Stratigraphic Uncertainty Research Articles

The influence of major faults and fractures on the development of non-matrix porosity system in a pre-salt carbonate reservoir, Santos Basin – Brazil

Faults and fractures are central for characterizing the permeability distribution in carbonate reservoirs since they act as pathways for diagenetic fluids that often favor intense rock dissolution and permeability. Usually, high permeability zones and fractures are not easily recognized in seismic data due to limited resolution and they are often associated with higher concentrations of hydrocarbons or even significant fluid losses during drilling, thus creating a challenge for hydrocarbon exploration. In the Santos Basin, southeast Brazil, the pre-salt carbonate reservoirs from the Barra Velha Formation (BVE) are the main hydrocarbon producers in Brazilian Atlantic margin and well-known for being extremely heterogeneous, exhibiting complex dual-porosity systems. In this study, we built a conceptual model of these fracture zones and non-matrix porosity formation that helped narrowing the understanding of these complex systems. The characterization of faults and fractures was carried out using seismic, well-logs, and borehole image data to understand the influence of these structures in the porosity formation along the Barra Velha Formation. In the study area, three fault sets were defined (F1, F2, and F3) from seismic data. F1 represents to the larger faults, while the F3 fault set represents the smaller faults related to the reactivation of F1; both sets being oriented NE-SW. The F2 fault set is associated with the rift formation and is oriented to NNE-SSW. These three fault sets compartmentalized the studied area into different domains, each exhibiting distinct fracture sets. The natural open fractures were formed during the reactivation of rift faults and are oriented mainly NW, NNE-NNW, NE, and ENE and were identified across the entire study area, but with different intensity values. The fracture intensity closely relates to the distance from major faults where the wells with the highest fracture intensity occurs located 150–590 m from the larger F1 fault set. Scan-lines were conducted throughout the area to determine the fault width, and an average value of 1.2 km was established. Seven non-matrix porosity classes were characterized and classified into stratigraphically concordant and discordant non-matrix pore types at well scale through borehole image interpretation. The Barra Velha Formation exhibit higher occurrence of stratigraphically discordant non-matrix porosity related to fractured zones while stratigraphically concordant non-matrix porosity is mainly controlled by the paleotopography of the study area. Overall, non-matrix porosity formation tends to follow an orientation that suggests a preferential dissolution flow towards NE and ENE directions. Intervals with higher silica content shows a positive correlation with both fracture intensity and stratigraphically discordant non-matrix porosities. This work provides a conceptual model about the fractures and non-matrix porosity distribution in pre-salt carbonate rocks that can help address some of associated structural and stratigraphic uncertainties during field appraisal and development.

Real-time chemostratigraphy at wellsite; removing drilling uncertainties

Real-time chemostratigraphy is a workflow deployed at wellsites to remove stratigraphic uncertainty and support drilling operations. This study presents the chemostratigraphic wellsite workflow which, during drilling operations, support either the placement of casing points (using predefined geochemical markers to target a depth above a specific stratigraphic feature) or aid geosteering. Geosteering is typically employed during the drilling of a horizontal well and is done to keep the drill within a certain zone or horizon to maximise production. Wellsite chemostratigraphy has been effectively deployed to wellsites within multiple basins around the globe and has recently been utilised in the Perth Basin. The wellsite workflow starts before the deployment, with offset wells analysed by inductively coupled plasma (ICP) spectroscopy within a laboratory-based setting. This builds the correlative element-based framework and identifies the trends required to assist with either core placement, casing point, or geosteering. Samples are analysed by energy dispersive x-ray fluorescence units (these data are then used to confirm the zone previously defined) and can be identified at wellsite using the onsite tool, the selected elements of which are those less likely to be affected by drilling mud contamination and loss of circulation material (LCM). However, during drilling operation, there may be instances where LCM must be added to the drilling muds, which may have an effect on the data quality for certain elements. With a dedicated ICP spectrometry elemental database behind the wellsite operation, machine learning tools may be employed to ‘repair’ correlation critical elements.

Local coupled Markov chain model for simulating varied stratigraphy

The coupled Markov chain (CMC) model is widely employed to simulate geological uncertainty based on sparse borehole data, owing to its simplicity, parameter efficiency, and computational effectiveness. However, the existing CMC methods exhibit limitations in accurately predicting the orientation and significant dip angle changes of soil stratigraphy. These limitations stem from their reliance on a globally constant Walther constant, resulting in generated layer boundaries converging to a fixed angle. To address these issues, this study develops a local CMC model to enhance the accuracy of predicting layer sequences and describe the geological uncertainty of complex geological structures using limited borehole and surface data. In the proposed local CMC method, the global geological profile is decomposed into interconnected fragments, guided by reference boreholes. For each CMC fragment, the likelihood of observational scenarios is determined to identify the layer orientation and optimal Walther’s constant corresponding to the local geological profile. By combining the most likely realizations from all CMC fragments, the final geological profile and stratigraphic uncertainty are obtained. Illustrative case studies demonstrate that the proposed approach effectively addresses the limitations of conventional CMC models in accurately simulating significant changes in layer orientation and dip angle variations, given the same conditions of sparse borehole data. The employed localization strategy enables the prediction of geological profiles that encompass special layers such as soil interlayers, anticlines, and synclines, which cannot be well achieved by conventional CMC models. The local CMC model significantly improves the matching accuracy of geological model in situations with limited boreholes. Therefore, it plays a crucial role in the analysis of geological structures under data-scarce conditions.

Characterization of geological uncertainties from limited boreholes using copula-based coupled Markov chains for underground construction

This paper proposes an efficient method for quantifying the stratigraphic uncertainties and modeling the geological formations based on boreholes. Two Markov chains are used to describe the soil transitions along different directions, and the transition probability matrices (TPMs) of the Markov chains are analytically expressed by copulas. This copula expression is efficient since it can represent a large TPM by a few unknown parameters. Due to the analytical expression of the TPMs, the likelihood function of the Markov chain model is given in an explicit form. The estimation of the TPMs is then re-casted as a multi-objective constrained optimization problem that aims to maximize the likelihoods of two independent Markov chains subject to a set of parameter constraints. Unlike the method which determines the TPMs by counting the number of transitions between soil types, the proposed method is more statistically sound. Moreover, a random path sampling method is presented to avoid the directional effect problem in simulations. The soil type at a location is inferred from its nearest known neighbors along the cardinal directions. A general form of the conditional probability, based on Pickard's theorem and Bayes rule, is presented for the soil type generation. The proposed stratigraphic characterization and simulation method is applied to real borehole data collected from a construction site in Wuhan, China. It is illustrated that the proposed method is accurate in prediction and does not show an inclination during simulation.

Stability of unsaturated soil slope considering stratigraphic uncertainty and rotated anisotropy of soil properties

AbstractThe stratigraphic uncertainty and rotated anisotropy of soil properties exist widely in nature. Recent studies have shown that the slope stability was significantly influenced by these two uncertainties. However, there is no proper method for simulating these two uncertainties at the same time, and the influence of the two uncertainties has not been considered in previous unsaturated soil slope stability analysis. This paper aims to propose a coupled method for characterizing the stratigraphic uncertainty and rotated anisotropy of soil properties, and investigate the unsaturated soil slope stability considering the two uncertainties. Through a slope case, the proposed method for characterizing the two uncertainties is examined. The effect of rotational angle on the slope stability and groundwater table is studied. In addition, four different uncertainty considerations are chosen to compare their influence on the slope stability and groundwater table. The results show that the proposed method can well characterize the two uncertainties at the same time. The rotational anisotropy of soil properties has a substantial impact on the slope stability and groundwater table. The rotational angles corresponding to the maximum and minimum reliability index of slope depend on the uncertainty considerations in the slope stability analysis. The slope reliability index only considering stratigraphic uncertainty is the highest, and the slope reliability index considering stratigraphic uncertainty and rotated anisotropy of soil properties is the lowest.

Data-driven multi-stage sampling strategy for a three-dimensional geological domain using weighted centroidal voronoi tessellation and IC-XGBoost3D

Conventional site investigation schemes typically use an empirical sampling strategy, which involves equal sampling spacing with regular grid patterns. However, this approach assigns equal importance to the entire site and does not account for subsurface stratigraphic variations and site constraints. In this study, a data-driven multi-stage sampling strategy is proposed to adaptively optimize the borehole locations for a three-dimensional (3D) geotechnical site, considering the subsurface stratigraphic uncertainties and irregular site geometries. The initial sampling plan is determined based on weighted centroidal Voronoi tessellation, which assigns various sampling densities to zones depending on their importance. Measurements obtained in the previous sampling stage are combined with prior geological knowledge to establish and update a 3D geological domain using a physics-informed geological modelling method. The next optimal location for sampling is adaptively determined with the objective of maximizing the reduction in the stratigraphic uncertainty. The proposed method represents the first data-driven 3D site planning method that explicitly considers 3D subsurface stratigraphic variations. The performance of the proposed multi-stage sampling strategy is illustrated using a simulation study. The results indicate that the proposed method efficiently identifies the optimal sampling locations while comprehensively considering the 3D subsurface geological uncertainties and irregular site geometries.

Stability of Unsaturated Soil Slope Considering Stratigraphic Uncertainty

Stratigraphic uncertainty is widely present in nature, but it has not been well considered in the stability analysis of unsaturated soil slopes in the past. In this study, the stability of the unsaturated soil slope is evaluated based on borehole data considering stratigraphic uncertainty. Firstly, an enhanced coupled Markov chain model is used to simulate stratigraphic uncertainty. Then, a finite element algorithm for automatically calculating the safety factor (FS) and the average groundwater table (AGT) of the unsaturated soil slope is developed. At last, a hypothetical slope located in the stratum from Perth, West Australia is analyzed using the proposed algorithm under different borehole schemes. The results show that with the increase in the borehole number, the statistics of FS and AGT will not monotonically increase or decrease. But the trend is that the mean values of FS and AGT gradually approach and eventually converge to the real values, and the standard deviations of FS and AGT decrease. There is a linear relationship between the standard deviation of FS (or AGT) and the average information entropy. The FS and AGT are negatively correlated considering stratigraphic uncertainty.

A rigorous random field-based framework for 3D stratigraphic uncertainty modelling

Sustainable development goals for site-specific and customized design have brought needs to assess and quantify the stratigraphic uncertainty in three-dimensional (3D) modelling. The interpretation of uncertainty distribution with limited boreholes has always been challenging in 3D space, which is curial in engineering applications such as borehole sampling. In this study, we propose a rigorous random field-based framework to model the 3D stratigraphic uncertainty distribution with improved physical interpretation. Rather than offering constants or physically ambiguous models for Scales of Fluctuation (SoFs) as previous research, the framework adopts a series of SoFs calibrated by fitting stratum characteristics, to cope with local site-specificity in large geological sites. To evaluate the performance of the proposed framework, hypothetical cases are firstly used to demonstrate its applications in both 2D and 3D scenarios. The effectiveness of the proposed model is verified by borehole layout scheme optimization. Then, the proposed framework is applied to the 4 km-long geological site of the immersed tunnel of Hong Kong-Zhuhai-Macao Bridge using 246 boreholes, which is to date the first study on large 3D stratigraphic uncertainty modelling. Finally, a comprehensive discussion is presented for correct physical interpretation of SoFs, comparative studies and determination of important parameters, aiming at evaluating the compatibility in all scenarios. The novel method is proved to be effective in 3D uncertainty modelling for geological stratigraphy with improved physical interpretability, which provides guidelines for underground design.

Data-driven analysis of soil consolidation with prefabricated vertical drains considering stratigraphic variation

In coastal cities such as Hong Kong, rapid reclamation using prefabricated vertical drains (PVDs) is preferred as it can accelerate land supply to meet the urgent demand for houses. A robust PVD design relies on the correct identification of permeable soil layers and accurate delineation of their stratigraphic connectivity with surrounding drainage boundaries. The current engineering practice often ignores the physical locations of minor drainage boundaries (e.g., sand lenses) in the subsurface stratigraphy and might lead to a false interpretation of potential drainage and consolidation mechanisms. In this study, a data-driven analysis framework that takes stratigraphic uncertainty into consideration is proposed to investigate the spatiotemporal consolidation of PVD-improved ground using sparse site investigation data often encountered in engineering practice. The method adaptively develops multiple geological cross-sections from limited measurements and prior knowledge that is reflected by a single training image. The resulting multiple geological realizations serve as the input for PVD analysis. The proposed data-driven framework allows for a probabilistic evaluation of soil spatiotemporal behavior in terms of the degree of consolidation and future ground settlement. More importantly, the proposed method accurately predicts the spatial distribution of stratigraphic boundaries with quantified uncertainty, which significantly influences the consolidation mechanism.

Triangulated irregular network-based probabilistic 3D geological modelling using Markov Chain and Monte Carlo simulation

Three-dimensional geological modelling based on triangulated irregular networks (TINs) often results in deterministic models that do not capture the uncertainty of stratigraphic attributes. This study presents a method for building a 3D TIN geological model that incorporates probability analysis and random simulation to account for this uncertainty. First, we generated the Markov chain sequence of the stratum and combined it with the correlation of boreholes to obtain the transition probability matrix of the virtual borehole. This matrix was then used to describe the probability of changes in the stratum. Second, we constructed a probabilistic model of stratum thickness and generated the stratum state of virtual boreholes through Monte Carlo simulation. Using the TIN algorithm, we built an overall probabilistic model and added validation boreholes to adjust the virtual boreholes and obtain the maximum probabilistic model of the study area. Finally, we performed a case analysis in South China to verify the modelling method, including visualisation, probability analysis and comparative analysis. Our results demonstrate that the proposed modelling method effectively visualises the stratigraphic configuration and quantifies the stratigraphic uncertainty, providing a reference for 3D geological modelling.

A New Method for Determination of Optimal Borehole Drilling Location Considering Drilling Cost Minimization and Sustainable Groundwater Management.

Drilling boreholes for the exploration of groundwater incurs high cost with potential risk of failures. However, borehole drilling should only be done in regions with a high probability of faster and easier access to water-bearing strata, so that groundwater resources can be effectively managed. However, regional strati-graphic uncertainties drive the decision of the optimal drilling location search. Unfortunately, due to the unavailability of a robust solution, most contemporary solutions rely on physical testing methods that are resource intensive. In this regard, a pilot study is conducted to determine the optimal borehole drilling location using a predictive optimization technique that takes strati-graphic uncertainties into account. The study is conducted in a localized region of the Republic of Korea using a real borehole data set. In this study we proposed an enhanced Firefly optimization algorithm based on an inertia weight approach to find an optimal location. The results of the classification and prediction model serve as an input to the optimization model to implement a well-crafted objective function. For predictive modeling a deep learning based chained multioutput prediction model is developed to predict groundwater-level and drilling depth. For classification of soil color and land-layer a weighted voting ensemble classification model based on Support Vector Machines, Gaussian Naïve Bayes, Random Forest, and Gradient Boosted Machine is developed. For weighted voting, an optimal set of weights is determined using a novel hybrid optimization algorithm. Experimental results validate the effectiveness of the proposed strategy. The proposed classification model achieved an accuracy of 93.45% and 95.34% for soil-color and land-layer, respectively. While the mean absolute error achieved by proposed prediction model for groundwater level and drilling depth is 2.89% and 3.11%, respectively. It is found that the proposed predictive optimization framework can adaptively determine the optimal borehole drilling locations for high strati-graphic uncertainty regions. The findings of the proposed study provide an opportunity to the drilling industry and groundwater boards to achieve sustainable resource management and optimal drilling performance.

An Uncertainty-driven Peak-integration (UP) Strategy for 3D Borehole Layout Planning

Equal-spaced borehole drilling is commonly adopted in practice for site investigation, which ignores spatial uncertainty and might result in insufficient site characterization. Model-driven precise drilling is regarded as a potential solution. In this study, an uncertainty-driven peak-integration (UP) strategy is proposed to improve decision-making in 3D borehole planning. The UP strategy consists of stratigraphic uncertainty modelling and peak-integration method. The random field-based uncertainty modelling approach has been improved by employing stratum-informed Scales of Fluctuation and a novel site division method to capture the site-specific uncertainty distribution with sufficient interpretation. The peak-integration method is proposed to consider the uncertainty around peaks, as improvement of existing peak uncertainty-based borehole planning strategy. To provide an objective evaluation for borehole layout planning, a new index “Contribution of Borehole” (CoB) is proposed for the first time, using the Earth Mover’s Distance, to quantify the net changes of stratigraphic distribution revealed by additional boreholes and offer validation for every decision-making. The feasibility and practicability of the proposed strategy are demonstrated by applying to an artificial geological model created based on real borehole data. The superiority of the proposed strategy is elucidated by comparing with equal-spaced strategy, via both CoB and conventional entropy reduction with detailed discussion.

An efficient model to estimate the soil profile and stratigraphic uncertainty quantification

Comprehensive geological information is very important and needed in underground construction and underwater infrastructure. However, it is quite difficult to reveal the configuration of different soil features of underground with sparsely measured data. In this study, a new algorithm named General Regression-Markov Chain (GR-MC) was proposed by coupling the general regression neural network and the Markov chain method together to reproduce the characteristics of underground stratification based on rare borehole information. The general regression neural network method was introduced to evaluate the geological variation along the inclined orientation, while the Markov chain method was applied to examine the change in soil types along the depth. The spatial coordinates of the treated construction site were designed as the input variables, while a vector including the probability information of different soil types was set as the output. The proposed GR-MC method is very intuitive and can be efficiently implemented to predict the geological model using only the coordinates as inputs. Moreover, the proposed method can be smoothly extended into a three-dimensional case with low computational resources. Compared to the existing methods, the proposed GR-MC algorithm is computationally efficient and robust. Three cases from real practice were evaluated and the calculated results were consistent with the known borehole information. Besides, the stratigraphic uncertainty was quantified based on information entropy theory. The results can visually reveal the zones of the estimated soil profile with relatively large uncertainty. The practical application is that possible additional borehole locations were identified to reduce the stratigraphic uncertainty of a construction site.

A two-dimensional approach to quantify stratigraphic uncertainty from borehole data using non-homogeneous random fields

In many instances, conditions in the subsurface are highly variable, while site investigations only provide sparse measurements. Consequently, subsurface models are usually inaccurate. These characteristics reflect uncertainty and mean significant engineering and environmental hazards. Such uncertainty should be quantified, in order, ultimately, to be reduced. To this end, in this paper, a new two-dimensional approach to quantify stratigraphic uncertainty is proposed and described. The approach is based on non-homogeneous random fields and considers categorical quantities to represent geological structures. Unlike other reported approaches in related literature, we provide evidence on the usefulness of the approach to: (i) comprehensively explore the many diverse and potential geological structures' configurations to exhaust uncertainty quantification; (ii) facilitate the representation of non-homogeneous fields in a computational inexpensive fashion; and (iii) ameliorate the complication of producing spatial Markov chains or Bayesian models to quantify uncertainty using finite difference equations. The proposed approach is demonstrated using a case analysis.

Development of training image database for subsurface stratigraphy

ABSTRACT Image-based stochastic simulation methods, such as multiple point statistics (MPS), can be viewed as a physics-informed Bayesian learning approach, which samples typical stratigraphic patterns from a single training image for onward conditional modelling of subsurface stratigraphy. A training image is essentially a prior geological model, which comprises representative stratigraphic connectivity at the site of interest. One key difficulty hindering wide application of image-based geological modelling methods is the lack of qualified training images. In this study, a systematic framework is proposed to develop training image databases for conditional simulations of subsurface stratigraphy. Collected training images can be further categorised based on three key factors, namely, geological origin, site location and application scenario. As a pilot study, a total of 54 geological cross-sections, mainly interpreted by experienced engineering practitioners, for weathered granite and tuff slopes in Hong Kong are collected and compiled as two training image databases. To demonstrate value and application of the established training image databases, subsurface stratigraphy for real weathered granite slope examples are used as illustrative examples, and stratigraphic uncertainty is also quantified. Results indicate that training image databases are of great significance for subsurface stratigraphy and uncertainty quantification, particularly when only limited site-specific data are available.

Revised age and stratigraphy of the classic Homo erectus-bearing succession at Trinil (Java, Indonesia)

Obtaining accurate age control for fossils found on Java (Indonesia) has been and remains challenging due to geochronologic and stratigraphic uncertainties. In the 1890s, Dubois excavated numerous faunal fossils—including the first remains of Homo erectus—in sediments exposed along the Solo River at Trinil. Since then, various, and often contradictory age estimates have been proposed for the Trinil site and its fossils. However, the age of the fossil-bearing layers and the fossil assemblage remains inconclusive. This study constructs a chronostratigraphic framework for the Trinil site by documenting new stratigraphic sections and test pits, and by applying 40Ar/39Ar, paleomagnetic, and luminescence (pIRIR290) dating methods. Our study identifies two distinct, highly fossiliferous channel fills at the Trinil site. The stratigraphically lower Bone-Bearing Channel 1 (BBC-1) dates to 830–773 ka, while Bone-Bearing Channel 2 (BBC-2) is substantially younger with a maximum age of 450 ± 110 ka and an inferred minimum age of 430 ± 50 ka. Furthermore, significantly younger T2 terrace deposits are present at similar low elevations as BBC-1 and BBC-2. Our results demonstrate the presence of Early and Middle Pleistocene, and potentially even late Middle to Late Pleistocene fossiliferous sediments within the historical excavation area, suggesting that Dubois excavated fossils from at least three highly fossiliferous units with different ages. Moreover, evidence for reworking suggests that material found in the fossil-rich strata may originate from older deposits, introducing an additional source of temporal heterogeneity in the Trinil fossil assemblage. This challenges the current assumption that the Trinil H.K. fauna –which includes Homo erectus-is a homogeneous biostratigraphic unit. Furthermore, this scenario might explain why the Trinil skullcap collected by Dubois is tentatively grouped with Homo erectus fossils from Early Pleistocene sediments at Sangiran, while Trinil Femur I shares affinities with hominin fossils of Late Pleistocene age.

Application of Supersplining to the Mesozoic and Paleozoic Geologic Time Scales

Methods to determine the ages of period and stage boundaries of the geologic time scale (GTS) have a long history and continue to be steadily improved. Stage boundary age estimates are now accompanied by error bars showing stratigraphic uncertainty. Most GTS2004, GTS2012 and GTS2020 results involved cubic spline-curve fitting, and in this study, Ordovician through Cretaceous smoothing splines of GTS2020 are spliced together to construct a Paleozoic–Mesozoic superspline. This methodology, its advantages and its results are outlined.

Quantification of Stratigraphic Transition Location Uncertainty for Tunneling Projects

Quantification of Stratigraphic Transition Location Uncertainty for Tunneling Projects

A decentralized operating performance assessment for geological drilling process via multi-block total projection to latent structures and Bayesian inference

Geological drilling process is an industrial process that contains a lot of variables, and the relationships and characters among them are also complex. Accordingly, implementing operating performance assessment by conventional methods has always been a problem that operators are dedicated to solving. In fact, geological drilling process is a complex industrial process involving multiple systems and stratigraphic uncertainties, which makes assessing its operating performance difficult. A decentralized operating performance assessment based on multi-block total projection to latent structures (T-PLS) and Bayesian inference is proposed for the geological drilling process. Utilizing the variational trends of the detection variables, the most related variables can be grouped in the same block, and the process capability index determines the operating performance grade. This is followed by a T-PLS algorithm-based operating performance assessment model and Bayesian inference. Each sub-block’s assessment results are integrated to achieve a comprehensive performance assessment. Last but not least, drill data show that the proposed method is effective and superior. The proposed method provides better accuracy and generalizability in assessing drilling performance. The novelty of the assessment scheme involves that a decentralized framework is proposed for operating performance assessment by identifying normal operating conditions first and then constructing the multi-block T-PLS-based monitoring model on the local sub-blocks.

Stochastic stratigraphic modeling using Bayesian machine learning

Stratigraphic modeling with quantified uncertainty is an open question in engineering geology. In this study, a novel stratigraphic stochastic simulation approach is developed by integrating a Markov random field (MRF) model and a discriminant adaptive nearest neighbor-based k-harmonic mean distance (DANN-KHMD) classifier into a Bayesian framework. The DANN-KHMD classifier is effective for extracting anisotropic patterns from sparse and heterogeneous spatial categorical data such as borehole logs. The MRF parameters can be initially estimated roughly or customized (if site-specific knowledge is available). Later these parameters can be updated and regularized in an unsupervised manner with constraints from site exploration results in a Bayesian manner. Throughout the learning process, both the soil profile and the MRF parameters are updated in a probabilistic manner. The advantages of the proposed approach can be summarized into four points: 1) inferring stratigraphic profile and associated uncertainty in an automatic and fully unsupervised manner; 2) reasonable initial stratigraphic configurations can be sampled and hence lower the computational cost; 3) both stratigraphic uncertainty and model uncertainty are taken into consideration throughout the inferential process; 4) relying on no training stratigraphy images. To illustrate the effectiveness of the developed approach, two synthetic cases and three real-world cases are studied and the advantages of the new approach over existing approaches are demonstrated.