# SAR Imaging Distortions Induced by Topography: A Compact Analytical Formulation for Radiometric Calibration

## Abstract

**:**

## 1. Introduction

## 2. Analytical Formulation

#### 2.1. Cylindrical Coordinate System

#### 2.2. Surface Parametric Representation

#### 2.3. Metric Properties

#### 2.4. Geometric Interpretation

#### 2.5. Local Incidence Angle

#### 2.6. Analytical Consistency

#### 2.7. Singular Behavior

## 3. Discrete Mapping

#### 3.1. Mapping in Digital Image Processing

#### 3.2. Inverse Cylindrical Mapping

## 4. Discrete Implementation

#### 4.1. Range-Doppler Backward Georeferencing

#### 4.2. Look-Angle Function Regridding

#### 4.3. Image Domain Processing

## 5. Experimental Results

^{2}. The characteristic “V” shaped pattern, which is clearly recognizable in Figure 9a, is indeed associated with the surface lake.

## 6. Conclusions

- (1)
- The adopted formalism is rigorously derived by using the fundamental concepts of differential geometry of surfaces [27].
- (2)
- The expression of the local radiometric distortion has been established in analytical form, in terms of the magnitude of the gradient of a scalar function defined in the SAR image space. Such a function is indeed the look-angle function, which analytically captures in a convenient form the information on the topographic reliefs as seen by the sensor.
- (3)
- The proposed formalism turns out compact and expressive; further, accordingly, the inherent look-angle based encoding scheme can be valuable because of its ease of implementation. As a matter of fact, the novel formulation reduces the problem to a 2D domain calculation, with relevant computation carried out on a regular grid in the image domain, thus requiring only scalar functions handling, with important implications in terms of computational efficiency.
- (4)
- The proposed formulation maintains also the local consistency with the classical approach in [17]. Nonetheless, the area-stretching concept might be more insightful with respect to the projection angle notion.
- (5)
- The proposed method completely circumvents the drawback of the pixel-area fragmentation issue, because it follows an approach based on a cylindrical inverse mapping. Conversely, the existing forward-mapping-based approaches [24] entail the burden of pixel-area fragmentation and relevant integration [20].

## Funding

## Conflicts of Interest

## Appendix A

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**Figure 3.**3D geometrical scheme of a ground surface patch: ${\widehat{n}}^{\prime}$ is the surface unit normal, ${\chi}_{l}$ is local incidence angle, $\mathsf{\omega}$ is the projection angle, $\widehat{z}$ and $\widehat{a}\left(=\widehat{x}\right)$are the vertical and azimuth directions, respectively.

**Figure 4.**Schematic illustration of the discrete mapping for spatial transformation $q=\tau (s$), with $s=\left(u,v\right)$ and $q=\left(r,a\right)$.

**Figure 7.**Elevation (m) of the DEM: representation in the image space. The range direction is from left to right; the azimuth direction is from bottom to top.

**Figure 8.**The range direction is from left to right; the azimuth direction is from bottom to top: (

**a**) Look Angle Function (LAF) (degree):$\theta =\theta \left(r,a\right)$; (

**b**) a mask identifying (red) layover and (blue) shadow areas.

**Figure 9.**Magnitude of the (range-weighted) partial derivative of look-angle function along: (

**a**) the azimuth direction (dB), $\left|r\frac{\partial \theta}{\partial a}\right|$; (

**b**) the range direction (dB), $\left|r\frac{\partial \theta}{\partial r}\right|$.

**Figure 10.**(

**a**) Simulated radiometric-distortion image (dB) associated with the ground surface area; (

**b**) local incidence angle (LIA) (degree): ${\mathsf{\chi}}_{l}={\mathsf{\chi}}_{l}\left(r,a\right)$.

**Figure 11.**(

**a**) ${\tilde{\sigma}}_{}^{0}$ (dB) image obtained from SAR data without compensation of topography-induced radiometric distortions; (

**b**) ${\mathsf{\sigma}}_{}^{0}$ (dB) image obtained from SAR data including the compensation of topography-induced radiometric distortions. A mask identifying (red) layover and (blue) shadow areas is superimposed.

**Figure 12.**Distribution of the backscattering coefficient ${\mathsf{\sigma}}_{}^{0}$ (dB): (

**a**) obtained without compensation of topography-induced radiometric distortions; (

**b**) obtained by including the compensation of topography-induced radiometric distortions.

**Figure 14.**Backscattering coefficient ${\mathsf{\sigma}}_{}^{0}$ [dB] without the compensation of topography-induced distortions vs. local incidence angle [degree].

**Figure 15.**Backscattering coefficient ${\mathsf{\sigma}}_{}^{0}$(dB) after the compensation of topography-induced distortions vs. local incidence angle (degree).

SAR Sensor | CSK | |
---|---|---|

Acquisition Date | 12 April 2009 | |

Observation Direction | Right looking | |

Polarization | HH | |

Orbit | 7260 | |

Orbit Direction | Ascending | |

Carrier Frequency (GHz) | 9.60 | |

Off-nadir Angle (degree) | 35.90 | |

Sampling Frequency (MHz) | 112.50 | |

Chirp Bandwidth (MHz) | 42.00 | |

PRF (Hz) | 3104.31 | |

Range Pixel Spacing (m) | 1.33 | |

Azimuth Pixel Spacing (m) | 2.34 | |

First near | (latitude (deg), longitude (deg)) | (42.145, 13.202) |

First far | (42.215, 13.755) | |

Last near | (42.562, 13.103) | |

Last far | (42.632, 13.660) |

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Imperatore, P.
SAR Imaging Distortions Induced by Topography: A Compact Analytical Formulation for Radiometric Calibration. *Remote Sens.* **2021**, *13*, 3318.
https://doi.org/10.3390/rs13163318

**AMA Style**

Imperatore P.
SAR Imaging Distortions Induced by Topography: A Compact Analytical Formulation for Radiometric Calibration. *Remote Sensing*. 2021; 13(16):3318.
https://doi.org/10.3390/rs13163318

**Chicago/Turabian Style**

Imperatore, Pasquale.
2021. "SAR Imaging Distortions Induced by Topography: A Compact Analytical Formulation for Radiometric Calibration" *Remote Sensing* 13, no. 16: 3318.
https://doi.org/10.3390/rs13163318