1Analysis of hyperbolic responses from point source reflectors in Ground Penetrating Radar (GPR) profile data can provide a useful method to establish the velocity of the wave-front in the subsurface. An accurate determination of the velocity field can be used to better estimate the apparent depth of reflectors and allow appropriate parameters to be chosen when applying migration algorithms to process the data. Lateral variation or changes of the velocity with depth may also help with the interpretation of the data, perhaps indicating different material properties or changes in local water content.

2Determination of the velocity is often made by visual comparison of the GPR response to a hyperbolic curve where the velocity is adjusted to provide a “best fit”. Figure 1 shows an example of how a manual determination of velocity was used to identify areas water ingress in the walls of the Captain’s chamber at Portland Castle, Dorset. Degraded ferrous wall ties (Fig. 1A) have allowed cracks to develop in the Portland stone walls leading to subsequent water ingress. A GPR survey was conducted with a Sensors & Software PE1000 GPR unit and 900 MHz centre frequency antenna over a survey grid established over the vertical face of the wall (Fig. 1B). Hyperbolic responses were identified in the GPR profiles (e.g. Fig. 1C) and the velocity determined manually by curve fitting at each point. The velocity field through the wall section was then estimated by an inverse distance weighted interpolation from the location and depth from the surface for each point where the velocity had been determined from a hyperbolic response. Figure 1D shows the interpolated velocity map between 0.9 to 1.0 m from the surface of the wall. The location of the hyperbolic responses are shown as green (within the current depth slice and a higher) and red (outside the current depth slice) crosses.

3Conducting the velocity determination manually through curve fitting can be time consuming, particularly on larger data sets and the process can be automated, for example using Hough transform to determine the best fit to the hyperbolic response or in combination with a computer vision algorithm to isolate hyperbolic responses from field GPR data sets (e.g. Dou et al., 2017; Maas & Schmalzl, 2013; Mertens et al., 2016; Rivera-Rios & Flores-Marquez, 2012; Simi et al., 2008; Verdonck, 2019). An alternative approach is to apply a constant velocity migration to the data and then assess how well the hyperbolic response has been collapsed back to a point source reflector, for example through local maximum energy (Dang, et al., 2018). Figure 2 shows an example of this approach using synthetic data with the migration repeated over a range of trial velocities to determine the value that provides the best fit. In this case the synthetic data was calculated from a circular reflector of radius 0.1 m located at a position of x = 10 m and y (depth) = 1.0 m, with a relative dielectric permittivity k = 4.5 compared to the model background k = 9.0 (Bitri & Grandjean, 1998).

4This paper proposes the analysis of the migrated data in a manner analogous to the detection of pit-type anomalies from area GPR time slices by extracting closed vector polygons following the application of an amplitude threshold to the image (Linford & Linford, 2017). Each extracted polygon can be described in terms of a central coordinate, the area enclosed and morphology used to determine how circular the anomaly appears to be. In theory, as the estimated value used for the constant velocity migration gets closer to the best fit for a particular hyperbolic response, the area enclosed by the extracted vector polygon should reach a minimum and the morphology should become more circular.

5For a circle of radius r, the area A and circumference C will be of the A = πr² and C = 2πr respectively. The degree to which the morphology M of an extracted vector polygon is assessed to be circular can be estimated by:

6M = 4πA / C²

7Where values of M → 1 are more circular.

8Beyond the best fit value the hyperbolic anomalies will be either over or under-migrated, altering both the morphology of the extracted polygon and increasing the enclosed area of high amplitude response. Figure 2F shows the closed vector polygon extracted from the data following migration with a best fit velocity estimate of v = 0.1122 m/ns that confirms the location and depth of the synthetic model (estimated depth 1.00419 m versus a model depth of 1.0 m).

9Constant velocity migration can be performed through batch processing of field GPR data over an appropriate range of velocities, for example from between 0.05 and 0.15 m/ns. Closed vector polygons are then extracted from the migrated data at each velocity step and each polygon can then be assessed in terms of its size, location and morphology. Preprocessing of the data prior to migration, through the application of background subtraction and trace difference, can be applied to suppress horizontal anomalies and enhance any hyperbolic responses present. The closed vector polygons are then extracted from the amplitude enveloped data to better define the energy associated with each point reflector.

10The morphology M of the vector polygons also varies with the migration velocity and becomes most circular at 0.07 m/ns before decreasing as the velocity is increased Figure 3D. There is some instability with this parameter as the velocity increases and the data becomes over-migrated. However, comparison between the minimum area enclosed by the vector polygon and the peak value for the morphology confirms the most appropriate velocity estimate in the region of 0.07 m/ns shown as a hyperbola fitted to the raw data in Figure 3A. It is hoped that the determination of velocity can, to a certain degree, be automated by this method for application to much larger data sets.