0025 The oversampling of the SLCs is measured from the speckle correlation¶

Status¶

Accepted

Date¶

2026-10-05

Context¶

Neighbouring pixels of an SLC are correlated (oversampling of the focused data, interpolation of the coregistration), so n SHPs are fewer independent looks. Every noise level of a coherence estimate depends on the effective number of looks nₑ, not on n: incoherent image pairs have E|γ̂|² = 1/nₑ (the noise floor removed by the weights of decision 0024), the EMI coherence magnitude matrix needs nₑ ≳ M/2 to be positive definite (decision 0023), precision bounds scale as 1/√nₑ.

For a circular Gaussian speckle with complex correlation coefficient ρ(Δ) between pixels at lag Δ, the variance of a second order estimate over a set S of n pixels is proportional to Σ_{p,q∈S} |ρ(p − q)|² / n², so nₑ = n² / Σ_{p,q∈S} |ρ(p − q)|² ≈ n / f for a compact S, with the oversampling f = Σ_Δ |ρ(Δ)|² (sampling rate over the equivalent noise bandwidth of the spectrum, a product of an azimuth and a range factor). ρ is the response of the system and of the resampling, the same for every scatterer, so f is one number per data set.

Measured 2026-10-05 (central 1000 x 1000 pixels, 10 images): Campi Flegrei f = 2.59 (azimuth 1.9 x range 1.4, |ρ| 0.64 at the azimuth and 0.43 at the range neighbour), Xinpu 2.75; between images ± 3 %. With 121 SHPs at most, nₑ ≤ 47 for Campi Flegrei: M / (2nₑ) ≥ 1 for its 92 images, which is why 97 % of its DS candidates had a coherence magnitude matrix that is not positive definite.

Decision¶

slc_correlation(slc, max_lag=(4, 6)) (API) returns |ρ(Δ)|² of one SLC, (2·max_lag + 1) lags per axis:
each pixel divided by the square root of its local mean power (15 x 15 pixels), so that bright extended structures, whose neighbours are more correlated than speckle, do not dominate (without it Campi Flegrei gave 2.87);
per azimuth line pair |Σ a b*|² / (Σ|a|² Σ|b|²) − 1/m, averaged over the line pairs: an azimuth phase ramp (TOPS) is constant along a line pair, while an average over a block of lines cancels the azimuth correlation (it gave f_az = 1.00);
the remaining bias of the correlated pixels of a line (about (f_range − 1)/m per lag, 4 % of f in total) is removed with the mean of the outermost lags, where the correlation is 0.
The command slc-correlation writes the median table over evenly spaced images of a central block, with the sum as attribute oversampling, the value for oversampling of emperical-co-emi-temp-coh-pc (nₑ = number of SHPs / f; decision 0024). (Removed by decision 0027: the commands measure the table themselves, per raster chunk.)
shp_n_looks(pc_is_shp, rho2) returns the exact nₑ of each SHP set, n² / Σ_{p,q∈S} |ρ(p − q)|² (negative table values as 0, lags outside the table as 0); emperical-co-emi-temp-coh-pc --rho2 uses it for the weighted temporal coherence instead of n / oversampling (decision 0027: always, internally) (CPU numba, GPU numba.cuda with a warp per point: 200 000 points in 0.18 s on 128 cores, 27 ms on an A100).

Consequences¶

For compact SHP sets n / f is 7-11 % below the exact nₑ (full window, compact blobs, strips), for scattered SHPs up to 40 % below (50 scattered pixels: 19.6 against 32.3), i.e. the noise floor of the weights is then set a little too high; rho2 avoids it.
Xinpu with the measured f = 2.75 instead of 2 (decision 0024): weighted temporal coherence >= 0.8 for 46 % instead of 41 % of the candidates, the points it adds to the uniform one (uniform < 0.6) 49 000 instead of 33 000 with about the same spatial consistency (0.48 against 0.51).
On synthetic speckle with a known kernel the estimate is 1.5 % below the truth; texture, holes and azimuth phase ramps do not change it. 0.06 s per image of 1000 x 1000 pixels (numba).
A window that keeps EMI without regularization for M images needs about 2 M f SHPs (490 for 92 images and f = 2.6, a 23 x 23 window).

Do not¶

Do not estimate the azimuth correlation of TOPS SLCs over blocks of lines.
Do not use the number of SHPs as the number of independent looks.
Do not use the intensity (power weighted) correlation without normalizing the local power: urban structures bias it upwards.