Application of the tilt derivative transform to bathymetric data for structural lineament mapping

Abstract High-resolution bathymetry surveys provide an opportunity to analyse local geological structure where onshore areas afford limited exposure. Semi-automated lineament detection methods are necessary for areas of large coverage where a manual analysis would be subjective and time-consuming. However, semi-automated approaches are dependent on effective feature extraction methods to identify genuine lineaments. This study offers solutions to common problems that can impede processing methods where sharp steps in the seafloor (e.g. palaeocoastlines) are present. Directional gradient, Sobel and Laplacian filters are explored as well as the hillshade and tilt derivative transform for feature extraction prior to applying an object-based image analysis lineament detection approach. The filtered datasets generally perform poorly with a marked improvement when using the hillshade transform. However, we find the azimuth-invariant tilt derivative, which incorporates a convolved vertical derivative, to be the most successful, identifying lineaments in a range of orientations and across a sharp step in the seafloor.


Introduction
. They form the target for semi-automated lineament detection in this study and are of particular 48 importance for understanding the post-Variscan structural evolution of the region.

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Manual lineament extraction studies can be effective at identifying structural features and creating maps of fault 51 systems (e.g. Nixon et al., 2012). These studies often produce maps with long lineament traces that appear robust but 52 can be subjective and dependent on the visualisation method (Scheiber et al., 2015). Biases can exist in various aspects 53 of a manual analysis including lineament length and the scale/detail of fractures mapped, although user experience 54 appears to be less important (Andrews et al., 2019). Semi-automated methods can mitigate these biases but the data 55 often still require enhancement via feature extraction methods; thus requiring careful consideration. Directional filters 56 such as gradient and Sobel kernels are effective at finding lineaments where the orientation is known; the same holds for 57 the hillshade transform. Where this is not the case, the method must be azimuth-invariant and weight all orientations 58 equally (e.g. Laplacian filters). Changes in the vertical plane of the source data can also influence the outputs, which 59 is why the tilt derivative (TDR) transform is investigated in this study and compared to the aforementioned filters and 60 hillshade transform. to the nearshore coves with small pockets found along the palaeocoast. The area was featured as part of a manual 71 lineament analysis by Nixon et al. (2012) who determined a series of NW-SE and NNE-SSW trending fault sets that 72 exhibit dextral and sinistral offsets, respectively. The area is also used as a case study site to showcase the NetworkGT 73 plug-in for QGIS software, which consists of a suite of tools for geometric and topological analysis of two-dimensional fracture networks (Nyberg et al., 2018). Both Nixon et al. (2012) and Nyberg et al. (2018) have demonstrated that the 75 area provides an excellent site for studying fault networks and this study will aim to extend this into deeper water. 76 2.2. Filters and transforms 77 Geospatial data, even after all processing steps have been completed, almost always require some further ma-78 nipulation to enhance certain features prior to further analysis; for image or raster data, this often involves a filter or 79 transform. There are a broad range of enhancements that can be tailored to the task and, when used with an appropriate 80 semi-automated algorithm, a high degree of accuracy can be achieved (Sukumar et al., 2014). However, determining 81 a suitable image enhancement can be difficult and potentially subjective especially depending on the target structure 82 and the signal-to-noise ratio (Smith and Clark, 2005;Rahnama and Gloaguen, 2014). 83 Band pass filters, such as the gradient and Sobel operators, are effective at selecting a particular range (based on 84 directionality). Low-pass and high-pass filters are useful for mitigating noise and enhancing the sharpness of features, 85 respectively (Rahnama and Gloaguen, 2014). Transforms do not preferentially select data but convert the whole dataset 86 to derive a new variable.

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In this study, the directional gradient, Sobel and Laplacian filters as well as the hillshade and TDR transforms 88 have been selected to demonstrate various feature extraction methods. These operators have been selected as they 89 are commonly applied in lineament detection studies, be it manual or semi-automated, to enhance features prior to 90 detection on a variety of datasets (Smithurst, 1990;Morris, 1991;Maini and Aggarwal, 2009;Airo and Wennerström, 91 2010;Mallast et al., 2011;Grebby et al., 2012;Hashim et al., 2013;Rahnama and Gloaguen, 2014;Sukumar et al., 92 2014;Middleton et al., 2015;Mwaniki et al., 2015;Scheiber et al., 2015;Sedrette and Rebaï, 2016;Šilhavý et al., 93 2016;Thiele et al., 2017;Yeomans et al., 2019;Xu et al., 2020). It is worth noting that other methods are available  98 Directional filtering of spatial data is a well-established tool used to highlight features for lineament detection and 99 structural mapping. The filter uses a weighted kernel to accentuate particular-oriented features, where features are 100 perpendicular to the overall gradient of weights within the kernel. The use of directional filters was detailed by Moore 101 and Waltz (1983) who provided a five-step framework for lineament enhancement that included smoothing, directional 102 filtering, smoothing directional components, lineament extraction and scaling. For a 3 x 3 matrix, the process takes 103 the focal pixel, 0 , and surrounding pixels ( , ... ) from the input data in Equation 1:

Directional filters
Or, for a 5 x 5 matrix, Equation 2: The values in are convolved by a directional kernel in Equation 3 containing, in this case, a northwest gradient: or a northeast gradient using Equation 4: The results of these orthogonal filters can be combined as a magnitude using Equation 5: The weights used here have been chosen to emphasise the main directions of known faults in the study area.

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However, directional filtering can vary considerably depending on the task in hand, but generally take the form of a 3 110 x 3 kernel where the direction of positive-to-negative weighting provides the orientation of the kernel.

Sobel filter 112
The Sobel filter is a commonly used edge detector technique and allows the calculation of the and derivatives 113 with a level of smoothing imparted via the kernel (Sobel and Feldman, 1973;Favalli and Fornaciai, 2017). It is another 114 directional gradient-based method where the and derivatives for the Sobel filter are calculated using Equation 6 115 and Equation 7, respectively.
These two first-order derivatives can then be combined into a gradient magnitude image using Equation 8: The Sobel filter is most sensitive to lineaments in the X and Y directions and diagonal components can be sup-118 pressed (Sobel and Feldman, 1973). The Sobel filter is essentially a modification of the Prewitt filter which does not 119 account for smoothing. The introduction of a -2 weight to the filter (compared to a -1 for the Prewitt filter) adds a more 120 'circular' operation to the kernel that is advantageous over the Prewitt filter (Davies, 1986).

Laplacian filter 122
The Laplacian filter is a second-order derivative, non-directional filtering tool that has been widely applied for 123 detecting structural lineaments from remotely sensed data (e.g. Grebby et al., 2012;Rahnama and Gloaguen, 2014;124 Al-Azemi and Divi, 2017). The Laplacian can be derived using Equation 9, which can be approximated by convolving 125 the matrices described in Equation 10 and Equation 11 for a 3 x 3 kernel and 5 x 5 kernel, respectively.
The Laplacian filter is useful as it returns a smoother image where edges are located at the zero-contour (Marr and 127 Hildreth , 1980). Being a second-order derivative, the Laplacian filter is more sensitive to noise in the data and may The TDR transform is a useful tool for preserving low amplitude signals which may be attenuated over the dynamic 153 range in the presence of a larger amplitude signal (Miller and Singh, 1994;Verduzco et al., 2004;Fairhead et al., 154 2004). Values are restricted to ± ∕2 by the arctangent function, regardless of the derivative magnitudes, preserving 155 low amplitude signals and reducing the effect of noise. Additionally, this feature assists the interpretation where the 156 continuity of a body may vary due to lateral changes in signal (Verduzco et al., 2004). Furthermore, the zero-contour 157 passes over or near the edge of bodies (Miller and Singh, 1994). These features make the TDR transform an effective 158 tool for mapping edges or mapping minima/maxima. that lineaments between 060°and 120°were purposefully not included in the detection ranges for the two phases as 180 these equate to bedding surfaces that varied due to tight fold axes.

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In this section, we present visualisations using each of the filters and transforms introduced above and the subse-

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The operations performed on the data are presented in Figure 2, over the magnified area (illustrated in Figure 1b).

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This small subset of the study shows the edge of the platform and provides a good comparison of how the filters and 190 transforms perform across this pronounced change in depth. It can be seen from Figure  The use of a total horizontal derivative in the denominator means that there is no azimuthal bias to highlight particular 207 orientations of lineaments in the data, as is the case with the hillshade transform.

Data quality assessment 215
A key aspect of semi-automated lineament detection is identifying where false positive lineaments are being intro-216 duced into the final product. This can be difficult to quantify in absolute terms without manual inspection of individual 217 lineaments; an onerous task for large datasets. Spurious lineaments can be introduced by either similarity between 218 desirable and undesirable features or due to artefacts in the data. Improved feature extraction methods can assist in 219 correctly targeting desired geological lineaments, however, once introduced, false positive occurrences must be filtered 220 out using either through statistical, geometric or spatial methods.

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Statistical methods are often easy to apply and can include identifying a threshold in the population at which to 222 discard lineaments. This approach can be crude and is not particularly insightful, often resulting in the removal of 223 genuine data, however, it can also be a question of scale and usability. For example, it is common practise to use 224 Tobler's rule to derive a minimum "map scale" from an image by multiplying the pixel resolution by 2000 (Tobler, 225 1988). In this case, for a 2 m pixel resolution, the minimum map scale is 1:4000 where 1 mm represents 4000 mm 226 (4 metres). Lines of 1 mm length on a map result in data saturation and, for any map, one could argue that mapped 227 lineaments shorter than 4 mm are too detailed. In this study, a 4 mm line at our minimum map scale of 1:4000 would 228 make lineaments shorter than 16 m redundant. Therefore, a threshold has been applied to remove lineaments with 229 lengths less than 16 m. Finally, entries that resulted in anomalous NaN values in the final datasets were removed 230 resulting in 6, 2 and 1 lineaments being removed from the TDR-derived data and the 3 x 3 and 5 x 5 Laplacian filters, 231 respectively; these were likely generated when calculating the average depth of a lineament.

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Geometric filtering can include looking at shape of the image object and/or its relationship to others. OBIA methods 233 are particularly powerful at geometric filtering due to the combination of vector-type metadata for image objects and 234 the internel raster-based statistics which is enhanced by the topological analysis of different image objects. These have been incorporated into the workflow adopted from Yeomans et al. (2019) where the asymmetry, internal mean and area 236 of image objects are considered as well as their relationship to adjoining image objects. These help to merge adjacent 237 image objects that are similar whilst removing unusual geometries that are likely to not represent true lineaments.

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Spatial filtering requires a supplementary spatial dataset that can be used to identify co-located features. It can be 239 a particularly effective method over onshore areas for removing lineaments that pertain to roads or field boundaries, 240 should these data be readily available, however such datasets are less prevalent in offshore areas. Creating masks from 241 the original bathymetric data can provide an efficient spatial filter for identifying likely false positive lineaments. In this 242 study, smooth areas of the seabed were interpreted as sediment cover obscuring potential seafloor lineaments. These 243 areas can be identified by deriving a textural Terrain Ruggedness Index (TRI) raster layer from the original bathymetric 244 model, where smooth areas (low values) can be selected using a threshold to highlight likely areas of sediment cover.

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The resulting mask can then be used to post-process lineament datasets and remove likely false positive lineaments in 246 these areas. The threshold is user-defined and requires careful consideration so as not to mask viable areas but provides 247 an efficient means of removing likely spurious data. The full mask for this study is presented in the Supplementary

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By applying the spatial filter using the TRI mask to the lineament data across the whole study area, lineaments 250 were removed over these potentially problematic areas for each method; precise numbers are given in Table 1. Due 251 to the mask covering areas largely in off-platform areas, the greatest number of lineaments were selected from the 252 TDR-derived population. However, it is worth noting that comparatively few lineaments were detected in off-platform 253 areas for the other operators.

Visual lineament assessment 255
The six lineament sets derived from different feature extraction methods are displayed in Figure 3. These highlight 256 the performance of each operator from on-platform to off-platform areas, where the water depth increases >10 metres.

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The off-platform area has some sedimentary cover producing low-quality data but does display NW-SE structures that 258 correlate with on-platform structures.
The number of lineaments derived from gradient-filtered data across Zone 1 is not substantial, with the majority 260 of lineaments found over on-platform areas. Figure 3a weakly defines some NW-SE features in the data but the lack 261 of contiguous segments make interpretation more difficult. The off-platform areas perform even more poorly and 262 this is likely a function of the lack of smoothing (as mentioned for Zone 2) but also the more subtle features in off-263 platform areas being masked by the significant gradient caused across the step in the seafloor. In Figure 3b  show a fairly consistent picture of the depths at which lineaments are identified but there is a marked difference in the 306 depth at which lineaments are detected for the TDR transform, which clearly shows a greater mean depth and higher 307 standard deviation. This is further illustrated by the kernel density estimation in Figure 4 where the TDR shows a 308 distinctly different shape at depths deeper than -28 m (i.e. off-platform areas). It can be seen that between depths of -28 m and 0 m the trend in lineament density is similar for all methods, however, the large population of lineaments 310 noted for the 5 x 5 Laplacian kernel in Table 1 is focused between depths of -16 m to -20 m. The KDE function shows 311 relative density of lineaments, thus, the lower values within this range for the TDR transform do not represent fewer 312 lineaments at these depths but demonstrate that the method captures lineaments across the depth range.

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Axial data for lineaments derived for each method have been calculated and presented in rose diagrams in Figure 5.

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The data quality assessment using the TRI identified areas sediment cover assuming these remain smooth and free 335 of artefacts. By using these areas as a mask, likely spurious lineaments can be spatially filtered and removed as false 336 positive results. Due to the low numbers of selected lineaments for all methods apart from the TDR transform, only the 337 TDR-derived lineaments are used here to analyse the effects of false positives on the population. The total number of 338 false positives removed from spatial filtering from the TDR-derived dataset was 549 (Table 1). These are presented and 339 compared to the remaining lineament population in Figure 6 where equal-area wedge rose diagrams are implemented.

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A rose diagram of the unfiltered data is shown in Figure 6a where a strong modal axis can be identified in both a NW-341 SE and NNE-SSW orientation. Figure 6b shows a strong modal axis approximately NNE-SSW and are concentrated 342 over areas that were interpreted as sediment-covered due to their smoothness when applying the TRI mask. The TDR 343 transform is often heralded for its ability to give equal weight to minor features on a surface whilst in the presence of 344 larger structures ((Miller and Singh, 1994;Verduzco et al., 2004), however, the strong population in this orientation is 345 likely caused by small artefacts that are exaggerated by the TDR transform. We therefore suggest caution when using 346 the TDR transform in areas of sediment cover. Furthermore, the rose plot of data generated by the 3 x 3 kernel of the

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Laplacian filter in Figure 5c shows a similarly strong NNE-SSW trend which may imply that similar artefacts are being 348 identified by this filter but are not found of areas of sediment cover.

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The different feature extraction techniques tested in this study show markedly different lineament populations that 359 are variably affected by the sharp break in the seafloor. Gradient-based filters are the least effective, despite the initial 360 kernels being selected to emphasise NW and NE gradients and the rose diagrams show that the directional filtering has 361 forced the lineament population towards these major trends. The Sobel filter underperforms, relative to other methods, 362 even with smoothing incorporated into the kernel but does allow greater flexibility away from the major modal axes.

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Neither of these gradient-based filters are capable of capturing lineaments in the off-platform areas.

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The Laplacian filters successfully identify structures, albeit discontinuously, with a distinct improvement when

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The bathymetric data used in this study show a network of NW-SE and NNE-SSW faults sets that can be identified 398 using semi-automated lineament detection techniques. Semi-automated approaches have been demonstrated to produce 399 markedly different lineament populations based on different feature extraction tools. Thus, testing over a small area is 400 an important step prior to using semi-automated methods on regional scale.