GRA calculates the relationship between the system and the influencing factors, aiming to determine the main factor with the closest relationship [5]. The main mechanism is to convert the variable factor value into a geometric curve and compare the shape of the curve. A higher similarity of the shape indicates a greater relational degree. With high similarity and great relational degree, the corresponding influencing factor can be concluded as the main factor of the turbidite channel. The intervals of sand bodies where a turbidite channel is drilled are set as the positive sample series of the system, and those of sand bodies where no turbidite channel is found are taken as the negative sample series. The specific process is as follows: (1)

It is assumed that the characteristic series of the system (based on known well series) are

Series participating in the comparison are (2)Xi={xi(1),…,xi(n)} (2)

Dimensionalization of influencing factors. Because the dimensions and range of many seismic attributes are different, the calculation results may seriously affect the prediction accuracy. Therefore, it is necessary to dimensionalize the attributes of the original series.

SVM is applied to nonlinear high dimension and local minima of small samples. Its principle is to construct an optimal decision function, thus mapping the training samples into a high-dimensional feature space through nonlinear mapping, converting the regression of nonlinear function into that of linear function with high-dimensional feature, and then completing the transformation process from nonlinear to linear function [9].

The sample set of attributes participating in the training is established as {x_i, y_i}, (i = 1, ..., n), where n is the number of spatial samples. The input of the optimized attribute sample is selected as x_i, while the output of the sample is selected as y_i. For the desired function, (6)f(x)=ωϕ(x)+b

The optimization function is defined as (7)minω,b,ξ,ξ∗⁡12ωTω+C∑i=1n⁡(ξi+ξi∗)

Constraint condition s· t (8)yi−[ωTϕ(xi)+b]≤ε+ξi (i=1,…,n) (9)[ωTϕ(xi)+b]−yi≤ε+ξi∗ (i=1,…,n) (10)ξi,ξi∗≥0 (i=1,…,n)where ϕ is the non-mapping function; ω is the normal vector of the function plane; b is the bias of the function plane; T is the transpose of matrix; ε is the error probability; ξi,ξi∗ is the upper limit of training error under the constraints of error ξ; ξ is a positive number and can be set in advance according to the desired accuracy, and C is the penalty factor. The constraint conditions are set as follows: (11){∑i=1k⁡yi(ai−ai∗)=00≤ai,a∗≤C

The Lagrange function is established as (12)L(ω,b,a)=12||ω||2−∑i=1lai(yi((xi⋅ω)+b)−1)

The dual form of the optimization method is (13)maxai,ai∗=−12∑i,j=1n(ai−ai∗)(aj−aj∗)K(xi−xj)−ε∑i=1n(ai−ai∗)+∑i=1nyi(ai−ai∗)where K(xi,xj)=ϕ(xi)⋅ϕ(xj) is the kernel function; αi and αi∗ are the Lagrange operator, and j = 1, …, n.

The complementary conditions corresponding to Eq. (12) are

αi(yi−ωϕ(xi)−b−ε−ξi)=0 (i = 1,…, n)

αi∗(ωϕ(xi)+b−yi−ε−ξi)=0 (i = 1,…, n)

ξiξi∗=0, αiαi∗=0 (i = 1,…, n)

(ai−C)ξi=0,(ai∗−C)ξi∗=0 (i = 1,…, n)

The final regression function obtained is (14)f(x)=∑i=1n(αi−αi∗)K(xi,x)+b

Based on the above principles, a GRA-SVM model was established to achieve the method and process introduced above. The detail is shown in Fig. 1.

(1)

Determine reference series and comparison series

According to the GRA theory, the signal that reflects the stratum attribute extracted from the seismic attribute is a gray signal, and the analysis of the attribute of the river channel and the thickness of sandstone through the seismic attribute signal is a grey system process. Combined with the seismic forward analysis results, seismic attribute signals of known wells in the experimental zone were extracted to form a reference series. Other Seismic attribute signals of the target layer were selected as the comparison series for building a response model of the seismic attribute and the thickness of sandstone. (2)

Nondimensionalize reference series and comparison series

Due to the different physical significance of each seismic attribute in the system, the dimensions of the data are not always the same, which leads to inconvenient comparison and difficulty in obtaining correct conclusions during comparison. Therefore, when GRA is conducted, data are generally required to be nondimensionalized. This study processed the original data through the average standardization approach. (3)

Calculate grey relational coefficient

For reference series X0, there are several comparison series such as X1, X2, and Xn. The relational coefficients between each comparison series and the reference series at different time points (i.e., each point in the curve) can be calculated through GRA. (4)

Sort relation order

The degree of relation between the sample and the reference series is mainly described in the greatness and ranking of the relation degree. A larger relation degree value and higher ranking indicate a greater relation degree with the reference series. On the contrary, a smaller relation degree and lower ranking indicate a lower relation degree with the reference series. This study analyzed the relation degree between sandstone thickness and seismic attributes through the GRA method, calculated the relation degree between these seismic attributes and sandstone thickness, and selected the typical attribute variables as the data samples for SVM classification and combination. (5)

Classify samples

SVM is a learning classification method under supervision. Therefore, for the given training samples, it is necessary to clarify the classification of each sample, which is either 0 or 1. In other words, each sample should be labeled with an exact category label for SVM training. (6)

Set SVM parameters

The commonly used settings for SVM parameters include the SVM type, the kernel function type, the termination conditions of the algorithm, relaxation variables, etc. The kernel function is to map the training samples to a sample set that is easier for linear segmentation. The mapping, completed through kernel functions, results in an increase in the dimension of the sample vector. As the SVM training process is an iterative solution to the quadratic optimization issue under constraint conditions, a combination of maximum iterations and allowable errors can be set as the termination condition of the algorithm to allow the algorithm to stop computing under appropriate conditions. (7)

Train SVM

As SVM is a supervised learning classification method, for the given training samples, it is necessary to clarify whether the classification of each sample is 0 or 1, or in other words, each sample needs to be labeled with an exact category label for SVM training. SVM has no limitations on the features and dimensions of the samples, which means that different features can be used as representations of training samples to participate in SVM training. (8)

Segment sample area

There is an intuitive way to judge the classification of samples in the test sample space through the above SVM model, namely, to color the space where different classifications of samples are located. Based on red, green, and blue (RGB), different color zones can represent different sample spaces. The junction of the color zones is the classification hyperplane. By this method, the sample area can be segmented.

The Lingshui Depression is located at the southeast edge of the Qiongdongnan Basin, between the central depression of Qiongdongnan Basin and Hainan Island, and it belongs to the northern lower continental slope zone of the basin. Controlled by boundary faults, the Lingshui Depression mainly develops NE-SW faults. The stratigraphic sedimentation in the Lingshui Depression is a multi-source sedimentary system, jointly controlled by the boundary faults on both sides of the depression and the antithetic faults inside the contemporaneous slope. The Lingshui 17-2 Gas Field is located in the central canyon in the northern part of the low uplift in the south of the Lingshui Depression, with a water depth of 1250–1550 m and an average effective permeability of 596 mD. It is a high-quality and large-scale deep-water gas field with high porosity, high permeability, and high hydrocarbon production. As per the historical well analysis, the Lingshui 17-2 Gas Field is a typical deep-water, high-temperature, and high-pressure gas reservoir. The formation of the Neogene Huangliu Formation-Yinggehai Formation is mainly composed of submarine fans and central canyon reservoir cap association, with lithologic traps distributed in strips along the canyon. The stratigraphic dip is small, with the closure amplitude between 83 and 215 m. The Huangliu Formation is the main gas-bearing target layer, which is divided into four layers from the bottom to the top [18], with the Gas Layer Groups I and II being the main gas-bearing sand bodies [19]. Having the largest production potential, the Gas Layer Group I is further subdivided into two sets of sand bodies, namely, the Gas Layer Group I upper and Gas Layer Group I lower. Each gas group is a different turbidite sand body cut by watercourses in the later period under isochronous deposition. The major gas layer group is mainly composed of well-sorted, argillaceous-cemented, and loose gray fine sandstone, fine siltstone, and siltstone, making it a typical deep-water turbidite sedimentary environment.

Based on the GRA-SVM method, this study conducted practical research on the seismic geological data of the Lingshui Depression, extracting more than 60 kinds of along-layer seismic attributes, covering amplitude, phase, frequency, etc. [20]. After that, this study used the homogenization method to dimensionalize these attributes, then used the GRA method to identify the main factors of the attributes, and found the root mean square (RMS) of the amplitude. The max peak, the instantaneous frequency, and the instantaneous phase have high main relational factors (Fig. 2). This study then accumulated the top 20 attributes with the highest relational factors to obtain a first-order accumulated sequence, which was used as the input training sample of the SVM model. Then the SVM model was used to establish a prediction model and obtain the prediction results. To restore the prediction results through consecutive subtraction, the attribute prediction diagram of the main oil layer of the Huangliu Formation (Top interface S30, bottom interface S40) was ultimately obtained (Fig. 3).

The Gas Layer Group I lower of the Huangliu Formation was taken as an example. Based on the GERA-SVM prediction attribute diagram, the sample area segmentation indicates the difference in sedimentation, or in other words, erosion is dominant on the left side, while sedimentation is dominant on the right side, and curved NE-SW channels can be seen clearly on the plane. As proven by the drilling results, the attribute anomaly body is the sedimentation in the deep-water turbidite channel. The channel has developed to a large scale, with a width of about 2 km and a length of 18 km on the plane. During the drilling, seven wells encountered this turbidite sedimentary body. The sandstone coincidence rate was 93% among the drilled water channels, with an average sandstone thickness of 30 m. In conclusion, a greater relationship degree of each seismic attribute indicates a higher probability of responding to the attributes of the channels, revealing that the turbidite channel method based on GRA-SVM is feasible.