An application for the earthquake spectral and source parameters and prediction using adaptive neuro fuzzy inference system and machine learning

Abstract

Parameters related to earthquake origins can be broken down into two broad classes: source location and source dimension. Scientists use distance curves versus average slowness to approximate the epicentre of an earthquake. The shape of curves is the complex function to the epicentral distance, the geological structures of Earth, and the path taken by seismic waves. Brune’s model for source is fitted to the measured seismic wave’s displacement spectrum in order to estimate the source’s size by optimising spectral parameters. The use of ANFIS to determine earthquake magnitude has the potential to significantly alter the playing field. ANFIS can learn like a person using only the data that has already been collected, which improves predictions without requiring elaborate infrastructure. For this investigation’s FIS development, we used a machine with Python 3x running on a core i5 from the 11th generation and an NVIDIA GEFORCE RTX 3050ti GPU processor. Moreover, the research demonstrates that presuming a large number of inputs to the membership function is not necessarily the best option. The quality of inferences generated from data might vary greatly depending on how that data is organised. Subtractive clustering, which does not necessitate any type of normalisation, can be used for prediction of earthquakes magnitude with a high degree of accuracy. This study has the potential to improve our ability to foresee quakes larger than magnitude 5. A solution is not promised to the practitioner, but the research is expected to lead in the right direction. Using Brune’s source model and high cut-off frequency factor, this article suggests using machine learning techniques and a Brune Based Application (BBA) in Python. Application accept input in the Sesame American Standard Code for Information Interchange Format (SAF). An application calculates the spectral level of low frequency displacement (Ω₀), the corner frequency at which spectrum decays with a rate of 2(f_c), the cut-off frequency at which spectrum again decays (f_max), and the rate of decay above f_max on its own (N). Seismic moment, stress drop, source dimension, etc. have all been estimated using spectral characteristics, and scaling laws. As with the maximum frequency, fmax, its origin can be determined through careful experimentation and study. At some sites, the moment magnitude was 4.7 0.09, and the seismic moment was in the order of (107 0.19) 1023. (dyne.cm). The stress reduction is 76.3 11.5 (bars) and the source-radius is (850.0 38.0) (m). The ANFIS method predicted pretty accurately as the residuals were distributed uniformly near to the centrelines. The ANFIS approach made fairly accurate predictions, as evidenced by the fact that the residuals were distributed consistently close to the centerlines. The R2, RMSE, and MAE indices demonstrate that the ANFIS accuracy level is superior to that of the ANN.

Keywords

Artificial neural networks brune based application adaptive neuro fuzzy inference system source dimension earthquake occurrence prediction

1 Introduction

For the purposes of seismic design at target location, it is necessary to make a quantitative estimate of characteristics for strong ground motion. Stochastic approaches are the most often used, and they necessitate familiarity with spectral and source factors (stress drop and corner frequency) in the area under investigation. However, recent outcomes have been excellent in both recorded and unrecorded regions. In order to gain insight into the nature of the mechanisms that cause earthquakes and as a guide to the simulation of strong ground motion for engineering applications, a significant amount of discussion has been focused on the shape of the seismic spectrum as well as the scales of earthquake size. This discussion has been going on for quite some time [1]. This seismic spectrum structure and its relationship to earthquake magnitude has been an interesting research issue. In this context, Aki [2] studied the scaling relation of the earthquake source spectra based on the ω² model by analysing the dependence of seismic wave amplitude spectrum on the source size. This was done to further investigate this direction. On the basis of the hypothesis of similarity, there was discovered to be satisfactory agreement with the observations. It has been noted that scaling law does not hold true, if stress drop varies which contradicts the previously held assumption that it does. It has also been pointed out that multiple scaling laws can be built for various different environments, if the stress drop changes systematically with regard to the environmental characteristics such crust and mantle structure focal depth, and fault plane direction. For computers to acquire new skills and knowledge without being explicitly taught them, researchers have developed a field known as machine learning (ML) [3]. Supervised learning and unsupervised learning are the two main categories of ML algorithms. To identify a pattern or make broad predictions in supervised learning, it is necessary to have access to the labelled dataset in the first place. In contrast, the goal of unsupervised learning is to draw conclusions about the underlying structure of a data set without using labels. A basic ω² shape characterises the displacement source spectrum, source-corner frequency (f_c) and decline of ω^-2 at higher frequencies. Similar to the acceleration spectrum has ω² shape below source corner frequency (f_c) and a level above f_c the acceleration spectrum also has a ω² shape for frequencies below f_c and a flat level above f_c. Above the certain frequency, which Hanks [4] calls the max cut-off frequency f_max, the amplitudes of acceleration spectral decrease dramatically. From an earthquake engineering perspective, the cut-off frequency, f_max, is crucial since it determines the maximum ground acceleration. Its provenance is a matter of some debate. Many researchers, including Hanks [4] and Anderson and Hough [5], think that f_max is an artefact of the recording environment. Most of the research, however, attributes this to the origin [6–13].

Neural networks and fuzzy logic expert system have become popular due to their near-human reasoning ability. These expert systems are modelled after the work and research of experts in their respective fields, who carry out and refine a number of technical procedures. Not much research was done utilising ANFIS to predict earthquake occurrence and their significant properties like size, frequency of occurrence, amount of energy produced during earthquake, etc. Almost all of the research into earthquake forecasting has focused on ANN. ANFIS used in a novel way to foretell the energy released during earthquakes [14]. The earthquake magnitude was predicted using ANFIS [15] and then standardising the data with Principal Component Analysis (PCA). In this article, we do something that no other researcher has done before: we compare grid partitioning with subtractive clustering to see which method is better for estimating earthquake magnitude. An initial stage in earthquake risk assessment is locating the seismogenic zone. By using seismological criteria, it is possible to divide it into smaller subzones. Zoning for seismic risk can be accomplished with the help of professionals. The clustering method is widely used for this kind of zonation or grouping. In order to recover the spatiotemporal pattern produced by events, a clustering approach is used. Therefore, a more precise model of earthquake clustering is needed to describe the pattern or grouping tendency [17]. Each clustered region contains unique historical earthquake data. The dataset is used to estimate the likelihood of earthquakes occurring. The Gutenberg-Richter law is the empirical relationships used extensively in long term forecasting. Seismic metrics like b-value can be analysed more effectively with clustering. Clustering’s end product can be used to speculate on the nature of future occurrences. Therefore, measures for reducing the risk of earthquakes can be implemented by employing an ANFIS model with a magnitude that is either equal to or greater than the threshold that is specified in the selected cluster [18]. The total number of clusters is calculated automatically. Authorities can use information about anticipated aftershocks to disseminate safety measures. This research provides a simple and effective method for estimating earthquake magnitude using the three most significant input factors, including time, epicentre distance, and peak ground acceleration. Results show that this method is quick and simple to implement. According to the current literature, there has only been a minimal investigation into using ANFIS to forecast earthquake magnitude. An application based on machine learning methods is developed to estimate the following source spectrum parameters: the spectral level of low frequency displacement (Ω₀), the frequency at which the spectrum begins to decay at a rate of 2(f_c), maximum frequency at which spectrum begins to decay again (f_max), and the rate of decay above f_max (N). These spectral parameters are employed in parameter estimation and scaling law construction at the source level [19, 20]. When selecting input parameters for an application, it is important to provide robust justifications for each parameter. This helps ensure that the application produces accurate and reliable results. For the application described, the input parameters are: Spectral level of low frequency displacement (Ω₀), Corner frequency at which spectrum decays with a rate of 2(fc), Cut-off frequency at which spectrum again decays (fmax), Rate of decay above fmax (N). In the case of low frequency displacement, it may be important to consider the size and weight of the object being displaced, as well as any external factors that could affect the displacement. The Ω₀ parameter should be chosen based on the expected displacement level of the system being analyzed. The corner frequency (fc) and cut-off frequency (fmax) parameters are also important to consider. These parameters help to define the shape of the frequency response curve and can have a significant impact on the results of the analysis. Finally, the rate of decay above fmax (N) should also be chosen based on the physical properties of the system being analyzed. This parameter helps to determine the rate at which the spectral level decreases above the cut-off frequency. In summary, when selecting input parameters for an application, it is important to consider the physical properties of the system and should be provided for each parameter to ensure accurate and reliable results. Using Brune’s source model and high cut-off frequency factor, this article suggests using machine learning techniques and a Brune Based Application (BBA) in Python. Application accept input in the Sesame American Standard Code for Information Interchange Format (SAF). An application calculates the spectral level of low frequency displacement (Ω₀), the corner frequency at which spectrum decays with a rate of 2(f_c), the cut-off frequency at which spectrum again decays (f_max), and the rate of decay above f_max on its own (N). Seismic moment, stress drop, source dimension, etc. have all been estimated using spectral characteristics, and scaling laws [21]. The spatial-temporal changes in seismicity characteristics for the September 10th, 2008, Qeshm earthquake in south Iran were investigated in this work [22]. Artificial neural networks and the Adaptive Neural Fuzzy Inference System (ANFIS) were used to achieve this goal. Because of its effectiveness in classification and prediction challenges, the supervised Radial Basis Function (RBF) network and ANFIS model were applied. To investigate the spatial and temporal seismicity pattern, the eight seismicity parameters were determined. Prior to feeding the data into the RBF network and ANFIS model, data preprocessing techniques such as normalisation and Principal Component Analysis (PCA) were used [22–26]. To create models using the ANFIS structure, three techniques were used: grid partitioning (GP), subtractive clustering (SC), and fuzzy C-means (FCM). Because the earthquake data for the defined region was recorded on several magnitude scales, appropriate relationships were developed to convert the magnitude scales to moment magnitude, and the records were unified based on the relationships. The uniform data were utilised to calculate seismicity indicators, and ANFIS was built using thought-out methods. The results demonstrated that ANFIS-FCM could predict earthquake magnitude with good accuracy [23, 27, 28]. Extreme learning machine (ELM) concepts are used to modify the general structure of the group method of data handling (GMDH). In fact, using ELM as inspiration, a novel GMDH method called GMDH network based on using extreme learning machine (GMDH-ELM) is proposed in which the weighting coefficients of quadratic polynomials used in conventional GMDH are no longer required to be updated using back propagation or other evolutionary algorithms during the training stage [29]. Average flow velocity, critical threshold velocity of sediment movement, flow depth, median particle diameter, geometric standard deviation, un-contracted and contracted channel widths are input parameters that affect the scour phenomenon in the Adaptive Neuro-Fuzzy Inference System (ANFIS) and Support Vector Machines (SVM) [30, 31].

2 Architecture of ANFIS

According to its proponents, ANFIS is a type of hybrid system that may produce results that are analogues to those produced by humans using sets of inputs and desired outcomes. This sort of humanlike behaviour is trainable in dataset. The development of thinking capabilities comparable to those of humans is possible through the utilisation of an optimal combination of neural networking and fuzzy logic [18]. Figure 1(a) explain the three layer architecture of ANFIS. Following the completion of a five (5) layer training process that was mostly predicated on input-output data, a Sugeno-type fuzzy system was implemented. Because they use a set of fuzzy IF-THEN rules to describe the input-output relationship of a real system, fuzzy inference systems are non-linear. This is because fuzzy inference systems are not deterministic.

Fig. 1(a)

Architecture of ANFIS.

Fig. 1(b)

Notepad File for Application input Earthquake Source Dimension Estimation.

An adaptive strategy is developed in order to learn the data set and compute the membership function. This is done so that the strategy can follow the input/output and produce the required outcomes. There does not seem to be a consistent method for determining the membership function’s parameters. Finer tuning and membership functions are established using either forward propagation, reverse propagation, or a combination with the least-squares method. In this paradigm, input is mapped using a membership function, which is then linked to a set of parameters, and the resulting output is produced using another membership function. Modelling the desired result using rules for a fuzzy inference system and initial membership functions takes lot of expertise. ANFIS if-then rules & membership functions described the input/output characteristics to the complex system. At the outset of data set training, fuzzy inference system is generated using grid partitioning technique. In this method, membership functions with arbitrary parameters are used to construct the grid with regular partitions. The ANFIS model is trained with the combination of hybrid optimization technique and back-propagation algorithm. Distinctive clustering by subtraction Many automated techniques, including subtractive clustering, have been created to aid in the determination of the most important processes, which are the identification of forms of fuzzy rules and optimal numbers. The method operates on the premise that each and every data point represents possible cluster of data and calculates data points’ density around each data point as a proxy for the probability that it serves as the cluster’s epicentre.

3 Methodology

The earthquake dataset that is mentioned in Table 1 having the latitude and longitude of the earthquake, focal depth, hypocentre, magnitude, corner frequency f_c, amplitude’s spectrum diminishes abruptly for frequencies which is greater than f_max, kappa values, zone of earthquake, p and sl values with FEMA classes.

Table 1
Earthquake dataset

Eq source Latitude Eq Source Long. Focal depth Hypo-centre FEMA Mag SL fc fmax p kappa Zone

33.6808 132.39 55 63 B 4 1.0 4.5 22.0 6.3 0.010 4

33.775 132.53 47 61 B 4 2.0 5.3 18.7 5.5 0.015 4

33.4907 132.37 49 66 B 4.3 3.0 4.3 19.5 5.3 0.015 4

38.6415 141.45 106 113 B 4 23.0 6.6 24.2 5.2 0.008 3

39.1302 141.4 79 122 B 4 24.0 4.6 23.6 4.9 0.011 3

38.5735 141.28 111 117 B 4 25.0 4.5 21.5 4.2 0.011 3

38.9415 141.62 74 104 B 4 26.0 8.4 17.2 4.4 0.024 3

42.1862 144.78 27 119 C 6.1 332.0 0.4 6.1 3.9 0.035 2

41.8612 144.44 28 144 C 6.1 333.0 1.4 6.8 3.2 0.039 2

41.7318 143.72 40 162 C 6.2 334.0 0.3 3.6 1.9 0.062 3

42.5627 144.67 51 91 C 6.4 335.0 0.3 9.4 4.3 0.051 2

Eq source Latitude	Eq Source Long.	Focal depth	Hypo-centre	FEMA	Mag	SL	fc	fmax	p	kappa	Zone
33.6808	132.39	55	63	B	4	1.0	4.5	22.0	6.3	0.010	4
33.775	132.53	47	61	B	4	2.0	5.3	18.7	5.5	0.015	4
33.4907	132.37	49	66	B	4.3	3.0	4.3	19.5	5.3	0.015	4
38.6415	141.45	106	113	B	4	23.0	6.6	24.2	5.2	0.008	3
39.1302	141.4	79	122	B	4	24.0	4.6	23.6	4.9	0.011	3
38.5735	141.28	111	117	B	4	25.0	4.5	21.5	4.2	0.011	3
38.9415	141.62	74	104	B	4	26.0	8.4	17.2	4.4	0.024	3
42.1862	144.78	27	119	C	6.1	332.0	0.4	6.1	3.9	0.035	2
41.8612	144.44	28	144	C	6.1	333.0	1.4	6.8	3.2	0.039	2
41.7318	143.72	40	162	C	6.2	334.0	0.3	3.6	1.9	0.062	3
42.5627	144.67	51	91	C	6.4	335.0	0.3	9.4	4.3	0.051	2

Figure 1(b) shows the coordinates of the point of origin, as well as the time and date of its creation. Information saved in the file includes altitude, latitude, recording station code, recorded depth in kilometres, recorded magnitude on the Rector scale, and recorded depth at a certain location. There is a detailed log of the date and time, the maximum.

acceleration, the sampling rate in hertz (Hz), the length of time, and the scale factor.

Time records extract the SH-component of the ground motion after being compensated for instrument response. The SH spectrum is adjusted for route attenuation. In order to estimate the source parameters, a two-stage methodology was employed, involving spectral analysis of S-waves and Levenberg-Marquardt non-linear inversion. After considering the high frequency diminution factor and Brune’s source model, which both result in fall-off of corner frequency, we find that a Butterworth high-cut filter works for the frequencies higher to f_max in the acceleration spectrum that is observed Equation (1) and the displacement spectrum Equation (2). $A (R, f) = \begin{matrix} \frac{\begin{matrix} {(2 π f)}^{2} & Ω_{0} \end{matrix}}{\begin{matrix} 1 & + & {(\frac{f}{f_{c}})}^{2} \end{matrix}} & {[\begin{matrix} 1 & + & {(\frac{f}{f_{\max}})}^{N} \end{matrix}]}^{1 / 2} \end{matrix}$ (1) $D (R, f) = \begin{matrix} \frac{Ω_{0}}{\begin{matrix} 1 & + & {(\frac{f}{f_{\max}})}^{2} \end{matrix}} & {[1 + {(\frac{f}{f_{\max}})}^{N}]}^{1 / 2} \end{matrix}$ (2)

An application automatically chooses the parameters (spectral):

low frequency displacement spectral level, Ω₀,

spectrum’s corner frequency (decays at the rate of 2, f_c),

spectrum’s frequency (again decays, f_max),

decay rate f_max, N.

In order to calculate the SH component of ground motion, we must first rotate the time histories along the axis of azimuth. This is done to take into consideration the instrument response, which is accounted for by using an approximated transfer function based on zero-pole values.

Figure 2 depicts a typical instance of an applied instrument response yielding a selected SH-component and a time-history acceleration. To acquire the SH-component spectrum, the Fast Fourier Transform (FFT) is applied to a subset of the time series containing the SH-component. Due to the route effect, the spectrum of the SH-component has been corrected with a frequency-dependent attenuation correction, Q_c = 110f^1.02.

Fig. 2

The response of instrument during the time-series of accelerations and the velocities of SH component.

Fig. 3

Estimating fc with a log-log plot of the velocity spectrum (left) and a log-linear plot of the velocity spectrum (right) can help you determine what frequency f_c.

Fig. 4

Estimating the Acceleration spectrum for Mo & Mw estimation.

When calculating f_c, the peak frequency of the velocity spectrum is used since it provides a good approximation of the true value of f_c. Figure 3(a) shows a log-log plot depicting this, while Fig. 3(b) illustrates how this peak on a log-linear scale becomes apparent when the two plots are compared.

The fquency at which the spectral peak occurs is then used to estimate a Magnitude value. You can see this in Fig. 4. The estimation of seismic moment from value of Ω₀ in Equation 3: $M_{o} = \frac{4 Π ρ β^{3} R Ω_{0}}{\begin{matrix} R_{θ φ} & . & S_{a} \end{matrix}}$ (3)

The seismic moment (M_o), The Average Density (= 2.7 g/cm³), S-wave velocity (= 3.21 km/s source zone), Hypocentral distance(R), Radiation pattern avg (R_θφ = 0.6) and Surface amplification free (S_a = 2).

The magnitude of the moment is calculated in Equation 4: $M_{w} = \frac{2}{3} log (M_{o}) + 10.7$ (4)

Using the corner frequency f_c and the phase velocity β, the Brunes model provides an estimate for the source radius (r). In this case, it is written as Equation 5: $r = \frac{2.34 β}{2 Π f_{c}}$ (5)

The following relationship can be used to calculate the stress drop (Δσ) in Equation 6: $Δ σ = \frac{7 M_{0}}{16 r^{3}}$ (6)

The objective is to minimise an error function E, which is defined in Equation 7 as follows: $E = \sqrt{\frac{\sum_{i = 1}^{n} {[\begin{matrix} D_{o}^{i} (f) & - & D_{t}^{i} (f) \end{matrix}]}^{2}}{n}}$ (7)

$D_{o}^{i} (f)$ observed displacement spectra, $D_{t}^{i} (f)$ theoretical displacement spectra and the data points represent as n.

Information gathered in Himachal Pradesh’s Himalaya (Fig. 5). Source parameters have been estimated using data from the Strong motion network (GSR-18, 200 Hz (sampling re); see www.pesmos.in) and the Guralp, CMG 40T-1, 12-station seismological network (sampling rate 100 Hz).

Fig. 5

Fault map with the location of cities and free surface amplification Seismotectonic map of Himachal [22].

4 Earthquake location estimation

To approximate this complex nonlinear function with the goal of minimising error in position estimate, ANN is a useful machine learning tool [23, 24]. The neurons Connections in output, hidden, and input layer can be thought of as directed edges in a weighted directed graph. Each neuron takes the qualities as input, which are often presented as a vector, and determines the weighted sum of those attributes. After that, an activation function is applied to the weighted sum. Connections are made between the outputs of one layer and the inputs of the following layer. Input and output layers contain neurons whose total number is based on the total number of the input-output variables required for optimal solution of the problem. The difficulty of the job will decide the proper number of hidden layers, as well as the number of neurons that will be included within each hidden layer. During the training of an ANN utilising the provided training data, the proper weights’ edges that are necessary to generate the intended output are learned. After the network has been trained, it can be utilised to provide accurate results from unlabelled data. In order to train multilayer neural networks, the back-propagation technique is commonly utilised due to its high computing efficiency and widespread adoption. After each layer’s output error has been backpropagated, the cost function’s partial derivatives are computed relative to the network weights. [25].

4.1 Acquisition of data

Both the training and testing datasets are comprised of observed and predicted latencies based on earthquake hypocentres and station location. This study’s earthquake data comes from the United States Geological Survey and the website https://strongmotioncenter.org (USGS). Earthquake epicentres are kept between 379 and 4 kilometres away. Fifty distinct earthquake ground motions are recorded from various places throughout the world.

4.2 Procedure

In Fig. 3, 20 neurons are spread throughout the first and second hidden levels of a 4-layer neural network (Hl1and Hl2 respectively). The input layer receives the slowness parameter, and the output layer generates the desired output using a single neuron. Since the sigmoid activation function is non-linear and differentiable, it is employed by the neurons in input & hidden layers. However, the linear activation function used at the output layer. Output error, as measured by mean square error (MSE), reduces with training iterations as shown in Fig. 7. Output error quickly decreases for the first thirty (30) iterations, going through 2 or 3 local minima as seen in the inset of Fig. 7. After around 150 iterations, the rate of gain in ANN performance slows to a plateau, with further iterations providing little to no benefit. This curve is shown in greater detail for the first iteration in the inset.

Fig. 6

Schematic illustrating the method used to calculate earthquake origin characteristics.

Fig. 7

Statistical analysis of output error versus number of iterations (epochs).

5 Result and discussion

5.1 Epicentral distance estimation

The trained ANN receives slowness values from this dataset at its input layer and produces an estimate of slowness at its output layer as computed Δ is output of network. The suggested ANN-based approach is compared to a polynomial fitting approach by first fitting polynomials of varied degrees to the test dataset in a least-squares manner. It is shown that the fit does not better for orders higher than seven as the polynomial order is increased from one to seven. The desired and calculated Δ value difference is called absolute error. Figure 8 compares the computed and desired Δ as a function of slowness using ANN and polynomial fit for whole dataset in the Appendix 1. As can be observed, training and testing events Δ computed (in red) closely tracks by ANN desired Δ (in blue) within an error of roughly±1°, but the polynomial fit technique (in solid line) demonstrates higher scatter between two values across both testing and training sessions.

Fig. 8

Comparisons of intended and calculated Δ using ANN and 7th order of polynomial fit.

Fig. 9

Seismic Waveforms with Three Components.

5.2 Source parameter & dimension estimation

A vertical line denotes the three-channel vertical, transverse, and radial components of earthquake seismograms recorded from the main shock and P start of the event. By analysing the seismogram at a certain p phase, we have been able to determine the earthquake’s source dimension. Appendix 2 provides estimated source parameters in addition to spectral parameters acquired from velocity & acceleration records at numerous places. It has been determined that the moment magnitude for this event was 4.7±0.09 at several locations, and the seismic moment was in order of (107±0.19)×1023 (dyne.cm). A stress reduction of 76.3±11.5 (bars) and a source-radius of (850.0±38.0) (m) are determined. The maximum frequency of this earthquake was measured to be 9.1±1.7 (Hz) using data collected from a wide range of stations in a variety of geological settings. Some short period instruments show a shift in the spectral fall-off above fmax, while strong motion instruments maintain a constant value.

Time histories from the seismogram and the strong motion instrument, after being rotated, and the portion of the SH component that was analysed are displayed in Fig. 10. Additionally, graphics depicting the fitted model and the acceleration and displacement spectra are provided below. Source model fitting to acceleration and displacement spectra is displayed at the bottom.

Fig. 10

Time history of an earthquake, including the SH component, as recorded by a short-period seismograph.

5.3 Earthquake prediction

For more precise forecasts, it is common practise to input training data to an expert system; in this example, forty earthquakes are used as training data, while the rest of used as testing and validation data. The validation and testing data is useful for verifying the modelled ANFIS’s performance [26]. Obtaining the magnitude of an earthquake in the future or even in the past is possible by considering the three most relevant variables. The magnitude of the earthquake is desired outcome, and the three assumed input variables are the epicentre distance, year of occurrence, and the earthquake’s peak ground acceleration. This part of the process ensures that the ANFIS model is accurate, efficient, and capable of saving time. It explains how to use triangular membership functions in the event of grid portioning and compares the outcomes to those obtained using the subtractive clustering method. The output is determined by a membership function of a triangular kind, with 5 and 7 neurons, respectively. Data for training evaluation has improved from Fig. 11(a) to (c) as a result of increasing the number of neurons from three to five (b). While these models perform well in the training data, they forecast earthquakes of an unsatisfactory size in the testing data, which is unacceptable if we want to improve the quality of our predictions. The testing data set comparing observed and forecasted earthquake magnitudes is depicted in Fig. 12(a) and (b).

Fig. 11

Analysis of the relationship between (a) 5no’s and (b) 7no’s and the triangle membership function’s training data set.

Fig. 12

Analysis of the relationship between (a) 5no’s and (b) 7no’s and the triangle membership function’s testing data set.

When employing subtractive clustering, the user is not given the option to select the membership function used in the process. This study’s data set was well predicted by subtractive clustering even without the use of any normalising approach. Subtractive clustering was found to be the most effective predictor of the desired outcome in this investigation. The ANFIS method predicted pretty accurately as the residuals were distributed uniformly near to the centrelines. Predicted and observed earthquake magnitudes versus sample size are shown in Figs. 13 and 14, respectively. Accuracy on both training and test data was used to gauge a network’s performance, with values for R and RMSE calculated. Different combinations of input parameters were used to determine the accuracy of the four models in terms of R and RMSE values for both shear parameters (c and f). For a subset of neural networks, where the neurons in the hidden layer are allowed to change, we compare the accuracy of all the models in training and testing. As neural network designs shift, it becomes clear that different models have varying degrees of success with different types of data in training and in testing. Table 2 and Fig. 15 displays the R, RMSE, and MAE values observed during training and testing with various neural networks and models.

Table 2

Artificial neural network for evaluating the accuracy of predictions

Model	Stimulating Function	R²	RMSE	MAE
ANN	Sigmoid	0.9325	0.52056998	0.130
ANFIS	Gaussmf	0.9815	0.81269547	0.039

Fig. 13

Clustering-induced data comparisons in training experimental settings.

Fig. 14

Clustering-induced data comparisons in testing experimental settings.

Fig. 15

Artificial neural network for evaluating the accuracy of predictions.

6 Conclusion

ANFIS for determining earthquake magnitude can be game-changing. ANFIS can learn like a person using only the data that has already been collected, resulting in more accurate forecasts without the need for elaborate machinery. In order to create the FIS for this investigation, a computer running Python 3x on a core i5 11th generation with NVIDIA GEFORCE RTX 3050ti GPU processor is employed. The study also shows that it is not always preferable to assume a high membership function input count in order to obtain a desired outcome. Different data arrangements can have a significant impact on the quality of the conclusion that can be drawn. The magnitude of earthquakes can be predicted with high accuracy using subtractive clustering, which does not require any sort of normalisation. Therefore, this study has the potential to contribute in the forecasting of earthquakes of a magnitude larger than 5. While this research cannot provide definitive solutions, it may provide practitioners a direction to go in.

The Brune Based Application (BBA) written in Python programming, is predicated on Brune’s source model and frequency factor (high cut-off). The Sesame ASCII Format (SAF) is used for input. Spectral level low frequency displacement (Ω₀), over which spectrum’s corner frequency decays (at the rate of 2(f_c)), rate of decay (above f_max), and at which spectrum again decays (cut-off frequency (f_max)) are all chosen automatically by the software (N). Source parameters (such as stress drop, seismic moment, source parameter, source dimension, and etc.) have been estimated and scaling laws have been developed using the derived spectrum parameters. The frequency cut-off origin, denoted with the symbol “ f_max,” can also be deduced by observation and analysis. The ANFIS approach made fairly accurate predictions, as evidenced by the fact that the residuals were distributed consistently close to the centerlines. Figures 13 and 14 represent, respectively, the predicted and observed earthquake magnitudes vs the sample size of the data set. The performance of a network was evaluated based on its accuracy on both training data and test data, and values for R² and RMSE were calculated. It was determined, using a variety of different combinations of input parameters, how accurate each of the four models were in terms of the R and RMSE values for both shear parameters (c and f). We evaluate the accuracy of all of the models during both the training phase and the testing phase for a subset of neural networks in which it is permissible for the neurons in the hidden layer to undergo change. It is now plainly clear that different models have varying degrees of success with different types of data when being trained and tested, and this is something that has become increasingly clear as neural network designs continue to advance. The R2, RMSE, and MAE values that were observed throughout the training and testing phases of the project. The ANFIS technique produced predictions that were reasonably accurate, as shown by the fact that the residuals were distributed in a manner that was consistent with being close to the centerlines. The R2, RMSE, and MAE indices all point to the fact that the accuracy level of the ANFIS is significantly higher than that of the ANN.

Footnotes

Data for examination of epicentral distances

Sr. No.	Slowness (Sec/Degree) I/P	Distance (Δ∘) O/P
1	5.306	74.93
2*	5.206	73.46
3	5.286	74.35
4	5.347	69.09
5	5.375	71.92
6*	5.521	72.12
7	5.578	74.2
8	5.626	66.67
9	5.683	69.14
10*	5.749	67.78
11	5.8	72.45
12	5.909	73.55
13	6.07	66.65
14*	6.152	62.19
15	6.195	59.6
16	6.195	62.7
17	6.195	62.91
18*	6.3	59.78
19	6.476	64.7
20	6.476	59.12
21	6.484	64
22	6.586	49.97
23*	6.601	58.01
24	6.612	50.3
25	6.612	48.89
26*	6.691	54.39
27	6.692	58.34
28	6.753	50.02
29	6.753	54.61

Source estimates and spectral measurements

Eq source Latitude	Eq Source Longitude	f_c (Hz)	f_max (Hz)	N	Mw	Δσ (bars)	M_o (dyne-cm)	Ω_o (cm)	R (m)
33.6808	132.39	1.40	12.0	6	4.7	92.6	1.30*10²³	0.0130	850.9
33.775	132.53	1.30	9.0	6	4.7	68.1	9.60*10²³	0.0110	850.9
33.4907	132.37	1.30	8.0	6	4.8	78.0	1.37 *10²³	0.0140	916.4
38.6415	141.45	1.50	12.0	6	4.6	67.7	7.75 *10²²	0.0063	794.2
39.1302	141.4	1.40	9.0	6	4.6	56.1	7.90 *10²²	0.0060	850.9
38.5735	141.28	1.50	10.0	6	4.7	99.7	1.14*10²³	0.0110	794.2
38.9415	141.62	1.40	7.0	6	4.7	70.7	9.96*10²²	0.0062	850.9
42.1862	144.78	1.40	8.0	6	4.7	71.9	1.012*10²³	0.0120	850.9

References

Joyner

W.S.

, A scaling law for the spectra of large earthquakes, Bulletin of the Seismological Society of America 74(4) (1984), 1167–1188.

Aki

, Scaling law of seismic, Journal of Geophysical Research 72(4) (1967), 1217–1231. doi: 10.1029/JZ072i004p01217 spectrum.

Samuel

A.L.

, Some studies in machine learning using the game of checkers, IBM J. Res. Dev. 3 (1959), 535–554.

Hanks

T.C.

, Fmax, Bulletin of the Seismological Society of America 72 (1982), 1869–1879.

Anderson

J.G.

, and Hough

S.E.

, A model for the shape of the fourier amplitude spectrum of acceleration at high frequencies, Bulletin of the Seismological Society of America 74(5) (1984), 1969–1993.

Papageorgiou

A.S.

, and Aki

, A specific barrier model for the quantitative description of inhomogeneous faulting and the prediction of strong ground motion. I1. Application of the model, Bulletin of the Seismological Society of America 73(4) (1983), 953–978.

Aki

, Magnitude-frequency relation for small earthquakes: a clue to the origin of fmax of large earth-quakes, Journal of Geophysical Research 92(B2) (1987), 1349–1355. doi: 10.1029/JB092iB02p01349.

Toshiaki

, and Irikura

, Meaning of source con- trolled fmax in empirical green’s function technique based on a T2-scaling law,, Disaster Prevention Research Institute, Kyoto University, 1991.

Papageorgiou

A.S.

, On two characteristic frequencies of acceleration spectra: patch corner frequency and fmax, Bulletin of the Seismological Society of America 78(2) (1988), 509–529.

10.

Kinoshita

, Local characteristics of the fmax of bed- rock motion in the Tokyo metropolitan area, Japan, Journal of Physics of the Earth 40(3) (1992), 487–515. doi: 10.4294/jpe1952.40.487

11.

Tsurugi

, Kagawa

, and Irikura

, Study on high- cut frequency characteristics of ground motions for Inland Crustal Earthquakes in Japan, 14 World Conference on Earquake Engineering, Beijing, 12-17 October 2008, N0 02-0036.

12.

Tsurugi

, Tai

, Kowada

, Tatsumi

, and Iri-kura

, Estimation of Empirical Site Amplification Effects Using Observed Records, 12 World Conferenc on Earquake Engineering, Auckland, 30 January-4 Febuary 2000, pp. 1–6.

13.

Wen

, and Chen

, Variations in fmax along the Rup-tured Fault during the Mw 7.9 Wenchaun Earthquake of 12 May, Bulletin of the Seismological Society of America 102(3) (2012), 991–998. doi: 10.1785/0120110105.

14.

Shibli

, A novel approach to predict earthquakes using adaptive neural fuzzy inference system and conservation of energy-angular momentum, Int J Computer Information Systems and Industrial Management Applications 3 (2011), 371–390.

15.

Zamani

, Sorbi

M.R.

, and Safavi

A.A.

, Application of neural network and anfis model for earthquake occurrence in Iran, Earth Sci Inf 6(2) (2013), 71–85.

16.

Mirrashid

, Givehchi

, Miri

, and Madandoust

, Performance investigation of neuro-fuzzy system for earthquake prediction, Asian Journal of Civil Engineering (BHRC) 17(2) (2016), 213–223.

17.

Shodiq

M.N.

, Barakbah

A.R.

, and Harsono

, Spatial analysis of earthquake distribution with automatic clustering for prediction of earthquake seismicity in Indonesia, KCIC2015, 2015.

18.

Pandit

, and Biswal

K.C.

, Prediction of earthquake magnitude using adaptive neuro fuzzy inference system, Earth Sci Inform 12 (2019) 513–524. https://doi.org/10.1007/s12145-019-00397-w

19.

Gaurav

Rajguru

, Bhadauria

Y.S.

and Mukhopadhyay

, Estimation of earthquake source parameters using machine learning techniques, 2018 9th International Conference on Computing, Communication and Networking Technologies (ICCCNT). IEEE, 2018.

20.

Mustafa Abdul

Salam

, Ibrahim

Lobna

and Abdelminaam

Diaa Salama

, Earthquake prediction using hybrid machine learning techniques, International Journal of Advanced Computer Science and Applications 12.5 (2021), 654–6652021.

21.

Kumar

, Kumar

, Mittal

, Kumar

, and Bhardwaj

, Software to estimate earthquake spectral and source parameters, International Journal of Geosciences 3(5) (2013), 1142–1149. doi: 10.4236/ijg.2012.35116.

22.

Zamani

, Sorbi

M.R.

, and Safavi

A.A.

, Application of neural network and ANFIS model for earthquake occurrence in Iran, Earth Sci Inform 6 (2013), 71–85. https://doi.org/10.1007/s12145-013-0112-8

23.

Mirrashid

, Earthquake magnitude prediction by adaptive neuro-fuzzy inference system (ANFIS) based on fuzzy C-means algorithm, Nat Hazards 74 (2014), 1577–1593. https://doi.org/10.1007/s11069-014-1264-7.

24.

Prabhu

Muthuganeisan

and Raghukanth

S.T.G.

, Site-specific Probabilistic Seismic Hazard Map of Himachal Pradesh, India. Part II. Hazard Estimation, Acta Geophysica 64 (2016), 853–884.

25.

Shang

Xueyi

, Li

Xibing

, Morales-Esteban

and Chen

Guanghui

, Improving microseismic event and quarry blast classification using artificial neural networks based on principal component analysis, Soil Dynamics and Earthquake Engineering 99 (2017), 142–149.

26.

Rana

, Vaidya

, and Hu

Y.-C.

, A Comparative Analysis of ANN and ANFIS Approaches for Earthquake Forecasting, 2022 Second International Conference on Computer Science, Engineering and Applications (ICCSEA) (2022), pp. 1–6, doi: 10.1109/ICCSEA54677.2022.9936133.

27.

Rana

Anurag

, Kumar

Arjun

and Sharma

Ankur

, Neural Network Radial Basis Function classifier for earthquake data using aFOA, Int. J. of Adv. Res 4(Aug) (2016), 537–540.

28.

Mohammad Nur

Shodiq

, et al. Adaptive neural fuzzy inference system and automatic clustering for earthquake prediction in indonesia, JOIV: International Journal on Informatics Visualization 3.1 (2019), 47–53.

29.

Saberi-Movahed

, Najafzadeh

, and Mehrpooya

, Receiving More Accurate Predictions for Longitudinal Dispersion Coefficients in Water Pipelines: Training Group Method of Data Handling Using Extreme Learning Machine Conceptions, Water Resour Manage 34 (2020), 529–561. https://doi.org/10.1007/s11269-019-02463-w

30.

Najafzadeh

, Etemad-Shahidi

, and Lim

S.Y.

, Scour prediction in long contractions using ANFIS and SVM, Ocean Engineering 111 (2016), 128–135. https://doi.org/10.1016/j.oceaneng.2015.10.053

31.

Rana

Anurag

, Gupta

Gaurav

, Vaidya

Pankaj

, Salehi

Waleed

, Basheer

Shakila

and Bhatia

Madhulika

, Techniques based on metaheuristics combined with an adaptive neurofuzzy system and seismic sensors for the prediction of earthquakes, Journal of Sensors 2023 (2023), Article ID 5063981, 14. https://doi.org/10.1155/2023/5063981