Fractal-Based Computation of Heat Source Depths and Temperatures for the Soutpansberg Basin, South Africa

The fractal-based approach was used for computing magnetic depths and temperatures for the Soutpansberg Basin in South Africa. The average depth to the top Zt and basement depth Zo for the Soutpansberg Basin were 4.36 ± 0.28 km, 10.43 ± 0.65 km, respectively. The average temperature at depth Zt was 184.69 ± 7.66 oC. Magnetic source depths and basal temperatures that were in the Curie point range were determined, to be within 20.35 km to 21.68 km and 549.34 oC to 585.24 oC, respectively. Increasing the value of the fractal parameter β from 0 to 4, had an effect of retaining deeper depths and higher temperatures. The fractal parameter values of β > 3 retained Curie point depths and temperatures that indicated basal rock types with an igneous predisposition. The fractal-based approach proved to be an improved technique as compared to the conventional centroid method.


Introduction
Depths and temperatures of magnetic sources were computed for the Soutpansberg Basin in northern part of South Africa, from power spectra of the airborne magnetic data from square data blocks with sides with dimensions L of 51 km, 103 km, and 129 km ( Figure 1). Ledwaba et al. (2009) reported that the airborne magnetic data for the study area was collected in 1973 along survey lines that were 1000 m apart, maintaining a mean sensor elevation of 150 m. The aim of the research was to characterise the heat sources and investigate the geothermal potential of the Soutpansberg Basin. Jones (2017) reported that the formation heat for the northern part of South Africa was due to radiogenic elements such as uranium, thorium or potassium. Jones (2017) attributed the anomalous heat flow in the study area, to the concentration of geothermal heat by the deep-seated circulation of meteoric water.
The conventional computation of magnetic source depths from the analysis of Fourier Transform computed spectra data was reported to result in the overestimation of Curie Point Depth in studies by Bansal et al. (2016) and Khojamli et al. (2017). Bansal et al. (2016) mentioned that conventional methods assumed that magnetic sources had distribution that were random and uncorrelated, and further stated that the sources follow random and fractal distributions. Nyabeze & Gwavava (2016) used the conventional centroid and spectral peak methods and achieved basal depths and temperature that were below the Curie point values for the Soutpansberg Basin. Nwankwo et al. (2009) defined the Curie point as having a temperature of 580 °C. Pilkington and Todoeschuck (1993) reported that the magnetic sources displayed random and fractal distributions. Bansal et al. (2016) however reported that the simultaneous estimation of depth and scaling exponents was a limitation of the fractal-based approach for computing depth. The correction of the power spectrum before computing the depth to the top of magnetic sources using magnetic data was reported by Bouligand et al. (2009) and Salem et al. (2014). Bansal et al. (2011) corrected the power spectrum for scaling distribution of magnetic sources for the computation of both depth to the top Z t and the centroid Z o . Khojamli et al. (2017) and Ravat et al. (2007) stated that fractal source distributions had power spectra proportional to k -β , where k represented the wavenumber and β denoted the fractal parameter. Values of β where reported to be varying between 1.5 to 5.8 for lithology types ranging from sedimentary to igneous (Khojamli et al., 2017). The value of β was found to be equal to 4 for data from South Africa by Maus et al. (1997). Ravat et al. (2007) stated that the power spectrum for random depth source variations was associated with a β value of 2.9. Akbar & Fathianpour (2016) applied the Fractal Based Approach and computed depths for blocks with dimensions of L of 10 km and 51 km for the Sabalan geothermal field in Iran, and obtained depths in the 5.2 km range, that were closer to the actual well depth and comparable to those obtained using an L of 100 km. Bektas et al. (2007) mentioned that the Curie Temperature for magnetite was 580 °C and that the Curie Point depth was determined from the temperature versus depth graph. Curie Point values were reported as being 573 °C for quartz, 585 °C for magnetite, 770 °C for iron, and that values for the upper lithosphere varied between 550 °C and 580°C (Arnaiz-Rodriguez & Orihuela, 2013). Temperatures below the maximum depth of magnetic source of approximately 40 km should be above the Curie point value of approximately 550 °C (Telford et al., 1990). Campbell et al. (2016) reported the existence of sedimentary basins in the south and south-eastern part of South Africa with temperatures ranges above 200°C to 230°C at depths below 1 km to 3.5 km. Figure 1. Total field magnetic data showing the outline of the Soutpansberg Basin, hot springs as red symbols, block centres 1-36, A1-A14, and B1-B6 for blocks with dimensions L of 51 km, 103 km and 129 km, respectively and mapped geological faults (after Barker et al., 2006).

Method
Airborne data window sizes with L dimensions of 51 km, 103 km, and 129 km were utilised for spectral analyses of data for depth determinations. The number of data magnetic blocks with L dimensions of 51 km, 103 km, and 129 km were 36, 14, and 6 respectively. The largest window size with L of 129 km was chosen to ensure the preservation of spectral signatures. The computation of depth to the top Z t and depth to the centroid Z o , after Bansal et al. (2016) and Khojamli et al. (2017), involved the correction of power spectrum for the distribution of sources and obtained the following relationships, as Equation 1: and the depth to the centroid Z o is obtained, as Equation 2: where k is the wavenumber, β is the scaling component, and k -β being a scaling factor.
The basal depth, Z b is computed using Equation 3 below, where Z o is the depth to the centroid, and Z t is the depth to the top of a geological body Eletta and Udensi, 2012). The Curie temperature is computed, as in Equation 4: where, Ɵ is the Curie temperature and / is the geothermal gradient and is the Curie depth (Kasidi and Nur, 2013). The depth to the top Z t was determined from the gradient 2Z t of the graph of spectral energy versus wavenumber k (km -1 ) for wavelength between 0.50 and 0.20 (km -1 ). The depth to the basement or centroid Z o was determined from the gradient 2Z o of the graph of spectral energy versus the wavenumber k for wavelength between 0.10 and 0.5 (km -1 ). The depth to the basal Z b and corresponding Curie point depths were computed using depths Z t and Z o . All depths and basal temperatures were computed by applying fractal parameter values of β ranging between 0 to 4.
The subsurface formation temperature is calculated, as in Equation 5: where is the temperature at depth, is the temperature at the surface, is the geothermal gradient and is the source depth (Tiab & Donaldson, 2015). The geothermal gradient, for the study area was 27 °C/km (Jones, 1992) and an average temperature of water at the hot spring T s was 67.5 °C (Brandl et al., 2001 andShabalala et al., 2015).

Results
The Fractal based approach for depths Z t , Z o and Z b and corresponding formation temperatures T t , T o and T b for L values of 51 km, 103 km and 129 km for fractal parameters β = 0 to 4, are presented in Table 1, Table 2 and Table  4, respectively. Results showed wider standard deviations from the mean for data obtained using different fractal parameter values β of 0 to 4. The standard deviation for depths Z t , Z o and Z b were 14.00% to 15.50%, 27.10% to 27.70% and 30.40% to 31.04%, respectively.
Depths and temperatures were computed for data points, N of 36, 14 and 6 for magnetic blocks with dimensions L of the of 51 km, 103 km and 129 km, respectively. Table 1 has standard deviations of the mean showing the comparison of results obtained from the application of the fractal parameter β on spectral data with specific data points N and window size L. The variation of mean depth and temperature results were 0.6% to 6.6% and 0.5% to 6.6%, respectively (Table 4). Table 5 has a comparison of depths and temperatures obtained for blocks with L values of 51 km, 103 km and 129 km, respectively with standard deviations between 6.20% and 6.50%.
The depths to the top Z t , for L dimensions of 51 km and 103 for β = 3, are in the range 4.70 km to 4.80 km ( Figure  2) and 4.40 km to 4.60 km (Figure 3), respectively. Figure 4 has an illustration of the depth to the top Z t , basement depth Z o and basal depth Z b for blocks with L= 129 km fractal from application of fractal parameter β = 3 showing depths in the range 5.20 km to 5.37 km, 13.26 km to 13.50 km and 21.24 km to 21.67 km, respectively.      Std.Dev (%) 6.48% 6.23% 6.28% 4.15% 5.04% 6.28%

Discussion
The value of computed depths and temperature increased with the application of higher fractal parameters values of β from 0 to 4 for blocks with L = 51 km, 103 km and 129 km. Depths Z t , Z o , and Z b and basal temperatures were consistent within a unique value of β, with the low standard deviations below 1.2%. The variation of the average depths and temperature values for the three block sizes was below 6.5%, indicating that any block size with an appropriate fractal parameter can be used compute the source depth and temperature. A comparison of depths and temperatures for a fractal parameter range of β = 0 to 4, retained a large standard deviation between 14.80% to 31.04%, showing that changing β values influences depths and temperatures. The correction of depths using the fractal parameter was reported by Bansal et al. (2011), Bouligand et al. (2009), and Salem et al. (2014. Obtaining of different depths and temperatures for unique values of β was reported to be due to different types of lithologies (Khojamli et al., 2015). The results showed that there was an improvement in the standard deviation of the mean for depths and temperature L = 51 km to L = 129 km. The Fractal based approach retained basal depths and corresponding temperatures that were within the Curie point range (Nwankwo et al., 2009) for fractal parameter values of β > 3.

Conclusion
The average depth to the top Z t and basement depth Z o for the Soutpansberg Basin were 4.36 ± 0.28 km, 10.43 ± 0.65 km, respectively. The shallower depth to the top Z t is important for geothermal exploration. The average temperature at depth Z t was 184.69 ± 7.66 ºC, indicating geothermal potential. The application of a higher fractal parameters β > 3 achieved basal depths and temperatures within the Curie point range of 20.35 km to 21.68 km and 549.34 ºC to 585.24 ºC, respectively. The Curie depth at which basement rocks lost their magnetisation due to heat, the Curie Point Depth was only achieved using the Fractal based approach. The depth and temperature results from the application of the same fractal parameter β on data with different window sizes L were found to be comparable and below 10%. Changing the window size did not have a significant effect on depth and temperature values. The application of different values of the fractal parameter β on the same or different window sizes had an effect of increasing the deviation of results from the mean by more than 15%. The use of data windows with different sizes does not have a significant effect on the computed depths and temperatures. Fractal based approach results, showed that the conventional Centroid based approach retained results that were comparable to application of the lower fractal parameter β values of 0 to 1, that represent sedimentary type formations. The fractal-based approach is a better technique for determination of magnetic heat source depths and temperatures. The higher fractal parameter β > 3 retained depths and temperatures in the Curie point range that were indicative of rock formations with an igneous bias.