The Formation of SAR Images

A SAR system illuminates a scene with microwaves and records both the amplitude and the phase of the back-scattered radiation, making it a coherent imaging process. An inherent feature of SAR imaging is speckle which arises because each resolution cell contains many scatterers; the phases of the return signals from these scatterers are randomly distributed and speckle is caused by the resulting interference.

The received signal is sampled and converted into a digital image. Note that the pixel spacing in the final image and the resolution of the system may not be the same. If a system with a resolution of 5m is set to oversample the signal and produces an image in which each pixel corresponds to 3m, you do not get a more detailed image -- the information is just smeared out over several pixels. In this case, the image is correlated and is much more difficult to process.

The field recorded at pixel $x$, denoted $E(x)$, can be written

$$E(x) \ = \ \sum_s a(s) \; \exp(i \, \phi(s) ) \; h(s,x)$$

where the summation ranges over the scatterers, $a(s)$ and $\phi(s)$ are the amplitude and phase (respectively) of the signal received from scatterer $s$, and $h$ is the instrument (or point-spread) function. The value of $h$ is near 1 when $s$ is in or near the resolving cell corresponding to pixel $x$, and near zero otherwise. If we assume that $h$ is translation-invariant (does not depend on $x$) then it can be written as a one-parameter function: $h(s-x)$.

Thus the detected field $E$ is an array of complex numbers. The square of the modulus of the field at $x$ is called the detected intensity at $x$; the square-root of the intensity is called the envelope or the amplitude. Note that this is not the same as the amplitude of the received signal because the received field is perturbed by the instrument function. The amplitude of the received signal, $a(s)$, is called the reflectivity, and its square is called the surface cross-section. It is this that we try to recreate when reducing the speckle.

Datasets for SAR images can be stored as either intensity or amplitude data. It is important that you know what type of data you are using. Many images are stored as amplitude since the speckle is less obtrusive when these images are displayed.