IEEE Geoscience and Remote Sensing Magazine - December 2020 - 112

The purpose of this article
is to derive the two points'
THERE ARE MANY OPEN
distances on the ground usSOURCE OR COMMERCIAL
ing a radar principle and visAPPLICATION SOFTWARE
ibly reveal the characteristics
PROGRAMS AVAILABLE FOR
of SARIF. During the process
SARIF PROCESSING.
of mathematical formulation,
we e x press geomet r y a nd
physics in equations with
known and unknown variables. The number of equations must be equal to that
of the unknown variables (solutions), and the infinite
approximation is applied in many places. In this article,
we make the mathematical derivation process as transparent as possible.

z

S (xs, zs)

M (xm, zm)

ρm 2
ρm 1

ρs1

ρs2

DISTANCE MEASUREMENT BY RADAR
SYNTHETIC APERTURE RADAR
INTERFEROMETRY GEOMETRY
We can identify one point on the ground surface by conventional radar measurements. Suppose we have two radar
satellites, i.e., primary (M) and secondary (S). The radar
measurement geometry is depicted in Figure 1. To find
the distance of two points on the ground, we need four
distances, i.e., t m1, t m2, t s1, and t s2. By solving the four radar equations, we obtain the coordinates of points P1 and
P2: x 1, z 1, x 2, and z 2 . We can then calculate the point distance by the four coordinates. For this purpose, we need
the distance values for all t m1, t m2, t s1, and t s2. However,
we cannot obtain all four distance values directly from
the radar measurement, so we have to find a new method,
e.g., SARIF.
We introduced one coordinate system with the center at
M, the z-axis along M-P1, and the x-axis perpendicular to
the z-axis. Point S is expressed as (B x, B z). And we introduce
new variables: the parallel and perpendicular components
of the P1-P2-distance, Tl < and Tl =. The SARIF geometry is
shown in Figure 2.
The four distances are expressed as
	

P1
(x1, z1)

	

P2
(x2, z2)

	
x

	

x
S (Bx, Bz)

	

M

t 2s1 = ^t m1 - B zh2 + B 2x , (3)
t 2s2 = ^t m1 + Tl < - B zh2 + ^Tl = - B x h2. (4)

^ t m2 - t m1 h^ t m2 + t m1 h = 2t m1 Tl < + Tl <2 + Tl =2 . (5)

Because the second and third terms on the right-hand side
are far smaller than the first term, they can be ignored.
Thus, we obtain Tl <:

ρm 2
ρm 1

t 2m2 = ^t m1 + Tl < h2 + Tl 2=, (2)

There are only two unknown variables, i.e., Tl < and Tl =.
Equation (1) is a trivial equation, and we can ignore it;
but we need t m1 as a known variable. The value of t m1
should be given outside the radar measurement. Equation (3) does not include Tl < and Tl =, so it too may be
ignored. However, we need (3) for the SARIF derivation.
From (2), we find

FIGURE 1. The radar measurement geometry.

ρs 1

t 2m1 = t 2m1, (1)

ρs 2

	

Tl < =

t 2m2 - t 2m1
2t m1 + t m2 - t m1. (6)

Subtracting (3) from (4), we achieve
P2
∆l
∆l⊥

P1
∆l

FIGURE 2. The SARIF geometry.

112

z

	

t 2s2 - t 2s1 = 2t m1 Tl < - 2B z Tl < - 2B x Tl = + Tl <2 + Tl =2 . (7)

Subtracting (5) from (7), we obtain
	

^ t 2s2 - t 2s1 h - ^ t 2m2 - t 2m1 h = - 2B z Tl < - 2B x Tl =. (8)

The left-hand side of (8) is further approximated as
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

DECEMBER 2020
IEEE Geoscience and Remote Sensing Magazine - December 2020

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