# Uniqueness

Tikhonov (1965) was the first to consider the uniqueness theorem for the 1-D MT inverse problem. He proved that different, piecewise analytic, conductivity profiles correspond to different response functions, and also indicated that the converse may be proved in a similar manner. He therefore established a one-to-one relationship between every complete data set and a distinct conductivity profile. Hence, surface measurements are, in principle, sufficient to determine the true Earth *σ*(*z*). In fact, responses known precisely on any open interval *ω** _{1}* <

*ω*<

*ω*

*or at an infinite number of equally spaced frequencies are sufficient to uniquely define*

_{2},*σ*(

*z*) (Parker, 1983).

Other uniqueness proofs, valid for different types of conductivity models, also exist. Bailey (1970) proved a uniqueness theorem for the class of nonzero, bounded, infinitely differentiable conductivity functions. He showed that complete knowledge of a single surface harmonic mode is sufficient to give the unique σ*(T)* of a radially symmetric Earth. MacBain and Bednar (1986) extended Bailey's work to prove that *c(w)* can be uniquely inverted when σ*(z)* is from the class of all homogeneous layered models, all piecewise polynomials, and all piecewise finite trigonometric-series functions. Loewenthal (1975) proved that models consisting of homogeneous, isotropic, parallel layers each with equal field attenuation are uniquely determined by the surface impedance.

MT uniqueness theorems assume that an infinite number of responses are known precisely. In contrast, for all practical data sets consisting of a finite number of inaccurate data the inverse problem is always nonunique. The nonuniqueness arises for two reasons. First, there are an infinite number of ways to interpolate and extrapolate a finite number of accurate data, and each such realizable response corresponds to a different conductivity profile. Second, since any practical algorithm must permit misfits to the noisy observations, errors in the data make the space of acceptable interpolations even larger.

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