esa-sceva/satcom-chunk-collection huggingface dataset:
## 4 Analysis of Results  
### Effect of \(L_{t}\) on the Stability of Ventilation Network  
According to the discriminant Equation (10), it can be seen that the air flow direction in the cross passage completely depends on the air flow-resistance ratio of the advance side tunnel and the undercut side tunnel, and has nothing to do with the air flow resistance of the cross passage itself.  
The air flow resistance of each branch:  
\[R_{\text{m}}=h_{\text{m}}/{Q_{\text{m}}}^{2} \tag{16}\]  
Among them, \(Q_{\text{m}}\) is the flow rate of each branch of the double-hole tunnel, and \(h_{\text{m}}\) is the resistance of each branch.  
Figure 9 shows the comparison of the flow rate of each section of the tunnel under different \(L_{t}\) conditions. In Figure 9, if \(Q_{a}>Q_{\text{u}}\), when the \(L_{t}\) is 80 m, 130 m, and 180 m, the air flow through the traffic cross passage decreases by 0.9, 1, and 0.23 m\({}^{3}\)/s (\(Q_{1}-Q_{3}\)), respectively, and the cross-section flow rate of the undercut side tunnel increases by 2, 1.1, and 0.3 m\({}^{3}\)/s, respectively (\(Q_{4}-Q_{2}\)). With the increase of \(L_{t}\), the dynamic energy dispersion rate in the traffic cross passage increases, and the flow rate near the working face of the advance side tunnel continues to increase. If \(Q_{a}<Q_{\text{u}}\), when the \(L_{t}\) is 80 m, 130 m, and 180 m, the air flow decreases by 16.7, 1.8, and 0.17 m\({}^{3}\)/s, respectively after the air flows through the cross passage, and the cross-section flow rate of the undercut side tunnel increases by 18.7, 2.1, and 0.19 m\({}^{3}\)/s, respectively.  
Figure 10 shows the comparison of the resistance of each branch of the tunnel under different \(L_{t}\) conditions. In Figure 10, the resistance \(h_{\text{m}}\) of each branch decreases as \(L_{t}\) increases. Combined with the data in Figures 9 and 10, the air flow resistance \(R_{\text{m}}\) of each branch can be calculated, and then \(K\) is obtained according to the discriminant Equation (10), which are summarized in Table 3.  
It can be seen from Table 3 that the \(R_{\text{m}}\) of each branch decreases with the increase of \(L_{t}\), but the degree of decrease is different, resulting in an obvious difference in the \(K\). When \(L_{t}=80\) m, \(Q_{a}>Q_{\text{u}}\), \(K=0.69<1\); \(Q_{a}<Q_{\text{u}}\), \(K=0.009<1\), it indicates that the air flows from the advance side tunnel to the undercut side tunnel. When \(L_{t}=130\) m, \(Q_{a}>Q_{\text{u}}\), \(K=2.49>1\); \(Q_{a}<Q_{\text{u}}\), \(K=1.26>1\), it indicates that the air flows from the undercut side tunnel to the advance side tunnel. When \(L_{t}=180\) m, if \(Q_{a}>Q_{\text{u}}\), \(K=4.19>1\), the air flows from the undercut side tunnel to the advance side tunnel. If \(Q_{a}<Q_{\text{u}}\), \(K=0.77<1\), the air flows from the advance side tunnel to the undercut side tunnel.  
Figure 8: Vector diagram of tunnel air flow at \(L_{t}=130\) m.
## 3 Results and Validation
### Parameterization Results
#### 3.3.3 Effect of the Length of the Piezo Patch on Transmission Force and Deformation  
As shown in Figure 20, the \(k\) factor remarkably changes for piezo patches with small angular extensions (diminishing by about 40% in the extension range of 10-30 degree), while it remains practically constant as the extension increases.  
A linear trend for the extension was also estimated for the radial/tangential pin force ratio (Figure 21). Finally, a parabolic trend for the beam-transmitted axial strain over the free strain of the piezoelectric patch was observed (Figure 22).  
Figure 21: Fr/Ft percentage vs. angular position of the piezoelectric.  
Figure 20: The \(k\) factor vs. the angular extension of the piezoelectric.
## 9 Relation with Other Formal Models
### A Trade-off between Grammar and Rewrite Rules
#### Rules  
\[a{\sim}b{\sim}c{\sim}X \leftarrow \mathsf{del}(a,\mathsf{del}(b,\mathsf{del}(c,X))) \rightarrow\ X \tag{20}\] \[X \leftarrow \mathsf{ins}(a,\mathsf{ins}(b,\mathsf{ins}(c,X))) \rightarrow\ a{\sim}b{\sim}c{\sim}X \tag{21}\]  
This set of rules defines a partial rewrite relation. A core term with singleton deletions, such as e.g. \(\mathsf{rep}(a,b,\mathsf{del}(c,\mathsf{rep}(e,f,X)))\) cannot be rewritten at all. This is intended, as this term encodes an alignment that is not authorized in the edit scheme.  
The alternative is to specify a more restrictive grammar for the core terms:  
Grammar \[S^{*} \rightarrow \mathsf{del}(a,D_{1})\mid\mathsf{ins}(a,I_{1})\mid\mathsf{rep}(a,b,S)\mid\mathsf{mty}\] \[D_{1} \rightarrow \mathsf{del}(a,D_{2})\] \[D_{2} \rightarrow \mathsf{del}(a,S)\] \[I_{1} \rightarrow \mathsf{ins}(a,I_{2})\] \[I_{2} \rightarrow \mathsf{ins}(a,S)\]  
This has the same effect as the partial rewrite relation above: indels come in groups of three, and alignments where this is violated are no longer in the search space. Here we can live with the original, simpler rewrite rules such as \(\mathsf{del}(a,X)\to X\) and a complete rewrite relation, because the malformed gap terms are forbidden by the grammar.  
Given this trade-off, one might ask if the tree grammar is strictly necessary (aside from being convenient) for the modeling power of ICOREs.  
**Fact 4**.: _Tree grammars are a necessary concept in ICOREs._  
Proof.: We give a specific example where the search space restriction imposed by the grammar cannot be expressed via the rewrite relation. Recall sequence alignment under the affine gap model, ICORE Affine of Section 5. We have for some \(n>1\)  
\[\mathtt{a}^{n+1}\leftarrow\mathsf{open\_del}(a,\{\mathsf{del}(\mathtt{a},\}^{n }\,\mathsf{mty}\{\})^{n}\rightarrow\varepsilon\]  
which implies (With a finite rewrite system, we can require that up to \(k\) nested \(\mathsf{del}\) operations can only be rewritten together with a leading \(\mathsf{open\_del}\), but \(k\) would have to be a constant while \(n\) is arbitrary.) that  
\[\mathtt{a}^{n}\leftarrow\{\mathsf{del}(\mathtt{a},\}^{n}\,\mathsf{mty}\{\})^ {n}\rightarrow\varepsilon\]  
Hence, without the grammar, this mal-formed term is in the search space, and under the usual kind of scoring, it scores even better than the legal  
\[\mathsf{open\_del}(a,\{\mathsf{del}(\mathtt{a},\}^{n-1}\,\mathsf{mty}\{\})^ {n-1}\]  
Therefore, the grammar is required to ban malformed alignments from the search space.  
We conjecture that grammars could be dropped from the ICORE framework by defining a more sophisticated rewrite relation. However, we find it better to use two simple and well known mechanisms in combination, rather than a single but more sophisticated one.
## 4 Pre-Evaluation Based on the Spatio-Temporal Model
### Passenger Path Comparison  
The path with the smallest \(\delta\) is used as the optimal selection path, while a random method is used to determine multiple paths when \(\delta\) values are completely equal. Compared with the original travel route, one can analyze the changes in transfer number, distance, and travel time of each passenger after a route adjustment. The passenger section changes of an affected line can also be compared. The specific process is shown in Figure 8 below.  
The key for passenger path comparison is the real OD identification of passengers. This mainly includes transit OD analysis, basic data preparation, transfer identification, and original path matching. In engineering applications, one can not infer all passengers'  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Group**} & \multirow{2}{*}{
\begin{tabular}{c} **Time** \\ **Period** \\ \end{tabular} } & \multicolumn{8}{c}{**Weight Coefficient**} \\ \cline{3-13}  & & \(\alpha_{1}\) & \(\alpha_{2}\) & \(\alpha_{3}\) & \(\alpha_{4}\) & \(\alpha_{5}\) & \(\alpha_{6}\) & \(\alpha_{7}\) & \(\alpha_{8}\) & \(\alpha_{9}\) & \(\alpha_{10}\) & \(\alpha_{11}\) \\ \hline \multirow{2}{*}{working} & Peak & 91 & 80 & 70 & 3 & 5 & 13 & 1 & 33 & 40 & 64 & 44 \\ \cline{2-13}  & Off-peak & 97 & 90 & 31 & 1 & 2 & 8 & 1 & 6 & 10 & 19 & 34 \\ \hline \multirow{2}{*}{elderly} & Peak & 87 & 88 & 21 & 9 & 12 & 67 & 1 & 4 & 8 & 2 & 51 \\ \cline{2-13}  & Off-peak & 56 & 97 & 46 & 2 & 13 & 37 & 3 & 1 & 24 & 8 & 93 \\ \hline \multirow{2}{*}{student} & Peak & 81 & 80 & 30 & 3 & 5 & 25 & 1 & 8 & 77 & 8 & 25 \\ \cline{2-13}  & Off-peak & 97 & 76 & 7 & 19 & 12 & 70 & 5 & 14 & 30 & 50 & 35 \\ \hline \multirow{2}{*}{disabled} & Peak & 87 & 97 & 73 & 4 & 28 & 94 & 3 & 27 & 7 & 24 & 29 \\ \cline{2-13}  & Off-peak & 39 & 82 & 75 & 10 & 1 & 50 & 1 & 68 & 31 & 11 & 35 \\ \hline \multirow{2}{*}{other} & Peak & 99 & 85 & 91 & 7 & 4 & 59 & 42 & 49 & 50 & 59 & 85 \\ \cline{2-13}  & Off-peak & 77 & 97 & 24 & 11 & 3 & 40 & 4 & 6 & 40 & 14 & 22 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Weight coefficient.  
Figure 8: The process of passenger distribution and path comparison.  
boarding and alighting stops, so the gravity model method can be used to fix the missing OD. On this basis, one can identify the transfer trip so as to obtain the'real OD' of passengers for a complete trip. The pre-evaluation of line adjustments must be carried out with'real OD' to avoid repetition errors.
## 5 Discussion
### Evolution of Sedimentary Facies and Its Response to the Guangxi Movement  
The tectonic movement may be the primary driver of the evolution of sedimentary facies for the Wufeng Formation. In the Late Ordovician, the southern Sichuan Basin was situated in the mid to low latitudes of the Gondwana Continent of the SouthernHemisphere. Influenced by the Guangxi Movement, the Yangtze and the Cathaysian Plates converged, and the Yangtze region transitioned from a passive continental margin basin to a foreland basin. The Guangxi Movement can be divided into three stages: (1) thrusting and flexural subsidence, (2) bending under tectonics and sedimentary loading, viscoelastic deflection, and equilibrium subsidence, and (3) late-stage slow subsidence, stress relaxation, erosion, rebound, and uplift [12; 47]. The Wufeng period possibly corresponds to the first and second stage. The strong tectonic loading not only caused an angular unconformity between the Wufeng Formation and the underlying Baota Formation but also led to a shift from a clear-water carbonate sedimentary facies to a muddy-water fine-grained clastic sedimentary facies in the southern Sichuan Basin.  
During the early Wufeng period, the Guangxi Movement was very active, leading to a rapid uplift of surrounding plates and strong volcanic eruptions. This movement produced sufficient sediment flux and volcanic ash, forming widespread fine-grained turbidite deposits intercalated with volcanic ash layers in the basin. During the middle Wufeng period, the basin evolved into a stage of equilibrium subsidence because of the waning Guangxi Movement. The reduced terrestrial supply and waning volcanic eruptions favored the formation of hemipelagic deposits. During the late Wufeng period, the basin evolved into a stage of slow subsidence because of the increasingly quiescent Guangxi Movement. As a result of continuous supplies of terrestrial flux and volcanic ash, the basin was dominated by shallow shoal deposits. During this period, the shallow water biological community flourished, forming abundant bioclasts and limestone.  
Previous studies showed three origins of the volcanic ash: (1) the northward subduction of the Qinling ancient oceanic crust beneath the North China block on the north side of the Yangtze block, (2) extension and detachment along the northern margin of the Yangtze block, and (3) fragments of the Gondwana continent that subducted and collided from the southeast into the South China block. The volcanic ash produced by volcanic eruptions contributed to the enrichment of organic matter in shale, global sea level fall, reduced seawater temperatures, and upwelling of bottom water in the basin.
# Urban Road Collapse Risk Assessment Study Based on InSAR Spatiotemporal Data +
## 3 Assessment of Road Collapse Risk  
### Model Configuration  
This study leveraged the LSTM network to achieve temporal information mining of collapse accidents and subsidence data. LSTM has demonstrated exceptional performance in sequence task processing and plays a crucial role in the field of deep learning. In the domain of risk assessment, LSTM has also been widely applied. The key advantage of LSTM lies in its ability to dynamically adjust the transmission and processing of information flow, thereby adapting to different types of data and task requirements. LSTM effectively addresses the problems of vanishing and exploding gradients through its unique gating mechanism. These gating structures enable LSTM to retain important information when necessary and discard it when it is no longer relevant. This flexible information management mechanism makes LSTM highly suitable for processing sequence data with long-term dependencies or extended time intervals.  
The LSTM network is composed of a series of LSTM units, each of which contains four layers, three sigmoid layers and one tanh layer, the specific structure of which is shown in Figure 4. The information flow between units is controlled by three gates. The specific unit structure is shown in the figure: At time step t, the values of the forget gate \(F_{t}\), input gate \(I_{t}\), output gate \(O_{t}\), and candidate memory \(\widetilde{C}_{t}\) are calculated as follows:  
\[F_{t}=\sigma\Big{(}X_{t}W_{xf}+H_{t-1}W_{hf}+b_{f}\Big{)}\] \[I_{t}=\sigma\big{(}X_{t}W_{x1}+H_{t-1}W_{hl}+b_{t}\big{)}\] \[O_{t}=\sigma\big{(}X_{t}W_{x0}+H_{t-1}+b_{O}\big{)}\] \[\widetilde{C}_{t}=tanh\big{(}X_{t}W_{xc}+H_{t-1}W_{hc}+b_{C}\big{)}\]  
where \(W_{xf},W_{x1},W_{xC},W_{x0}\in R^{d\times h}\), \(W_{hf},W_{hl},W_{hc},W_{hO}\in R^{h\times h}\) are weight parameters, \(b_{f},b_{t},b_{C},b_{O}\in R^{1\times h}\) are bias parameters, and \(\sigma\) and \(tanh\) are activation functions. In LSTM, the core component is the memory cell \(C_{t}\), which persists throughout the chain-like system of LSTM. Within each LSTM unit, there exist two gates, the forget gate \(F_{t}\) and the input gate \(I_{t}\), which control the forgetting and updating of the input memory, respectively. The forget gate \(F_{t}\) determines the extent to which the content of the past memory cell \(C_{t-1}\) is retained, while the input gate \(I_{t}\) regulates the amount of the candidate memory \(\widetilde{C}_{t}\) value to be adopted.  
\[C_{t}=F_{t}\odot C_{t-1}+I_{t}\odot\widetilde{C}_{t}\]  
Figure 3: InSAR imagery in the study area.  
Finally, the output gate \(O_{t}\) is employed to update the hidden state information:  
\[H_{t}=O_{t}\odot\tanh(C_{t})\]  
When the output gate approaches 1, it can effectively transmit the information from the memory cell to the prediction component, yielding accurate prediction results.
## 3 Algorithm for Clustering Problem
### Metaheuristic GGA+ELM Algorithm Application for Spectral Analysis  
As already mentioned, the spectral data have 655 values in the frequency range (20 Hz-22.05 kHz). Obviously 655 characteristics is far too high for classification purposes. The target of the optimization algorithm is to reduce the number of features useful for classifying liquid samples. This means a wrapper feature selection [57] where the GGA maximizes the classification accuracy. The solution is composed of a collection of features varying in length and composition. The set of features extracted by the GGA among the 655 total will constitute the classifier applicable on the spectra of the liquid samples. The challenge is to not exceed more than 10 features and to achieve a classification accuracy of more than 95%.  
The training and testing data sets are disjoint sets randomly selected from the total of samples. The usual ratio is 80/20, with the training set having the largest number (80%) and the test data the remaining 20%. The population size usually used in the literature varies between 20 and 100 [58; 59]. The pair composition of individuals for the crossover operation is randomized. This method has also provided good results in previous research [60; 61]. The crossover operation generates a population increase of 50% (a single offspring from each pair), on which a 10% mutation is applied [47; 59]. This percentage is higher than usual in genetic algorithms, with the purpose of quickly exploring multiple areas of the solution space. The survival population for the next generation is composed of the winners of pairwise tournaments among the total population. The matches are chosen randomly. The fitness function value of the fighters determines the winner of each tournament.  
As already described, an individual is coded as a set of groups, where each group is a collection of features that can be a valid classifier of the input data. Not all groups of an individual are useful for classification, but only those with better accuracy. Note that considering a specific individual, each feature of the 655 is only present in a single group. In the GGA fitness function, the ELM algorithm is applied over each group of the individual to classify the testing set data from the knowledge of the training data. The group with the best classification accuracy is selected as a candidate classifier. The fitness of the individual takes the value of the classification accuracy of this highlighted group, which is the best accuracy obtained among all the groups of the individual.  
The stopping condition employed in optimization is the maximum number of generations. To ensure the high-quality solutions are found within a reasonable computation time, the maximum number of genera considered is \(\rm{G_{max}}=50\).
# TETES: Trust Based Efficient Task Execution Scheme for Fog Enabled Smart Cities
## 4 Proposed Scheme
### Trust Management
#### 4.2.4 Trust Probability  
Trust probability of all nodes present in a cluster are calculated by assigning weighted metric on trust values calculated for above mentioned three parameters. The highest weight is applied to the accumulated trust values collected from the directly and indirectly connected neighbouring nodes for the trust-finding node. Weights assigned to the trust values of the node itself \(T^{tf}\) is the lowest as it may have false information due to malicious node and it will be collective trust value of \(T_{nd}^{tf}\) and \(T_{id}^{tf}\).  
The trust probability is calculated through a sigmoid function (\(\sigma Z\)) for each trust-finding node in a cluster as:  
\[\sigma Z=\frac{1}{1+e^{-[\alpha(T_{nd}^{tf}+T_{id}^{tf})+T^{tf}(T_{nd}^{tf}+T _{id}^{tf})]}} \tag{9}\]  
The nodes with trust values less than a threshold limit are considered malicious nodes. A complete trust-finding mechanism is shown in Figure 3.  
Fog nodes after determining the trust value of each of its attached clustering nodes, compare it with a threshold value. In case, the trust value of a node is less than the predetermined threshold value then it is considered as a malicious node, otherwise, it will be considered as a trusted node. A complete procedure is mentioned in the Algorithm 1.  
```
1 Trust calculating policy
2 Input:
3 Number of nodes \(N\)
4 Threshold Value \(V_{th}\)
5 Self-trust value of nodes \(T_{1}^{tf},T_{2}^{tf},T_{3}^{tf},\ldots,T_{N}^{tf}\)
6 Direct trust of neighboring nodes for node \(x=T_{ndx}^{tf}\)
7 Indirect trust of nodes for node \(x=T_{idx}^{tf}\)
8 for\(i=1\) to \(N\)do
9 Calculate \(\sigma Z_{i}\) for all nodes
10 if\(\sigma Z_{i}\leq V_{th}\)then
11 Node is Malicious
12
13 end if
14 else
15 Node is legitimate
16
17 end if
18
19
20 end for
```  
**Algorithm 1**Trust Evaluating Algorithm for Fog Node
## 3 System Model
### Node Hidden Communication Model
#### 3.4.2 Anonymity Analysis  
In the anonymous forwarding network, the sender sends a pseudonym \(S\), and only one precursor node exists in the collection of the entry nodes. Although it is easy to infer the sender, the inference is only a pseudonym, and the sender's true identity is not leaked, and the sender's anonymity is guaranteed.  
In an anonymous forwarding network, the recipient's exit node is also its precursor node, and there are \(m\) in subsequent nodes, so the probability that the recipient is inferred by the exit node is only \(1\) m. The anonymity of the receiver is guaranteed.  
In the process of constructing anonymous forwarding network, for each relay node, its information range only includes the last hop and the next hop node, and the information of other nodes is not known. In this situation, even if the attacker can intercept routing information, it is also impossible to infer the relevant information of the receiver and sender. Anonymity of communication relationship is guaranteed.  
In the node hidden communication model, the anonymity of the forwarding network, sender, and receiver determines the anonymity of the model. The use of recipient pseudonym ensures that the identity of the node cannot be guessed, and the sender has the anonymity of \(d_{S}=1\); The number of \(n\) hop adjacent points determines the anonymity of the receiver, so the sender has the degree of \(d_{R}=\frac{m-1}{m}\) anonymity; The number of forwarding nodes and the length of forwarding path determine the anonymity of forwarding network. If an attacker wants to infer the communication relationship between the receiving and sending sides of the information, he must have a detailed grasp of all the \(n\) forwarding paths. Combining the probability of an attacker, one of the forwarding paths can be obtained, and the expectation of \(k\) routing and forwarding paths can be expressed by \(E=kp=\frac{k}{m^{n}}\). The forwarding network has \(d_{N}=1-\frac{k}{m^{n}}\) anonymity. Thus, it can be concluded that the hidden communication path of the whole node has \(d=d_{N}\times d_{R}\times d_{N}=1\times\left(1-\frac{k}{m^{n}}\right)\times \frac{m-1}{m}=\frac{(m^{n}-k)(m-1)}{m^{n}}\) anonymity.  
Figure 1: Group-based self-organizing anonymous forwarding network.
# Monitoring and Forecasting Green Tide in the Yellow Sea Using Satellite Imagery
## 3 Method
### Growth Curve Area Proportion Calculation Method  
As shown in Figure 6, the area and the percentage points corresponding to the start and dissipation times of the green tide, i.e., the statistic P related to the proportion of integral area at the corresponding time (d), was calculated according to the definite integral of the Gaussian fitting result of the first derivative, as shown in Equation (7):  
\[P(x_{\alpha})=\frac{s_{\alpha}}{s_{all}}=\frac{\int_{x_{1}}^{x_{\alpha}}y^{ \prime}dx}{\int_{x_{1}}^{x_{2}}y^{\prime}dx}\qquad\qquad 0\leq P\leq 1 \tag{7}\]  
where \(x_{\alpha}\) is the start and dissipation time of the green tide bulletin; \(x_{1},x_{2}\) are day 1 and day 365, respectively; and \(S_{\alpha},S_{all}\) are the integral areas corresponding to day \(x_{\alpha}\) and the whole year, respectively. When \(x_{\alpha}\) is the start time of the green tide, \(P(x_{\alpha})\) is the starting percentile, defined as \(P_{start}\); when \(x_{\alpha}\) is the end time of the green tide, \(P(x_{\alpha})\) is the dissipation percentile, defined as \(P_{dissipation}\).  
\begin{table}
\begin{tabular}{c c c} \hline
**Curve Model** & **Gompertz** & **Logistic** \\ \hline Formula & \(y=ae^{-bc^{-cx}}\) & \(y=\frac{a}{1+bc^{-cx}}\) \\ \hline Straightening of the curves & \(\ln(\ln\frac{y}{a})=-\ln b+cx\) & \(\ln(\frac{a-y}{y})=\ln b-cx\) \\  & \(y^{\prime}=\ln(\ln\frac{y}{a}),bb=\ln b\) & \(y^{\prime}=\ln\frac{a-y}{y},bb=\ln b\) \\  & \(y^{\prime}=bb+cx\) & \(y^{\prime}=bb-cx\) \\ \hline The constant term \(a\) & substitute \((x_{1},y_{1}),(x_{2},y_{2}),(x_{3},y_{3})\) into & substitute \((x_{1},y_{1}),(x_{2},y_{2}),(x_{3},y_{3})\) into \\  & \(\ln a=\frac{\ln y_{1}\ln y_{2}-\ln y_{3}^{2}}{\ln y_{1}+\ln y_{2}-\ln y_{3}}\) & \(a=\frac{2y_{1}y_{2}y_{3}-\ln y_{3}^{2}y_{1}-y_{3}^{2}y_{2}}{y_{1}y_{2}-y_{3} ^{2}y_{2}}\) \\ \hline \end{tabular}
\end{table}
Table 2: Approximate nonlinear method fitting formula.