Inverse Problems in Physical Diagnostics
Unlike direct problems related to the solution of differential equations, inverse problems are typically expressed by integral equations. These equations relate media parameters to parameters of a measured signal, or input parameters to output ones in various measurement systems. The solution of integral equations is, with few exceptions, an ill-posed problem, and additional a priori information about the exact solution should be used to solve such problems. The specific character of the a priori information determines various regularisation methods that are in use here to obtain an approximate solution: Tikhonov's method, statistical regularisation method, methods based on the use of additional equations or restrictions or of models with limited number of unknown parameters. The main point of this book is the study of convergence properties of each method and the wide application of numerical modelling that gives the accuracy of the solution in a chosen metric. It is an unaccustomed procedure for physicists, but, because there is no proportionality between data and solution errors in ill-posed problems, such approach is inevitable.
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altitude antenna approximation atmosphere boundary layer brightness temperature calculated components considered contact measurements convolution corresponding covariance function dashed line dependence depth determined dielectric discrepancy dynamics electric ﬁeld elevation angles error level estimation evaporation exact solution expressed ﬁrst ﬂuctuations Fredholm equation Fredholm integral equation frequency heat ﬂux height ill-posed problem inﬂuence inhomogeneities inverse problem kernel l-st kind laser level of measurement linear magnetic ﬁeld measurement accuracy measurement error measurements data medium microscopy microwave modulation near-ﬁeld numerical modeling observed obtained optical parameters possible priori information probe proﬁle retrieval radiobrightness radiometer reconstruction reﬂection coefﬁcient refraction index refraction measurements region retrieved proﬁle scanning shown in Figure SNOM soil solid line solved speciﬁc spectral subsurface surface temperature taking into account temperature proﬁle thermal conductivity thermal ﬁlm thermal radio emission thickness Tikhonov’s method transfer function turbulent variations Volterra equation Volterra integral equation water surface wavelength