
2
R e 10 R e 20 D e t
x1 0
R e10 D e t 2x 2 2y 2 2 M 0 D 33 D13 D 31 M 0
2 D n 2
Dn R e 20 D e t R e10 R e 20 D e t
1 D 22 2 1 D 33 2
2 y1 0 2
Dn Im 1 0 D e t Dn
x 20
R e 20 D e t 2k 2x
y2 2 k2 y x2
Dn
2k x 2k y 2 M 0 D11 D13 D 31 M 0
1x 1 1y 1
2 x1 0 Define the matrix B :
Dn R e10 D e t R e 10 R e 20 D e t
b1 1 0 b1 3 0
2k y 2k x
1x 1 1y 1 2k 2x
y2 2 k 2 y x2 0 b22 0 b24
y10 y 20 B ,
Dn Dn b31 0 b33 0
D13 D 31 1 D 44 2 0 b42 0 b44
x1 0 x 2 0 2 y 20
R e10 R e 20 D e t D
n Im 2 0
D e t 1 D11 1 D 22
where b1 1 , b22 ,
2k x 2k y R e10 D e t
2 2
2x 2 2y 2 2 M 0 D 33 D13 D 31 M 0 Dn D
n Im 1 0
D e t
2 x 20
Dn R e 20 D e t R e 10 R e 20 D e t 1 D 33 1 D 44
b33 , b44 ,
R e 20 D e t Im 2 0 D e t
2 2
D 24 D 42 Dn Dn
y 1 0 y 2 0 d x1 0 d y 1 0 d x 2 0 d y 2 0
Im 1 0 Im 2 0 D e t D13 D 31 D 24 D 42
b1 3 b 3 1 , b24 b42 .
We calculate the resulting integral. R e 10 R e 20 D e t Im 1 0 Im 2 0 D e t
We write
As you can see, it is a multidimensional probability
1 1
density of a combination of random variables, then p I1 , I 2 | h
1 6 D n R e 1 0 Im 1 0 R e 2 0 Im 2 0
4 2
D et
1 1
... e x p x a B x a d x 1 ,
T
2
(9) 1 x 2 y 2 x 2 y 2 D11 M 0
2
2 d e t B exp 2
n 1 1 1 2 2
2 Dn R e10 D e t
x1 0
D 33 M 0
2
D13 D 31
2
y10 M 0
where x , a is the vector of mean values, B is the R e 20 D e t
2
R e10 R e 20 D e t
x 20
y 20
1
covariance matrix. e x p x B x C x d x
T
2
Since the matrix B is symmetric, we can write: Finally, we obtain an algorithm for estimating the height
of the terrain, considering the random nature of signal
x a B x a x Bx 2a Bx a Ba propagation in the Earth’s atmosphere in a form that does not
T T T T
contain multiple integrals:
x B x C x D
T
1 1 h m a x p x1 , y 1 , x 2 , y 2 | h
C 2a B
T
T
where a B C then h
2
1 1
1 1
B CB
T
1 1 1
D a Ba
T T
CB CB C . 4 D n R e 1 0 Im 1 0 R e 2 0 Im 2 0
2 2
D et det B
4 4
1 1 x1 y 1 x 2 y 2
2
D11 M 0
2
2 2 2
We will receive exp D 2
2 2 Dn R e10 D e t
1 1 1
... e x p 2 x B x C x d x 2 d e t B e x p 2 D .
T n
D 33 M 0
2
D13 D 31
M 0 .
2
R e 20 D e t R e 10 R e 20 D e t
2
Define a vector C :
The main question that arises in this case is the
advisability of considering the atmosphere in the algorithm
for determining altitude. Will there be a gain in the correct
accounting of the statistical model of the atmosphere. To
answer this question, mathematical modeling was carried
out.
VI International Conference on "Information Technology and Nanotechnology" (ITNT-2020) 128
Image Processing and Earth Remote Sensing
III. MATHEMATICAL MODELING RESULTS IV. CONCLUSION
Figure 1 show the results of calculations for different From the data obtained it follows that the greater the
values of the true correlation coefficient with the following correlation coefficients between the real and imaginary parts
initial data: of two images (interferometric pair), the greater the value of
the gain from the application of the proposed algorithm,
- signal to noise ratio 23 dB considering statistical data on the state of the ionosphere.
- interferometric base 10 km.
- the angle of inclination of the base is zero degrees. REFERENCES
- angle of sight 45 degrees.
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c)
Fig. 1. The true value of the correlation coefficient is 0.7 (a), 0.8 (b) and
0.9 (c), respectively.
VI International Conference on "Information Technology and Nanotechnology" (ITNT-2020) 129