Note

The latest example has been generated using GG3D v6.0.2. The exercise can be completed using previous versions. However some functionality has been added since v5.0, and improvements in performance since v6.0 and v6.0.1 may result in slightly different recovered models.

5.4. Least-Squares Inversion

Here, we use gzsen3d_60.exe to compute the sensitivity matrix required for the inversion; which is scaled by distance weighting provided. We then use gzinv3d_60.exe to perform a least-squares inversion and recover a density contrast model. To keep the example simple, we added Gaussian noise with a standard deviation of 5e-4 mgal to all data points. We then assigned uncertainties of 5e-4 mgal to all gravity anomaly data. In practice, the noise on the data is not trivial to quantify and choosing appropriate uncertainties is very important for successful inversion.

Files relevant to this part of the example are in the sub-folder inv_L2 . Before running this example, you may want to do the following:

5.4.1. Sensitivities

Here, the code gzsen3d_60.exe and the input file sens.inp (see format) are used to construct the sensitivity matrix and scale it using distance weighting. The distance weighting is applied to the sensitivity matrix to counteract the inversion’s natural tendancy to incorrectly place anomalous structures near the observation locations.

To compute the sensitivities, the following input file was used. Since we are performing a least squares inversion, a flag of 1 is entered on the last line of the input file.

Note

If using GRAV3D v6.0.2, the last line in the input file is unused and the sensitivity matrix can be used for either L2 or sparse-norm inversion.

../../_images/sensitivity_L2_input.png

5.4.2. Inversion

Here, the code gzinv3d_60.exe and the input file inv.inp (see format) was used to recover a susceptibility model. You cannot perform the inversion until you have created the sensitivity matrix.

../../_images/inv_L2_input.png

The true model (left) and the final recovered model (right) are shown below. The least-squares inversion almost always recovers a smooth structure that underestimates the amplitude of the target. With distance weighting however, the center location of the recovered body is consistent with the true model.

../../_images/model_L2.png