3D seismic analysis of mine planning using Aczel–Alsina aggregation operators based on T-spherical fuzzy information

3D seismic attributes analysis can help geologists and mine developers associate subsurface geological features, structures, faults, and ore bodies more precisely and accurately. The major influence of this application is to evaluate the usage of the 3D seismic attributes analysis in gold mine planning. For this, we evaluate the novel theory of complex T-spherical hesitant fuzzy (CTSHF) sets and their operational laws. Furthermore, we derive the CTSHF Aczel–Alsina weighted power averaging (CTSHFAAWPA) operator, CTSHF Aczel–Alsina ordered weighted power averaging (CTSHFAAOWPA) operator, CTSHF Aczel–Alsina weighted power geometric (CTSHFAAWPG) operator, and CTSHF Aczel–Alsina ordered.com weighted power geometric (CTSHFAAOWPG) operator. Some properties are also investigated for the above operators. Additionally, we evaluate the problems of 3D seismic attributes analysis to mine planning under the consideration of the proposed operators, for this, we illustrate the problem of the multi-attribute decision-making (MADM) technique for the above operators. Finally, we demonstrate some examples for making the comparison between prevailing and proposed information to improve the worth of the derived operators.


Preliminaries
In this section, we discussed the HFSs, CTSFSs, and Aczel-Alsina norms and their related properties.Definition 1 22 A HFS H h is initiated by: Observed that � H h (x) = T H h (x) : j = 1, 2, 3 . . ., n represents the positive grade where Definition 2 21 A CTSFS H p is computed in the form: We can observe that the positive grade is computed in the shape: , where the ne g at ive g r a d e is i nve nte d by : . Moreover, the neutral grade is the following: .
Definition 3 21 Some algebraic laws for any two CTSFNs are stated below: 1. We Propose the CTSHFSs and their operational laws.
2. We Propose the Aczel-Alsina power operators for CTSHF numbers.
4. We Propose the MADM technique for evaluating the problems of gold mines.
5. We compare the proposed work some existing works.www.nature.com/scientificreports/Definition 4 36 Some Aczel-Alsina operational laws for any two CTSFNs are evaluated below: Vol.:(0123456789) www.nature.com/scientificreports/Definition 5 21 Some score and accuracy values are stated below: Definition 25 The PAO is initiated in the shape: where,

CTSHFSs
In this section, we investigate the theory of CTSHFS and their related information such as algebraic laws, and Aczel-Alsina operational laws.

Definition 6
A CTSHFS H p is initiated in the form: We observed in positive grade , Further, we describe the main purpose of neutral grade such as: and the purified for m of t h e C T SH F nu mb e r ( C T SH F N ) i s d e du c e d by : Vol.:(0123456789) we have Under the consideration of data in Eq. ( 22) and Eq. ( 23), we express few properties, such as:

Aczel-Alsina power aggregation operators for CTSHFSs
This section introduces the theory of CTSHFAAWPA operator, CTSHFAAOWPA operator, CTSHFAAWPG operator, and CTSHFAAOWPG operator.Further, we also investigate the idea of idempotency, monotonicity, and boundedness based on the weight vectors, such as where n j=1 j s = 1.

Definition 10 The mathematical information
Represents the CTSHFAAWPA operator, such as Shown the power operator which is used as a weight vector.(10), we derive that the aggregated information of the above theory is again a CTSHFN, such as Proof For n = 2 we prove that the above information, we have Vol.:(0123456789) www.nature.com/scientificreports/Hence, we can prove our result successfully under the above consideration.Furthermore, we expressed the idea of idempotency, monotonicity, and boundness under the given techniques.
Proposition 1 For any CTSHFNs, we have Then we observe that

. Id e m p o t e n c y : W h e n � H
Then we observe the following properties.

Idempotency: When
The mathematical information Represents the CTSHFAAWPA operator, such as Shown the power operator which is used as a weight vector.
Theorem 4 Information in Def.(13), we derive that the aggregated information of the above theory is again a CTSHFN, such as Proof Straightforward.Further, we observed the main purpose of idempotency, monotonicity, and boundness for the given techniques.□

Proposition 4 Under the presence of CTSHFNs � H
Proof Straightforward.By using the above operators, our target is to simplify it with the help of some practical examples to compute the efficiency and exactness of the invented techniques.□

Decision-making procedure based on proposed operators
In this section, we elaborate on the procedure of MADM techniques under the consideration of the proposed operators such as the CTSHFAAWPA operator and CTSHFAAWPG operator to evaluate the supremacy and validity of the derived information.
For this, we demonstrate the procedure of MADM techniques with alternatives H 1 p , H 2 p , . . ., H n p and for each alternative, we have some attributes H 1 AT , H 2 AT , . . ., H m AT with weight vectors j s in the shape of power operators.Based on the above information, we compute the matrix by including the CTSHFNs, such as we observed in , and a negative grade with the following characteristics: , Further, we describe the main purpose of neutral grade such as: and the purified form of the CTSHFN is deduced by : � Using the above information, we compute the procedure of MADM techniques to evaluate the above problems, such as.
Step 1: Derive the matrix by putting the values of CTSHFNs, if we have cost type of data, then But if we have a benefit type of data, then do not normalize the matrix.
Step 2: Aggregate the matrix based on the CTSHFAAWPA operator and CTSHFAAWPG operator, such as  www.nature.com/scientificreports/

Decision
Step 3: Evaluate the Score values of the above-aggregated information, such as Step 4: Derive the ranking values according to the score function to examine the best one.The geometrical shape of the proposed algorithm is listed in the shape of Fig. 2.
To simplify some real-life problems, we use the above procedure based on the MADM technique to improve the reliability and supremacy of the proposed information.

Application of 3D Seismic Attributes analysis to Mine Planning Based on Proposed Methods
In this application, we evaluate the best way in which 3D seismic attribute analysis can be used or applied to mine planning.3D seismic attributes analysis can help geologists and mine developers associate subsurface geological features, structures, faults, and ore bodies more precisely and accurately.The major influence of this application To depict the best one, we use the following features such as growth analysis, social impact, political impact, and environmental impact.Using the above information, we compute the procedure of MADM techniques to evaluate the above problems, such as Step 1: Derive the matrix by putting the values of CTSHFNs, see   But if we have benefit type of data, then do not normalize the matrix., however, the data in Table 1 is not required to be evaluated.

CTSHFAAPAWA operator CTSHFAAPAWG operator
Step 2: Aggregate the matrix based on the CTSHFAAWPA operator and CTSHFAAWPG operator, see Table 2 for ∅ = 2.
Step 3: Evaluate the Score values of the above-aggregated information, see Table 3.
Step 4: Derive the ranking values according to the score function to examine the best one, see Table 4.
The most preferable and the most dominant optimal is η A−2 p according to the theory of CTSHFAAWPA operator and CTSHFAAWPG operator.Additionally, we evaluate the comparison between proposed operators with some existing operators to show the supremacy and validity of the proposed operators.

Comparative analysis
In this section, we compare the proposed operators with our considered prevailing operators based on FSs and their extensions, because without comparison every paper has no worth.For comparing our ranking results, we are required to collect some existing operators and then try to evaluate our data with the help of existing operators for comparing it with our ranking results to show the validity of the proposed information, therefore, we select the following operators, such as: AAOs for HFSs 26 , AAOs for IFSs 27 , AAOs for PFSs 28,29 , AAOs for CPFSs 30 , AAOs for IFSs 31 , geometric AAOs for IFSs 32 , AAOs for PyFSs 33 , AAOs for QROFSs 34 , AAOs for TSFSs 35 , and AAOs for CTSFSs 36 .Using the data in Table 1, the comparative analysis is listed in Table 5.
The most preferable and the most dominant optimal is η A−2 p according to the theory of CTSHFAAWPA operator and CTSHFAAWPG operator.After the investigation in Table 5, we observed that the existing operators have no worth or capability to evaluate the data in Table 1 because these all operators are the special cases of the proposed operators.

Conclusion
The major theme of this manuscript is listed below: 1. We evaluated the novel theory of the CTSHF set and its operational laws.2. We investigated Aczel-Alsina operational laws for CTSHF information.3. We derived the CTSHFAAWPA operator, CTSHFAAOWPA operator, CTSHFAAWPG operator, and CTSH-FAAOWPG operator.Some properties are also investigated for the above operators.4. We evaluated the problems of 3D seismic attributes analysis to mine planning under the consideration of the proposed operators, for this, we illustrated the problem of MADM technique for the exposed operators.5. We demonstrated some examples for making the comparison between prevailing and proposed information to improve the worth of the derived operators.
Table 5. Comparative analysis of the proposed and existing techniques.

Figure 1 .
Figure 1.Graphical abstract of the proposed information.

Table 2 .
Table 1, if we have cost type of data, then CTSHFAAPO aggregated matrix.