Multiscale analysis of GPS velocity fields in the oblique collision zone of Arabia-Eurasia tectonic plates using spherical wavelet

GPS velocity fields consist of a set of geodetic observations of displacement with an irregular spatial distribution on the sphere. In this study, multiscale analysis based on spherical wavelet is used to estimate the GPS velocity field in the oblique collision zone of Arabia-Eurasia tectonic plates. Multiscale velocity field estimation is well suited for dense geodetic networks and is straightforward to implement.We show that for DOG wavelet a scale of 3 to 8 is suitable for GPS velocity field analysis in the study area. Estimation output can be used as a data layer in GIS analysis. Regularization is required to obtain a smooth estimated velocity field from the discrete observations. This is achieved through two possible actions. First, one can cull the set of possible spherical wavelets based on the coverage of observations. If each spherical wavelet has a sufficient number of observations constraining its coefficient, then no regularization is needed (λ = 0). Second, if all spherical wavelets are used for the inverse problem, then extensive regularization will be needed, since most wavelets will have zero observations for constraining their corresponding coefficients. In this research, we have chosen something in between these two extreme cases, where we at the outset eliminate many candidate spherical wavelets based on data coverage, but we still require a moderate amount of explicit regularization in the inversion.As the adopted spherical wavelets are analytically differentiable, spatial gradient tensor quantities such as strain rate, dilatation rate and rotation rate can be directly computed using the same coefficients. The gradient quantities are then calculated directly from the estimated field to identify potential deformation signals. The first factor controlling the estimation is the distance between the network stations. Wherever stations are dense, short-scale spherical wavelets participate in the estimation; and where stations are sparse, only long-scale spherical wavelets is used to do the estimation. As we allow shorter length-scale frame functions to be used in estimating the velocity field, the residual field vectors decrease in magnitude. The smallest residuals occur at observation points that are dense enough to fall within the support of the smallest length-scale frame functions which also have the smallest estimated uncertainties in the data. The largest residuals overall are associated with the largest uncertainties in the data.From the perspective of monitoring a GPS network, the residual map may be helpful in detecting spurious behaviour of singles at stations. If unusual strain, dilation, or rotation are observed around a station, such an observation would warrant additional analysis of the GPS time-series and error estimate. We remove a rotational field from the observation set to obtain the velocity field. We then estimate the horizontal velocity field. Once we have estimated the multiscale velocity field, we can readily compute other scalar quantities, such as dilatation, strain and rotation. The high density of stations near the fault system could capture the spatial gradient in the velocity field, and give rise to estimate strain-rates with a maximum of 1.410×10-7, an average of 1.786×10-8 and a standard deviation of 1.626×10-8 per year. The dilation rate is obtained with a maximum of -8.684×10-8, an average of -3.487×10-9 and a standard deviation of 1.144×10-8 per year and the rotation rate is obtained with a maximum of 7.771×10-8, an average of 7.720×10-9 and a standard deviation of 7.040×10-9 radians per year in the study area.Due to the shorter distance of observation stations in the southern part of central Alborz and northwestern Iran, the values of strain, dilatation and rotation rate can be observed in these areas on large scales.In multiscale estimation, the residual field between the original field and estimated field may reveal two key features. First, if there are systematic residuals in a particular region, then it is probable that one needs to include shorter-scale wavelets in the estimation. Second, if there is a strong residual at a single station, then the station is anomalous and is either malfunctioning or is capturing a signal that is not spatially resolved. The advantages of the multiscale approach are its ability to localize the deformation field in space and scale, as well as its ability to identify outliers in the set of observations. This approach can also locally match the smallest process obtained with the local density of observations, thus maximizing both the amount of information extracted and the possibility of comparing the resulting quantities in different regions of a scale. Multiscale estimation of the three-dimensional GPS velocity field is also possible using spherical wavelet frame functions. The vertical component, if any, should be used to estimate the velocity field, as deformation may not be predominant in horizontal directions. This formulation may be easily applied either regionally or globally and is ideally suited as the spatial parametrization used in any automatic time-dependent geodetic transient detector.