EW | NS | Vertical Gradient Weighting

Use Constraints > Add Constraint > EW | NS Vertical Gradient Weighting or EW | NS | Vertical Gradient Weighting > Modify to define a confidence level associated with various regions of physical discontinuity depicted by the Gradient Reference Model. This weighting can be applied separately for each cardinal direction.

Modify EW|NS|Vertical Gradient Weight Constraint dialog box options

Constraint type	The voxel elements of this constraint vary from >0 to 1, indicating respectively how abrupt or smooth the transition between adjacent cells could be. The Default simply indicates that, in the specified direction, the model will display a smooth transition from cell to cell. Choose Constant to assign a constant value to all voxel elements. The closer this constant is to 1, the smoother the transition from cell to cell. Choose Voxel to assign weighting per voxel element.
Filename	If the Voxel option has been chosen, you are prompted to provide the directional weighting voxel covering the volume of interest. This voxel model is re-gridded to coincide with the padded VOXI document mesh. The range of values should preferably be between >0 to 1, indicating less or more smoothness along the specified direction in the transition between adjacent cells.
Weighting	The input gradient weighting model need not be normalized or scaled prior to the inversion. You can specify a weighting factor that will scale the input voxel and thus eliminate the need for storing an additional scaled voxel model.

Application Notes

The contacts between different lithological units may manifest an abrupt physical change, while the variation within a lithological unit remains continuous. If a priori knowledge of such contacts is available, it can be used to generate these constraints and force the inversion to honour discontinuities.

The directional weight constraints act on the Gradient Reference Model, in the absence of which a voxel model whose elements are all zeros is assumed.

The values of a directional gradient weight voxel model should preferably be normalized, with a data range between a very small value, such as 10.0^-4, and 1.0. If the data distribution is not normalized or if you would like to vary its relative discontinuity, you can make use of the weighting factor.

To force discontinuity along the x axis, load a NS gradient weight constraint model, whose elements have a value near 1 everywhere except along the contact, where they approach 0. For a horizontal contact, load a vertical gradient weight constraint model of a similar nature. If the contact has a component in all three cardinal directions, load a weight voxel for each direction. Use Constraints > Constraint builder > Gradient Weight Model... to build the directional gradient weight constraints from a lithological/geophysical voxel model.

Assuming that the weight values are normalized, the voxel elements within the lithological units should be close to 1.0, indicating a smooth transition between adjacent voxel elements. The voxel elements along the contact surface should be set to a very small positive value, indicating an abrupt change along that contact. You may also sharply vary the weight in a narrow range immediately on either side of the contact to suggest the effects of geologic stress and strain.

Once the weight voxel models are ready, select them under the Constraints item of the Tree Viewer. If a supporting gradient reference model is available, it should also be loaded. The inversion then will result in a voxel model that honours the contacts.

Sample Scenario

This example assumes a voxel model with a single contact. The contact surface is a linear plane defined by the equation Ax + By + Cz + D =0. Using voxel math, set the voxel element values along this surface to 10.0^-4. Set the value of the three voxel elements on either side of the contact to 10.0^-2. All other elements will be set to 1.0. The math expression is developed below:

1. Create a voxel with the same coverage and element size as the initial mesh and set all the voxel elements to 1.0. Use voxel math equation:

   V2 =VI*0.0+1.0

2. Set all the voxel elements within approximately three voxel elements of the contact plane to 0.01:

   V3= ((Ax+By+Cz)<(D+2.5* E) && (Ax+By+Cz)>(D- 2.5 * E) )? 0.01: V2;

3. Set all the voxel elements on the contact plane to 0.0001:

   VG= ((Ax+By+Cz)<(D+ E/2) && (Ax+By+Cz)>(D- E/2) )? 0.0001: V3;

Where:

• VI is an input voxel defining the volume of interest. You can use the output voxel of an earlier inversion session.

• VG is the output gradient voxel.

• E is the average voxel element size.

• x,y,z are the cardinal axis of the voxel.

• V2, V3 are interim voxel models. Note that the above equations can be concatenated into a single voxel math expression:

• @V2 =VI*0.0+1.0

• @V3= ((Ax+By+Cz)<(D+2.5* E) && (Ax+By+Cz)>(D- 2.5 * E) )? 0.01: @V2;

• VG= ((Ax+By+Cz)<(D+ E/2) && (Ax+By+Cz)>(D- E/2) )? 0.0001: @V3;