GAMMA Interferometric Point Target Analysis Software (IPTA): Reference Manual

mb_pt

ANSI-C programs: mb_pt.c

NAME
mb_pt -Calculate phase time series from a set of multi-reference unwrapped  point interferograms (mb stands for multi-baseline (temporal and spatial))

SYNOPSIS
mb_pt <plist> <pmask> <pSLC_par> <itab> <pdiff_in> <np_ref> <sigma> <itab_out> <pdiff_out> [pdiff_sim] [psigma_out] [hgt_flg] [hgt_out] [gamma] [SLC_ref_par]

DESCRIPTION
This program addresses the problem where you want to solve for the time-series of the deformation phase given a set of multi-reference unwrapped deformation phases. The input set of deformation phases are typically over specified such that more than one interferogram contains information from each of the SLCs in the input data set and that the observations can be linked together. Implicit is the idea that the total deformation phase at time t₂ is the sum of deformations from t₀ to t₁ and from t₁ to t₂.  Therefore the total deformation phase at some time  time t_n can be expressed as a linear combination of  measurements in the input data if tn is  exists in the set of input deformation phases and the the set of measurements is connected.

The program mb_pt generates a times-series of unwrapped deformation phase given a multi-temporal reference stack of unwrapped phases. The deformation phase data are derived from pairs of SLC images covering the time intervals of interest.  mb_pt estimates the deformation phase time-series using a weighted least-squares algorithm that minimizes the sum of squared weighted residual phases. The residuals are the differences between input  phases (the observations) and the differential phases derived from the time-series solution. Smoothing of the time series solution is acheived by introduction of constraints on the change in velocity as described by Schmidt and Burgmann (Journal of Geophysical Research,  VOL. 108, NO. B9, 2416, doi:10.1029/2002JB002267, 2003). Higher values of gamma lead to smoother solutions. Values of gamma =1 are moderately smoothed, while gamma=100 leads to an essentially linear deformation curve.

We also include estimation of a DEM height correction parameter (hgt_flg=1). The phase model includes a height related term proportional to the derivative of the interferometric phase with respect to height. This additional degree of freedom permits calculation of a height correction for all the interferograms that minimizes the phase residuals in a least-squares sense. If the value of the smoothing parameter is set to 0, then the value of the height correction will be very close to 0.0 since the phase can then be fully attributed to motion of the surface or atmosphere. Only when the smoothing parameter is > 0 can height corrections be calculated.  The height corrections are consistant over a range of values of the smoothing parameter gamma > 0.5.

Input to mb_pt is the multi-reference itab file that provides information on the pairs of SLCs used to create each of the interferogram layers in the diff_in stack. Also input is the stack of SLC parameters for the images referenced in the itab, and the corresponding unwrapped deformation phases diff_in. Typically the input deformation phases are from differential interferograms with the topographic and residual orbital phases removed. 

An  example of a multi-reference itab is:

1   3    1  1
1   4    2  1
1   5    3  1
3   5    4  1
4   7    5  1
4   8    6  1
5   7    7  1
5   9    8  1
6   8    9  1
6  10   10  1
6  11   11  1
  ...
22  25   35  1
22  27   36  1
23  27   37  1
24  27   38  1
25  27   39  1
25  28   40  1

The entries in the multi-reference itab were selected such that they satisfy specific criteria with respect to perpendicular baseline and time interval. The program  base_calc, can be used to construct the multi-reference  itab with entries that fulfill baseline, and time interval criteria. Note that the minimum time interval should be set to 1 day to avoid having auto-interferograms that contain no phase information.

Gaps in interferometric times series occur if there is no interferogram connecting different dates in the input data set.  In  a connected set of observations it is possible to synthesize an interferogram for any set of dates covered by the input measurements by taking a linear combination of the existing interferograms. If this is not possible then the network is not fully connected. The gap can be interpolated under the assumption that the phase rate does not change discontinously. Setting the smoothing parameter gamma > 0  adds a cost associated with a change in the deformation rate.  The least-squares minimization will then interpolate across the gap.  If no smoothing is performed, then the deformation rate will be 0 across the gap and lead to errors in the total deformation.

An example of an input itab with a temporal gap would be:

1   2    1  1
1   3    2  1
4   5    3  1
4   6    4  1

As can be seen by inspection  there is no interferometric phase information covering the period between the times when SLC-3 and SLC-4 were acquired. There is no connection between the interferometric data for acquisitions {1, 2, 3} and acquisitions {4, 5, 6} and hence there is no change in the output  time series phase during the time interval from SLC-3 to SLC-4. When the smoothing of the time series is performed then the gap interpolated with a function that minimizes the change in deformation rate before and after the gap.

The output deformation phase time-series is calculated at each acquisition time of SLCs used to create the input interferogram stack. The solution is only available for times covered in the input interferogram data set. The phase time-series values are obtained by summing up the phase contribution for each time interval from the start of the series. In the example, the output single reference itab_out has entries:

1   3    1  1
1   4    2  1
1   5    3  1
1   6    4  1
...
1  27    23  1
1  28    24  1

Note that each differential interferogram in the output is referenced to the time of the earliest scene in the series. By definition, the deformation is set to 0 at the initial epoch for the series. If phase values are desired for other times, then these can be obtained by interpolation using the program t_interp_pt. The phase series values can be converted to deformation along the line of sight (LOS) using the program dispmap_pt.

The program mb_pt uses Singular Value Decomposition (SVD) (see  http://en.wikipedia.org/wiki/Singular_value_decomposition ) to obtain the least-squares solution for the phase time-series. As part of the screen output mb_pt generates a list of the singular values of the design matrix constructed from the itab and observation times. The design matrix contains the coefficients of the set of linear equations that express the relationships between the observed phase data and the desired times series values. The number of singular values is equal to the number of  times in the output time series. If a value is less than the largest singular value/1.e5, the value is set to 0. A singular value indicates that the constraints in the design matrix are insufficient to fully define the output time series. Setting the singular value to zero forces the time-series output  to remain constant over the unconstrained gap in the input observations. 

mb_pt also calculates the set of simulated deformation phase for each of the input interferograms  using  the estimated deformation phase-time series. Subtracting the simulated deformation phase from the input interferogram leaves a combination of  residual baseline phase, noise, and phase unwrapping errors. This suggests the use of mb_pt  for iterative improvement of the processing results.  

Redundancy in the the differential interferogram input data reduces errors in the time-series due to phase errors due to baseline, topographic phase correction, and interferometric decorrelation. The effect of atmospheric phase on the other hand is not reduced by this estimation procedure. The reason for this is that the atmospheric phase cannot be distinguished from random deformation since it is directly related to path delay. The smoothing parameter gamma introduces an additional cost related to rapid changes in the phase. This permits independent estimation of a height correction for each point.

Individual interferogram layers in the input stack can be switched on and off using the switch_flag value (last column) in the input itab. Setting the switch_flag to 0 removes that particular layer from the solution. Note that if a particular acquisition occurs only in this layer, then it will not appear in the solution. Also realize that removing a particular interferogram has a good chance of destroying the connectivity of the input interferogram network causing a gap in the output  time-series. However even if the the switch_flag is set to 0 for a particular layer, the deformation phase for that layer is simulated when possible. The simulated phase can then be subtracted from the input interferogram and help with unwrapping that layer.  If the unwrapped phase for a single point is 0.0 in a layer  that is turned on (switch_flag = 1), then no time-series solution is generated for that point.

The output stack psigma_out contains an estimate of the standard deviation of time series phase values obtained from the residuals simulated phases pdiff_out and the input observations pdiff_in. For each point sigma is evaluated over the number of layers in the diff_in stack Contributions to the standard deviation include interferometric decorrelation, residual baseline error, and possibly phase unwrapping errors.

It is also possible to perform a linear regression of the the time-series phase. The slope of the linear regression is stored in the the prate file, the phase constant at relative time t=0 is stored in pconst, and the standard deviation of the phase relative to the linear fit in psigma_fit.

SEE ALSO
base_calc, dispmap_pt, spf_pt, tpf_pt, t_interp_pt, mb, ipta.h.