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The basic configuration of InSAR is shown in figure D.1.

Figure D.1: Geometric configuration for InSAR. $ R_i$ are the range vectors to the corresponding resolution element. The statevector of the reference satellite is denoted by $ \rho _1$. (h denotes the satellite height, $ \mu $ the location angle of the co-registered resolution cell in the interferogram.)
There are different representations for the baseline, see Figure D.2.
Figure D.2: Definition of the baseline parameters. (a) parallel/perpendicular; (b) horizontal/vertical; (c) length/orientation; Position 1 is the reference position. $ {B_{\parallel }}$ $ >$ 0 when $ R_1 > R_2$, where $ R_i$ is the corresponding slant range. The angle $ \ensuremath {\alpha }$ is defined counter-clockwise from the reference satellite (1), starting from the horizontal at the side of the look direction.

Conversions between baseline representations:

Table D.1: Conversion between baseline representations (note that the four quadrant arctangent should be used).
  $ [B_h, B_v]$ $ [B, \alpha]$ $ [{B_{\perp}}, {B_{\parallel}}]$
$ [B_h, B_v]$ - $ B_h=B\cos\alpha$ $ B_h={B_{\perp}}\cos\theta + {B_{\parallel}}\sin\theta$
  - $ B_v=B\sin\alpha$ $ B_v={B_{\perp}}\sin\theta - {B_{\parallel}}\cos\theta$
$ [B, \alpha]$ $ \alpha=\arctan(B_v/B_h)$ - $ \alpha=\theta-\arctan({B_{\parallel}}/{B_{\perp}})$
  $ B = \sqrt{B_h^2+B_v^2}$ - $ B=\sqrt{{{B_{\parallel}}}^2+{{B_{\perp}}}^2}$
$ [{B_{\perp}}, {B_{\parallel}}]$ $ {B_{\parallel}}=B_h\sin\theta-B_v\cos\theta$ $ {B_{\parallel}}=B\sin(\theta-\alpha)$ -
  $ {B_{\perp}}=B_h\cos\theta+B_v\sin\theta$ $ {B_{\perp}}=B\cos(\theta-\alpha)$ -

The baseline parameters can be computed when the statevectors of the points M, S and P (master ,slave and point on surface) are known. (The distance between the points x and y is denoted by $ {\rm d(x,y)}$; and the sharp angle between two vectors x and y by $ \angle(x,y)$.)

$\displaystyle B = {\rm d}(M,S)$ (D1)

$\displaystyle {B_{\parallel}}= {\rm d}(M,P) - {\rm d}(S,P)$ (D2)

Now the perpendicular baseline has to be computed. The definition states that $ {B_{\perp}}$ is positive if the slave satellite is to the right of the slant range line of the master. Which yields for a mountain an increasing phase from foot to summit(?) We had some trouble finding a simple expression to find out the correct sign but at the moment we do something like the following.

$\displaystyle {{B_{\perp}}}^2 = B^2 - {{B_{\parallel}}}^2$ (D3)

$\displaystyle \vec{r}_1 = \vec{M} - \vec{P}$ (D4)

$\displaystyle \vec{r}_2 = \vec{S} - \vec{P}$ (D5)

$\displaystyle \gamma_1 = \angle(\vec{P},\vec{r}_1)$ (D6)

$\displaystyle \gamma_2 = \angle(\vec{P},\vec{r}_2)$ (D7)

$\displaystyle {\rm sign} = \left\{ \begin{array}{ll} -1 & , \gamma_1 < \gamma2 \\ \phantom{-}1 & , \gamma_1 > \gamma2 \end{array}\right.$ (D8)

$\displaystyle {B_{\perp}}= {\rm sign} \sqrt{{{B_{\perp}}}^2}$ (D9)

$\displaystyle \theta = \angle(\vec{M},\vec{r}_1)$ (D10)

$\displaystyle \ensuremath{\alpha}= \theta - \arctan2({B_{\parallel}},{B_{\perp}})$ (D11)

$\displaystyle {B_{h}}= B \cos(\ensuremath{\alpha})$ (D12)

$\displaystyle {B_{v}}= B \sin(\ensuremath{\alpha})$ (D13)

next up previous contents
Next: Interferogram Up: Definitions Previous: Constants   Contents
Leijen 2009-04-14