synth

synth is a package to generate synthetic stacks of single-look complex (SLC) SAR data for InSAR time series analysis. Realistic results are achieved by using the global seasonal coherence dataset to create realistic decorrelation, and by using the troposim package to generate tropospheric turbulence noise.

The overall workflow for generating a stack of SLCs is as follows:

• Choose an area of interest (AOI) and set the parameters for the noise sources
• Generate the "truth phase"/"propagation phase" sources, which may include deformation, tropospheric turbulence, and phase ramps
• Download the coherence and backscatter data to use for correlation models
• For each pixel, create a coherence matrix using the decorrelation model and the truth phase differences
• Generate correlated noise samples for this pixel and multiply by the average backscatter

The SLCs are saved in GeoTIFF format, one raster per requested date. The propagation phases are also saved as 3D data cubes in HDF5 files.

Note that the "truth" phase sources include tropospheric noise and phase ramps. One common use is to compare phase linking algorithms, which estimate a wrapped phase time series; in this case, the desired outputs include all "propagation phases".

Propagation phase sources

Tropospheric Turbulence

The tropospheric turbulence noise uses the troposim package (Staniewicz, 2024), which uses an isotropic representation of the power spectral density (PSD) (cite: Hanssen, 2001).

Phase ramp

To simulate longer-wavelength noise sources, we allow simple planar phase ramps to be added at each date. The maximum amplitude may be set, and the orientation of each date's phase ramp is randomized

Deformation sources

Currently, the following deformation sources are supported:

• Gaussian blob
• (to move from troposim): Okada model
• (to move from troposim): Geertsma reservoir source model

Decorrelation Noise

At each pixel, we use some decorrelation model to create a coherence matrix $T$, where element $|T_{ij}| = \gamma \exp(-1j \phi)$ contains the coherence magnitude $\gamma$ and interferometric phase $\phi$.

To generate correlated random samples,

1. Find $L$ such that $T = L L^H$ using the Cholesky factor¹.
2. Generate an instance $\mathbf{y}$ of Circular Complex Gaussian (CCG) noise.
3. Multiply $\mathbf{z} = T \mathbf{y}$ to obtain the correlated sample.
4. (optional) If we want the amplitudes to match real data, we can multiply $A \mathbf{z}$, where $A = \text{diag}(\mathbf{\boldsymbol{\sigma} })$.

Here, we use the coherence matrix, which is power-normalized. The relation to the sample covariance matrix covariance matrix $C$ is

$$ C = T \circ \boldsymbol{\sigma}\boldsymbol{\sigma}^{H} $$

where $\circ$ is the Hadamard product, and $\boldsymbol{\sigma}= [\sigma_{1}, \dots, \sigma_{N}]^{T}$ is the vector of SLC standard deviations: $\sigma_{k} = \sqrt{ \mathbf{E}[\mid x_{k}\mid^{2} ] }$ .

Making the decorrelation realistic

Choosing random decorrelation parameters for each pixel in space would not be conducive for running real workflows that rely on averaging pixels in space. Therefore, we use the global coherence dataset to estimate the decorrelation parameters for each pixel.

The coherence dataset, publicly available on AWS, contains seasonal coherence models and average backscatter. The model for coherence $\gamma$ was

$$ \gamma(t) = (1 - \rho_{\infty})\exp(-t / \tau) + \rho_{\infty} $$ where $\rho_{\infty}$ is the long-term coherence, $\tau$ is the decorrelation time, and $t$ is the time in days. The backscatter model $\sigma_0$ was also estimated for each season.

Using the decorrelation model parameters

Two example $\rho_{\infty}$ maps are shown below over the Central Valley in California. The first is for winter, and the second is for fall.

To create an exponentially decaying correlation matrix, we could attempt to blend all four seasons' model parameters; however, for the current version, we have simply chosen the minimum $\rho_{\infty}$ to use for the entire pixel's stack. For certain regions, even using the minimum $\rho$ did not produce very strong decorrelation noise. Therefore we also added an option to use, as the long-term coherence value

$$ \begin{cases} \rho^2 & \text{if } \rho > 0.2 \\ \rho^4 & \text{otherwise} \end{cases} $$

This is a heuristic to shrink all long-term coherences toward zero, where the $\rho^{4}$ case is to add more noise in low-coherence regions to combat the upwardly biased coherence estimator².

Modeling seasonal decorrelation

Since certain regions show large differences for $\rho_{\infty}$ we create a "seasonal" map for pixels which have a peak-to-peak change greater than some threshold (e.g. 0.5). For pixels with $\max_{k}(\rho^{k}{\infty}) - \min{k}(\rho_{\infty}^{k})>0.5$, we model decorrelation similar to (cite Even and Schulz, 2018). The correlation $\gamma$ for the interferogram formed using images from time $t_{m}$ and $t_{n}$ is

$$ \gamma(t_{m}, t_{n}) = \left( A + B\cos\left( \frac{2\pi t_{n}}{365} \right) \right) \left( A + B\cos\left( \frac{2\pi t_{m}}{365} \right) \right) $$

where $A, B$ are related to $\rho_{\infty}$ and $\rho_{0}$ (the initial correlation) by $\gamma_{0} = (A + B)^{2}$ and $\gamma_{min} \triangleq \rho_{\infty} = (A - B)^{2}$. We can solve for the $A, B$ parameters by inverting this relation: Since $\gamma_{0}=1$ in the (Kellndorfer, 2020) model, then $B = (1 - A)$, and thus

$$ \begin{gather} A = \frac{1}{2} (1 + \sqrt{ \rho_{\infty} }) \\ B = \frac{1}{2} (1 - \sqrt{ \rho_{\infty} }) \end{gather} $$

Footnotes

1. Any factorization such that $\Sigma = L L^{T}$ would work. The multivariate_normal function has three options: Cholesky, eigenvalue decomposition with eigh, and SVD. The Cholesky is the fastest, but least robust if your covariance matrix is near-singular. ↩

2. While it's true that the Kellndorfer paper says the bias is very low due to the number of looks they took, my initial Central Valely test had very few regions that looks like pure noise even after 1 year. ↩