Depth From Extreme Points Research Articles

Magnetic and GPR Data Modelling via Multiscale Methods in San Pietro in Crapolla Abbey, Massa Lubrense (Naples)

ABSTRACTWe performed magnetic and GPR measurements to image the buried ruins of the Middle Age abbey San Pietro in Crapolla, on the Sorrento‐Amalfi Coast (Massa Lubrense, Southern Italy). The site represents an important religious location, which is nowadays partially buried along the cliff. An integrated study was necessary to map the buried structures and address the archaeological excavation. For this reason, we carried out the surveys on two main grids in order to reconstruct the structures of the abbey and of its related church. The magnetic data were filtered through the discrete wavelet transform (DWT) and then transformed to total gradient maps. The obtained maps were interpreted with depth from extreme points (DEXP) imaging method to assess the horizontal and depth positions of the top. The GPR data were processed and time‐depth converted. Results from the integrated interpretation of these data suggest the possible presence of different vaulted rooms and an elongated structure at 0.3‐m depth from ground surface. This latter is interpretable in terms of perimetral and internal walls of the abbey and its church. These outcomes were crucial to successfully address archaeological excavations, which targeted one of the modelled areas and unearthed a wall at the predicted depths.

Crustal structure and lateral variations in Moho beneath the Delhi fold belt, NW India: Insight from gravity data modeling and inversion

The Proterozoic Delhi fold belt (DFB) in NW India shows a prominent NNE-SSW trending gravity high, but its northward extension beneath Gangetic alluvium lack understanding. We model the gravity anomaly using radially averaged power spectrum, depth from extreme points (DEXP) method, wavelength filtering, 3D structural inversion to constrain crustal structure of the DFB. The radially averaged power spectrum analysis suggests three interfaces and the DEXP method indicates presence of a horizontal cylindrical source below the DFB. The 3D structural inversion shows depth variation from 39 to 46 km in the study region and presence of an upwarp below the DFB and Sandmata complex. Our modeling suggests that an underplated mafic material possibly a residual plume head produces gravity high. The transverse structure of DFB is absent beneath Gangetic alluvium and show notable westward deflection in line with the NW-SE trending Delhi-Sargodha ridge. The transverse DFB attains westward deflection possibly due to NW indentation and counterclockwise rotation of Indian plate after Himalayan collision in Eocene. With continued convergence, this DFB structural element at the leading edge of Indian peninsula got involved in flexural upwarp of underthrusting Indian plate beneath the Himalaya.

UXO Detection by Multiscale Potential Field Methods

The choice of an effective interpretative methodology in terms of management costs, timing, reliability, safety and efficiency is a key issue for the identification of unexploded ordnance (UXO). Among geophysical methods, magnetometry plays an important role in detection and interpretation of UXO objects. We present a study aimed at the identification and characterization of buried UXO in Rohoznik, Slovakia, by a multiscale analysis on high-resolution magnetic data. The area is located close to a former army shooting range and is characterized by several buried UXO. When a UXO hits the ground but does not explode, the impact causes a re-organization of the magnetic domains within the armature (shock demagnetization). Due to shock demagnetization the object gets a magnetization oriented in the direction of the actual Earth magnetic field. Furthermore, the elongated shape of the UXO of Rohoznik causes their anomalies to be oriented along the longitudinal axis of the projectiles. Our multiscale analysis was performed through methods such as depth from extreme points (DEXP) and the multi-ridge method. The upward continuation employed by these two methods allowed reducing the influence of the high-frequency noise related to non-UXO buried iron objects. Also, the use of high-order derivatives in the multiscale approach decreased interference effects related to close-by sources and avoided any pre-filtering. Our approach was first validated on a test site located close to Rohoznik, where unexplored projectiles were buried with different orientations at a known depth. Then we focused on the real data of the former army shooting range. For all our analyses we employed a structural index value of 2.5, which was estimated through the scaling function analysis and the derivative euler deconvolution algorithm. Both the (DEXP) and multi-ridge depth-estimation methods proved to be effective and rapid with respect to more expensive and time-consuming methods such as inversion. Our analysis revealed average depths for the analysed UXOs matching rather well with what was found by subsequent excavations, with a maximum discrepancy of a few decimetres.

Improving the local wavenumber method by automatic DEXP transformation

In this paper we present a new method for source parameter estimation, based on the local wavenumber function. We make use of the stable properties of the Depth from EXtreme Points (DEXP) method, in which the depth to the source is determined at the extreme points of the field scaled with a power-law of the altitude. Thus the method results particularly suited to deal with local wavenumber of high-order, as it is able to overcome its known instability caused by the use of high-order derivatives. The DEXP transformation enjoys a relevant feature when applied to the local wavenumber function: the scaling-law is in fact independent of the structural index. So, differently from the DEXP transformation applied directly to potential fields, the Local Wavenumber DEXP transformation is fully automatic and may be implemented as a very fast imaging method, mapping every kind of source at the correct depth. Also the simultaneous presence of sources with different homogeneity degree can be easily and correctly treated. The method was applied to synthetic and real examples from Bulgaria and Italy and the results agree well with known information about the causative sources.

A fast interpretation of self-potential data using the depth from extreme points method

We used a fast method to interpret self-potential data: the depth from extreme points (DEXP) method. This is an imaging method transforming self-potential data, or their derivatives, into a quantity proportional to the source distribution. It is based on upward continuing of the field to a number of altitudes and then multiplying the continued data with a scaling law of those altitudes. The scaling law is in the form of a power law of the altitudes, with an exponent equal to half of the structural index, a source parameter related to the type of source. The method is autoconsistent because the structural index is basically determined by analyzing the scaling function, which is defined as the derivative of the logarithm of the self-potential (or of its [Formula: see text]th derivative) with respect to the logarithm of the altitudes. So, the DEXP method does not need a priori information on the self-potential sources and yields effective information about their depth and shape/typology. Important features of the DEXP method are its high-resolution power and stability, resulting from the combined effect of a stable operator (upward continuation) and a high-order differentiation operator. We tested how to estimate the depth to the source in two ways: (1) at the positions of the extreme points in the DEXP transformed map and (2) at the intersection of the lines of the absolute values of the potential or of its derivative (geometrical method). The method was demonstrated using synthetic data of isolated sources and using a multisource model. The method is particularly suited to handle noisy data, because it is stable even using high-order derivatives of the self-potential. We discussed some real data sets: Malachite Mine, Colorado (USA), the Sariyer area (Turkey), and the Bender area (India). The estimated depths and structural indices agree well with the known information.

Understanding imaging methods for potential field data

Several noniterative, imaging methods for potential field data have been proposed that provide an estimate of the 3D magnetization/density distribution within the subsurface or that produce images of quantities related or proportional to such distributions. They have been derived in various ways, using generalized linear inversion, Wiener filtering, wavelet and depth from extreme points (DEXP) transformations, crosscorrelation, and migration. We demonstrated that the resulting images from each of these approaches are equivalent to an upward continuation of the data, weighted by a (possibly) depth-dependent function. Source distributions or related quantities imaged by all of these methods are smeared, diffuse versions of the true distributions; but owing to the stability of upward continuation, resolution may be substantially increased by coupling derivative and upward continuation operators. These imaging techniques appeared most effective in the case of isolated, compact, and depth-limited sources. Because all the approaches were noniterative, computationally fast, and in some cases, produced a fit to the data, they did provide a quick, but approximate picture of physical property distributions. We have found that inherent or explicit depth-weighting is necessary to image sources at their correct depths, and that the best scaling law or weighting function has to be physically based, for instance, using the theory of homogeneous fields. A major advantage of these techniques was their speed, efficiently providing a basis for further detailed, follow-up modelling.

2D Continuous Wavelet Transform of potential fields due to extended source distributions

We analyse the real Continuous Wavelet Transform 2D (CWT2D) of potential fields for the investigation of potential field singularities. We focus our attention to extended geological sources, in order to verify the reliability of this method with realistic fields. 3D space-scale representation (3D Scalogram) related to synthetic models were generated, showing the Wavelet Transform Modulus Maxima (WTMM) at each scale. The WTMM are related to the shape of the source, so defining some sort of source boundary analysis through the CWT. Wavelets of different order may help to gain resolution and define source features. Selecting a range of scales where the sources behave as if they are approximately isolated, the depth to the source may be estimated basing on the property that the lines joining the modulus maxima of the wavelet coefficients at different scales (WTMML) intersect each other at the edges of the causative body. Therefore, it is possible to manage the information contained in the wavelet transform of fields related to extended sources. In the real case of the anomaly gravity map of the Vesuvius area (Italy), we estimated the depth of the Mesozoic carbonate basement in the Pompei Basin. We showed also how the WTMML information can be integrated to that of another multiscale method, the Depth from Extreme Points (DEXP) transformation, which is also related to the source density distribution of a given region.

DEXP: A fast method to determine the depth and the structural index of potential fields sources

We show that potential fields enjoy valuable properties when they are scaled by specific power laws of the altitude. We describe the theory for the gravity field, the magnetic field, and their derivatives of any order and propose a method, called here Depth from Extreme Points (DEXP), to interpret any potential field. The DEXP method allows estimates of source depths, density, and structural index from the extreme points of a 3D field scaled according to specific power laws of the altitude. Depths to sources are obtained from the position of the extreme points of the scaled field, and the excess mass (or dipole moment) is obtained from the scaled field values. Although the scaling laws are theoretically derived for sources such as poles, dipoles, lines of poles, and lines of dipoles, we give also criteria to estimate the correct scaling law directly from the data. The scaling exponent of such laws is shown to be related to the structural index involved in Euler Deconvolution theory. The method is fast and stable because it takes advantage of the regular behavior of potential field data versus the altitude [Formula: see text]. As a result of stability, the DEXP method may be applied to anomalies with rather low SNRs. Also stable are DEXP applications to vertical and horizontal derivatives of a Newtonian potential of various orders in which we use theoretically determined scaling functions for each order of a derivative. This helps to reduce mutual interference effects and to obtain meaningful representations of the distribution of sources versus depth, with no prefiltering. The DEXP method does not require that magnetic anomalies to be reduced to the pole, and meaningful results are obtained by processing its analytical signal. Application to different cases of either synthetic or real data shows its applicability to any type of potential field investigation, including geological, petroleum, mining, archeological, and environmental studies.