Forward and inverse modelling

There are two different modelling techniques for analysing geophysical data: forward and inverse modelling. The main differences between these is the level of human interaction required to obtain a satisfactory match between the observed and computed responses. Both can be applied to cell- and shape-based models.

In forward modelling, the model parameters are adjusted by the interpreter until a match is obtained. This is an iterative process that requires the model and both the observed and computed responses to be displayed graph- ically so that the result can be assessed by the interpreter.

The model parameters are adjusted interactively and the model response recomputed. The number of iterations required correlates closely with the expertise of the inter- preter, the complexity of the model and the knowledge of the subsurface geology. Obtaining a suitable match with the observed data can be quite straightforward, provided the computed response is easy to anticipate and that the observed anomaly has a fairly simple form. Responses from gravity and SP models are usually quite easy to predict, but magnetic and electrical and electromagnetic responses can be very difficult to predict, especially when the model contains several sources. Forward modelling also has an important role in survey design which is discussed inSection 2.6.3.1.

With inverse modelling, also known as inversion, the iterative modelling process is automated; it is done by a computer algorithm so that, from the interpreter’s view, it appears as though the model parameters are obtained directly from the set offield observations, with or without some level of human intervention. Inverse modelling is a far more difficult proposition for the software engineer than forward modelling, but for the interpreter the pro- cess is normally simpler, in some cases apparently redu- cing the interpretation procedure to a‘touch of a button’. However, inversion is deceptively simple. In most cases the algorithm will produce a result but, unfortunately, mathematical limitations of the inversion algorithm and the phenomenon of geophysical non-uniqueness

(see Section 2.11.4) combine to produce many possible

models that willfit the data. The result from the inversion is one of what may be an infinite number of possibilities, and choosing the best one is often a major challenge in itself for the interpreter. Forward modelling may be slower and require more operator time and skill but, consciously or not, the process gives the interpreter a better understanding of the relationships between the data and the subsurface, and a better appreciation of the uncertainties in the resulting model.

There are situations when inverse modelling is essential: when the link between the model and its geophysical response is difficult to anticipate; when the model com- prises a great many parameters (as is usually the case with cell-based models); and when a large volume of data needs to be analysed quickly. Also, forward and inverse model- ling can be used in combination. Inversion of a large dataset will produce an initial model of the subsurface which can be refined using either closely controlled inver- sion or detailed forward modelling. Conversely, forward

modelling may be used to produce an approximate match to the data and inversion used to refine the fit.

2.11.2.1 Inverse modelling methods

The mathematics of geophysical inversion is complex and beyond our scope. Only the basic principles are described here to provide the reader with some insight into the source of possible problems when using inversion tech- niques. Oldenburg and Pratt (2007) provide a comprehen- sive description of inversion and its applications to mineral exploration. Additional examples are presented by Olden- burg et al. (1998).

Inverse modelling requires the interpreter to specify a starting model (although this may be no more complex than a half space) which is systematically refined by the inversion process. The difference between the calculated and observed responses at each data point is called the residual. Fundamental to the inversion process is the over- all match, or the‘degree of fit’, between the two responses, which is mathematically represented by an objective func- tion that describes the degree of match as a function of the model parameters. The task for the inversion process is to adjust the model parameters so as to minimise the object- ive function, i.e. to minimise the residuals.

There are various mathematical methods for minimising the residuals, but the most common involves determining how the residuals vary as each model parameter is altered, which in turn indicates how the parameters should be adjusted to minimise the objective function. Normally in geophysical inversion, it is not possible to directly predict the optimal value for a particular parameter; the problem is said to be non-linear. Non-linear inverse problems are solved using an iterative strategy that progressively alters the model parameters, calculates the objective function and then, if necessary, adjusts the parameters again until a satisfactory match between the observed and calculated responses is obtained. The inversion algorithm is described as converging on a solution, i.e. a model that produces a satisfactory match to the observed data.

Figure 2.48 illustrates how a gradient-based inversion

algorithm, a commonly used strategy, models a set of gravity observations. The source is spherical, and the par- ameters that can be varied are its depth, lateral position and density contrast with its surrounds. A useful analogy is to visualise the objective function as a terrain, with the algorithm seeking to ‘roll downhill’ until it reaches the lowest point in the topography, i.e. the place where the objective function and the residuals are minimised. This is

a reasonable approach provided the form of the objective function is not overly complex. The problem is that non- linear inverse problems normally have objective functions which have extremely complicated forms and, conse- quently, it can be very hard to find the overall minima. An algorithm that goes‘down-hill’ is likely to be caught in one of many ‘valleys’ or ‘basins’, which are called local minima, and has little chance of‘rolling to’ the absolutely lowest point in the terrain, i.e. the global minimum, which is the ultimate objective.Figure 2.48also demonstrates the reason for non-uniqueness in geophysical modelling, i.e. more than one model willfit the data (seeSection 2.11.4).

In Fig. 2.48d, where the depth and physical property

contrast are allowed to vary, the lowest part of the topog- raphy representing the objective function is not localised; rather than being a‘basin’, it is instead a ‘valley’. Once the inversion algorithm has found the valley floor, moving across the floor creates little change in the value of the objective function. The example shows that multiple com- binations of depth and property contrast values can pro- duce the samefit to the observed data, i.e. the result is not unique.

The terrain analogy is a simplification, since mathemat- ically the objective function will be defined in more than three variables (dimensions), i.e. it is a hyperspace. Math- ematically exploring the hyperspace in an efficient and effective manner can be exceptionally difficult because the gradient-based search algorithm can get hopelessly confused between local and global minima and will not converge at all. Even when a minima is found it may be impossible to tell if this is the global minimum.

If local minima are expected then global search methods are required; these‘see through’ local features in the func- tion to seek the overall minima. A well-known global search algorithm is the Monte Carlo method, which ran- domly assigns values to model parameters, usually within defined bounds. The match between the computed and observed responses is then determined. If the match is acceptable, according to some defined criteria, the model is accepted as a possible solution and becomes one of a family of solutions. The process is then repeated to find more possible solutions. Monte Carlo methods require a large number of tests, hundreds to millions depending on the number and range of model parameters being varied, and so are computationally demanding. Another disadvan- tage is that it is still not possible to determine whether all possible models have been identified, a particular problem when very different solutions can exist. Global search

a) b) c) Observed variation Objective functions Low High Model Correct

Ideal convergence path Correct solution X_T XT XT Z_T ZT Z_T Z = Z_T X = X_T P = P_T PT PT P_T Source X Location (X) Correct solution _P Z X Z P d) Amplitude Location (X) Amplitude Depth ( Z ) Location (X) Correct solution Amplitude Location (X) Correct solution Amplitude Location (X) Amplitude Location (X) Amplitude Location (X) Amplitude Location (X) Amplitude Location (X) Amplitude Location (X) Amplitude Location (X) Amplitude Location (X) Amplitude

Figure 2.48 Illustration of geophysical inversion based on the gradient of the objective function. (a) Data from a traverse across a body with a

positive contrast in some physical property. The inversion seeks the true location of the source (XT, ZT) and the true physical property contrast

(ΔPT). (b) Inversion constrained by setting depth (Z) equal to ZTwhilst the lateral position (X) and physical property contrast (ΔP) are allowed

to vary. The objective function shows a single minimum coinciding with the correct values XTandΔPT. Also shown are the observed and

computed responses for selected pairs of X andΔP. (c) Inversion constrained by setting ΔP equal to ΔPTwhilst X and Z are allowed to vary.

Again, the objective function has a simple form with a single minimum. (d) Inversion constrained by setting X equal to XTwhilst Z andΔP are

allowed to vary. In this case, the ability to balance variations inΔP and Z between each other produces a broad valley of low values in the

objective function. Note that the inverse problem is greatly simplified when there are only three variables, one of which is held constant and set to the correct value, whilst the other two are allowed to vary.

methods that retain and use information about the object- ive function when searching for the global minima are usually more efficient than Monte Carlo methods. Algo- rithms of this type include simulated annealing and genetic algorithms; see Smith et al. (1992).

Constrained inversion

Inversion can be directed towards a plausible solution by including known or inferred information about the area being modelled into the inversion process. This is known as constrained inversion and the information helps direct the inversion algorithm to that part of the objective function hyperspace where the global minimum exists. Provided the information is accurate, the result will usually be a more useful solution than that from an unconstrained inversion (see Section 3.11.2). One way to constrain the inversion is to set bounds, or limits, on the values of selected parameters (with or without probability compon- ents). For example, the variation of physical property may be constrained; non-negativity being an obvious constraint for a property such as density, and one that greatly improves the likelihood of obtaining a geologically realistic result from the inversion. The possible locations of parts of the model may also be constrained: forcing them to below the ground being an obvious constraint; or to honour a drilling intersection. Particularly important in many instances are distance weighting parameters which counter the tendency of inversion algorithms to place regions with anomalous physical properties close to the receiver, which equates to close to the surface for airborne and ground surveys. Other common constraints include geometric controls, for example the source should be of minimal possible volume; it should be elongated in a particular direction (useful in layered sequences); it should not con- tain interior holes; and adjacent parts of the model should be similar, i.e. a smoothing criterion. Smooth inversion tends to define zones with gradational or ‘fuzzy’ boundar- ies and with properties that are less accurately resolved, rather than as compact sharp-boundary zones whose prop- erties are more accurately determined, but with greater probability of uncertainty in their locations and shapes. The inversion process can also be restricted to adjust one or a few of the possible model parameters whilst fixing others, e.g. to invert for dip, in other words to find the ‘best-fitting’ dip if the source is assumed to be a sheet.

It is preferable to direct the inversion towards more likely solutions through the use of an approximate, but appropriate, forward solution as a starting model; see for

example Pratt and Witherly (2003) and the modelling example described in Inverse modelling in Section 3.11.2. In practice, there are often limited data available to con- strain the nature and distribution of the geological units in the model. Starting models are invariably built from a combination of geological observations and inferences. Petrophysical data, if available, may be incorporated. A relatively small number of constraints can make a big difference to the result (Farquharson et al., 2008; Fullagar and Pears,2007).

Joint inversion

The reliability of a solution may be improved by modelling two or more types of data simultaneously in a process known as joint inversion, which is becoming increasingly common. These data have properties containing common or complementary information about the subsurface. The incremental results for one data type guide the changes made to the model during the inversion of the other data type, and vice versa. Early forms of joint inversion tended to assume a correlation between, say, the density and magnetism in equivalent parts of the model. This assump- tion is rarely justified, and more recent work has concen- trated on correlating regions where the physical properties are changing rather than an explicit correlation of their magnitudes (Gallardo,2007).

A joint inversion model may be more accurate as it is the ‘best fit’ to two disparate data types; and it has the add- itional advantage of reducing the time and effort needed to analyse two different datasets independently. A disadvantage is that more computational effort is required and that the results critically depend on the assumed relationship between the physical properties being inverted for. If the assumption is good then the results will probably be more reliable. If it is not, then the results could be less reliable than individual inversions of each dataset. When jointly inverting different types of geophysical data, it is important to account for the differ- ent physics of the methods. For example, and as shown in

Fig. 3.65, magnetic data are more influenced by the shallow

subsurface than are gravity data. A joint inversion of these two data types will be mostly influenced by the gravity data in the deeper parts of the model, so in this sense the inversion is not‘joint’.

Tomography

An important, but specialised, type of inverse modelling is based on tomography (Dyer and Fawcett, 1994;

Wong et al.,1987). In tomographic surveys, measurements are made‘across’ a volume of rock in a range of directions. Usually measurements are restricted to a plane or near plane. A signal is generated by a transmitter, and some property of the signal, usually strength, is measured on the other side of the volume. Any change to the signal, after accounting for the source–receiver separation, depends on the properties of the rock through which it has passed. The method is the geophysical equivalent of computer-axial tomography (CAT scans) which uses X-rays to create images of‘slices’ through the interior of the human body.

The volume-slice is represented by a cellular model, and by combining many measurements it is possible, using inversion, to determine the physical property within each cell. The model is usually 2D, i.e. it is only one cell‘deep’, so the results represent a ‘slice’ through the volume of interest. Good results require the signal to cross each cell over a wide range of directions, otherwise the results tend to show anomalies elongated in the measurement direc- tion. This restricts this kind of survey to areas with a suitable number of drillholes and underground access. Even then the directions may be limited to a less than optimal range.

Tomographic surveys are relatively specialised, being mostly restricted to in-mine investigations. Surveys of this type using seismic and electromagnetic waves are described

inSection 6.8.2and onlineAppendix 5.