Three-Dimensional (3D) Modelling

6.5.1 3D R

T

3D ray tracing solutions have a long history in seismic (Kline and Kay (1965), Cerveny and Ravindra (1971)) and electromagnetics (Luneburg (1964), Borne and Wolf (1980)). In these cases, the signals propagate large distances through the material compared to the wavelength and the ray optics approximation is quite suitable in many instances.

For GPR, 3D ray tracing solutions have seen limited use and development. There are two reasons for this. First, GPR signals are strongly attenuated limiting the conditions where the geometric optics assumptions are fully applica-ble. Secondly, by the time the demand for 3D modelling started to occur, computer power had reached the level where it was possible to consider full numerical solutions and hence the use of approximate solutions, such as ray tracing which becomes complex in 3D, were perceived to have little benefit.

6.5.2 3D F

D

M

The majority of 3D GPR modelling uses numerical finite difference style techniques. Depending on the particular application, the solution may be set up as finite element or finite difference but the general implication is that one is solving the full vector field at a mesh of points in 3D space.

The initial 3D modelling for GPR was reported by Roberts & Daniels (1996). 3D modelling for EM induction was also becoming available Alumbaugh and Newman (1994). This modelling required use of a Cray super computer to generate results. Similar modelling results are reported by Bergmann et al (1996), Wang & Trip (1996), Carcione (1996) and Holliger & Bergmann (2000).

It is beyond the scope of this brief discussion to get into the details of numerical simulation. Referring to the above papers is the best way to learn about 3D GPR modelling. There are a number of different techniques which are used to approach these problems and considerable skill and modelling experience is required to confidently enter this area of GPR at present. One must solve the vector field equation for the field components with full variation of the mate-rial properties.

One can solve for the fields for particular sinusoidal excitation frequency. This approach was the most common method to start with as commonly returned to as FDFD (finite difference frequency domain) modelling.

Using a frequency domain solution, the time domain solution is obtained by modelling at a number of different fre-quencies and then fourier transforming the frequency response into the time domain to obtain the transient response.

More common now is the FDTD (finite difference time domain) solution. The approach here is to by-pass the fre-quency domain and solve the full transverse vector wave equation in the time domain. These results require both spa-tial and temporal finite differencing or numerical discretization. To date, however, the complexity of the responses and the length of time required to compute them has limited widespread use. As computer power increases and com-puter codes become more efficient, more and more utilization of the numerical solutions is occurring.

The key reason why full 3D numerical solutions are needed is that they provide the basis for inverting or extracting quantitative information about the subsurface in an automatic fashion.

Figure: 6-8 This grid pattern shows the concept of the staggered grid for electromagnetic modeling. The electric and magnetic fields are computed independently at grid points which are off set from one another. The magnetic fields at points around an electric field point are used to update the electric field. Similarly the electric field points

around an individual magnetic observation point are used to update a magnetic field.

The general approach to 3D modelling is to establish a grid. If a single vector field (i.e. electric or magnetic) is used then a second order spatial difference equation is required. More commonly, a staggered grid, where both the electric and magnetic field are computed alternately in time using first order derivatives rather than second derivatives as pro-posed by Yee (1965), has become the standard in electromagnetic modelling.

The example grid shown in Figure 6-8 shows a staggered grid. The electric field is computed on one set of mesh nodes while the magnetic field is computed at another set of mesh nodes which are offset.

Once these solutions are computed they can be displayed as fields versus time such as the 3D wavefront shown in Figure 6-9. These examples were drawn from the paper by Giannopoulos (1997).

Figure: 6-9 An example of using a finite difference time domain (FDTD) computational algorithm based on a stag-gered grid to compute the response of a reinforcing bar buried in concrete.

Other topics which are addressed in FDTD models are the inclusion of dispersive electric properties (Bergmann et al (1999), Young and Nelson (2001)). In all numerical models, the edges or boundaries of the model space can create artificial events. Delving into the modelling field requires learning about devising absorbing boundary conditions (Holtzman and Kastner (2001), Oguz, Levent (2001)).

6.5.3 I

E

– E

S

The equivalent source method is an elegant and physically intuitive means of formulating EM responses (Hohmann (1987), Annan (1974)). Instead of numerically solving for all the fields, the approach was the known response and formulates the response for localized targets which can greatly reduce the size of the numerical calculation.

Figure: 6-10 The conceptual idea of an embedded localized inhomogeneity (a) being transformed into an equiva-lent signal source (b) in a known background.

Details of this methodology and beyond this presentation. The concept is explained in scalar form here. Referring to Figure 6-10, a localized target with differing properties is present in a media of a known response which can be char-acterized by the green’s function

(6-2)

If a source is present and described in space and time as s(r,t), the fields created, f(x,t), are mathematically expressed as the convolution

(6-3)

By describing the target as a difference in physical properties p = (p-pB) incorporated in the background response greens function, the excitation field f(r,t) will cause an apparent source signal

. (6-4)

This new equivalent source generates the response

(6-5)

with the source signal satisfying an integral equation of the form