A well transient electromagnetic weak signal reconstruction method based on bidirectional time sequence Kolmogorov-Arnold network

Abstract

This invention discloses a method for reconstructing weak transient electromagnetic signals in wells based on a bidirectional temporal Kolmogorov-Arnold network. This method addresses the problems of extremely low amplitude and susceptibility to noise in the late-stage transient electromagnetic signals in wells. First, the noisy response sequence undergoes temporal channel unification, baseline correction, drift suppression, outlier suppression, normalization, and optional amplitude logarithmic mapping. The late-stage window is then determined based on diffusion attenuation laws. Subsequently, a bidirectional temporal network consisting of a forward TKAN encoder and a backward TKAN encoder is constructed. Bidirectional hidden features are fused for each time channel, and a reconstructed sequence is output. A learnable spline function based on B-spline basis functions is introduced into the TKAN nonlinear mapping branch to enhance cross-scale nonlinear expression capabilities. Through this structural design, the method can stably recover weak transient electromagnetic signals in wells during the late-stage phase with ultra-low signal-to-noise ratios, improving the effective late-stage window and the availability of deep information.

Description

Technical Field

[0001] This invention relates to the field of geophysical electromagnetic exploration signal processing, specifically a method for reconstructing transient weak electromagnetic signals in wells based on a two-way time-series Kolmogorov-Arnold network. Background Technology

[0002] Downhole transient electromagnetic methods involve deploying excitation and reception systems in or near the well, recording the decay sequence of the electromagnetic response over time after the excitation is turned off. Compared to surface transient electromagnetic methods, downhole observations offer advantages such as closer proximity to the target body, less susceptibility to surface interference, and greater sensitivity to deep and lateral anomalies. These methods have been widely applied in fields such as deep ore body detection, fault zone and water-bearing structure identification, well connectivity assessment, and underground space and engineering geological exploration.

[0003] However, the transient electromagnetic response in wells exhibits a typical characteristic of multi-order-of-magnitude attenuation: the amplitude is high in the early stage but is significantly affected by coil turn-off transients, instrument bandwidth, and near-field effects; the middle stage reflects the main formation electrical information; and the amplitude is extremely low in the late stage, containing information from deeper or more distant fields, but is also most easily overwhelmed by noise. Common noise sources in the late stage include: instrument thermal noise and quantization noise, downhole electromechanical noise, power frequency and harmonic interference, cable coupling interference, pulse anomalies caused by poor contact, and environmental slow drift. Due to the requirements of field efficiency and stability, data acquisition is often completed under limited superposition and observation duration, making the insufficient signal-to-noise ratio problem more prominent in the late stage.

[0004] Existing reconstruction methods for transient electromagnetic signals mainly include wavelet thresholding, empirical mode decomposition, singular value decomposition and low-rank decomposition, smoothing filtering and spectral domain filtering, and curve fitting based on splines or polynomials. These methods typically rely on some separability assumption or parameter selection. When the signal amplitude is close to the noise floor in the late-night period, the signal and noise characteristics highly overlap, making it difficult to set thresholds and parameters stably. Excessive smoothing or filtering can easily change the late-night attenuation rate, causing distortions such as "lifting," "stepping," and "bounce" in the late-night tail, directly affecting subsequent inversion and interpretation, and even causing systematic biases in the judgment of the depth, size, and electrical properties of anomalies.

[0005] In recent years, data-driven sequence models have been used for electromagnetic signal reconstruction, enabling them to learn, to some extent, the mapping from noisy signals to a reference. However, when using only ordinary mean square error as the reconstruction objective, the large amplitude in the early stages leads to the loss being dominated by the early stages, while the error in the later stages accounts for a very small proportion of the overall loss, resulting in the problem of "good fitting in the early stages but insufficient recovery in the later stages." Furthermore, fixed-form nonlinear mappings have limited ability to express cross-scale attenuation and nonlinear responses under different operating conditions, easily leading to attenuation rate shifts, tail-stage morphological distortion, or noise-driven rebound in the later stages.

[0006] Furthermore, the multilayer perceptron (MLP) commonly used in existing data-driven models typically employs a stacked approach of "linear transformation + fixed activation function" to characterize nonlinear mappings. The nonlinearity of this type of structure is mainly provided by preset activation functions (such as ReLU, SiLU, tanh, etc.). Since the activation function shape is fixed, parameter learning is primarily concentrated in the weight matrices of each layer, leading to strong coupling of the influence of different input dimensions through matrix multiplication. This makes it difficult to explicitly separate the contribution of a particular input dimension to the output. Therefore, in scenarios such as transient weak electromagnetic signals in wells, characterized by "order-of-magnitude attenuation + ultra-low signal-to-noise ratio," MLP-type structures often require deeper / wider networks to improve fitting ability. They are also prone to non-physical artifacts such as local rebound, oscillations, or attenuation rate shifts in the later stages, and it is difficult to form interpretable function-level analyses of the model output. Summary of the Invention

[0007] The purpose of this invention is to provide a method for reconstructing transient weak electromagnetic signals in wells based on a bidirectional time-series Kolmogorov-Arnold network, comprising the following steps:

[0008] Step 1) Obtain the response sequence from the transient electromagnetic measurement in the well. t is the time channel index, and T is the total number of time channels;

[0009] Step 2) Analyze the response sequence Preprocessing is performed to obtain the preprocessed transient electromagnetic response sequence. ;

[0010] Step 3) Determine the late-time window based on the diffusion and decay law of the transient electromagnetic response in the well. ;

[0011] Step 4) Construct a bidirectional temporal Kolmogorov-Arnold network (Bidirectional Temporal KAN), including a forward TKAN encoder and a backward TKAN encoder;

[0012] Step 5) Introduce learnable spline functions into the nonlinear mapping branches of the forward and backward TKAN encoders. And the bidirectional temporal KAN network was trained to obtain a transient electromagnetic weak signal reconstruction model.

[0013] Among them, learnable spline functions As shown below:

[0014] (1)

[0015] in, , For activation function, Let i be the i-th order B-spline basis function. These are the corresponding learnable control coefficients. and These are the weight parameters for the basis function part and the spline part, respectively;

[0016] Step 6) Transient electromagnetic response sequence The transient electromagnetic weak signal is input into the transient electromagnetic weak signal reconstruction model to obtain the transient electromagnetic weak signal reconstruction sequence used to characterize the diffusion decay response. .

[0017] Furthermore, the response sequence is any one of the induced voltage of the receiving coil, the magnetic field, or the time derivative of the magnetic field.

[0018] Furthermore, regarding the response sequence The preprocessing operations include at least one of the following: time channel normalization, baseline offset correction, slow drift suppression, abnormal impulse point suppression, amplitude normalization, and amplitude logarithmic mapping;

[0019] Among them, time channel unification refers to mapping the original sampling time channel sets of different measuring points or different working conditions to a unified time grid {τ} using monotonic interpolation or piecewise interpolation methods. t ;

[0020] Abnormal impulse point suppression refers to: identifying a set of abnormal points using sliding median filtering, Hampel filtering, or mutation detection based on first-order difference thresholds, and replacing the abnormal points using neighborhood interpolation or local robust regression;

[0021] The magnitude logarithmic mapping operation is as follows:

[0022] (2)

[0023] in, It is a positive constant; The response at time t; The magnitude logarithm of the response.

[0024] Furthermore, the late-night window is determined using the end-proportion method, noise floor method, slope method, or combined rule method;

[0025] The end-point ratio method determines the late-time window by the proportion of end-point channels or the number of end-point channels;

[0026] The noise floor method determines the late-time window by using a threshold value relative to the noise floor of the response amplitude;

[0027] The slope method determines the late window by using the criterion that the slope of the logarithmic field enters the smooth interval;

[0028] The joint rule method determines the late window by the proportion or number of end channels, the threshold of the response amplitude relative to the noise floor, and the criterion that the logarithmic domain slope enters the smooth interval.

[0029] Furthermore, both the forward and backward TKAN encoders include at least an input gate, a forget gate, and an output gate, and learnable spline functions are introduced into the candidate state branches or gated mapping branches. .

[0030] Furthermore, the B-spline basis functions are second-order or higher B-splines;

[0031] Spline nodes are set at equal intervals or adaptively based on the input distribution, and spline coefficients are subject to sparse regularization or smoothing regularization.

[0032] The learnable spline function is configured in the form of a univariate function connecting the edges on the connection relationship between the input vector and the hidden features, so that the nonlinear contribution of each input dimension to the output can be independently represented by the corresponding univariate spline function.

[0033] Furthermore, transient electromagnetic weak signal reconstruction sequence ; Reconstructed output ; ; , Forward hidden features output by the forward TKAN encoder and the backward TKAN encoder Backward hiding features; , For weights and biases.

[0034] Furthermore, the bidirectional temporal KAN network is trained using training samples;

[0035] Training samples are obtained in the following way:

[0036] A reference response sequence is generated based on the transient electromagnetic theory model or numerical forward model in the well. Noise is superimposed on the reference response sequence to form a noisy sequence. A training sample set covering different working conditions is formed by changing formation resistivity, layer thickness, well diameter, casing parameters, excitation intensity, coil geometric parameters, transmitter-receiver spacing and receiver position.

[0037] Furthermore, a segmented signal-to-noise ratio (SNR) control strategy is adopted for noise superposition, so that the SNR in the later period is lower than that in the earlier period, in order to simulate the more difficult noise distribution in the later part of the actual well measurement.

[0038] The noise includes Gaussian noise, 1 / f noise, power frequency interference, impulse noise, or a combination thereof.

[0039] A well transient electromagnetic weak signal reconstruction device using the method described above includes a data acquisition module, a time channel unification module, a preprocessing module, a late-time window determination module, a bidirectional TKAN reconstruction module, and an output module.

[0040] The data acquisition module is used to acquire the response sequence obtained from transient electromagnetic measurements in the well. :

[0041] The time channel unification module is used to map response sequences with different sampling settings to a unified time grid;

[0042] The preprocessing module is used to perform baseline correction, drift suppression, outlier suppression, normalization, and magnitude logarithmic mapping on the response sequence;

[0043] The late-time window determination module is used to determine the late-time window based on the attenuation law and preset rules. ;

[0044] The bidirectional TKAN reconstruction module uses the bidirectional temporal Kolmogorov-Arnold network BidirectionalTemporal KAN to perform forward TKAN encoding, backward TKAN encoding, bidirectional feature fusion and linear mapping output to obtain the reconstructed sequence;

[0045] In the bidirectional temporal KAN network (TKAN), a learnable spline function is introduced into the nonlinear mapping branch of the TKAN encoder. ;

[0046] The output module is used to output the reconstructed sequence and late-time quality metrics, including LateSNR Improvement, SlopeRelativeError, and LateDistortionOnset Ratio.

[0047] The technical effects of this invention are undeniable, and this invention has at least the following beneficial effects:

[0048] (1) Bidirectional temporal coding enables the high signal-to-noise ratio information in the early time to provide contextual priors for the late time recovery, while the tail information inversely constrains the local morphology, thereby improving the stability of the late time recovery;

[0049] (2) Learnable spline nonlinear mapping enhances the model’s ability to express cross-scale diffusion decay and changes under different working conditions, reducing the risk of decay rate shift and morphological distortion;

[0050] (3) Improve the recoverability of the late-time effective window, thereby improving the availability and reliability of deep information interpretation. Attached Figure Description

[0051] Figure 1 : Schematic diagram of transient electromagnetic response decay over time in well, and schematic diagram of high signal-to-noise ratio region, signal recoverable region and noise-dominant region;

[0052] Figure 2 The end-to-end training process of Bi-TKAN;

[0053] Figure 3 The conceptual diagram of KAN illustrates that KAN uses learnable spline functions instead of fixed activation functions to parameterize nonlinear mappings;

[0054] Figure 4 The proposed TKAN unit schematic diagram shows that it embeds a spline-based KAN module to generate candidate unit state updates.

[0055] Figure 5 The effect of reconstructing BTEM using different methods on noise power improvement;

[0056] Figure 6 On representative samples of the well transient electromagnetic dataset, the time decay curves of BTEM are reconstructed using different methods;

[0057] Figure 7 : A schematic diagram of the process for reconstructing transient electromagnetic weak signals in wells provided by this invention. Detailed Implementation

[0058] The present invention will be further described below with reference to embodiments, but it should not be construed that the scope of the present invention is limited to the following embodiments. Various substitutions and modifications made based on ordinary technical knowledge and common practices in the art without departing from the above-described technical concept of the present invention should be included within the scope of protection of the present invention.

[0059] Example 1:

[0060] See Figures 1 to 7 A method for reconstructing transient weak electromagnetic signals in wells based on a bidirectional temporal Kolmogorov-Arnold network includes the following steps:

[0061] Step 1) Obtain the response sequence from the transient electromagnetic measurement in the well. t is the time channel index, and T is the total number of time channels;

[0062] Step 2) Analyze the response sequence Preprocessing is performed to obtain the preprocessed transient electromagnetic response sequence. ;

[0063] Step 3) Determine the late-time window based on the diffusion and decay law of the transient electromagnetic response in the well. ;

[0064] Step 4) Construct a bidirectional temporal Kolmogorov-Arnold network (Bidirectional Temporal KAN), including a forward TKAN encoder and a backward TKAN encoder;

[0065] Step 5) Introduce learnable spline functions into the nonlinear mapping branches of the forward and backward TKAN encoders. And the bidirectional temporal KAN network was trained to obtain a transient electromagnetic weak signal reconstruction model.

[0066] Among them, learnable spline functions As shown below:

[0067] (1)

[0068] in, , For activation function, Let i be the i-th order B-spline basis function. These are the corresponding learnable control coefficients. and These are the weight parameters for the basis function part and the spline part, respectively;

[0069] Step 6) Transient electromagnetic response sequence The transient electromagnetic weak signal is input into the transient electromagnetic weak signal reconstruction model to obtain the transient electromagnetic weak signal reconstruction sequence used to characterize the diffusion decay response. .

[0070] Example 2:

[0071] A method for reconstructing transient electromagnetic weak signals in a well based on a bidirectional time-series Kolmogorov-Arnold network, with the same technical content as in Embodiment 1, further wherein the response sequence is any one of the induced voltage of the receiving coil, the magnetic field, or the time derivative of the magnetic field.

[0072] Example 3:

[0073] A method for reconstructing transient electromagnetic weak signals in wells based on a bidirectional temporal Kolmogorov-Arnold network, with the same technical content as any one of embodiments 1-2, further comprising the following steps for reconstructing the response sequence. The preprocessing operations include at least one of the following: time channel normalization, baseline offset correction, slow drift suppression, abnormal impulse point suppression, amplitude normalization, and amplitude logarithmic mapping;

[0074] Among them, time channel unification refers to mapping the original sampling time channel sets of different measuring points or different working conditions to a unified time grid {τ} using monotonic interpolation or piecewise interpolation methods. t ;

[0075] Abnormal impulse point suppression refers to: identifying a set of abnormal points using sliding median filtering, Hampel filtering, or mutation detection based on first-order difference thresholds, and replacing the abnormal points using neighborhood interpolation or local robust regression;

[0076] The magnitude logarithmic mapping operation is as follows:

[0077] (1)

[0078] in, It is a positive constant; The response at time t; The magnitude logarithm of the response.

[0079] Example 4:

[0080] A method for reconstructing transient electromagnetic weak signals in a well based on a bidirectional temporal Kolmogorov-Arnold network, with the same technical content as any one of embodiments 1-3, further wherein the late time window is determined by the end-proportional method, the noise floor method, the slope method, or the joint rule method;

[0081] The end-point ratio method determines the late-time window by the proportion of end-point channels or the number of end-point channels;

[0082] The noise floor method determines the late-time window by using a threshold value relative to the noise floor of the response amplitude;

[0083] The slope method determines the late window by using the criterion that the slope of the logarithmic field enters the smooth interval;

[0084] The joint rule method determines the late window by the proportion or number of end channels, the threshold of the response amplitude relative to the noise floor, and the criterion that the logarithmic domain slope enters the smooth interval.

[0085] Example 5:

[0086] A method for reconstructing transient electromagnetic weak signals in wells based on bidirectional temporal Kolmogorov-Arnold networks, with technical content identical to any one of embodiments 1-4, further comprising: the forward TKAN encoder and the backward TKAN encoder each including at least an input gate, a forget gate, and an output gate; and the candidate state branch or gated mapping branch introducing a learnable spline function. .

[0087] Example 6:

[0088] A method for reconstructing transient electromagnetic weak signals in a well based on a bidirectional temporal Kolmogorov-Arnold network, with the same technical content as any one of embodiments 1-5, further wherein the B-spline basis function is a second-order or higher B-spline;

[0089] Spline nodes are set at equal intervals or adaptively based on the input distribution, and spline coefficients are subject to sparse regularization or smoothing regularization.

[0090] The learnable spline function is configured in the form of a univariate function connecting the edges on the connection relationship between the input vector and the hidden features, so that the nonlinear contribution of each input dimension to the output can be independently represented by the corresponding univariate spline function.

[0091] Example 7:

[0092] A method for reconstructing transient electromagnetic weak signals in wells based on a bidirectional temporal Kolmogorov-Arnold network, with the same technical content as any one of embodiments 1-6, further comprising a transient electromagnetic weak signal reconstruction sequence. ; Reconstructed output ; ; , Forward hidden features output by the forward TKAN encoder and the backward TKAN encoder Backward hiding features; , For weights and biases.

[0093] Example 8:

[0094] A method for reconstructing transient weak electromagnetic signals in a well based on a bidirectional temporal Kolmogorov-Arnold network, with the same technical content as any one of embodiments 1-7. Furthermore, the bidirectional temporal KAN network is trained using training samples.

[0095] Training samples are obtained in the following way:

[0096] A reference response sequence is generated based on the transient electromagnetic theory model or numerical forward model in the well. Noise is superimposed on the reference response sequence to form a noisy sequence. A training sample set covering different working conditions is formed by changing formation resistivity, layer thickness, well diameter, casing parameters, excitation intensity, coil geometric parameters, transmitter-receiver spacing and receiver position.

[0097] Example 9:

[0098] A method for reconstructing transient electromagnetic weak signals in wells based on a bidirectional temporal Kolmogorov-Arnold network, with the same technical content as any one of embodiments 1-8, further wherein the noise superposition adopts a segmented signal-to-noise ratio control strategy to make the signal-to-noise ratio of the later period lower than that of the earlier period, so as to simulate the more difficult noise distribution in the later period of actual well measurements.

[0099] The noise includes Gaussian noise, 1 / f noise, power frequency interference, impulse noise, or a combination thereof.

[0100] Example 10:

[0101] A method for reconstructing transient electromagnetic weak signals in wells based on a bidirectional temporal Kolmogorov-Arnold network, comprising the following steps:

[0102] (1) Bidirectional temporal feature extraction: The response sequence is encoded using a bidirectional temporal TKAN structure. The forward branch extracts the decay evolution features from early to late time, and the backward branch extracts the global backtracking features from late to early time. The bidirectional hidden states are fused so that the high signal-to-noise ratio information in the early time and the global trend jointly constrain the late time recovery process, thereby reducing the drift and abnormal fluctuations in the late time tail segment and improving the reconstruction consistency.

[0103] (2) Learnable spline nonlinear mapping: In the nonlinear mapping stage of TKAN, learnable nonlinearity based on spline function family (linear combination of B spline basis functions) is introduced to replace the fixed shape activation function, so as to enhance the expressive ability of cross-scale decay pattern and improve the nonlinear fitting accuracy; at the same time, the local support and controllable smoothness characteristics of spline function are used to suppress high-frequency pseudo fluctuations, so that the reconstruction results of weak signals in the late time are smoother and more stable.

[0104] (3) End-to-end reconstruction and error accumulation suppression of channel-level sequence to sequence: The transient electromagnetic response in the well is organized into a sequence input according to the time channel, and the network outputs the same dimension reconstructed sequence in the same channel unit to realize end-to-end sequence to sequence mapping; through the synergistic effect of bidirectional encoding and nonlinear mapping, the error propagation and accumulation caused by unidirectional recursion are reduced, the recovery consistency of the tail channel is improved, and the local error in the early time is continuously amplified in the subsequent channel, resulting in distortion in the late time.

[0105] This allows for stable recovery of the electromagnetic attenuation response during the late-stage period of ultra-low signal-to-noise ratio.

[0106] Example 11:

[0107] A method for reconstructing transient electromagnetic weak signals in wells based on a bidirectional temporal Kolmogorov-Arnold network, comprising the following steps:

[0108] (a) Data Acquisition and Organization (S1)

[0109] The transient electromagnetic response in the wellbore measured in a single measurement is organized into a sequence according to the time channel. When multiple receivers or multiple component observations are present, they can be organized into multi-channel sequences. ,in This refers to the number of components or the number of receiving channels. To maintain consistency, the following description uses a single channel as an example; multi-channel scenarios can be achieved through parallel input and channel fusion.

[0110] (ii) Time channel unification and alignment (S2)

[0111] Different measurement points may exhibit differences in the number of sampling channels, channel intervals, start times, or shutdown delays. To ensure consistent model input dimensions and reduce mapping bias caused by channel differences, this invention maps the original time channel set to a unified time grid {τ}. t Unification can be achieved through interpolation on either the time axis or the logarithmic time axis: when emphasizing details in the later stages, interpolation on the logarithmic time axis can be used to give the later stages a more balanced representation density; when it is necessary to maintain local monotonicity and morphological stability, monotonic interpolation or piecewise interpolation can be used to avoid overshoot caused by interpolation. At the same time, time offsets can be estimated and aligned based on feature points in the earlier stages to reduce the difference between trigger jitter and shutdown delay.

[0112] (III) Pretreatment (S3)

[0113] 1) Baseline offset correction: Estimate the baseline offset during the post-shutdown delay window or late time tail segment. Perform full-channel calibration To improve stability, the baseline can be estimated using the median or truncated mean.

[0114] 2) Slow drift suppression: A trend term can be fitted to the slow rise or fall of the tail segment. And remove them; the trend term can be obtained from a low-order polynomial or spline curve.

[0115] 3) Abnormal pulse point suppression: Peak points are identified by using the moving median and robust scaling estimation, or abrupt changes are detected by using a first-order difference threshold, followed by neighborhood interpolation or robust regression replacement.

[0116] 4) Normalization and optional logarithmic mapping of amplitude: Normalization is performed according to the maximum value, root mean square, or quantile scale; when the amplitude decays significantly across orders of magnitude, logarithmic mapping can be used.

[0117] (1)

[0118] This is to enhance numerical stability and make the power-law decay more linear in the logarithmic domain.

[0119] (iv) Determining the evening window (S4)

[0120] like Figure 1 As shown, the response can be divided into early, middle, and late periods. Late window. The determination of the noise floor can be achieved using the end-proportion method, the noise floor method, the slope method, or the combined rule method. The end-proportion method uses the channel with the end proportion ρ as the late-time window; the noise floor method estimates the noise standard deviation. ,when When a certain number of channels are used, the channel is taken as the starting point for the late time; the slope method calculates the local slope in the logarithmic domain and finds the position that enters the smooth interval as the starting point; the joint rule method integrates multiple criteria and sets protective boundaries to avoid misclassification.

[0121] (V) Bidirectional Temporal TKAN Reconstruction Model (S5)

[0122] like Figure 2 As shown, construct Bi-TKAN:

[0123] Forward TKAN encoder edge Recursively obtained Capture the decay evolution from early to late;

[0124] Backward TKAN encoder edge Recursively obtained The tail segment and the global context are used to inversely constrain the local shape;

[0125] Feature fusion: Or weighted fusion;

[0126] The output layer is linearly mapped to obtain:

[0127] (2)

[0128] like Figure 3 , Figure 4 As shown, the TKAN unit adopts a gated recursive structure, introducing a learnable spline function in the candidate state generation branch. :

[0129] (3)

[0130] Where B-spline basis functions The order of the learnable function can be second, third, or higher, and the nodes can be set equidistantly or adaptively. Sparse or smooth constraints can be applied to the coefficients to control complexity and maintain smoothness. Through this family of learnable functions, the model can more flexibly express the nonlinear decay patterns under different diffusion scales and operating conditions.

[0131] (vi) Model Training (S6)

[0132] The Bi-TKAN network was trained using a sample dataset containing different geological models and noise levels. During training, the loss function was minimized using the backpropagation algorithm. Update network parameters including spline control coefficients. Weight , And network layer weights.

[0133] (vii) Signal reconstruction and application (S7)

[0134] The transient electromagnetic data from the actual well is input into the trained network model, which outputs a reconstructed sequence. Based on this reconstructed sequence, apparent resistivity calculation or inversion imaging can be performed to obtain more accurate information on the formation's electrical distribution.

[0135] Example 13:

[0136] A well transient electromagnetic weak signal reconstruction device using the method described above includes a data acquisition module, a time channel unification module, a preprocessing module, a late-time window determination module, a bidirectional TKAN reconstruction module, and an output module.

[0137] The data acquisition module is used to acquire the response sequence obtained from transient electromagnetic measurements in the well. :

[0138] The time channel unification module is used to map response sequences with different sampling settings to a unified time grid;

[0139] The preprocessing module is used to perform baseline correction, drift suppression, outlier suppression, normalization, and magnitude logarithmic mapping on the response sequence;

[0140] The late-time window determination module is used to determine the late-time window based on the attenuation law and preset rules. ;

[0141] The bidirectional TKAN reconstruction module uses the bidirectional temporal Kolmogorov-Arnold network BidirectionalTemporal KAN to perform forward TKAN encoding, backward TKAN encoding, bidirectional feature fusion and linear mapping output to obtain the reconstructed sequence;

[0142] In the bidirectional temporal KAN network (TKAN), a learnable spline function is introduced into the nonlinear mapping branch of the TKAN encoder. ;

[0143] The output module is used to output the reconstructed sequence and late-time quality metrics, including LateSNR Improvement, SlopeRelativeError, and LateDistortionOnset Ratio.

[0144] Example 14:

[0145] A method for reconstructing transient electromagnetic weak signals in wells based on a bidirectional temporal Kolmogorov-Arnold network, comprising the following steps:

[0146] Basis for organizing and zoning measurement data

[0147] The post-shutdown decay response obtained from transient electromagnetic measurements in the well can be discretized into a sequence along the time channel. Typically, the early time period is more significantly affected by instrument transients and near-field conditions, and the curve may exhibit short-term "plateaus" or rapid changes; the middle time period reflects the main formation electrical properties; and the late time period has low amplitude and is sensitive to deep diffusion. To ensure the stability of the late time window selection, the noise level of the sequence estimate can be lowered first (e.g., by taking a robust scale estimate in the tail section), and then the late time starting channel t can be determined by combining the criterion that the logarithmic domain slope enters the smoothing interval. s Under conditions of high resistivity or strong bushing influence, the starting point of late time can be relatively shifted later; under conditions of low resistivity or strong noise, the starting point of late time can be relatively shifted earlier.

[0148] Time channel unification and alignment

[0149] The sampling time channels at different measurement points may differ. The original time channel set... Mapping to a unified grid Logarithmic time axis interpolation can be used to obtain a more balanced channel density in the later time period. To reduce the difference between trigger jitter and shutdown delay, feature points (such as peak positions, inflection points, or threshold crossing points) can be selected in the early time period to estimate the time offset and align them, and then interpolated to a unified grid, thereby improving the comparability between different sequences.

[0150] Baseline drift and outlier suppression

[0151] Baseline offset can cause an overall rise or fall in the late tail segment. Baseline offset can be estimated within a delay window. And remove them; if slow drift exists, the trend g(t) can be fitted and removed. Outlier suppression can use the moving median and robust scaling to identify peaks, and replace them with neighborhood interpolation. This step can avoid peaks being mistakenly identified as "valid features" and causing non-physical spikes in the output.

[0152] Normalization and logarithmic field mapping

[0153] To improve the stability of decay data across orders of magnitude, normalization is first performed using the maximum value, root mean square, or quantile scale, followed by optional logarithmic magnitude mapping. Logarithmic mapping makes power-law decay closer to a linear trend in the logarithmic domain, which is beneficial for learning the decay pattern and avoids training insensitivity caused by excessively small late-time values. The output is then subjected to the corresponding inverse mapping to restore the physical dimensions.

[0154] Multi-rule determination of late window

[0155] Evening window The end-ratio method, noise floor method, slope method, or combined rule method can be used to determine it.

[0156] (1) End-point ratio method: ;

[0157] (2) Noise floor method: when And continue Take the starting point when there are multiple channels;

[0158] (3) Slope method: calculation in the logarithmic field The starting point is taken when the slope enters the smooth interval and the amplitude is close to the bottom of the noise.

[0159] (4) Joint rule method: First, take the candidate window and then fine-tune the starting point based on the noise level to avoid misclassification.

[0160] Bi-TKAN structure and bidirectional fusion

[0161] Bi-TKAN consists of two recursive paths: the forward path propagates early-time trend information to later-time states; the backward path propagates the tail segment and global final-state constraints back to the local interval, reducing bounce caused by local noise. The fusion method can be either concatenation or weighted fusion. Concatenation retains richer information, while weighted fusion maintains compactness when the number of parameters is limited. The output layer uses a linear mapping to maintain the controllability and stability of the channel-level output and facilitates the interpretation of hidden features as decaying trend representations.

[0162] TKAN units and learnable spline mappings

[0163] In the gated recursive structure, the candidate state generation branch is a learnable spline function. B-spline basis functions can balance expressive power and smoothness constraints by adjusting node density and order. To avoid overfitting, sparsity or smoothness constraints can be imposed on spline coefficients; penalties can also be imposed on the differences between adjacent coefficients to ensure function smoothness and reduce the tendency for high-frequency oscillations in the output.

[0164] Sample Construction and Noise Modeling

[0165] The reference response can be obtained from theoretical models or numerical forward modeling, or from the fusion of highly superimposed measurements. Noise superposition can include Gaussian noise, 1 / f noise, power frequency interference, and impulse noise. To closely approximate downhole conditions, segmented signal-to-noise ratio (SNR) control can be employed to lower the SNR in later periods; and the sample coverage can be expanded by changing formation resistivity, layer thickness, wellbore diameter, casing conditions, excitation intensity, and geometric parameters to improve adaptability to different operating conditions.

[0166] Typical parameter range

[0167] Number of time channels T: 50-300; late-night window ratio : 0.15-0.40; Number of spline basis functions K: 8-64; Spline order: 2-4.

[0168] Applicable Scenarios

[0169] It is applicable to deep ore body exploration, response quality improvement under complex well conditions, late-time recovery under low-overlay rapid measurement, and inter-well connectivity evaluation; it can be used as a data quality improvement step before imaging, inversion and interpretation to improve the availability and reliability of deep information interpretation.

[0170] Example 15:

[0171] Verification of a well transient electromagnetic weak signal reconstruction method based on a bidirectional temporal Kolmogorov-Arnold network, the steps of which include:

[0172] Dataset training results

[0173] (1) Data and training setup: The reference response sequence is generated using numerical forward modeling or theoretical modeling. The noisy sequence is obtained by superimposing noise according to the time-segmented signal-to-noise ratio control strategy. This ensures that the signal-to-noise ratio is significantly lower in the late period than in the early period, thus closely resembling the noise distribution in actual well measurements where "late-night measurements are more difficult." The samples are divided into training, validation, and test sets; the input during training is the pre-processed... The output is the reconstructed sequence. .

[0174] The proposed method is compared with traditional Empirical Mode Decomposition (EMD), Wavelet Threshold-Exponential Adaptive Window Width Fitting Filter (WEF), and Long Short-Term Memory (LSTM) networks:

[0175] (1) EMD: A traditional adaptive time-frequency decomposition and reconstruction method, which decomposes the input noisy signal into several intrinsic mode function components and residual terms, and suppresses, filters or weights the high-frequency noise related components based on the frequency band characteristics of the components to obtain the reconstructed signal, which is used as a non-learning class contrast baseline method.

[0176] (2) WEF: A traditional reconstruction method based on wavelet transform, which decomposes the input signal into multiple scales and performs thresholding and / or weighting on the scale coefficients obtained by decomposition to suppress noise, and then reconstructs the signal to obtain the reconstructed signal, which is used as a non-learning class contrast baseline method.

[0177] (3) LSTM: A standard Long Short-Term Memory (LSTM) network is used as the baseline for comparison in the context of deep temporal learning. This network utilizes special gating mechanisms (input gate, forget gate, and output gate) to address the vanishing gradient problem in standard recurrent neural networks, enabling it to capture long-range temporal dependencies in noisy signals. As a unidirectional modeling method, it reconstructs the decay sequence based solely on current and past observation information to evaluate the performance of the baseline deep temporal model in signal recovery tasks.

[0178] (4) Bi-TKAN: Based on TKAN, bidirectional temporal modeling is introduced to encode the noisy sequence forward and backward and fuse the output to reconstruct the response; the nonlinear mapping is implemented by a learnable spline function. Compared with the unidirectional model, Bi-TKAN can utilize the bidirectional context to improve the characterization of the decay process and the stability of late-time weak signal reconstruction.

[0179] like Figure 5 As shown in Figures 5(a) to (d), the distribution of noise power variations caused by different methods on the test set is compared. Overall, the scatter points for the four methods are mostly located below the contour lines, indicating that the output noise power is generally lower than the input noise power, and the noise suppression effect is relatively significant; however, there are significant differences in the stability of each method in the low-noise region. Specifically, Figure 5 (a) The point cloud distribution of EMD is relatively diffuse and has a large dispersion. In the low noise range, some samples can still be seen approaching or crossing the baseline, reflecting that its output fluctuation is relatively obvious under weak noise / weak signal conditions. Figure 5 (b) WEF can push some samples below the baseline, but the overall distribution is still relatively wide. Outliers in the low noise region are relatively prominent, suggesting that there is a certain risk of over-processing or noise re-amplification. Figure 5 (c) The LSTM further concentrates the samples below the baseline, indicating that depth temporal modeling can enhance noise suppression, but point clouds still exhibit some dispersion in the low-noise range. In contrast, Figure 5(d) shows that the point cloud of Bi-TKAN is the most compact, especially in the low-noise amplification region where the deviation from the reference value is the smallest. This indicates that it is less sensitive to noise re-amplification and overcorrection in the later weak signal scene and can achieve a more robust late recovery effect.

[0180] Figure 6 The reconstruction results of the time-domain decay curves of the same test sample under different methods are presented and compared with the reference signal (true value) to evaluate the ability of each method to preserve the main decay trend and the low amplitude tail segment in the late time. Figure 6 (a)–(d) correspond to EMD, WEF, LSTM and Bi-TKAN, respectively.

[0181] From the curve morphology, the distortion of EMD not only appears in the late stage but also shows significant deviation in the early to mid-stage: its reconstructed curve is generally lower than the true value throughout the early to mid-stage, manifested as a systematic suppression of amplitude and a "weakening" of the main attenuation trend, indicating that the decomposition-reconstruction process can also introduce morphological distortion in the high signal-to-noise ratio range. In contrast, WEF closely matches the true value in the early to mid-stage and can better preserve the main attenuation trend, but it shows a downward crossover and excessive attenuation in the late stage, suggesting that threshold / weighted multi-scale reconstruction is more likely to produce amplitude underestimation in extremely weak signal segments. LSTM fits well in the early to mid-stage, but some deviation and local fluctuations are still visible in the late stage, with slight instability in the tail stage (including signs of local "bounce" or increased deviation), reflecting that the fixed-form nonlinearity and gated recursion still have limited suppression of noise disturbances in the low signal-to-noise ratio tail stage. The Bi-TKAN curve shows the best fit to the true value throughout the entire time period, especially in the late stage where it maintains a smoother and more consistent decay pattern with significantly reduced tail-end fluctuations. This indicates that the learnable spline nonlinearity combined with bidirectional contextual information helps stabilize the recovery of weak signals in the late stage and reduces the distortion risk of low signal-to-noise ratio channels.

[0182] (2) Evaluation indicators: mean absolute error, peak signal-to-noise ratio, mean signal-to-noise ratio, signal-to-noise ratio improvement, late-time signal-to-noise ratio improvement, slope relative error, and late-time distortion onset ratio.

[0183] Late channel signal-to-noise ratio improvement is defined as:

[0184] (1)

[0185] (2)

[0186] exist After performing a logarithmic magnitude transformation, linear fitting yields the slopes k̂ and k of the reconstructed and reference responses, respectively. The relative slope error is defined as:

[0187] (3)

[0188] Late distortion start ratio via The relative error sequence in the equation is defined as:

[0189] (4)

[0190] make To meet > The earliest index of the first Z consecutive channels:

[0191] (5)

[0192] here, Z is the distortion threshold, and Z is the required continuous length. It is a small stability constant. yes The start time, yes The end time. If no index meets the condition, the ratio is set to 1, indicating that no systematic distortion was observed within the fitting window.

[0193] Table 1 presents the quantitative comparison results on the test set. The overall trend shows that as the model's temporal expressive power increases, various performance indicators gradually improve: among traditional methods, EMD performs poorly in both error and signal-to-noise ratio (SNR) related indicators; while WEF can reduce noise to some extent, the overall improvement is limited. After introducing bidirectional temporal modeling, Bi-LSTM achieves a significant leap in MAE, PSNR, and SNR improvement, indicating that deep temporal networks can effectively improve reconstruction quality. Building on this, Bi-TKAN further achieves superior MAE and PSNR, reaching the highest level in SNR improvement, demonstrating the beneficial effect of learnable spline nonlinearity on attenuation process modeling and reconstruction accuracy.

[0194] More importantly, late-time related metrics better characterize the core requirements of deep exploration scenarios. Since the amplitude of the early-time channel is much larger than that of the late-time channel, the single global error is often more dominated by the early-time segment, making it difficult to fully reflect the recovery capability of the low signal-to-noise ratio (SNR) late-time segment. As shown in Table 1, Bi-TKAN achieves the best global metrics while also obtaining the highest late-time SNR improvement, indicating its more significant enhancement of weak signal channels in the late time. Furthermore, its slope relative error remains at a low level, helping to stabilize key morphological information such as attenuation rate and reducing the risk of rebound, oscillation, or overcorrection during late-time reconstruction. In summary, Bi-TKAN performs best in both global fidelity and late-time reliability, demonstrating its comprehensive advantages for BTEM late-time weak signal recovery.

[0195] Table 1

[0196]

Claims

1. A method for reconstructing transient electromagnetic weak signals in a well based on a bidirectional temporal Kolmogorov-Arnold network, characterized in that, Includes the following steps: Step 1) Obtain the response sequence from the transient electromagnetic measurement in the well. t is the time channel index, and T is the total number of time channels; Step 2) Analyze the response sequence Preprocessing is performed to obtain the preprocessed transient electromagnetic response sequence. ; Step 3) Determine the late-time window based on the diffusion and decay law of the transient electromagnetic response in the well. To calculate evaluation indicators; Step 4) Construct a bidirectional temporal Kolmogorov-Arnold network (Bidirectional Temporal KAN), including a forward TKAN encoder and a backward TKAN encoder; Step 5) Introduce learnable spline functions into the nonlinear mapping branches of the forward and backward TKAN encoders. And the bidirectional temporal KAN network was trained to obtain a transient electromagnetic weak signal reconstruction model. Among them, learnable spline functions As shown below: (1) in, , For activation function, Let i be the i-th order B-spline basis function. These are the corresponding learnable control coefficients. and These are the weight parameters for the basis function part and the spline part, respectively; Step 6) Transient electromagnetic response sequence The transient electromagnetic weak signal is input into the transient electromagnetic weak signal reconstruction model to obtain the transient electromagnetic weak signal reconstruction sequence used to characterize the diffusion decay response. .

2. The well transient electromagnetic weak signal reconstruction method based on a bidirectional temporal Kolmogorov-Arnold network according to claim 1, characterized in that, The response sequence is any one of the following: the induced voltage of the receiving coil, the magnetic field, or the time derivative of the magnetic field.

3. The well transient electromagnetic weak signal reconstruction method based on a bidirectional temporal Kolmogorov-Arnold network according to claim 1, characterized in that, For response sequence The preprocessing operations include at least one of the following: time channel normalization, baseline offset correction, slow drift suppression, abnormal impulse point suppression, amplitude normalization, and amplitude logarithmic mapping; Among them, time channel unification refers to mapping the original sampling time channel sets of different measuring points or different working conditions to a unified time grid {τ} using monotonic interpolation or piecewise interpolation methods. t ; Abnormal impulse point suppression refers to: identifying a set of abnormal points using sliding median filtering, Hampel filtering, or mutation detection based on first-order difference thresholds, and replacing the abnormal points using neighborhood interpolation or local robust regression; The magnitude logarithmic mapping operation is as follows: )(2) in, It is a positive constant; The response at time t; The magnitude logarithm of the response.

4. The well transient electromagnetic weak signal reconstruction method based on a bidirectional temporal Kolmogorov-Arnold network according to claim 1, characterized in that, The late-night window is determined using the end-proportion method, noise floor method, slope method, or combined rule method. The end-point ratio method determines the late-time window by the proportion of end-point channels or the number of end-point channels; The noise floor method determines the late-time window by using a threshold value relative to the noise floor of the response amplitude; The slope method determines the late window by using the criterion that the slope of the logarithmic field enters the smooth interval; The joint rule method determines the late window by the proportion or number of end channels, the threshold of the response amplitude relative to the noise floor, and the criterion that the logarithmic domain slope enters the smooth interval.

5. The well transient electromagnetic weak signal reconstruction method based on a bidirectional temporal Kolmogorov-Arnold network according to claim 1, characterized in that, Both forward and backward TKAN encoders include at least an input gate, a forget gate, and an output gate, and learnable spline functions are introduced into the candidate state branches or gated mapping branches. .

6. The well transient electromagnetic weak signal reconstruction method based on a bidirectional temporal Kolmogorov-Arnold network according to claim 5, characterized in that, The B-spline basis functions are second-order or higher B-splines; Spline nodes are set at equal intervals or adaptively based on the input distribution, and spline coefficients are subject to sparse regularization or smoothing regularization. The learnable spline function is configured in the form of a univariate function connecting the edges on the connection relationship between the input vector and the hidden features, so that the nonlinear contribution of each input dimension to the output can be independently represented by the corresponding univariate spline function.

7. The well transient electromagnetic weak signal reconstruction method based on a bidirectional temporal Kolmogorov-Arnold network according to claim 1, characterized in that, Transient electromagnetic weak signal reconstruction sequence ; Reconstructed output ; ; , Forward hidden features output by the forward TKAN encoder and the backward TKAN encoder Backward hiding features; , For weights and biases.

8. The well transient electromagnetic weak signal reconstruction method based on a bidirectional temporal Kolmogorov-Arnold network according to claim 1, characterized in that, The bidirectional temporal Kolmogorov-Arnold network, Bidirectional TemporalKAN, is trained using training samples. Training samples are obtained in the following way: A reference response sequence is generated based on the transient electromagnetic theory model or numerical forward model in the well. Noise is superimposed on the reference response sequence to form a noisy sequence. A training sample set covering different working conditions is formed by changing formation resistivity, layer thickness, well diameter, casing parameters, excitation intensity, coil geometric parameters, transmitter-receiver spacing and receiver position.

9. The well transient electromagnetic weak signal reconstruction method based on a bidirectional temporal Kolmogorov-Arnold network according to claim 1, characterized in that, The noise superposition adopts a segmented signal-to-noise ratio control strategy to make the signal-to-noise ratio in the later period lower than that in the earlier period, so as to simulate the more difficult noise distribution in the middle and late stages of actual well measurements; The noise includes Gaussian noise, 1 / f noise, power frequency interference, impulse noise, or a combination thereof.

10. A wellbore transient electromagnetic weak signal reconstruction device using the method of any one of claims 1-9, characterized in that, It includes a data acquisition module, a time channel unification module, a preprocessing module, a late window determination module, a bidirectional TKAN reconstruction module, and an output module; The data acquisition module is used to acquire the response sequence obtained from transient electromagnetic measurements in the well. : The time channel unification module is used to map response sequences with different sampling settings to a unified time grid; The preprocessing module is used to perform baseline correction, drift suppression, outlier suppression, normalization, and magnitude logarithmic mapping on the response sequence; The late-time window determination module is used to determine the late-time window based on the attenuation law and preset rules. ; The bidirectional TKAN reconstruction module uses the bidirectional temporal Kolmogorov-Arnold network Bidirectional TemporalKAN to perform forward TKAN encoding, backward TKAN encoding, bidirectional feature fusion and linear mapping output to obtain the reconstructed sequence; In the bidirectional temporal KAN network (TKAN), a learnable spline function is introduced into the nonlinear mapping branch of the TKAN encoder. ; The output module is used to output the reconstructed sequence and late-time quality metrics, including LateSNR Improvement, SlopeRelativeError, and LateDistortionOnset Ratio.

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