Regular Article
Well log data superresolution based on locally linear embedding
College of Physics and Electronic Engineering, Northeast Petroleum University, Daqing 163318, PR China
^{*} Corresponding author: 2645073549@qq.com
Received:
7
February
2021
Accepted:
21
July
2021
Unconventional remaining oil and gas resources such as tight oil, shale oil, and coalbed gas are currently the focus of the exploration and development of major oil fields all over the world. Therefore, to make best understand of target reservoirs, enhancing the vertical resolution of well log data is crucial important. However, in the face of the continuous lowlevel fluctuations of international oil price, large scale use of expensive high resolution well logging hardware tools has always been unaffordable and unacceptable. In another aspect, traditional well log interpolation methods can always not realize high reliable information enhancement for crucial high frequency components. In this paper, in order to improve the well log data superresolution performance, we propose for the first time to employ Locally Linear Embedding (LLE) technique to reveal the nonlinear mapping relationship between 2timesscaledifference well log data. Several super resolution experiments with well log data from a given area of Daqing Oil field, China, were conducted. Experimental results illustrated that the proposed LLEbased method can efficiently achieve more reliable superresolution results than other stateoftheart methods.
© J. Han et al., published by IFP Energies nouvelles, 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
In recent years, most old oilfields in the world have entered the middle and late stages of development. However, due to the longterm waterflooding development of oilfields, several unavoidable complex cases happened to the conventional resources, such as frequent river channel reconstruction, complex vertical and horizontal changes of sand body, strong heterogeneity of lithology and lithofacies. Consequently, oil and gas exploration and development are becoming more and more difficult, and unconventional oil and gas resources such as tight sandstone, shale oil/gas, coalbed gas, etc, have become the most important target reservoirs. Obviously, the efficient exploration and development of unconventional oil and gas resources has become the main way to achieve stable production to increase and extend the life of oilfield development [1]. Therefore, in order to enhance the production capacity of the unconventional remaining oil and gas resources, the reservoir modelling resolution should be as high as possible. Meanwhile, the highresolution logging curves can better provide the possibility to predict the location of lost circulation [2], lithology identifies [3], and predict the productivity of heterogeneous reservoirs [4, 5]. In practice, because the vertical resolution of seismic data is too low and the cost of acquisition of core and other high resolution logging data is too high, it is an inevitable choice to employ conventional well log data to enhance the vertical resolution of the final reservoir models. However, in the conventional oil/gas era, interpolation algorithms based on simple models have been used for enhancing the vertical resolution of well log data for a very long time. Until the last decade or so, with the development of machine learning technology, a few data driven well log superresolution methods have been proposed in the literature. Furthermore, whether it is a simple modeldriven interpolation method or a datadriven superresolution method, they all try to establish a nonlinear mapping function between low resolution and highresolution data directly in the original amplitude space. Due to some uncontrollable factors, there are unavoidable data deficiencies existing in the raw well data, therefore, the quality of training data severely restricts the effectiveness and robustness of these superresolution methods. In order to deal with this problem to a certain extent, Locally Linear Embedding (LLE) technique which can explore intrinsic manifold of the raw data with sufficient outlierresistance and robustness was employed in this paper to perform well superresolution task. Therefore, in this paper, a novel LLE based well logs super resolution method was proposed in the literature for the first time.
2 Related works
In the actual survey work of the oil field, due to the influence of factors such as well diameter and measurement, the actual logging resolution that can be achieved at present is only 12.5 cm. Faced with complex geological conditions, the thickness of shale oil lamellae is less than 0.3 cm. Therefore, it is not realistic to accurately characterize the target reservoir using original logging data. Obviously, in order to enhance capacity of reservoir characterization accuracy, the resolution of well log data must be improved. However, due to the high cost of highresolution well logging tools, a series of methods to improve the logging resolution was proposed in the literature. In 1989, Flaum et al. [6] proposed the neutron density α factor method. By assuming that the α factor changes slowly, the highresolution neutron porosity curve can be calculated by using the counting rate of near detector. In 1991, Nelson, and Mitchell [7] deduced the expression of smoothing filter used for curve matching and proposed the resolution matching technology for high resolution processing of well log data. On this basis, [8] used genetic algorithm to enhance the resolution of well log data. Specifically, they first used genetic algorithm to determine the filter in the frequency domain and then used resolution matching technology to improve the resolution of the input well log data. Conaway [9] discussed the deconvolution technology of natural gamma ray data with point detector. Based on the threepoint deconvolution formula, and proposed a method for determining the shape constant α, which appropriately considers the influence of formation dip. Freedman and Minerbo [10] employed the maximum entropy deconvolution method to improve the vertical resolution of well log data. By taking the layer interface information into consideration, they formulated a Lagrange optimal function with maximum entropy constraint for super resolution, and reasonable vertical resolution enhancement can be achieved. Besides to the method processed in spatial domain, transformed domain based well log superresolution methods can achieve relative better performance. In 2005, Tai and Cao [11] used Walsh transform to improve the resolution of well log data. Furthermore, their method can also ensure that the error between the calculated value and the true value does not exceed the error of the raw well log data itself. However, this method is only suitable for linear response well log data and the response function H (τ) cannot be accurately determined. In 2015, Li et al. [12] proposed to employ window Fourier transform to transform the raw well log data from the spatial domain to the frequency domain. The relationship between highresolution and lowresolution well log data in the frequency domain can therefore be constructed.
Although the abovementioned modeldriven methods can improve the well log resolution to a certain extent, the original outliers or errors in the raw well log data will be certainly propagated or even enlarged to the corresponding highresolution data. To deal with similar problems, in the computer audio/vision fields, sparse representation and deep learning techniques have been successfully applied to the image superresolution task [13–17]. In 2017, Ledig et al. [18] proposed an image superresolution method based on generative adversarial network. Also in 2017, Volodymyr et al. [19] proposed a multilayer convolution neural network for highresolution audio signal processing. In 2018, Lim et al. [20] proposed a novel deep neural network structure to perform audio superresolution in timefrequency domain. Generally, promising superresolution results can be achieved in the computer audio/vision fields by employing datadriven superresolution techniques. However, compared with audio or image/video data, there are much more uncertainties existing in the well log data. Consequently, directly employing datadriven methods in spatial or classical transform domains cannot prevent the propagation of uncertainty to the superresolution version of data.
Based on the above review and analysis, we propose to employ manifold learning techniques, which can extract the intrinsic manifold of data with variable uncertainties, to enhance the superresolution performance of well log data. Specifically, the LLE algorithm is used in this paper to exploit the intrinsic manifold information of well log data to conduct the corresponding superresolution task.
3 Locally Linear Embedding (LLE) based superresolution
3.1 Locally Linear Embedding
LLE algorithm is originally a nonlinear dimension reduction algorithm, which belongs to the category of manifold learning. Manifold learning is to recover intrinsic low dimensional structure from highdimensional sampled data, that is, to find low dimensional manifold in highdimensional space and find the corresponding nonlinear mapping to achieve dimension reduction. Chang et al. [21] first proposed the use of LLE to deal with image superresolution and stable and promising results were illustrated. Specifically, for the LLE algorithm, suppose there are m ndimensional samples {X_{1}, X_{2}, X_{3}, ⋯ X_{m}}, the first step is to select the neighborhood size which is one of the hyperparameters used in the LLE algorithm. Without loss of generality, assuming that the neighborhood size is K. Therefore, for a given data point X_{i}, we assume that it can be represented by the weighted linear combination of its K nearest neighbors N_{i} = {X_{j}}, j = 1, 2, ⋯, K. And the corresponding mean square error is used as the loss function,
$$J\left(w\right)=\sum _{i1}^{m}\Vert {X}_{i}\sum _{j=1}^{K}{W}_{\mathrm{ij}}{X}_{j}\Vert .$$(1)
It should be noted that the weight coefficient W_{ij} is local normalized (for any data X_{i} ∉ N_{i}, the corresponding weight coefficient is zero). It means that the sum of the weight coefficients corresponding to X_{i}’s neighborhood {X_{j}}, j = 1, 2, ⋯, K is 1. Therefore, the weight coefficient should meet the following requirements:
$$\sum _{i=1}^{K}{W}_{\mathrm{ij}}={W}_{i}^{T}{1}_{k}=1.$$(2)
With equation (2) as constraint, the weight coefficients in equation (1) can be obtained by Lagrange multiplier method as follows:
$$\begin{array}{c}J\left(w\right)=\sum _{i=1}^{m}{\Vert {X}_{i}\sum _{j=1}^{K}{W}_{\mathrm{ij}}{X}_{j}\Vert}_{2}^{2}=\sum _{i=1}^{m}{\Vert \sum _{j=1}^{K}{W}_{\mathrm{ij}}{X}_{j}\sum _{j=1}^{K}{W}_{\mathrm{ij}}{X}_{j}\Vert}_{2}^{2}=\\ \sum _{i=1}^{m}\mathrm{}{\Vert \sum _{j=1}^{K}\mathrm{}{W}_{\mathrm{ij}}({X}_{i}{X}_{j})\Vert}_{2}^{2}=\sum _{i=1}^{m}\mathrm{}{\Vert ({X}_{i}{X}_{j}){W}_{i}\Vert}_{2}^{2}=\sum _{i=1}^{m}\mathrm{}{W}_{i}^{T}({X}_{i}{X}_{j}{)}^{T}({X}_{i}{X}_{j}){W}_{i}\end{array}.$$(3)
Let ${Z}_{\mathrm{i}}=({X}_{i}{X}_{j}{)}^{T}({X}_{i}{X}_{j})$, there will be:
$$J\left(w\right)=\sum _{i=1}^{m}\mathrm{}{W}_{i}^{T}{Z}_{\mathrm{i}}{W}_{i}.$$(4)
Then, the optimization objective can be constructed as below:
$$L\left(w\right)=\sum _{i=1}^{m}\mathrm{}{W}_{i}^{T}{Z}_{\mathrm{i}}{W}_{i}+\lambda \left({W}_{i}^{T}{1}_{k}1\right).$$(5)
Next, setting the derivative of L(W) to W is 0, the following results can be obtained:
$$2{Z}_{i}{W}_{i}+\lambda {1}_{k}=0.$$(6)
Let $\lambda \text{'}=\frac{1}{2}\lambda $ be a constant, then the final weight coefficient vector W_{i} can be obtained as follows:
$${W}_{\mathrm{i}}=\frac{{Z}_{i}^{1}{1}_{k}}{{1}_{i}^{T}{Z}_{i}^{1}{1}_{k}}.$$(7)
3.2 LLE based well log data superresolution
From Section 3.1, we can see that LLE method can obtain the intrinsic relationship between a given 1 × ndimensional local data patch and its K nearest neighbor patches. Inspired by the Sparse Representationbased image SuperResolution (SRSR) method proposed in [12], in which lowresolution and highresolution training data pairs were employed to share the same representation parameters. Therefore, combining the advantages of LLE and SRSR, two strategies were employed for well log superresolution task. Firstly, let the lowresolution and highresolution model data pairs share the same linear representation weights. Secondly, for the test data patches, let’s find their K nearest neighbors in the lowresolution model patches for LLE operation. With the guidance of these two strategies, the workflow diagram of the proposed well log superresolution method was illustrated in Figure 1.
Fig. 1 Flow chart of locally linear embedding algorithm. 
Specifically, the detail of the proposed well log superresolution method was given in Table 1.
Illustration of the detailed steps of the proposed method.
4 Experimental results
4.1 Geological background and data selection
In order to evaluate the performance of the proposed well log super resolution method, well log data from 10 wells in QijiaGulong depression, Songliao basin, China were selected. QijiaGulong depression illustrated in Figure 2 is one of the most important tight sandstone exploration area of Daqing oil field. The average permeability of this area is commonly less than 1 mD, and the average porosity is always less than 13%. Compared with these medium quality physical properties, the thickness of the target reservoir in this area is universally less than 3 m, and there are always too many complex thin interlayers existed [22]. Specifically, without loss of generality, wells, natural Gamma Ray (GR), and Deep Lateral Resistivity (LLD) were selected for conducting the following experiments for all selected wells.
Fig. 2 Illustration of the geological background of the selected well data: the basic phase map of the QijiaGulong depression (left), the histogram of the thickness of sand body (rightup), and the histogram of the porosity (rightbottom). 
4.2 Hyperparameter settings
As mentioned in Section 3, the local patch size and the number of neighborhood patches are important hyper parameters of the method. Therefore, in this experiment, the influence of these two parameters on superresolution results was discussed. Specifically, without loss of generality, we chose the GR curve as the test data. Firstly, using the same local patch size (r = 4), the performance of the proposed method was tested with neighborhood size varying from 2 to 16. As can be seen from Figure 3a, as the neighborhood size increases, the Peak SignaltoNoise Ratio (PSNR) value improves rapidly, and when K is set to 10, the result tends to be stable. In addition, it is obvious that when K is 11, the best superresolution result can be achieved. Meanwhile, by setting K to 11, as shown in Figure 3b, with the increase of the local patch size, the best superresolution result is found when r is set to 4.
Fig. 3 Comparison of the superresolution performance of the proposed method under a) different neighborhood sizes and b) different local patch sizes. 
4.3 Super resolution performance test
In this experiment, the standard 0.125 m GR and LLD logging data from 1550 m to 1600 m section of well_1 was selected as the test data, and the corresponding section of well_2 was selected as model data. The corresponding 0.25 m low resolution curve is obtained by downsampling. In order to verify the superiority of this method, the Bicubic Spline Interpolation method (BSpInterp for short), onedimensional Convolutional Neural Network (SRCNN for short) method, and the Sparse Representation method (SpR for short) were used for comparison.
Specifically, the comparison was conducted in three manners: (1) direct two times (2X) superresolution; (2) double two times (indirect 4X) superresolution; and (3) direct four times (4X) superresolution. Then, as discussed in Section 4.2, local patch size and the neighborhood size used in the proposed method were setting to 4 and 11, respectively. Experimental results of different comparison manners were illustrated in Figure 4 (2X), Figure 5 (indirect 4X), and Figure 6 (4X), respectively.
Fig. 4 Illustration of 2X superresolution results comparison of different methods for a) GR curve and b) LLD curve. 
Fig. 5 Illustration of indirect 4X superresolution results comparison of different methods for a) GR curve and b) LLD curve. 
Fig. 6 Illustration of direct 4X superresolution results comparison of different methods for a) GR curve and b) LLD curve. 
In order to quantitatively evaluate the objective performance of these comparison methods, the Mean Square Error (MSE), Peak SignaltoNoise Ratio (PSNR), and Pearson correlation Coefficient (Coeff) of the results acquired by each method were calculated. At the same time, the execution time required for each method is also recorded for comparison. Specifically, the detailed quantitative evaluation results of GR and LLD logging data were given in Tables 2 and 3, respectively. Our proposed method (LLE) achieves stateoftheart results compared to other methods. The corresponding timeconsuming data was given in Table 4.
Comparison of quantitative evaluation results of different superresolution methods for GR logging data.
Comparison of quantitative evaluation results of different superresolution methods for LLD logging data.
Timeconsuming comparison of different superresolution methods.
4.4 Super resolution comparison of logging data in different areas
In order to verify the robustness of the proposed method, using well_2 as model well, the GR curves of the other eight wells were selected as test data using the 2X superresolution manner. Specifically, for each well, 100 m log data was randomly extracted for processing. In this experiment, PSNR values of GR logs superresolution results of different wells in different areas were compared. The detailed comparison data were given in Table 5. Our method has achieved the best results in 7 out of 8 oil wells. This result verifies the effectiveness of our method in different regions. Taking Well_3 as an example, the final superresolution results was shown in Figure 7 with lithology information for reference. Among them, the black curve is the original highresolution curve, and the red curve is the superresolution result.
Fig. 7 Comprehensive interpretation diagram of Well_3 logging curve. 
Comparison of PSNR values of GR curves of different wells in different areas.
4.5 Discussion
In order to evaluate the performance of the proposed well log superresolution method, several experiments were conducted. In Section 4.3, the direct visual effect and quantitative evaluation results of different methods were given. Specifically, from Figure 4, for 2X superresolution, we can see that SpR and SRCNN methods cannot effectively reconstruct the detailed information of well log, and there are relatively large fluctuations. In addition, the result obtained by BSpInterp method looks pretty good, but there are two obvious shortcomings: (1) high frequency information gain is very small; and (2) the problem of peak shift. Compared with the above three methods, the proposed method is relatively stable and rich in preserving the overall contour and details of the curve. Obvious, the intrinsic local structure of the log data can be successfully reconstructed by using the LLE technique. In order to better reflect this result, taking GR log data as example, power spectrums of these methods were calculated and quantitative comparisons with the power spectrum of the original highresolution GR log data were calculated. Specifically, the corresponding power spectrums were illustrated in Figure 8. Then, each power spectrum was divided into three subbands, namely the low frequency band (1 Hz–100 Hz), the medium frequency band (100 Hz–300 Hz), and the high frequency band (300 Hz–500 Hz). The Pearson coefficient values between superresolution results of different methods and the original highresolution GR log data were given in Table 6. From Table 6, we can clearly see that the proposed method can recover more high frequency components than other comparison methods, and the recovered low frequency and medium frequency bands are also excellent. Quantitative evaluation results of different test manners given in Tables 2 and 3 can also verify that the proposed method can achieve the best superresolution results. In addition, we can see that unavoidable error propagation in the indirect 4X superresolution manner makes it worse than the 2X and direct 4X manners. And 4X superresolution results are worse than the results achieve in the 2X manner because of the original information loss existing in the 1/4 scale lowresolution data. Obviously, error propagation will lead to degradation of superresolution performance. Furthermore, we can see from Table 4 that the proposed LLEbased well log superresolution method is the fastest one among all the employed superresolution methods. In experiment shown in Section 4.4, the proposed method is also the best one with stable and robust superresolution performance. For the shortage of the proposed method, especially compared with BSpInterp method, the requirement of highresolution model data limits its application value. Fortunately, for a given target exploration area, a variety of highresolution well logs are commonly collected in some reference wells. Therefore, the shortage of the proposed method can be greatly alleviated.
Fig. 8 Illustration of the power spectrum comparison of different methods: a) Original GR; b) Bicubic Interpolation; c) Sparse representation; d) SRCNN; e) LLE. 
Comparison of the correlation coefficients of different subbands of the power spectrums between of original highresolution data and the superresolution ones of different methods.
5 Conclusion
In this paper, a novel well log superresolution technique was proposed. In order to preserve or recovery as much the detailed structure information as possible during the well log superresolution process, LLE technique was introduced for the superresolution task. From the experimental results we can see that commonly three kinds of problems occurred for other comparison methods: (1) detailed structure information cannot be recovered satisfactorily; (2) obvious peak shift problem occurred; (3) the error fluctuation cannot be handled well. For the proposed LLEbased superresolution method, its performance in experiments conducted with all the three test manners described in Section 4.3 is promising. Furthermore, the robustness of the proposed method is also verified in Section 4.4, which shows stable superresolution results for well log data from different test wells. In general, the results given in this paper illustrate that LLE technique can achieve stable and reliable superresolution of well log data.
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All Tables
Comparison of quantitative evaluation results of different superresolution methods for GR logging data.
Comparison of quantitative evaluation results of different superresolution methods for LLD logging data.
Comparison of the correlation coefficients of different subbands of the power spectrums between of original highresolution data and the superresolution ones of different methods.
All Figures
Fig. 1 Flow chart of locally linear embedding algorithm. 

In the text 
Fig. 2 Illustration of the geological background of the selected well data: the basic phase map of the QijiaGulong depression (left), the histogram of the thickness of sand body (rightup), and the histogram of the porosity (rightbottom). 

In the text 
Fig. 3 Comparison of the superresolution performance of the proposed method under a) different neighborhood sizes and b) different local patch sizes. 

In the text 
Fig. 4 Illustration of 2X superresolution results comparison of different methods for a) GR curve and b) LLD curve. 

In the text 
Fig. 5 Illustration of indirect 4X superresolution results comparison of different methods for a) GR curve and b) LLD curve. 

In the text 
Fig. 6 Illustration of direct 4X superresolution results comparison of different methods for a) GR curve and b) LLD curve. 

In the text 
Fig. 7 Comprehensive interpretation diagram of Well_3 logging curve. 

In the text 
Fig. 8 Illustration of the power spectrum comparison of different methods: a) Original GR; b) Bicubic Interpolation; c) Sparse representation; d) SRCNN; e) LLE. 

In the text 