A treatise on InSAR geometry and 3D displacement estimation

Abstract

It is well known that InSAR phase observations are only sensitive to the projection of the 3D displacement vector onto the radar line-of-sight (LoS) direction. We require at least three LoS observations to uniquely estimate the three displacement components, and the system of equations needs to have a full rank coefficient matrix. Unfortunately, in many practical situations, only two LoS observations are available at most (i.e., ascendingand descending), resulting in an underdetermined system with an infinite amount of possible solutions. Yet, this has not prevented many authors from performing LoS decompositions that are fundamentally flawed. Starting with a mathematical framework based on linear algebra, we introduce the concept of the null line, the direction in which no displacements can be observed, and identify the strict criteria to perform decompositions and projections. We propose using a null-line aligned (NLA) referenceframe, which results in bias-free estimates. Based on a literature survey, we identify the most common flaws in handling 3D InSAR geometry, and develop a taxonomy to label different classes of fallacies. This work results in recommendations for a more optimal and uniform handling of InSAR geometry, in terms ofclaims and results.

Estimating Three-Dimensional Displacements with InSAR: the Strapdown Approach

“…The actual (true) orientation of the TLN frame has a dominant impact on the precision of the final estimates. With two LoS observation geometries (ascending and descending), displacement components in the direction of the nullline (Brouwer and Hanssen, 2023b) cannot be observed: the more either the transversal or normal direction aligns with of the null-line, the less precise that parameter can be estimated. The most favourable option is therefore when the plane spanned by the transversal and normal axis (TN-plane) is orthogonal to the null-line, i.e., when Λ = ϕ and Φ = ζ, where ϕ and ζ are the azimuth angle and elevation angle of the null-line, respectively.…”

“…InSAR scatterers are typically not situated at ideal locations, and the observations have an imaging geometry that is not optimal for retrieving full three-dimensional (3D) displacements. Moreover, they are only sensitive to the projection of the 3D displacement vector onto the radar line-of-sight (LoS) direction, d LoS , along a plane orthogonal to the LoS (Massonnet and Feigl, 1998;Fialko et al, 2002;Hanssen, 2001;Wright et al, 2004;Brouwer and Hanssen, 2023b), i.e., T is the 3D displacement vector in east, north, and up direction, respectively. 1 P LoS ⊥ is the orthogonal projec-tor onto the LoS, where θ is the incidence angle towards the radar, and α d is the azimuth of its zero-Doppler plane (ZDP) at the position of the target, in the direction towards the satellite, see Brouwer and Hanssen (2023b, Fig.…”

“…Ideally, a decomposition of the LoS displacement vector in three orthogonal directions is enabled. Yet, this requires at least three independent LoS observations from significantly different viewing geometries, but since all current SAR satellites have very similar viewing geometries, the maximum number of available and effective observations often reduces to two, i.e., ascending and descending, resulting in an underdetermined problem with an infinite number of solutions along a solution line (Brouwer and Hanssen, 2023b).…”

“…Many contemporary InSAR information products, including most publicly available ones, 'circumvent' this problem by disregarding the north component of the deformation, and asserting a decomposition into the east and up components only (Crosetto et al, 2020). Yet, it is well known that this approach produces inherently biased estimates, particularly for the up component (Brouwer and Hanssen, 2023b). An alternative option is to use the null-line aligned (NLA) coordinate system as proposed in Brouwer and Hanssen (2023b), which ensures unbiased estimates, but produces results that can be more challenging to interpret for non-experts.…”

“…Yet, it is well known that this approach produces inherently biased estimates, particularly for the up component (Brouwer and Hanssen, 2023b). An alternative option is to use the null-line aligned (NLA) coordinate system as proposed in Brouwer and Hanssen (2023b), which ensures unbiased estimates, but produces results that can be more challenging to interpret for non-experts.…”