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Purpose

This panel is used to run the Non-Linear inversion of geodetic data, simultaneously combining an arbitrary number of observed datasets and sources.

An example involving this panel can be found in the tutorial.

Technical Note

This inversion approach is used to find the best-fit source parameters by minimizing a Cost Function based on the difference between observed and predicted geodetic data. In this inversion every source parameter can be inferred from geodetic data. Typical use of Non-Linear inversion is the retrieval of the fault parameters for an earthquake (source dimension, location, depth, mechanism, etc.), a magma chamber (source position, depth, volume variation, etc.) or a nuclear explosion (kilotons, position, depth).

The Non-Linear inversion is based on the Levemberg-Marquardt [Marquardt, 1963] minimization algorithm, implemented with multiple restarts to maximize the probability of the Cost Function to converge to the function global minimum. Details about  the inversion algorithm can be found in cited reference.

For this inversion, all the model parameters can be set free to vary between a minimum and maximum values, defined by the user; setting equal minimum and maximum for a parameter will keep it fixed during the inversion.  After the inversion, the Calculate and Draw Focal Mechanism panel is automatically invoked to inspect the moment tensor, the focal mechanism and the geodetic moment of the solution (if at least one seismirc source is involved is modeled).

For complex problem with multiple events or overlapping signals, there is also the possibility of introducing “fixed sources”, that are sources not retrieved by inversion (for instance coming from the inversion of other data) whose contribution is in any case accounted for in the inversion.

The panel is organized with folders:

 

The INPUT Dataset folder allows to add  the InSAR dataset and GPS dataset to invert. The “**CHECK PARAMETERS**” string indicates that some parameters must be set (‘EDIT DATASET’ – ‘Edit parameters…’ button).

The INPUT Sources folder allows to create, recover and setup the sources used to model the geodetic data. Three sources are available: Elastic dislocation source, used to model the surface displacement induced by a uniform dislocation for a rectangular source in an elastic and homogeneous half-space [Okada, 1985]; Point pressure source, used to model the surface displacement induced by an expanding or compressing point-source [Mogi, 1958]. The "Edit..." button must be used to properly set the source(s); Nuclear explosion (based on Mogi) to model the displacement induced by an underground nuclear explosion.

The Options folder allows to set processing parameters and the coordinate system of output sources.

After the processing, OUTPUT Datasets and OUTPUT Sources folders are added to the panel, with information on the modeled data and sources, with a Report folder containing text summary of the processing. The OUTPUT Statistics folder is created after running the Calculate statistics for the output sources.

 

In this panel the use of the XML Project File is mandatory.

 

INPUT Datasets

 

ADD DATASETS

 Add from file…: Add a geodetic (InSAR or GPS) dataset from shapefile

 From XML Project File…: Retrieve a dataset from an already existing XML Project File.

EDIT DATASETS

 Edit parameters…: Set the dataset properties with the Dataset Parameter Setting panel.

 Remove: remove the selected dataset(s) from the list

 

VIEW DATASETS

 Plot 2D:  open a Graphic Viewer to see the dataset

 

INPUT Sources

 

ADD SOURCES

New

 Elastic Dislocation (Okada): open the panel to set the parameters of an elastic dislocation/opening source

         Point Pressure (Mogi): open the panel to set the parameters of a point volume source

         Nuclear Explosion (based on Mogi): open the panel to set the parameters of a point source calculated in kilotons

 Import

         From Shapefile: add a source from an InSAR or GPS shapefile

         From XML Project File…: retrieve a source from an existing XML Project File

EDIT SOURCES

 Edit parameters…: set the source properties through a source dependent panel

 Remove: remove the selected source(s) from the list

VIEW SOURCES

 Plot 2D: open a Graphic Viewer to see the source in a geocoded map; for a non-linear source, two items are drawn, corresponding to the minimum and maximum values.

 

Options

 

Number of tests for global minimum

Number of times the current best fit configuration must be found to be considered the cost function global minimum. A default value of 3 is enough in most of the cases.

Cost function Tolerance

Threshold below which a Cost Function improvement is considered negligible. A default value of 0.001 is good in most of the cases.

Maximum Levemberg-Marquardt iterations

Maximum number of algorithm restarts; it is a convergence criterion used to terminate the inversion when geodetic data have poor quality and the cost function global minimum cannot be clearly identified 

Restarts for uncertainty/trade-off calculation

Number of inversion runs to calculate the uncertainty of every parameter free in the non-linear inversion and trade-offs between every pair of parameters. 

Restore Default

If pushed, all the previous parameters are set to default values.

Calculate standard deviation and trade-offs

If checked, statistics (parameter standard deviation and trade-offs)  are automatically calculated after the best-fit inversion. If unchecked, statistics can be calculated after the inversion in the ‘OUTPUT Sources’ folder (EDIT SOURCES – Calculate statistics)

Source coordinate system

Coordinate system used for the output sources. This is automatically set equal to the first dataset added, but it can be changed either among the available systems in the menu list of through the Change… button. When the coordinate system is changed, all the sources already set in the INPUT Sources folder are automatically modified accordingly.

 

OUTPUT Datasets

This folder is available after the inversion.

VIEW DATASETS

 Plot 2D: open a  Graphic Viewer to see the dataset (observed, modeled and residuals) in a geocoded map;

 View parameters…: open a panel with a detailed text report about the modeled dataset

 

OUTPUT Sources

This folder is available after the inversion.

EDIT SOURCES

VIEW SOURCES

 Plot 2D:  open a  Graphic Viewer to see the best fit sources

Plot 3D:  open a  Graphic Viewer to see the best fit sources in 3D

 View parameters…: read the parameters of the output sources

EXPORT SOURCES

 Save to shapefile…: allow the user to save the sources in a shapefile (it is automatically done after the inversion)

 

OUTPUT Statistics

This tab is created after the ‘Calculate statistics’ (EDIT SOURCES menu, in the OUTPUT sources folder) is run. It contains, for each source, a “cloud of solutions” corresponding to non-linear inversions of input datasets perturbed with ad hoc noise.

VIEW SOURCES

 Plot statistics: draw the scatter plots of trade-offs between parameters and the standard deviation of every free parameter. It can be saved to a pdf file.

 Plot 2D:  open a Graphic Viewer to see the best fit sources

Plot 3D:  open a Graphic Viewer to see the best fit sources in 3D

View parameters…: read the parameters of the output sources

EXPORT SOURCES

 Save to shapefile…: allow the user to save the sources in a shapefile (it is automatically done after the inversion)

 

Report

This folder is available after the inversion and it shows a summary of the non-inversion performances. Information are provided about the modeled sources (source parameters, geodetic moment, event magnitude, etc.) and datasets (RMS of the residuals, RMS of the original dataset). The symbols ‘***’ close to a source parameter indicates that a minimum or maximum value was found: in that case, the inversion should be repeated changing the allowed interval.

 

XML Project File section

New...

Create a new XML Project File

Open...

Open the XML Project File

Save

Save the XML Project File

Save as...

Save the XML Project File

Close

Close the XML Project File

 

Output Files

 *_nonlinear.shp

Modeled datasets and sources after the non-linear inversion.

 

General Functions

Start        

Start the processing.

Cancel

The window will be closed.
 

Coseismic automatic modeling

 

Click on this button to open the Automatic Inversion tool.
 
Help        

Specific help document section.

 

 

 

References

Marquardt, D. (1963), An algorithm for least-squares estimation of nonlinear parameters, SIAM J. Appl. Math., 11, 431 – 441, doi:10.1137/

0111030.

Mogi, K. (1958), Relations between eruptions of various volcanoes and the deformation of the ground surface around them, Bull. Earth Res. Inst., 36, 99–134.

Okada, Y. (1985), Surface deformation due to shear and tensile faults in a half-space, Bull. Seismol. Soc. Am., 75, 1135–1154.