Gravity field inversion

Therefore, the body will consist of two spherical layers the inner layer (ball) with a negative density, and the outer layer with a positive density. Thus we come to an idea of the existence of a density distribution that generates a zero external gravity field. This idea is the centerpiece of the non-uniqueness principle for gravity field inverse problems. One can add this kind of density distribution to any given density model and obtain another model generating the same gravity field. 

We apply the same technique to solve the minimization problem (7.58) that we used in gravity field inversion. 

II of this book, can be applied to gravity and magnetic inversion. As an illustration, we will show in this section how the conjugate gradient method, outlined in Chapter 5, works in this case. At the same time, we will use gravity field inversion to demonstrate the importance of the regularization technique in the solution of the inverse problems. 

The problem (1.4) is called an inverse source problem. In this case an assumption is made that the model parameters (the physical properties of the medium) are known. Typical examples of this problem are the inverse gravity problem and the inverse seismological problem. In the first case, the density distribution of the rock formation is the source of the gravity field. In the second case, the goal is to find the location and type of the earthquake sources from the observed seismic field. 

The inverse gravity problem is actually an inverse source problem, because the masses are the sources of the gravity field. Thus, it may have many equivalent solutions. For example, all spherical material balls with the same center and the same total mass (but with different radii) produce the same gravity field (outside these balls). Obviously, the densities Pi and of the balls Bj and B2 with the different radii R and i 2 must be different to produce the same total mass value M ... 

Novikov (1938) proved the uniqueness theorem for a star-type body with the given homogeneous density distribution p(r) = pg. The theorem states that if it is known that the gravity field is generated by a star-type body with the a given constant density, the gravity inverse problem has a unique solution. In the case of spherical balls, this result is obvious according to formula (1.39), if pj = P2, then R — H-2 ... 

Gravity field migration in the solution of the inverse problem... 

We have demonstrated in the previous sections that the solution of a gravity or magnetic field inverse problem in discrete form is reduced to a linear matrix equation. Therefore, the full arsenal of solutions to the linear inverse problem, developed in Part... 

The concept of a global gravity field is based on the basic principles of physics, which is at present largely Newtonian mechanics. Newton s Law of Gravitation states that the magnitude of the force between two masses M and m is inversely proportional to the square of the distance (r) between them and may be written as ... 

Repeating these calculations with different pairs of gx(x) we may increase the accuracy of the evaluation of h. Next, making use of the value of this component at any point, the mass m is evaluated. In the case when only the vertical component is known, the determination of the position of mass and its value is similar. Here it is appropriate to notice the following. Inasmuch as an arbitrary body, located at a large distance from an observation point p, creates a field, known always with some error, often it cannot be practically distinguished from that of an elementary particle, and for this reason we are able to determine only the product of volume and density, mass, but each of them remains unknown. It is the first illustration of the fact that the solution of the inverse problem in gravity, as well as in other geophysical methods, is an ill-posed one, because some parameters of a body... 

Later, we will add one more element, caused by the fact that the field containing information about the parameters of a body (useful signal), is never known exactly. Inasmuch as in the process of gravity interpretation every step is reasonably well defined, we may arrive at the impression that the solution of the inverse problem is... 

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. 

Formulae (6.2) and (6.6) provide the basis for the solution of forward and inverse geopotential (gravity and magnetic) field problems. 

The analytical representations derived above for anomalous gravity and magnetic fields provide a useful tool for the solution of the inverse problems. 

The only difference between the inverse problem formulation for gravity and magnetic fields is that now we use a complex Hilbert space of magnetization models /(C)-... 

Figure 3.15. Variation of the measured centre of gravity position of the MAS NMR spectrum from kaolinite as a function of the inverse square of the applied magnetic field.

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