Inverse Problems

Part of the Information Science and Statistics book series (ISS)


An understanding of forward and inverse problems [12, 301] lies at the heart of any large estimation problem. Abstractly, most physical systems can be defined or parametrized in terms of a set of attributes, or unknowns, from which other attributes, or measurements, can be inferred. In other words, the quantities \(\underline{m}\) which we measure are some mathematical function
$$\underline{m} = f(\underline{Z})$$
of other, more basic, underlying quantities \(\underline{z},\) where f may be deterministic or stochastic. In the special case when f is linear, a case of considerable interest to us, then (2.1) may be expressed as
$$\underline{m} = C\underline{Z} \quad {\rm or} \quad \underline{m} = C\underline{Z} + \underline{v}$$
for the deterministic or stochastic cases, respectively. Normally \(\underline{z},\) is an ideal, complete representation of the system: detailed, noise-free, and regularly structured (e.g.,pixellated), whereas the measurements \(\underline{m},\) are incomplete and approximate: possibly noise-corrupted, irregularly structured, limited in number, or somehow limited by the physics of the measuring device.


Clay Porosity Covariance Rubber Convolution 


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Copyright information

© Springer New York 2011

Authors and Affiliations

  1. 1.Department of Systems Design Engineering Faculty of EngineeringUniversity of WaterlooWaterlooCanada

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