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(a) Halfspace model

(b) One-dimensional layered model

(c) Two-dimensional pipe (d) Three-dimensional finite model length cylinder model

FIGURE 6.15 Dimensionality of models: (a) a homogenous half-space that extends to infinity in the x and y directions, (b) a layered-earth model that extends to infinity in the x and y directions, (c) a two-dimensional pipe model that extends to infinity in the y direction, and (d) a three-dimensional object that has finite dimensions in all three directions.

approximation to the boundary between two half-spaces. This brings us to a very important point about models: models are idealized approximations to the distribution of objects in the subsurface. Models make it possible to simulate geophysical measurements mathematically, and they are never an exact replication of conditions in the subsurface.

The layered earth (Figure 6.15b) is an example of a one-dimensional model. It is one dimensional because the physical properties change in only one dimension. A pipe that is infinitely long (Figure 6.15c) is an example of a two-dimensional object, and a spheriod (Figure 6.15d) is an example of a three-dimensional object.

Forward models like that shown in Figure 6.15 are used by geophysicists in a variety of ways, including the following:

• Presurvey prediction of the response expected from targets in the surface. Calculating models prior to a survey prevents using a particular technique when the theory predicts that the resulting field measurements will not detect a target.

• Design of field surveys to optimize line and station spacing.

• Inversion of field data.

Nearly all geophysical measurements that follow the basic laws of Newtonian and Maxwellian physics can be approximated by mathematical models. These mathematical models describe the spatial and temporal (time-dependent) "state" of the EM field. In other words, the models describe the physical concepts of changes the direction the EM field propagates as a function of time into the mathematical language by the use of derivatives and vectors. All of the fundamental equations in Newtonian and Maxwellian physics describe the physical phenomenon occurring at a particular point in space, and at a particular instant in time. The fundamental question that needs to be answered is: "What is the reaction to a propagating EM wave that is acting on a particular point at a particular instant in time?" These changes in the EM fields are described by a partial differential equation, which mathematically can be expressed in a general way as f(x,y,z,t), where f is the functional expression, x,y,z represent the spatial changes in Cartesian coordinates, and t is time. The differential equation is needed because fields (force and electric) change with time, and the derivatives are the mathematical way to express changes in time and space. It should be noted that geophysical forward models for all geophysical measurements that follow the laws Newton and Maxwell are related to, in the form of, or can be derived from the wave equation.

6.5.2.3 Inverse Modeling

Geophysical data are generally very simple measurements made at a single point below, on, or above the surface of the earth. These relatively simple measurements are generally used to infer some pretty complicated events, or physical property distributions in the subsurface. Inverse modeling involves the process of manipulating the parameters of a theoretical model until the values computed from the mathematical model match the field measurements. The process of inverse modeling for some geophysical methods (e.g., gravity, magnetic, resistivity, thermal) is often called curve matching, because the process involves matching, or fitting, the curve computed from mathematical modeling with the curve of values from field measurements. Figure 6.16 illustrates field data along a profile and the computed response from a hypothetical model. The generalized procedure that this used in interpretation (cut-and-try, analog curve match, or inversion) is shown in Figure 6.16c. The parameters of the model are adjusted in an iterative manner until the computed model response matches the field measurements.

Curve matching (or in the case of wave propagation, trace matching) can be achieved using several different approaches, including cut-and-try, analog overlays and templates, and automated inversion. The cut-and-try technique is a simple process that involves generating individual forward models, comparing the model to the field data visually, and iteratively changing the model parameters until the values for the theoretical model are close to the field measurement values. The analog overlay technique involves looking through a catalog of theoretical curves until the interpreter finds a curve that is very close to the field measurement curve. The analog overlay method was the only method available to interpreters prior to the widespread use of digital computers in the early 1960s. These are approximation techniques because of the simplifying assumptions made in arriving at a solution. Automated, or computer, inversion has become so standard in geophysics that it is nearly a specialization in itself. Most inverse modeling has been developed over the years for potential field and resistivity methods, and the following discussion reflects this fact. More recently, inverse models have been developed for EM methods.

Least-squares inversion is a numerical way to find the physical properties that generate a forward model response that most closely approximates the field measurements. Least-squares inversion is an automated way to implement the curve-matching flowchart shown in Figure 6.16c. Least-squares inversion can be applied to any data and model where changing the parameters in the mathematical equation changes the result of the equation in a linear fashion. Therefore, least-squares inversion can be applied to resistivity, gravity, and EM data by "linearizing" the model. An excellent explanation and summary of inverse modeling is provided at the UBC Geophysical Inversion Facility Web site (www.eos.ubc.ca/research/ubcgif/).

Inversion by the process of least-squares is a numerical problem of adjusting the physical properties of the model until the model curve matches (or comes close to matching) the curve traced

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