Forward and Inverse Problems

To find the electrical properties of file heart is an inverse problem, and in principle it is unsolvable, it is infinitely many source configurations that may result in the measured skin potentials. 

We therefore start with the forward problem from heart models in a conductive medium to surface potentials. Those problems are solvable, but it is difficult to model the heart and it is difficult to model the signal transmission from the heart up to the surface. The model may be an infinite homogeneous volume or a torso filled with saline or with a heterogeneous conductor mimicking the conductivity distribution of a thorax. 

The most basic electrical model of the heart is a bound vector with the variable vector moment m = iLcc see Eq. 6.10. Plonsey (1966) showed that a model with more than one dipole is of no use because it will not be possible from surface measurements to determine the contribution from each source. The only refinement is to let the single bound dipole be extended to a multipole of higher terms (e.g., with a quadrupole). 

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To illustrate Equation (1.8), consider a solution of the forward and inverse problems in the simplest possible case, when the field is caused by an elementary mass. Suppose that a particle with mass m q) is situated at the origin of a Cartesian system of coordinates. Fig. 1.2a, and the field is observed on the plane z — h. Then, as follows from Equation (1.8), the components of the attraction field at the point p(x,y,h) are... 

Formulation of forward and inverse problems for different geophysical fields... 

In this introductory section, I will give a mathematical formulation of several forward and inverse problems typical for geophysical methods. The definition of general forward and inverse problems can be described schematically by the following chart FORWARD PROBLEM model model parameters m data d. 

Now, let us turn to the analysis of typical formulations of the forward and inverse problems for major geophysical fields gravity, magnetic, electromagnetic and seismic wave fields. 

We can now summarize the operator equations for different geophysical forward and inverse problems ... 

The mathematical formulation of seismic forward and inverse problems in the simplest case of an acoustic model in the frequency domain is given by equation (1.25), which we will repeat here for convenience ... 

We examine some general properties of the forward and inverse problem (2.2) that can be treated as the sensitivity and resolution of the corresponding geophysical methods. 

This result was extended by Samokhin (1998) to the more general case of the linear operator equation in a complex Hilbert space M (see Appendix A), which is extremely important for many geophysical applications, for example, in electromagnetic forward and inverse problems. The main difference between the real and the... 

In summary, we have transformed a problem requiring the solution for six unknown variables (the six scalar components of the vectors E and H) to a problem requiring solution for four unknown variables (the three scalar components of the vector potential A and the single variable, the scalar potential U). As we will see below, this reduction in the number of variables allows us to simplify markedly the solution of electromagnetic forward and inverse problems in many cases. 

In practice we usually solve forward and inverse problems in the space of discrete data and model parameters. For a numerical formulation of QA inversion we can use the matrix formula for QA approximation (9.241), reproduced here for convenience ... 

Madden, T. R., 1972, Transmission system and network analogies to geophysical forward and inverse problems Report 72-3, Department of Earth and Planetary... 

In this section we consider a number of identities relating the wavefield values within a domain to its values on the domain boundary. These identities provide a tool for solving the boundary-value problem of the wavefield, which is extremely important in the solution of the forward and inverse problems. 

C.R. Johnson and R.S. MacLeod. Nonuniform spatial mesh adaption using a posteriori error estimates applications to forward and inverse problems. Appl. Numer. Math., 14 331-326, 1994. This is a paper by the author which describes the apphcation of the h-method of mesh refinement for large scale two- and three-dimensional bioelectric field problems. 

Critical aspects of the forward and inverse problems in electrocardiography... 

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