Simple Salt Dome¶

This example is devoted to demonstration of geometry optimization based on spring-based space warping applied to a simple synthetic model of a salt dome.

Description¶

Original model¶

The model contains four units, names and densities are given in the Table below.

Name	Density
S1	2.21 t/m $^3$
S2	2.57 t/m $^3$
S3	2.77 t/m $^3$
salt	2.31 t/m $^3$

The salt unit forms a salt dome starting at a depth of around 3.5 km and going up to around 1 km in the center of the model domain (see Figure below).

Vertical cross-section through the simple salt dome model.

The model has 4 working sections which are identical, so that the model is simply prolonged in the North direction (perpendicular to the working sections):

The 3D view of the simple salt dome model.

There is a horizontal extension of the bodies by 100 km in each side beyond the station boundaries (10 $\times$ 10 km):

The 3D view of the simple salt dome model with extensions.

The reference density is 2.43 t/m $^3$ .

Input data¶

The original model was used to create a gravity dataset calculated for a set of 441 stations regularly placed on the plane at a zero depth level (see Figure below).

Bouguer anomaly modelled above the simple salt dome model measured with a regularly spaced set of stations placed at a zero depth level.

Stations cover the area 40 $\times$ 40 km, exceeding the model edges by 10 km in each direction.
For simplicity, there is no extension of the bodies beyond the model boundaries, and the edge effect is clearly visible in the dataset.

The described dataset was used as an input measured gravity data for inversion.

Starting model¶

In order to have a starting model which would differ from the original model, the shape of the salt dome in the middle section has been manually modified (see Figure below).

Download¶

The input data required for this example as well as the two models are available for download here.

The share contains 3 files:

Simple_Salt_Dome_Measured_Gravity.csv: the input measured gravity data in CSV format
Simple_Salt_Dome_Original.zip: the original IGMAS+ model with a salt dome
Simple_Salt_Dome_Inversion.zip: the IGMAS+ model used for inversion consisting of two timeline steps:
- starting model: a model where the salt dome shape distorted
- final model - a model after application of the geometry optimization to the starting model

The input gravity data file Simple_Salt_Dome_Measured_Gravity.csv is in CSV format and has 4 data columns: "x" "y" "z" "measured z component" and 441 data rows corresponding to stations.
The values in columns are delimited with space.

The two models are zip archives with IGMAS+ projects. Simply unpack and load projects in IGMAS+.

Modelling¶

Goal

The goal of this modelling example is to apply geometry optimization to the starting model in order to demonstrate how it can be iteratively inverted to a final model somewhat close to the true one based on the input gravity data, assuming that densities of the units are known.

The geometry optimization functionality is provided by the Inversion plugin. With this plugin it is possible to iteratively optimize the geometry of a model by space warping using the misfit between the measured and calculated (at each iteration) field.

Open the starting model¶

Start IGMAS+
Select File → Open Project
Select the folder Simple_Salt_Dome_Inversion
In the Timeline window select the earliest timeline - this is the starting model
The starting model will open in the 3D view, already with the loaded input measured gravity data
Note that there are no working sections in the starting model, because 3D geometry optimization based on space warping will destroy the working sections
Now it is necessary to calculate the gravity field for the loaded.

Calculate gravity field¶

Calculate the anomaly of the model using Tools → Calculate Anomalies or by clicking
- Select "Calc Gz"
- Click Finish
Open 2D Maps View using Add View → 2D Maps View or by clicking
The program show the map views of the measured, calculated and residual gravity fields for the starting model:

Measured, calculated and residual gravity anomalies for the starting model.

Open Multiple Cutter View using Add View → Cutter View or by clicking
Draw a horizontal line and adjust the coordinates to be $X$ =0, $Y$ =5 for the start of the line and $X$ =0, $Y$ =10 for the end of it:

Vertical cross-section through the starting model and the misfit between the measured and calculated gravity fields.

Now, before starting the inversion, it is necessary to prepare the inversion lattice.

Tip

The model at this stage can be opened from the middle timeline of the Simple_Salt_Dome_Inversion model.

Prepare the inversion lattice¶

Create a lattice using icon
You will be asked to define the area for the lattice:
By default, the whole model volume is taken and the lattice consists of a single rectangular prism with 4 edges/nodes on the corners of the model:
With icon you can add more lattice nodes, i.e. make the lattice finer, step-by-step:
Similarly, with icon you can remove lattice nodes, i.e. make the lattice coarser
In case of mistakes, you can completely remove the lattice using icon
There are two lattice transformation modes controlled by swapping icons and :
- means that the matrix transformation mode is selected (default)
- means that the trilinear transformation mode is selected

After refining the lattice 7 times, one can obtain an optimal lattice for geometry optimization:

Now it is time to start geometry optimization.

Start geometry optimization¶

Open the geometry optimization wizard using the start icon
Adjust the parameters:
- The "Optimizer" is selected to be "SpringSystem Optimizer" and can't be changed, meaning that the spring-based optimization will be performed.
- Adjust the "Standard Deviation" to be in the range from 0.25 to 0.05 - this is the initial standard deviation for variation of depth coordinates of the lattice nodes. The more the value, the larger is the initial variation. We recommend to have 0.2 here.
- Adjust the "Stop-Quality SD" to be in the range from 0.1 to 0.01. The less the value is, the longer the inversion will last and the better the fit will be in the end. A recommended value for the "Stop-Quality SD" to reach an optimal accuracy is 0.05.
- Make sure the "Use Triangle Effect" is checked: it involves calculation of gravity effect for the triangulated bodies
- Make sure the "Use Voxel Effect" is unchecked: there are no voxel cubes and we don't need to involve it

Recommended geometry optimization settings.

Once ready, click Next
Click Finish
- Geometry optimization process will start

The Population Quality window will open automatically and will dynamically show the statistics on the optimization process:

Population quality statistics after reaching the "Stop-Quality SD" threshold.

To stop the inversion process before it reaches the stop quality threshold, use the stop icon
Once inversion is done, the stop icon will change back to the start icon

Final model¶

The final result, as explained earlier depends on the stop quality and the initial standard deviation. Besides, is not possible to get two identical final optimization results because of the random nature of the CMA-ES.

The result obtained for the stop quality of 0.05 and the initial standard deviation of 0.2 shows that the original salt dome shape is reconstructed with an excellent accuracy and the residual gravity field is minimal:

Result of geometry optimization: vertical cross-section through the final model and the misfit between the measured and calculated gravity fields.

Result of geometry optimization: measured, calculated and residual gravity fields for the final model.

The Figure below represents the 3D shape of the final reconstructed salt dome and the final distorted lattice:

Result of geometry optimization: 3D view of the final model with the lattice and the corresponding calculated gravity field.

Final model without lattice:

Result of geometry optimization: 3D view of the final model.

Tip

The final model can be opened from the latest timeline of the Simple_Salt_Dome_Inversion model.