Geophysical models of the subsurface The geological environment can be very complex, but by

necessity it must be represented by a manageable number of model parameters. This can result in a hugely simpli- fied representation of the subsurface. Greater geological complexity can be represented with more complex models, but these require more effort to define and require a greater degree of competence to make effective use of the additional complexities. Moreover, there is no advantage in using a geophysical model whose complex- ity is greater than the resolving capabilities of the data being modelled, which may be quite limited, especially for deeper regions.

To model a set of geophysical data the interpreter must select an appropriate model to represent the subsurface geology. There are two classes of geophysical model: shape-based models and cell-based models. In most cases, notably modelling of gravity and magnetic data, the model response is obtained by summing the responses of the ensemble of constituent shapes or cells. The situation is not as simple for electrical and electromagnetic methods where complex interaction between the different compon- ents of the model, and with the host rocks, strongly influ- ences the model’s overall response.

2.11.1.1 Shape-based models

Models consisting of geometrically simple bodies (Fig. 2.44atoi), each assigned a particular physical prop- erty value, are appropriate when a simple homogeneous form is considered to be a reasonable approximation of the shape of the anomaly source, or when too little is known about the subsurface geology and its physical properties to justify a more complex model. Models (a) to (e) and (j) in

Fig. 2.44 may be used for 2D and 2.5D modelling,

depending on the length L (seeSection 2.11.1.3). Multiple bodies can be arranged as an ensemble to simulate a wider variety of realistic 3D geological forms.

During modelling, a body’s parameters, i.e. its dimen- sions, position and physical property, are varied. These models are also known as parametric models and they make for quick and easy modelling, but their simplicity means a perfect match with the observed data may not be achievable. This is not necessarily a problem, because the results obtained are a good starting point for building more complex models.

Parametric models include sheets of different thickness (flat, dipping or vertical), dipping contact (sometimes called a step or fault) and a cylinder (vertical, horizontal or plunging) which when thin becomes a disc. The ellips- oid is a particularly useful model. It is an easy matter to stretch and rotate its axes to approximate a range of geological forms such as a plunging pipe-like ore shoot, a flat-lying stratiform orebody, or its axes extended equally to form a sphere to represent equidimensional bodies.

More complex models can be built using bodies whose surfaces are defined by a series of interconnecting planar elements, i.e. facets (Fig. 2.44jtol). Each body is defined by

specifying the coordinates of the vertices of the individual facets, which are adjusted during modelling. This provides moreflexibility for creating complex shapes, but manipu- lation of the model is then more complicated. By con- structing a model with multiple bodies of this type, complex and realistic geology can be modelled: see McGaughey (2007) and Oldenburgh and Pratt (2007). 2.11.1.2 Cell-based models

Complex physical property models can be created by rep- resenting the entire subsurface as a series of discrete cells or elements whose properties are homogeneous through- out the cell/element (Fig. 2.45aandb). A physical property value is assigned to each cell and the data are modelled by adjusting only these values; the cell geometry and positions remain unchanged. The interpreter can define complex geological features on sections and on layers of cells. It is important that models of this type extend well beyond the area of interest to prevent‘edge effects’ due to the abrupt change in physical properties at the edge of the model. The challenge of manually manipulating the large number of cells repetitively generally restricts these types of models to inverse modelling (seeSection 2.11.2.1).

A variant of this model type represents the subsurface as a series of cells whose dimensions are varied in only one a) g) b) h) d) i) c) f) e) j) k) l) L L L L L L

Figure 2.44Shape-based models. (a) Thin vertical sheet, (b) thick vertical sheet, (c) thin horizontal sheet, (d) step, fault or dipping contact model, (e) horizontal cylinder, (f) plunging cylinder, (g) disc, (h) sphere, (i) ellipsoid, (j) irregular prism, (k) and (l) facet models.

direction, usually the vertical (Fig. 2.45 candd). This kind of model is commonly used to simulate basement topog- raphy or sedimentary basins where all the cells are ascribed the same physical property value and the data modelled by adjusting the variable dimension.

Cell size and the number of cells are of fundamental importance in cell-based models. They determine the reso- lution of the model and the computational effort required to obtain the response. A sufficient number of sufficiently small cells are required to model the shorter-wavelength variations in the observed geophysical response, and to adequately represent areas where physical property vari- ations are more complex. However, too many cells results in a large number of model parameters and demands greater computational resources. Often smaller cells are used in the near-surface, becoming larger with depth where less resolution is acceptable. The cell size may also vary laterally, being smaller where more detail is required in the central part of the model or where the data suggest more complex structure.

2.11.1.3 1D, 2D and 3D models

In addition to selecting the type of model to use, the interpreter must also decide the number of directions, or dimensions, in which the model is to be defined. Both

shape- and cell-based models can represent a homoge- neous subsurface, or can be arranged into 1-, 2- and 3D forms. The dimensionality should reflect the geological complexity being modelled, which must be justified in terms of the available knowledge of the local geology, and the distribution and quality of the geophysical data.

The model response is usually computed above the surface, and topography may be included in the model. It is also possible to compute responses at locations below the surface, which is the requirement when modelling down- hole data.

Half-space model

The simplest representation of the subsurface is a homoge- neous volume with aflat upper surface, known as a half- space (the other half of the space being the air which, for practical purposes, is a medium with homogeneous phys- ical properties) (Fig. 2.46a). This model depicts those situ- ations where the ground’s physical properties are invariant in all directions, including to great depth, i.e. there are no physical property contrasts except at the ground–air inter- face. It is an important model for calculating the back- ground response of the host rocks of a potential target, and forms part of the response of discrete bodies in electrical and electromagnetic data

a) c) L d) b) L

Modelled cross section Modelled cross section

Lower Higher

Figure 2.45 Cell-based models. (a) The model comprises a series of voxels of the same size, with the magnitude of the physical property specified for each voxel; (b) 2D or 2.5D version of (a) depending on length L; (c) 3D cell-based model where one dimension of each cell is varied; and (d) 2D or 2.5D version of (c) depending on length L. The shading depicts variation in the magnitude of the physical property.

One-dimensional model

An overburden layer on a homogenous basement, and multi-layered geological environments, are represented as a series offlat-lying layers. Each layer has constant thick- ness and extends laterally to infinity in all directions, and each has its own (constant) physical property (Fig. 2.46b

andc). The model parameters are the number of layers and their thicknesses, and the physical properties of each layer and the underlying basement. Variation is only possible in one direction, i.e. vertically, so it is known as a one-

dimensional model. There is no change in the geophysical response laterally across the surface of the 1D model.

The 1D model is useful when geological features extend laterally well beyond the footprint (seeSection 2.6.2) of the geophysical survey. It is used to model (depth) soundings; where the variation of the response with depth is investi- gated (see Sections 5.6.6.1 and 5.7.4.3). As described in

Section 2.8.1, a series of 1D models from adjacent meas-

urements may be used to create parasections, or paravolumes.

Two-dimensional model

For geological features having a long strike length, a model consisting of a cross-section can be specified and the strike extent of the model set to infinity (Fig. 2.47). What consti- tutes ‘long’ depends on the geophysical method and the depth of the source but at least 5 times would be typical. The physical property distribution is specified in only two dimensions, i.e. depth and distance along the survey pro- file, so it is known as a two-dimensional model. When the geophysical data (and geology) are known to be constant along strike, or can reasonably assumed to be, a represen- tative profile perpendicular to the regional strike is selected and can be modelled using a 2D model.

Two-dimensional models have their longest (strike) axis horizontal and the cross-section modelled is vertical. For shape-based models the bodies in the cross-section can be of any shape, such as a rectangle (Fig. 2.44c) or a circle to represent a cylindrical source (Fig. 2.44e), or an arbitrary shape can be defined by specifying the coordinates of the nodes of straight-line segments defining the shape (Fig. 2.44j). For cell-based models the subsurface is defined

To To Position (y) Depth (z) Depth (z) Position (x) 2D 2.75D 2.5D 3D Modelled profile x (Modelling profile) z y

Figure 2.47Schematic illustration of prism models of different dimensionality.

a) b) c) Measurement Measurement Measurement

Figure 2.46One-dimensional models of the subsurface. (a) Half- space, (b) and (c) multiple layers of constant thicknesses and physical properties. All models extend laterally to infinity.

by cells with their long axes horizontal and perpendicular to the section being modelled (Fig. 2.45bandd).

Two-and-a-half-dimensional model

A very useful variation on the 2D model which removes the restriction of infinite strike length, and is easier to define than the more complex 3D model, is a model with constant cross-section extending over afinite strike length

(Fig. 2.47). This is known as a 2½ or 2.5D model. When

the source can have different strike extents on either side of the modelled profile, or the strike or plunge of the body is not perpendicular to the profile, this is sometimes called a 2.75D model. The 2.5D model gives the interpreter control of the third (strike) dimension without the complexity of defining and manipulating a full 3D model, so they are by far the most used models for analysing all types of geo- physical data in three dimensions.

Three-dimensional model

When the model of the subsurface can be varied in all three directions, it is known as a three-dimensional model. Shaped-based 3D models can be specified in a number of ways, but usually as a network of interconnecting facets

(Fig. 2.44kandl). Cell-based models comprise a 3D distri-

bution of uniform cells (Fig. 2.45aandc).

Three-dimensional models can take considerably more effort to define than 2D and 2.5D models and require computer systems to view and manipulate the geometry in three dimensions. The observed and modelled responses are usually displayed as a series of profiles across sections of interest. The simplest, and still very useful, form of 3D model is the ellipsoid (and the sphere– a special case of the ellipsoid). The model is very easy to manipulate and can adequately represent a wide range of source shapes.