The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bl. Beijing 2008 
respectively represent two positions of a SAR sensor where it 
takes images twice. The length of the baseline B is the 
distance between $ ] and _s 2 . The letter a stands for the angle 
between the baseline and horizontal line. The distance H is the 
length of line s^O, R e the earth’s radius and 6 the view angle 
when the sensor takes images of P at the position of s . Then 
we assumed r and r + Ar separately represent two routes when 
the sensor takes images at s ] and s 2 ■ In this paper, we mainly 
discussed repeating orbit InSAR carried by satellites. Thus, the 
interference phase <f> is presented by slant range difference Ar 
in Equ.2. 
An 
d) = —-/Sr 
A 
(2) 
Based on the information in Fig.2, we deduced two basic 
equations in the Precise Geometry Model (PGM). One is Slant 
Range Difference Equation and the other one is Phase 
Equation. 
d2 
Ar Bsin(9-a) 
2 r 
2.1.2 Phase Equation 
(5) 
0-0 o = arc cos( 
r +H -R. 
2rH 
-) - arccos( 
r 2 + H 2 - R e 
2rH 
-) (6) 
According to Equ.2 and Equ.5, we deduced the Phase 
Equation based on the geometric information from Fig.2. 
Then we gained the equation in Equ.6. From Equ.6 we knew 
that the value of 0-0 o is related to the height h at the 
position ofP. Taking JERS Satellites as an example, we 
calculated parameters with known data ( r - 80000m 
and H = 7151000m ) and reached conclusions as follows: 
When h = 6000 m, the value of 0-0 o is nearly one degree; 
when h = 1000 m, 0-0 o is about 0.173 degree. In 
consequence, higher positions accompany larger differences 
of view angles. 
2.1.1 Slant Range Difference Equation 
Because of the fact Ar « r ; slant range difference Ar can be 
approximately presented by Equ.3 with neglect of Ar 2 . 
2.2 Algorithm for Flattening Phase 
2.2.1 Principle of Flattening Phase 
Since SAR directly stored the information of slant range, the 
phase recorded includes flat ground phase shown in Equ.l 
which presents on the interferogram as periodical fringes 
under the name of “flat ground effect”. During the process of 
InSAR, only the phase concerning altitude is expected and 
thereby flattening phase is a needed step for further processing. 
There are two major reasons that cause flat ground effect— 
baseline change along time and curvature of the earth. 
However, different ranges of images taken by SAR influence 
the result of flattening phase. While small scale of images can 
be flattened by simple linear method, wide images can not 
achieve satisfactory results unless curvature of the earth is 
taken into account. To obtain better output, we proposed an 
algorithm based on Precise Geometry Model for the step of 
flattening phase. According to Equ.5, Equ.6 and geometric 
information in Fig.2, the phase difference between and 
P> on the interferogram can be presented in Equ.7. 
A r- fisin(#-«r) 
2 r 
(3) 
. . 2nB 2 A 2KB n {H 2 -R h 2 -r 2 ) K 
AS„= 7-Ar ^— h - — Ar 
Ar 2 AHr 2 sin 0 X 
(7) 
In the next step, we discussed whether the item b 2 / 2r can 
be neglected in different conditions. According to Cycle 
Slicing Limit (CSL) proposed by Massonnet [2], the slant 
range error ap brought by noises should be controlled 
within the range ofA/120, namely a p < a/120. Then the item 
B 2 Hr can be eliminated from the equation. When SAR 
carrier is a plane, its conditions satisfy the CSL and we gained 
the Slant Range Difference Equation shown in Equ.4. 
However, when SAR carrier is a satellite, its conditions are 
not subjected to CSL and that item b 2 Hr should be 
maintained in Equ.5 
Ar = -B sin(# - a) (4) 
2.2.2 Algorithm for Flattening Phase 
Based on the principle mentioned above, algorithm for 
flattening phase can be described in details as follows. 
Assuming an image with the width of M (slant range direction) 
and length of N (azimuth direction), steps of the algorithm can 
be implemented in the following order. 
1. Calculate basic parameters including B , a and H . 
(These letters are defined in chapter 2.1) 
2. Calculate slant range of every point on the image 
according to nearest range r lst and farthest range r lasl 
shown in Equ.8. 
r, = r. +jx (8) 
J \St J jy 
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