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# A mathematical perspective on radar interferometry

• * Corresponding author: Mikhail Gilman
• Radar interferometry is an advanced remote sensing technology that utilizes complex phases of two or more radar images of the same target taken at slightly different imaging conditions and/or different times. Its goal is to derive additional information about the target, such as elevation. While this kind of task requires centimeter-level accuracy, the interaction of radar signals with the target, as well as the lack of precision in antenna position and other disturbances, generate ambiguities in the image phase that are orders of magnitude larger than the effect of interest.

Yet the common exposition of radar interferometry in the literature often skips such topics. This may lead to unrealistic requirements for the accuracy of determining the parameters of imaging geometry, unachievable precision of image co-registration, etc. To address these deficiencies, in the current work we analyze the problem of interferometric height reconstruction and provide a careful and detailed account of all the assumptions and requirements to the imaging geometry and data processing needed for a successful extraction of height information from the radar data. We employ two most popular scattering models for radar targets: an isolated point scatterer and delta-correlated extended scatterer, and highlight the similarities and differences between them.

Mathematics Subject Classification: Primary: 78A46, 78A48; Secondary: 78A55, 86A30.

 Citation: • • Figure 1.  Traditional presentation of geometry for radar interferometry in the vertical cross-track plane. Points ${\mathit{\boldsymbol{z}}}$ and ${\mathit{\boldsymbol{T}}}$ are on the same circle centered at ${\mathit{\boldsymbol{x}}}^ {(0)}$ (dashed line)

Figure 2.  Calculation of the flat Earth phase in cross-track radar interferometry. The points ${\mathit{\boldsymbol{z}}}'$ and ${\mathit{\boldsymbol{T}}}$ have zero elevation, and both circles are centered at ${\mathit{\boldsymbol{x}}}^ {(0)}$

Figure 3.  The annulus in the vertical cross-track plane due to the main lobe of the $\mathop{\mathrm{{sinc}}}\nolimits$ in $(49)$. It is centered at ${\mathit{\boldsymbol{x}}}$ and has central radius $R_ {\mathit{\boldsymbol{z}}}$. Its thickness $2\Delta_ {\rm{R}}$ is defined by the system bandwidth $B$, see formula (50). Vertical localization of radar targets can be performed using either interferometry or external information about the elevation

Figure 4.  Radar interferometry with two coequal antennas. All coordinates are specified in the slant reference frame $(u,v)$. To illustrate formula (88), note that $l_1+l_2 = z_ {vb}-z_ {va}$

Figure 5.  Interferometry of an extended vertically stratified scatterer. In the analysis of (106), $V(z'_u) = \tau \mathop{\mathrm{{sinc}}}\nolimits (Bz'_u/c)$ is considered as a function of $z_u = y_u+z'_u$ (cf. Fig. 3)

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