modelling_field.md
Modelling field components and gradients
IGMAS+ allows modelling of the three components of gravity (G_x, G_y, G_z) of which G_z, the gravity field, is typically used for density modelling. In addition, the six independent tensor components of the gravity gradient (G_{xx}, G_{xy}, G_{xz}, G_{yx}, G_{yz}, G_{zz}) can be calculated. Gradients provide a higher resolution than the vertical component of the gravity field. The calculation of gravity mass effects require:
- a successfully triangulated model geometry,
- bodies to be assigned with density values, and
- stations to be located in the study area.
???+ note "Note on magnetic modelling"
As just described for the gravitational field, modelling of the Earth's magnetic field ($H$) is also possible if the modelling parameters are given (i.e. triangulated model bodies with defined magnetic susceptibility but also magnetic remanence). **IGMAS+** thus yields the three components of the magnetic field ($H_x$, $H_y$ and $H_z$) and the six independent gradients for the magnetic field components ($H_{xx}$, $H_{yy}$, $H_{zz}$, $H_{xy}$, $H_{xz}$ and $H_{yz}$).IGMAS+ uses the algorithm of Götze and Lahmeyer (1988) to calculate the effect of a homogeneous polyhedron on gravity by transforming the volume integral into a sum of line integrals by the application of theorems of potential theory. The fields are first calculated for each station and each interface (i.e. the set of triangles separating two bodies) separately and then the effects of all interfaces are summed up to obtain the total amount at a station. Likewise, the anomaly effect of a voxel model is calculated independently and then added to the effects of the remaining IGMAS+ model for each station. Thereby, each voxel is approximated by a sphere with its volume being identical to the volume of the voxel.
???+ note The default value of the gravitational constant used for any gravitational calculation in IGMAS+ is G = 6.67384 ⋅ 10^{−11} m^3 kg^{-1} s^{-2}. This value may be changed by the user following the recommendations of the CODATA.
Handling modelling shift
There is a general gap in magnitude of the measured gravity and the calculated gravity of an IGMAS+ density model. Gravity measured in the field is always caused by masses of the entire Earth. The modelled value in IGMAS+ is much smaller in spatial extent and therefore consists of less mass. To handle this problem, IGMAS+ operates with a shift value, thereby assuming that all far-field effects not considered by the IGMAS+ density model cause a constant offset in the calculated with respect to the observed gravity. By default, this shift value is derived from the gravity field values at all stations as follows:
{\mathrm{shift}} = \mathrm{mean} \mathrm{(observed~field)} – \mathrm{mean} \mathrm{(modelled~field)}
The derived shift value (alternatively, a user-defined one) is then added to the preliminarily calculated ones:
\mathrm{calculated~value} = \mathrm{modelled~value} + \mathrm{shift}
This correction is updated after each modification of the calculated anomaly. By introducing the shift value, the absolute differences between the observed and calculated anomalies are suppressed in support of the relative differences, which helps identifying and localising domains of mass deficit or mass excess, in the density model. Figuratively speaking, this means that the two fields are numerically merged so that their mean values are identical. This is necessary to make the phase (maxima and minima) and the magnitudes of these anomalies directly comparable in order to get information about the plausibility of the underground structures.
Handling edge effects
If the density of the space surrounding an IGMAS+ model was not defined and thus actually set equal to zero, the stations close to the model borders would reveal gravity edge effects according to the large density differences at the marginal interfaces. In IGMAS+ there are two solutions to this problem.
The first one is to introduce a reference density (refer to Body Manager) which in fact has two different meanings: first of all, it is a user-defined density assumed to be present wherever there is no model body, including the entire surroundings of the 3D model. Hence, an isolated body would automatically be surrounded by the reference density. Secondly, the reference density is subtracted from all defined densities (i.e., reference and body densities) and any gravity effects at the stations are calculated from the resulting density differences. Hence, if the reference density is chosen to correspond to an average density at the model borders, the unwanted edge effects can be substantially reduced. No reference susceptibility is needed. Further minimization of the edge effects may be obtained through the integration of a layered background reference model which accounts for general density trends, such as an overall increase with depth.
???+ note
The case with a layered background reference model has not been yet described in the documentation.The following sequence of inputs is recommended:
- Click on Model in the "Control Window"
- Click on "Property Editor"
- Select "Border Effect" and then "Border Algorithm"
- Select "Voxel Border Effect Kernel" and define a Density-Depth function
- This will define a layered background model.
The second strategy for reducing the edge effects of flat IGMAS+ models is to extend the model space for anomaly calculations beyond the initially defined model. Therefore, the four vertical border planes of the model are automatically mirrored to a set of new borders and the respective density structure is laterally continued in between. Per default, the amount of lateral extension is tenfold the total vertical depth range of the initial model. For example, we assume a vertical model extension of 100 km. Then the lateral model extension to each side should be larger or equal than 1000 km.