GNGTS 2018 - 37° Convegno Nazionale

728 GNGTS 2018 S essione 3.3 In order to analyze the LOS DInSAR measurements, the projection of the modeled deformation ( u LOS ) have to be simply calculated by combining the ground deformation field components ( u, v, w ) with the LOS unit vector as follows: (4) where c x , c y , and c z are the components of the LOS vector c . The operation of equation (4) also returns a harmonic function (Blakely, 1996), since we consider the mean values of LOS vector components. Consider, now, Euler’s equation (Blakely, 1996): (5) The function U , that satisfy equation (5) is said to be homogeneous with homogeneity degree n . By inserting equation (3) into equation (5), it is obvious that each component of u for Mogi’s source model is homogeneous of degree n = – 2. The homogeneity degree of the field ( n ) may be used to estimate the homogeneity degree of the source n s (Fedi et al., 2015) which, for a Mogi model, is given by: n s = – 1. (6) Since n s is a source parameter determined by a field parameter ( n ), it may be convenient to refer to its opposite, the so-called Structural Index ( N ): N = – n s . (7) We conclude that the Mogi model source is characterized by n s = – 3 and N = 3. Multiridge method. The Multiridge (Fedi et al., 2009) method is a multiscale method based on the analysis of so-called ridges, which are defined as lines passing through the maxima of a field and its derivatives at different scales. We emphasize that this method can only be applied in cases when the field can be expressed by harmonic functions. Specifically, the Multiridge method mainly consists of three phases: the creation of a multiscale dataset by performing upward continuation from the original measurement level to different levels; individuation and representation of the edges; representation and continuation of the ridges down to the source-region, to individuate the correct position of the source at the intersection of more ridges. ScalFun method. The ScalFun method is based on the properties of the scaling function, which was introduced into the framework of the DEXP theory (Fedi, 2007) to estimate the homogeneity degree of the observed field ( n ). For any p th vertical derivative for the Newtonian potential of a pole source f p ( z ) at x = x 0 and y = y 0, we define the scaling function τ p of order p as: (8) where n = – ( p + 1) represents the degree of homogeneity of f p and z = . Therefore, τ p ( q ➝ 0) = n and in a plot diagram of τ p as function of q , the intercept gives an estimate of the homogeneity degree n . Starting from the z 0 source depth retrieved by using the Multiridge method, we can use equation (8) to estimate n, n s , and N ; these values give us information about the geometry of the source. The real case of Okmok Volcano (Alaska). We analyze the Okmok volcano ground deformation pattern retrieved by processing the ENVISAT SAR images. The interferogram