Principles of RRINGG

Overview

RRINGG provides two main functionalities for combining GNSS and InSAR data:

  1. Correction of the InSAR velocity field

  2. Referencing of InSAR time series in a global reference frame (e.g., ITRF)

Both rely on fitting simple planar models (affine surfaces) to the differences between GNSS and InSAR observations. The GNSS network is used as the geodetic truth, ensuring that InSAR products are both bias-free and compatible with international standards.

Correction of the InSAR Velocity Field

Objective: Adjust InSAR velocities by removing systematic biases through GNSS calibration.

Mathematical Model

The bias between InSAR and GNSS velocities is assumed to vary linearly with geographic position:

\[\Delta v(x, y) = a \cdot x + b \cdot y + c\]
where:

  • \((x, y)\) = geographic coordinates of a station (longitude, latitude),

  • \(a, b, c\) = plane parameters estimated from GNSS–InSAR velocity differences.

The corrected InSAR velocity field is:

\[v_{\text{InSAR, corrected}}(x, y) = v_{\text{InSAR}}(x, y) - \Delta v(x, y)\]

Steps

  1. Compute GNSS–InSAR velocity differences at station locations.

  2. Fit a plane (\(a, b, c\)) using least-squares or RANSAC.

  3. Subtract the plane from the InSAR raster.

This ensures the InSAR velocity map is consistent with GNSS observations.

Referencing InSAR Time Series

Objective: Convert relative InSAR displacements into absolute displacements in a global frame. Two complementary methods are available.

Method 1: Mean Velocity Referencing

Principle - Assumes GNSS motion is linear over the observation period. - GNSS velocities (projected in LOS) are integrated to reconstruct displacements. - At each InSAR acquisition epoch \(t_i\), the GNSS-predicted displacement is:

\[G(x, y, t_i) = v_{\text{GNSS, LOS}}(x, y) \cdot (t_i - t_{\text{ref}})\]

where \(t_{\text{ref}}\) is the chosen reference date (usually first acquisition).

  • The bias is modeled as a plane:

    \[B(x, y, t_i) = D_{\text{InSAR, rel}}(x, y, t_i) - G(x, y, t_i) = a_i \cdot x + b_i \cdot y + c_i\]

  • The absolute displacement is then:

    \[D_{\text{InSAR, abs}}(x, y, t_i) = D_{\text{InSAR, rel}}(x, y, t_i) - B(x, y, t_i)\]

Key Assumptions - Deformation is essentially linear (tectonic drift, subsidence). - GNSS sampling can be sparse: only mean velocities are needed.

Advantages - Simple and efficient. - Robust even with limited GNSS temporal coverage.

Limitations - Cannot capture non-linear deformation (postseismic relaxation, volcanic inflation, seasonal cycles).

Method 2: Temporal Differential Referencing

Principle - Uses full GNSS time series (not just mean velocities). - A common reference epoch \(t_0\) is set where both GNSS and InSAR displacements are zero. - GNSS displacement at each epoch is:

\[G^*(x, y, t_i) = G(x, y, t_i) - G(x, y, t_0)\]

  • At each epoch \(t_i\), the bias is:

    \[B(x, y, t_i) = D_{\text{InSAR, rel}}(x, y, t_i) - G^*(x, y, t_i) = a_i \cdot x + b_i \cdot y + c_i\]

  • The absolute InSAR displacement is:

    \[D_{\text{InSAR, abs}}(x, y, t_i) = D_{\text{InSAR, rel}}(x, y, t_i) - B(x, y, t_i)\]

Key Assumptions - Requires dense GNSS time series (daily sampling preferred). - Designed to capture transient and non-linear deformation.

Advantages - Can correct for earthquakes, volcanic unrest, hydrological or seasonal deformation. - Provides high-fidelity referencing in dynamic environments.

Limitations - More computationally expensive. - Sensitive to GNSS time series noise.

Comparison of Referencing Methods

Method Comparison

Feature

Mean Velocity Referencing

Temporal Differential Referencing

GNSS Input

Mean velocities only

Full displacement time series

Deformation Assumption

Linear

Linear + Non-linear

GNSS Sampling Requirement

Sparse OK

Dense required

Computational Cost

Low

High

Best for

Tectonic drift, steady subsidence

Earthquakes, volcanoes, seasonal cycles

Limitations

Misses transients

Sensitive to GNSS noise