The start-stop approximation

Abstract

For the analysis of the SAR data inversion algorithm in Chapters 2 through 5, we have employed the start-stop approximation, which is considered standard in the literature, see, e.g., [25, 40, 76, 79] and also [86]. It assumes that the radar antenna is at standstill while it sends the interrogating pulse toward the target and receives the scattered response, after which the antenna moves down the flight track to the position where the next pulse is emitted and received.

Appendix 6.A Radiation by a moving source

Equation (6.3a) governs the radiation of waves by a moving source. It can be solved directly, i.e., by employing the fundamental solution in the original coordinates rather than by using the Lorentz-transformed coordinates (as done in Section 6.1). The fundamental solution of the d’Alembert operator on the left-hand side of equation (6.3a) (or equation (6.1)) is given by:

$$\displaystyle{ \mathcal{E}(t,\boldsymbol{z}) = \frac{\mathcal{H}(t)} {4\pi } \frac{\delta (\vert \boldsymbol{z}\vert - ct)} {t}, }$$

where $\mathcal{H}(t)\stackrel{\text{def}}{=}\left \{\begin{array}{@{}l@{\quad }l@{}} 1,\quad &t\geqslant 0,\\ 0,\quad &t <0, \end{array} \right.$ is the Heaviside function, and $\delta (\vert \boldsymbol{z}\vert - ct)$ is a single layer of unit magnitude on the expanding sphere of radius ct centered at the origin. Then, the solution of equation (6.3a) is obtained by convolution:

$$\displaystyle\begin{array}{rcl} u(t,\boldsymbol{z})& =& \frac{1} {4\pi }\int \limits _{-\infty }^{t}dt^{{\prime}}\iiint \limits _{ \mathbb{R}^{3}} \frac{\delta (\vert \boldsymbol{z} -\boldsymbol{ z}^{{\prime}}\vert - c(t - t^{{\prime}}))} {t - t^{{\prime}}} \\ & &\qquad \qquad \ \ \ \ \cdot P(t^{{\prime}})\delta (z_{ 1}' - \mathfrak{v}t')\delta (z_{2}' + L)\delta (z_{3}' - H)d\boldsymbol{z}' \\ & =& \frac{1} {4\pi }\int \limits _{-\infty }^{t}\frac{\delta (\vert \boldsymbol{z} -\boldsymbol{ x}(t^{{\prime}})\vert - c(t - t^{{\prime}}))} {t - t^{{\prime}}} P(t^{{\prime}})dt^{{\prime}} \\ & =& \frac{1} {4\pi }\int \limits _{-\infty }^{\mu (t)} \frac{\vert \boldsymbol{z} -\boldsymbol{ x}(t^{{\prime}})\vert \,\delta (\mu -ct)} {(c\vert \boldsymbol{z} -\boldsymbol{ x}(t^{{\prime}})\vert - (z_{1} - \mathfrak{v}t^{{\prime}})\mathfrak{v})(t - t^{{\prime}})}P(t^{{\prime}})d\mu.{}\end{array}$$

(6.111)

On the last line of (6.111), we have introduced a new integration variable: $\mu =\mu (t^{{\prime}}) = \vert \boldsymbol{z} -\boldsymbol{ x}(t^{{\prime}})\vert + ct^{{\prime}}\equiv \sqrt{(z_{1 } - \mathfrak{v}t^{{\prime} } )^{2 } + (z_{2 } + L)^{2 } + (z_{3 } - H)^{2}} + ct^{{\prime}}$. To evaluate the last integral on the right-hand side of (6.111), one needs to find the value of t ^′ for which μ(t ^′) = ct, or, equivalently, for which the argument of the δ-function becomes equal to zero. Finding such t ^′ also means that one can substitute $\vert \boldsymbol{z} -\boldsymbol{ x}(t^{{\prime}})\vert = c(t - t^{{\prime}})$ in the numerator, so that (6.111) reduces to

$$\displaystyle{ u(t,\boldsymbol{z}) = \frac{1} {4\pi } \frac{cP(t^{{\prime}})} {c\vert \boldsymbol{z} -\boldsymbol{ x}(t^{{\prime}})\vert - (z_{1} - \mathfrak{v}t^{{\prime}})\mathfrak{v}}\bigg\vert _{\mu (t^{{\prime}})=ct}. }$$

(6.112)

The equation μ(t ^′) = ct, which can be written as

$$\displaystyle{ \sqrt{(z_{1 } - \mathfrak{v}t^{{\prime} } )^{2 } + (z_{2 } + L)^{2 } + (z_{3 } - H)^{2}} + ct^{{\prime}} = ct, }$$

(6.113)

is a quadratic equation with respect to t ^′. It is very similar to equation (6.31). The solution of equation (6.113) determines the retarded moment of time at which the trajectory of the moving antenna intersects with the lower portion of the characteristic cone¹⁵(or light cone) that has its vertex at $(t,\boldsymbol{z})$. The appropriate root of equation (6.113) is

$$\displaystyle\begin{array}{rcl} t^{{\prime}}& =& \frac{1} {\beta ^{2}} \left (t -\frac{\mathfrak{v}z_{1}} {c^{2}} -\sqrt{\left (t - \frac{\mathfrak{v}z_{1 } } {c^{2}} \right )^{2} -\beta ^{2}\left (t^{2} -\frac{z_{1}^{2} + (z_{2} + L)^{2} + (z_{3} - H)^{2}} {c^{2}} \right )}\right ) \\ & =& \frac{\sigma } {\beta } -\frac{1} {\beta c}\sqrt{\frac{(z_{1 } - \mathfrak{v}t)^{2 } } {\beta ^{2}} + (z_{2} + L)^{2} + (z_{3} - H)^{2}} \\ & =& \frac{\sigma } {\beta } -\frac{\sqrt{\zeta _{1 }^{2 } + (z_{2 } + L)^{2 } + (z_{3 } - H)^{2}}} {\beta c} = \frac{\sigma } {\beta } - \frac{\rho } {\beta c}, {}\end{array}$$

(6.114)

where we have used the definition (6.2) of the Lorentz transform, as well as formula (6.5). Expression (6.114) for t ^′ is to be substituted into the numerator on the right-hand side of (6.112). As for the denominator of (6.112), taking into account that μ(t ^′) = ct we can write:

$$\displaystyle{ \begin{array}{rcl} & &c\sqrt{(z_{1 } - \mathfrak{v}t^{{\prime} } )^{2 } + (z_{2 } + L)^{2 } + (z_{3 } - H)^{2}} - (z_{1} - \mathfrak{v}t^{{\prime}})\mathfrak{v} \\ & =&c^{2}(t - t^{{\prime}}) - (z_{1} - \mathfrak{v}t^{{\prime}})\mathfrak{v} \\ & =&c^{2}t - z_{1}\mathfrak{v} - t^{{\prime}}(c^{2} - \mathfrak{v}^{2}) \\ & =&c^{2}\left (t -\frac{z_{1}\mathfrak{v}} {c^{2}} - t^{{\prime}}\beta ^{2}\right ) \\ & =&c^{2}\sqrt{\left (t - \frac{\mathfrak{v}z_{1 } } {c^{2}} \right )^{2} -\beta ^{2}\left (t^{2} -\frac{z_{1}^{2}+(z_{2}+L)^{2}+(z_{3}-H)^{2}} {c^{2}} \right )} =\beta c\rho, \end{array} }$$

(6.115)

where we have used formulae (6.2) and (6.114). Altogether, substituting (6.114) and (6.115) into (6.112), we see that the expression for $u(t,\boldsymbol{z})$ reduces precisely to what is given by formula (6.4), as expected.

Note that equation (6.113) that defines the retarded moment of time t ^′ is so simple (quadratic) only for the case of a uniform and straightforward motion of the antenna. When the motion is accelerated and/or non-straightforward, the resulting counterpart of equation (6.113) may become far more complicated; often, it can only be solved numerically as done, e.g., in [157]. In the theory of electromagnetism, the corresponding solutions of the d’Alembert equation can be interpreted as field potentials due to moving charges, and in this case they are referred to as the Liénard-Wiechert potentials, see [106, Chapter 8].

Equation (6.113) can also be solved approximately, as done, e.g., in [156]. Given that the square root on its left-hand side is the distance between the observation point $\boldsymbol{z} = (z_{1},z_{2},z_{3}) \in \mathbb{R}^{3}$ (location of the target) and the moving antenna at the moment of time t ^′, we can represent this distance using the law of cosines and then derive an approximate expression with the help of the Taylor’s formula. As in Section 6.1, let $\gamma _{\boldsymbol{z}}$ denote the angle between the velocity $\mathfrak{v}$ and the direction from the antenna $\boldsymbol{x}(t^{{\prime}})$ to the target $\boldsymbol{z}$, see Figure 6.1. Also recall that $r = \sqrt{z_{1 }^{2 } + (z_{2 } + L)^{2 } + (z_{3 } - H)^{2}}$, see formula (6.8). Then,

$$\displaystyle\begin{array}{rcl} \sqrt{ (z_{1} - \mathfrak{v}t^{{\prime}})^{2} + (z_{2} - H)^{2} + z_{3}^{2}}& =& \sqrt{r^{2 } + (\mathfrak{v}t^{{\prime} } )^{2 } - 2r\mathfrak{v}t^{{\prime} } \cos \gamma _{\boldsymbol{z}}} {}\\ & =& r\sqrt{1 - \frac{2r\mathfrak{v}t^{{\prime} } \cos \gamma _{\boldsymbol{z} } } {r} + \left (\frac{\mathfrak{v}t^{{\prime}}} {r} \right )^{2}} {}\\ & \approx & r - \mathfrak{v}t^{{\prime}}\cos \gamma _{ \boldsymbol{z}}. {}\\ \end{array}$$

In this case, equation (6.113) becomes:

$$\displaystyle{ r - \mathfrak{v}t^{{\prime}}\cos \gamma _{ \boldsymbol{z}} + ct^{{\prime}} = ct, }$$

so that its solution is

$$\displaystyle{ t^{{\prime}} = \frac{t -\frac{r} {c}} {1 -\frac{\mathfrak{v}} {c}\cos \gamma _{\boldsymbol{z}}}. }$$

Substituting this expression into formula (6.112), we arrive at the solution in the form (6.14).