3. Method
Observe that if
R is ignored, the correct inversion LHS (left-hand side) for (
3) is thaw depth difference:
However, only subsidence difference and
are measured. To see if subsidence difference can be related to thaw depth difference, (
1) is analyzed as a ratio:
For convenience,
is introduced. Next, a list of candidate
on some large interval of possible thaw depths is generated, and corresponding
is computed using (
7). The subsidence for each candidate
and
can be computed using (
2). A plot of candidate thaw depth differences and corresponding subsidence differences can then be constructed. See
Figure 2 for examples of this plot for two soil porosity models. Each of these models is explained in [
8]. As shown in the bottom two figures, a particular subsidence difference can unambiguously be mapped to a thaw depth difference. Intuitively, this is because (
7) enables multiplicative growth of
with respect to
. Consequently, the hope is that (
2) enables subsidence difference to grow strictly monotonically. Put another way, for each interferogram,
Q (which is known) acts as a constraint parameter that helps search the space of possible thaw depth pairs
and
. Corresponding subsidences can be computed and a graph can depict whether subsidence difference can be related to thaw depth difference. Of course, there is no guarantee this will work for all soil porosity models. A counterexample is discussed in
Section 3.1. Lastly, note that in
Figure 2 the bottom images demonstrate the effect of
Q on the method’s robustness. For the constant porosity model, there is no effect because the thaw depth difference is linearly related to the subsidence difference. For the “mixed soil model,” the slope is higher in the bottom-left image, making it easier to identify a particular thaw depth difference from a given subsidence difference. This is expected because larger
Q ratios make the thaw depth pair have smaller values for the same thaw depth difference. Smaller thaw values exhibit larger subsidence gradients, as the top-right plot shows.
This method is summarized in Algorithm 1. (
3) can be rewritten as:
Algorithm 1 Self-Consistent ReSALT LHS Retrieval |
 |
where
is the best-matching thaw depth difference reported for each
via Algorithm 1. This now forms a self-consistent modeling of thaw, temperature, and ground subsidence. Hence, this new method is called SCReSALT (Self-Consistent ReSALT). Least-squares inversion gives the best
N. This is the same
N as in (
1), which can then be used to find the thaw depth at any point in the thaw season.
To extend the formulation to multiple years, the long-term trend R must be discounted from before following the above approach. ReSALT jointly inverts for R and E, but a self-consistent model cannot do that. So, R must be estimated independently of and prior to N. This work does not explore R estimation with SCReSALT experimentally. Here are some possible solutions, but exploring them is left to future work:
Once
R is known, the multi-year problem can be solved using (
8) but instead of giving Algorithm 1 the observed subsidence difference
, give it
.
For completeness, it is worth noting that both ReSALT and SCReSALT do not need both SAR acquisitions to be during the thaw season. One acquisition could be before the thaw season, in which case
can be assumed for that acquisition. Hence, any measured subsidence difference can be modeled as the true subsidence at that point. ReSALT can directly use (
3). SCReSALT would need a slight modification because it operates using ratios. Algorithm 1 could be modified such that if
, simply report
. Similarly, if
, report
. The negative sign is because
is the subsidence underwent when going from
to
. That is expected to be positive if
and negative vice-versa.
3.1. SCReSALT Counterexample Discussion
In
Appendix C, a counterexample that does not meet the SCReSALT requirement is derived. Some possible solutions are described below, but exploring them is left to future work.
From multiple interferograms per year and obtain time-series measurements of subsidence difference. Attempt to interpolate the measurements onto a subsidence difference vs thaw depth difference graph and determine which thaw depth differences best explain the data.
Make sure to get a SAR acquisition before the thaw season starts. This allows one to treat the subsidence difference as the true subsidence. Handling this is easy and was described earlier in this paper.
Using a prior on thaw depth difference might isolate a thaw depth difference as the most likely candidate.
Importantly, any soil porosity model can be tested to see if it meets the SCReSALT requirement using Algorithm 1. Only then would exploring these options be necessary.
4. Experiments
To assess the practical improvement of SCReSALT, it is benchmarked against the results reported in [
9]. [
9] applies ReSALT to Utqiagvik (formerly Barrow), Alaska and uses the “mixed soil model” to predict ALT from 2006 to 2010. Metrics are extracted from the paper and the data product they release, which can be found at
https://daac.ornl.gov/ABOVE/guides/ReSALT_InSAR_Barrow.html. [
9] uses data from the U1 CALM
3 plot. It uses ALOS-PALSAR L-band data and generates interferograms using the ISCE2 framework
4. The paper defines the
metric:
where
e is the uncertainty in the measurement. In [
9],
cm. If
, it is called a “great match.” If the
is within the uncertainty of
, it is called a “good match." Otherwise, it is called a “bad match.” Uncertainty analysis is not described in this paper, but in brief, it could be computed by assigning an uncertainty distribution to each ReSALT/SCReSALT parameter and either analytically propagating that into an uncertainty in the prediction or more practically using numerical simulation. This work uses the fact that [
9] stated that the ReSALT uncertainty is roughly twice that of the measurement uncertainty. Hence,
cm. Other experimental details are described in
Appendix D. With those details explained, results are shown in
Table 1.
5. Results
The metrics reported in [
9] closely match the results computed from the data product. The “good match” category is a little worse because
is not precisely known. That then influences the “bad match” category. Otherwise, the alignment is high, meaning there is high confidence in the derived Pearson
R, mean absolute error (MAE), and RMSE values. The author also attempted to reproduce ReSALT results but generally had worse numbers. It is difficult to get precise replication without source code, which is not available for [
9]. Regardless, SCReSALT is shown to outperform ReSALT on virtually all metrics and handles outliers better given its reduced RMSE and
. If compared to the reproduced ReSALT, SCReSALT’s improvement is even stronger. It is also worth discussing the ReSALT data product’s low Pearson
R. It is not clear why, but given the high value of
, it is possible to have many “great matches” despite no correlation between predicted and ground-truth ALTs.