CSIRO PUBLISHING
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Exploration Geophysics, 2009, 40, 233–236
Blocking geophysical borehole log data using the continuous wavelet transform Gordon R. J. Cooper1,3 Duncan R. Cowan2 1
School of Geosciences, University of the Witwatersrand, Johannesburg, South Africa. Cowan Geodata Services, 12 Edna Road, Dalkeith, WA 6009, Australia. 3 Corresponding author. Email:
[email protected] 2
Abstract. The interpretation of geophysical log data is frequently difficult due to the noisy downhole environment. Blocking algorithms attempt to smooth the log data while leaving the boundaries between different geological units sharp. This paper introduces a method for the determination of the boundaries based on the zero contour of the continuous wavelet transform (CWT) of the data. The amount of blocking can be controlled by the choice of the scale of the wavelet used. The method is compared with results from the median filter and with discrete wavelet transform (DWT) blocking methods, and is here applied to log data from Australia. The application of the new CWT method overcomes the rounding and shifting of boundaries inherent in median filtering, and provides greater flexibility by overcoming the power of two limitations in the DWT log blocking. Introduction Geophysical log data measurements reflect changes in the physical properties of the rocks through which the borehole has been drilled. Local scale variability, the presence of layering effects at several scales, and aliasing frequently make the determination of the boundaries between different rock types difficult when interpreting log data. For that reason log data is commonly ‘blocked’ to aid in the location of the boundaries and to simplify log interpretation, especially in the case of multiparameter logs. Blocking involves detecting depth intervals for which parameters remain unchanged, corresponding to individual homogeneous formations, a process known as zoning (Kerzner, 1986). Blocking tends to be a subjective process with filter parameters selected to solve a particular problem. Blocking algorithms aim to smooth the log data while leaving the boundaries sharp. The most commonly applied blocking algorithm is the median filter which is robust against outliers, enabling it to remove noise spikes without blurring edges too badly. The median filter is applied using a sliding window, and the amount of smoothing is controlled by changing the window size. Figure 1A(a) shows a synthetic dataset and the result of applying two median filters of different widths to it (Figure 1A(b) and 1A(c)). An alternative method of blocking the logs is to use the discrete wavelet transform. Wavelet analysis is a powerful tool for examining the frequency content of data as a function of time or (in this case) distance (Strang and Nguyen, 1996; Mallat, 1998). There are two main areas of wavelet analysis – the discrete wavelet transform (DWT) and the continuous wavelet transform (CWT). The DWT analyses the data using only scales which are powers of two, and has a fast and efficient forward and inverse transform. The scales on which the CWT analyses the data are arbitrary, and it has no efficient inverse transform. The CWT of a dataset f(t) is defined as (Mallat, 1998, p. 5); ð¥ t u 1 dt f ðtÞ pffiffi C* Wf ðu; sÞ ¼ ð1Þ s s ¥ where s is scale, u is position, C is the wavelet used, and * indicates the complex conjugate. The CWT is most commonly ASEG 2009
used to examine visually how the frequency content of a dataset varies with time, whereas the DWT is used whenever data is to be filtered (denoised or compressed, for example). Cowan and Cooper (2003) denoised borehole log data by computing its DWT approximation using the Haar wavelet. This approach had the advantages that all segments of the denoised log were perfectly flat, unlike the median filtered data. The size of the segments was controlled by adjusting the level of the wavelet approximation used. Figures 1A(d) and 1B(e) show a synthetic log which has been denoised using the DWT approximation at two different levels. Denoising borelog data using the CWT Figure 1B( f ) shows the CWT of the synthetic log data computed using the ‘Mexican hat’ wavelet. This is the second derivative of a Gaussian function, and is defined as (Mallat, 1998, p. 77); 2 t2 2 t ð2Þ pffiffiffiffiffiffi 2 1 e2s2 CðtÞ ¼ p0:25 3s s The boundaries between the different geological units are characterised by maxima of the absolute value of the first derivative of the log data and consequently by zero values in the second derivative of the log data. Because of the noise in the data, a simple calculation of its second derivative yields results which are far too noisy to be useful in practice. However the application of the CWT using the ‘Mexican hat’ wavelet to the data effectively computes the second derivative over a range of scales (chosen by the interpreter). On small scales the result is noisy and edges appear almost everywhere, whereas on larger scales only the more significant edges are present. Figure 1B( f ) overlays contours of the zero value of the CWT on the image of the CWT values themselves. It can be seen that there are many contours at small scales, but fewer at larger scales. To perform the blocking of the log, a scale is chosen and boundaries are selected only using the contours which intersect this scale (marked as a vertical line in Figure 1B( f )). The intersection points themselves are not used as the boundaries because their position will vary depending on the scale that is chosen. Instead, the location of the contour at a scale of 1 unit is used. Figures 1B(g) 10.1071/EG08127
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Scale Fig. 1. (A) (a) Synthetic log data with boundaries at 20, 148, 190, and 255 m. Between each pair of boundaries the data consists of a constant value to which uniformly distributed random noise has been added. (b) Result of applying a median filter with a width of 21 points (20 m). (c) Result of applying a median filter with a width of 45 points (44 m). (d ) Discrete wavelet transform (DWT) approximation at level 4 using the Haar wavelet. (B) (e) DWT approximation at level 5 using the Haar wavelet. ( f ) Continuous wavelet transform of the data using the ‘Mexican hat’ wavelet and with its zero contour plotted. Large positive values are shown in white while large negative values are shown in black. Black vertical lines show the scales used in Figures 1g and 1h. (g) Data blocked using boundaries picked from Figure 1f at a scale of 8 units. (h) Data blocked using boundaries picked from Figure 1f at a scale of 45 units.
and 1B(h) show the result of using two different scales to identify the boundaries of the log. Note that any given boundary will either be present or not depending on the scale chosen, but its position cannot vary. Once the boundaries have been identified then the smoothed values of each portion of the log can be computed from the mean, median, or whatever other function (such as a polynomial fit) is deemed appropriate. A comparison of the results of the three different methods discussed shows that when the window size of the median filter
used is small then the method does not blur the boundaries very much but does not smooth the data much either. Conversely, when the window size is large then the result is smoother but sharp boundaries become blurred or lost altogether. The DWT method produces a smoothed log consisting of flat sections with sharp boundaries, but the boundaries do not always lie in the correct locations because they can only occur at positions which are multiples of 2L, where L is the scale used. However when the CWT method is used the boundaries are located accurately and
Blocking geophysical borehole log data using CWT
Exploration Geophysics
the data between them can then be smoothed in any desired manner.
susceptibility and natural gamma radiation that are easy to measure may provide useful information on local scale variations. The lithological log for the drill hole section studied is shown in Figure 2A(a). A raw susceptibility log of the Upper Dales Gorge and Whaleback Shale sequence is shown in Figure 2A(b). Readings were taken every 5 cm so variations in magnetic susceptibility at the microband to mesoband level are clearly aliased. The drill section is an alternating assemblage of BIFs and shale macrobands of the Palaeoproterozoic Hamersley Group. The BIFs are characterised by very high anisotropic magnetic susceptibilities and high remanent magnetization, whereas the shale macrobands are only weakly magnetic. The Hamersley Group BIFs show examples of layering over a wide range of scales from 1 mm up to ten of metres. Mesobanding is centimetre scale banding of chert/oxide layers, which produces the characteristic light/dark banding of the BIFs. The scale of layering ranges from 1 mm to 100 mm thickness with a mean around 10 mm. Microbanding, or aftbanding, is the internal zonation within the chert mesobands with iron-oxide layer thickness ranging between 0.1 mm and 1.5 mm. There are wide variations in the number and thickness of these aftbands. Macrobanding refers to the BIF/shale interlayering. Bulk magnetic susceptibility varies considerably throughout the BIF units due to variations in thickness and composition of oxide/ silicate mesobands, variations in the proportion of microbands
Application to susceptibility data from Hamersley Basin, Western Australia The new CWT method has been used to ‘block’ drill hole susceptibility data from Hamersley Basin, Western Australia (Cowan and Cooper, 2003). The Dales Gorge and Whaleback Shale Members of the Brockman Iron Formation (the subject of this study) consist of alternating assemblage of banded iron formation (BIF) and shale macrobands, which are laterally persistent throughout the province, although individual macroband thickness varies. In the past, it has been assumed (Harmsworth et al., 1990) that the BIFs have uniform composition. Recent analysis of susceptibility and natural gamma logs suggests some local-scale facies variation which may be important, as a possible control on the phosphorus content of BIFs, and the location of high-grade ore deposits. Krapez et al. (2003) applied sequence stratigraphic principles to Hamersley BIFs. Their sedimentological approach to the origin of BIFs and associated lithofacies provide new insights into depositional conditions and emphasise the need for very detailed log analysis to understand precursor sediments and protores. As metamorphic textures make it difficult to distinguish primary from secondary effects in drill core, parameters such as magnetic (A)
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Fig. 2. (A) (a) Lithological log, BIF, banded iron-formation; SHL, shale. (b) Raw susceptibility data. (c) Result of applying a median filter with a width of 55 points. (d ) Result of applying a median filter with a width of 99 points. (e) Discrete wavelet transform (DWT) approximation at level 3 using the Haar wavelet. ( f ) DWT approximation at level 5 using the Haar wavelet. (B) (g) Continuous wavelet transform (CWT) of the data using the ‘Mexican hat’ wavelet. (h) Blocking using CWT zero contour locations at scale 5 to select boundaries. (i) Blocking using CWT zero contour locations at scale 8 to select boundaries.
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and changes in the haematite/magnetite ratio. The raw susceptibility data in Figure 2A(b) show variation at several scales with the large-scale pattern of very high susceptibility values for BIF macrobands and low values for shale macrobands overprinted by rapid fluctuations in susceptibility, caused by local-scale variations in oxide/silicate and magnetite/haematite ratios. The local scale fluctuations in susceptibility mask differences between individual BIF macrobands, so some form of smoothing needs to be applied to the raw data to highlight differences and highlight sequence boundaries and facies variation at the 1–2 m range. Ideally, the data should be subdivided into relatively homogeneous segments or zones with sharp boundaries. Separating within-segment variance from between segment variance is a subjective process, and choice of window size for moving window techniques and signal to noise thresholds is critical. Figures 2A(c) and 2A(d) show the application of 55-point and 99-point window median filters to the raw susceptibility data. Although the median filter has simplified the spatial series, boundaries between segments are rounded and blurred and the filtered log is not particularly useful. Cowan and Cooper (2003) show that the Haar DWT can be an appropriate filtering method producing results very similar to the blocking technique developed in the petroleum industry to restore the original square shape of a log (Kerzner, 1986). Figures 2A(e) and 2A( f ) shows the Haar filter results for approximation levels 3 and 5, corresponding to scales of 8 and 32 samples respectively. As the approximation level increases, the within-segment variance is progressively reduced, and longer wavelength features become more obvious. Approximation level 3 still shows high within-segment variance whereas approximation level 5 has removed too much signal and only broad formation boundaries are left. The DWT method is superior to median filtering, because it produces a smoothed log consisting of flat sections with sharp boundaries. The limitation is that boundaries do not always lie in the correct locations because they can only occur at positions which are multiples of the scale used. The application of the new CWT method provides greater flexibility in the choice of approximation levels by overcoming the power of two limitations in the DWT log blocking. Lithological boundaries are no longer restricted to having thicknesses and starting locations that are located on data samples that are powers of two. Figure 2B(g) shows the
continuous wavelet transform of the susceptibility log data computed using the ‘Mexican hat’ wavelet. As in Figure 1, the log has been blocked by selecting boundaries using the contours which intersect the selected scales. In Figure 2B(h), the log has been blocked using a scale of 5 units. In Figure 2B(i), the log has been blocked using a scale of 8 (shown as a vertical black line on the CWT plot). The blocked logs give a much clearer picture of differences between BIF macrobands, and also suggest different patterns of internal zonation at the 1–2 m scale for different macrobands. Conclusions A new method for blocking borehole data, based on CWT, has been introduced. It has been compared with the median filter and a technique based on the discrete wavelet transform, and appears to give improved results. The application of the new CWT method overcomes the rounding and shifting of boundaries inherent in median filtering provides greater flexibility by overcoming the power of two segment size limitations of the DWT log blocking. References Cowan, D. R., and Cooper, G. R. J., 2003, Wavelet analysis of detailed drillhole magnetic susceptibility data, Brockman Iron Formation, Hamersley Basin, Western Australia: Exploration Geophysics, 34, 87–92. doi: 10.1071/EG03087 Harmsworth, R. A., Kneeshaw, M., Morris, R. C., Robinson, C. J., and Shrivastava, P. K., 1990, BIF-derived iron ores of the Hamersley Province. In F. E. Hughes, ed., Geology of the Mineral Deposits of Australia and Papua New Guinea, Monograph 14, The Australasian Institute of Mining and Metallurgy, 617–642. Kerzner, M. G., 1986, Image Processing in Well Log Analysis, D. Reidel Publishing Company. Krapez, B., Barley, M. E., and Pickard, A. L., 2003, Hydrothermal and resedimented origins of the precursor sediments to banded iron formation: sedimentological evidence of the Early Palaeoproterozoic Brockman Supersequence of Western Australia: Sedimentology, 50, 979–1011. doi: 10.1046/j.1365-3091.2003.00594.x Mallat, S., 1998. A Wavelet Tour of Signal Processing: Academic Press. Strang, G., and Nguyen, T., 1996, Wavelets and Filter Banks, WellesleyCambridge Press.
Manuscript received 22 September 2008; revised manuscript received 17 February 2009.
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