Baseline

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 $\angle(x,y)$.)

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

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

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

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

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

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

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

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

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

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

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

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

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