Electricity: measuring and testing – Of geophysical surface or subsurface in situ – Using electrode arrays – circuits – structure – or supports
Reexamination Certificate
2000-06-08
2002-10-15
Strecker, Gerard R. (Department: 2862)
Electricity: measuring and testing
Of geophysical surface or subsurface in situ
Using electrode arrays, circuits, structure, or supports
C324S335000, C324S345000
Reexamination Certificate
active
06466021
ABSTRACT:
FIELD OF THE INVENTION
The invention relates, in general, to the field of magnetotelluric (MT) inversion methods, a collection of electromagnetic geophysical prospecting techniques used to inverse image underground conductivity variations. More specifically, the invention provides an apparatus and method that works to minimize resolution loss due to the kind of output stabilization and smoothing commonly used in MT inversion methods. The technique is a particular type of inverse input conditioning that filters out noise effects but, in principle, involves no loss of resolution.
BACKGROUND
If one considers electromagnetic techniques for imaging underground conductivity variations that employ ideal steady-state, far-field plane wave excitations as input across a range of driving frequencies and restricts attention to inversion methods that are cast in the frequency domain, then—in the context of practical noise considerations—one must define all non-steady state and non-plane wave excitations as inadmissible input excitations to the inversion problem. For the purposes of this discussion, the MT inversion problem will be defined by these assumptions.
Every inversion problem involves a model describing all that is assumed about the forward problem of interest. A collection of model pieces in general, this model can be called the prior model because it is based exclusively on what is taken as fact, and to what degree, before any data is observed. No inversion problem is solved or can be solved without a prior model specification, although it is not always set out in an obvious manner. Every inversion problem also involves observed data that may be defined at the output of a forward system, i.e., the input to the inversion process. The prior model includes the specification of such observers. In what follows, an attempt will be made to make the prior specification of the MT problem apparent so as to render clear the contribution of the present invention.
In the ideal noise-free case, the MT forward problem is governed by the steady-state Maxwell's equations involving plane wave excitation of ground media for a collection of frequencies. The conductivity properties of common ground media require the use of low frequency plane waves to obtain significant depth of penetration. In such media, the required frequencies are typically low enough that, to a good approximation, Maxwell's equations reduce to diffusion equations and not to wave equations as is more common. This may be emphasized as the reason why MT methods do not enjoy the kind of resolution that is comparable, for example, to radar techniques. Nevertheless, there are existence and uniqueness theorems, e.g., for the one-dimensional inversion problem, that guarantee exact and unique inversion, in principle, for suitably well-behaved conductivity profiles.
It is in the context of this dichotomy—exact, unique inversion is possible in principle, while practical algorithms typically deliver poor resolution—that the concept of “ill-posedness” usually arises. Well-posed problems, in particular inversion problems, may be defined as having three properties:
1) A solution exists;
2) The solution is unique; and,
3) The solution depends (Lipschitz) continuously on the data (with a Lipschitz constant that is not too big), i.e., small changes in the input data (small with respect to some input reference) result only in small changes in the solution (small with respect to some output reference).
In the conductivity inversion problem, it is the third condition that presents real difficulty with respect to well-posedness. Indeed, inversion algorithms that do not properly address this third condition often exhibit wild variation in their solution output.
Handling ill-posed problems often involves the use of so-called regularization techniques that essentially “re-pose” the problem so that all three conditions are satisfied. It is interesting to note that the initial development of regularization theory was motivated by the MT problem itself. Unfortunately, the use of regularization usually costs resolution since dealing with highly variable solutions, i.e., avoiding solutions characterized by high-pass spatial variation, or noise, equivalently amounts to some kind of spatial low-pass filtering. As a result, properly addressing an ill-posed problem, in particular one requiring significant attention to resolution, means that whatever technique is used to render the problem well-posed, it should employ minimum low-pass filtering. Proper address therefore demands a clear definition of an objective component to minimize that can deliver such minimal filtration. In physical problems, such objective functions are ideal when they can be cast directly in terms of the physics of the problem. They are otherwise uncomfortably referred to as ad hoc, though often still necessary for stabilization purposes.
As defined above, the MT inversion problem assumes steady-state plane waves as input. Practically speaking, however, measured electromagnetic fields always have a portion involving time-varying and/or non-planar wave effects. As a result, a central problem is estimating the usable part of the total electromagnetic field on-site, namely, that due to steady-state plane wave excitation and response. Indeed, only this part of the total measured field constitutes physically justifiable input to an MT inversion algorithm proper; the remainder is noise or interference.
Dealing with the steady-state plane wave input requirement involves two basic approaches, one emphasizing source power and the other signal processing. The first concerns the ability of a given source to deliver to the measurement site plane waves of sufficient power, across a broad and dense spectral band, such that any on-site interference is relatively weak in comparison. The second approach emphasizes signal processing methods to derive from the measured signals the maximum content due exclusively to steady-state plane wave input.
Consider the first approach. However powerful the source, wave planarity still depends on justifiable far-field assumptions which in turn depend on the type of source, the source-to-site proximity, and, in the purely spectral approach taken here, the driving frequencies involved. Source types can be divided into natural sources and artificial/man-made sources; the latter can be further broken down according to controlled or uncontrolled sources. Plane waves due to natural sources can be used for MT imaging, but their random nature emphasizes proper signal processing. Some uncontrolled artificial sources offer significant steady-state plane wave power but have a frequency spacing too sparse be used alone. Ground-based controlled sources typically have the problem that either they cannot guarantee the delivery of sufficient power at a measurement site, or, that such a guarantee leads to source-to-site proximities so small as to violate the far-field, plane wave assumption. These difficulties have led to the investigation of controlled source techniques that attempt to include the more complicated near-field model. These methods are therefore not MT techniques and will not be discussed further. More recently, the controlled source problem has been addressed using ionospheric sources that can—by design—reliably deliver steady-state plane waves over global scales. Such sources once again place the emphasis on signal processing techniques to deal with non-plane wave and time-varying noise interference.
Signal processing to address the MT problem relies on the prior model restriction that valid input excitations consist of steady-state plane waves. This means that signal interference for the MT problem as defined consists of:
1) Non-steady state excitations; and,
2) Non-plane wave excitations.
In general, processing field data to filter out steady-state, non-plane wave interference requires the use of both on-site and remote reference sensor measurements at locations far from the primary site. The approach relies on the prior knowledge that such interference canno
Apti Inc.
Rossi & Associates
Strecker Gerard R.
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