Dynamic probability prediction method for physical precursors of prestressed steel wire fracture caused by high-temperature droplet loss of water delivery pipeline

Abstract

The application discloses a dynamic probability prediction method for physical precursors of high-temperature molten drop damage fracture of prestressed steel wires of water delivery pipelines, and comprises the following steps: acquiring an infrared temperature field time sequence of a target area and detecting a molten drop impact event; extracting impact cooling characteristic evolution and calculating thermal physical response degradation characteristics; extracting a precursor anomaly index based on physical baseline detrending in an adjacent impact silent period; constructing a single-event damage contribution, combining the degradation characteristics and the precursor anomaly index to dynamically and targetingly update damage fracture probability distribution parameters; calculating a spatial damage fracture probability distribution based on damage accumulation and updated parameters, and outputting a closed-loop control instruction in combination with the precursor index. The application decouples deep delamination and surface false interference, extracts micro precursors, and realizes adaptive early warning in the whole life cycle.

Description

Technical Field

[0001] This invention relates to high-temperature welding thermal damage monitoring and long-distance water conveyance structure safety assessment technology, and in particular to a dynamic probability prediction method for the physical precursors of high-temperature molten droplet damage fracture of prestressed steel wire in water conveyance pipelines. Background Technology

[0002] Among various water transmission pipe materials, prestressed reinforced concrete (PCCP) pipes have become the preferred type for large-diameter pressure water transmission pipelines due to their high pressure resistance, good impermeability, and relatively economical manufacturing cost. A typical PCCP pipe consists of a steel cylinder, a concrete core, a mortar protective layer, and high-strength prestressed steel wires wound around the outer wall of the core. However, the safety of the prestressed steel wires needs attention. If the prestressed steel wires experience large-area corrosion, stress corrosion cracking, or sudden brittle fracture, the pipe will lose its structural integrity and burst instantly under the pressure of internal water, leading to serious secondary disasters such as water outages, road collapses, and farmland flooding.

[0003] During the on-site installation or subsequent repair of PCCP pipelines, steel pipe welding or thermoforming cutting operations may be involved. At this time, splashing high-temperature molten droplets easily adhere to or impact the surface of the prestressed steel wire, causing a rapid local temperature rise. This extreme and discrete droplet thermal shock can form a brittle martensitic transformation layer or micro-quenching cracks on the steel wire surface, becoming a fatal cause of brittle fracture and damage fracture of the prestressed steel wire. Therefore, achieving dynamic prediction of droplet thermal shock damage on the surface of PCCP steel wire is of engineering significance for preventing sudden pipe bursts and ensuring the safety of long-distance water transport lifelines. However, the on-site construction environment of PCCP pipelines is complex and the operating conditions are uncontrollable, making it difficult to directly conduct systematic research on the droplet impact damage mechanism. In existing technologies, a high-temperature droplet impact test platform is typically used to simulate the droplet thermal shock environment experienced by the prestressed steel wire: by controlling a droplet generating device to drip high-temperature melt onto the prestressed steel wire sample, infrared thermal imager is used to acquire surface temperature field evolution data, thereby inferring the material's thermal shock resistance and failure characteristics. Currently, for thermal damage assessment under this type of simulation experimental platform, the average temperature rise of a fixed zone is usually extracted as the only damage characterization quantity. An isotropic geometric smoothing kernel is used to spatially accumulate historical thermodynamic damage. The accumulated damage value is then substituted into a statistical lifetime curve with constant scale and shape parameters to calculate the failure probability.

[0004] However, existing static assessment frameworks struggle to reflect the physical processes of material degradation, leading to parameter distortion and delayed warnings in prediction models. Specifically, firstly, a single apparent temperature amplitude cannot decouple temperature measurement deviations caused by surface oxidation, easily triggering false warnings; secondly, the isotropic assumption and static, constant lifetime parameters mask the directional crack propagation driven by local thermal gradients, failing to adapt to the material's evolution towards unstable fracture; thirdly, conventional methods of extracting macroscopic main heat flow characteristics drown out the weak heat flow fluctuations caused by microscopic crack friction during the impact quiescent period, resulting in the system losing its ability to intercept early microscopic precursors. Therefore, a dynamic probabilistic inference scheme integrating deep physical mechanisms is needed to achieve adaptive early warning throughout the entire damage evolution lifecycle. Summary of the Invention

[0005] The purpose of this invention is to provide a dynamic probability prediction method for the physical precursors of high-temperature molten droplet damage and fracture of prestressed steel wires in water transmission pipelines, in order to solve the aforementioned problems existing in the prior art.

[0006] Technical solution: A dynamic probability prediction method for physical precursors of high-temperature molten droplet damage fracture in prestressed steel wires of water transmission pipelines, comprising:

[0007] Obtain a continuous time series of infrared temperature field in the target area;

[0008] By detecting droplet impact events based on infrared temperature field time series, a set of impact events in the target region is obtained.

[0009] Impact cooling features are extracted based on infrared temperature field time series and impact event set, and the thermophysical response of each impact event in the impact event set is tracked based on infrared temperature field time series to calculate the thermophysical response degradation features of the target region.

[0010] By utilizing the infrared temperature field time series and the set of impact events, precursory anomaly indicators of the target area are extracted during the quiet period of adjacent impacts.

[0011] Based on the characteristics of shock cooling, a single-event damage contribution is constructed. Combined with the characteristics of thermophysical response degradation and precursor anomaly indicators, the pre-configured damage fracture probability distribution parameters are dynamically updated in a targeted manner. Based on the cumulative results of the single-event damage contribution and the updated damage fracture probability distribution parameters, the spatial damage fracture probability distribution of the target area is calculated.

[0012] Closed-loop control commands are output based on the spatial damage fracture probability distribution and precursor anomaly indicators.

[0013] Beneficial effects: This invention decouples deep delamination from surface spurious interference, extracts microscopic precursors, and achieves adaptive early warning throughout the entire life cycle. Attached Figure Description

[0014] Figure 1 This is a diagram illustrating the basic framework and macroscopic scheduling process of dynamic Weibull cumulative damage probability prediction in the embodiments of this application.

[0015] Figure 2 This is a flowchart illustrating the steps for calculating the thermophysical response degradation characteristics of the target region in an embodiment of this application.

[0016] Figure 3 This is a flowchart illustrating the steps of obtaining the local apparent thermal diffusivity of the target region through least squares estimation in an embodiment of this application.

[0017] Figure 4 This is a flowchart illustrating the steps of extracting precursory anomaly indicators of the target area during the quiet period of an adjacent impact, as described in this application embodiment.

[0018] Figure 5 This is a flowchart illustrating the steps for calculating the spatial damage and fracture probability distribution of the target area in an embodiment of this application. Detailed Implementation

[0019] Example 1, such as Figure 1 As shown, this paper describes the basic framework for predicting the cumulative damage probability of dynamic Weibull and the macroscopic scheduling process of the system.

[0020] Step 101: Obtain the continuous infrared temperature field time series of the target area.

[0021] In this embodiment, the infrared temperature field time series is specifically represented as a three-dimensional tensor T(x,y,t), where x and y represent the two-dimensional spatial pixel coordinates within the target area, and t represents the time dimension. The system can acquire raw image data by real-time observation of high-temperature molten droplets using a high-frame-rate infrared thermal imager. To meet the requirements of subsequent transient heat conduction analysis and weak signal extraction, the sampling frame rate for acquiring the infrared temperature field time series can be set to greater than or equal to 25 Hz, while its spatial resolution can reach the 1 mm level. After acquiring the raw observation data, the system can use a checkerboard-based camera calibration method to correct radial distortion of the image and utilize Gaussian low-pass filtering technology to smooth the high-frequency shot noise introduced by the sensor, providing a basic data source with accurate physical properties for subsequent calculations.

[0022] Accordingly, before acquiring the continuous infrared temperature field time series of the target area, an offline parameter calibration stage is also included. This stage is used to predetermine the baseline values ​​required for calculating the damage contribution of a single event and updating the damage fracture probability distribution parameters. The offline parameter calibration stage includes:

[0023] Historical damage fracture experimental data is acquired and pre-stored. A temperature factor exponent is fixed based on the physical law of residual compressive strength degradation in materials and used as a priori physical parameter. Under the condition of fixed priori physical parameters, a likelihood function is constructed using the historical damage fracture experimental data. The initial baseline Weibull scale parameter and the initial baseline Weibull shape parameter are solved by joint maximum likelihood estimation. A damage fracture prediction model is constructed based on the initial baseline Weibull scale parameter and the initial baseline Weibull shape parameter. The actual damage fracture frequency in the pre-acquired long-term validation experimental data is used as the calibration target. Within the set parameter space, the parameter combination that minimizes the root mean square error between the predicted damage fracture frequency output by the damage fracture prediction model and the actual damage fracture frequency is scanned and extracted, serving as the online update hyperparameter for updating the damage fracture probability distribution parameter. The priori physical parameter, the initial baseline Weibull scale parameter, the initial baseline Weibull shape parameter, and the online update hyperparameter are pre-configured as components of the pre-configured damage fracture probability distribution parameter for real-time prediction.

[0024] Step 102: Detect droplet impact events based on infrared temperature field time series to obtain the set of impact events in the target area.

[0025] Specifically, since the high-temperature molten droplet falling onto the relatively low-temperature surface of an object will cause a rapid local temperature rise, the system uses a temporal adjacent frame difference method to calculate the temperature change rate of each spatial pixel. When the positive temperature difference of a pixel exceeds a preset dynamic threshold, the system initially marks it as a suspected impact response point. The dynamic threshold can be configured between 20 and 50°C. To eliminate random noise interference caused by equipment vibration or dust obstruction, the system further uses an eight-neighbor connected component analysis algorithm to check the spatial morphological connectivity of suspected impact response points. Only when the area of ​​the extracted connected component meets the condition of 3 to 5 pixels is the system confirmed as a valid molten droplet impact event, and its spatial centroid coordinates and temperature rise start time are recorded in the impact event set.

[0026] Furthermore, as an auxiliary mechanism for the system's macro-situational awareness, after acquiring the set of impact events, the system can also statistically analyze the impact frequency at each spatial grid point within the target area based on a fixed-length sliding time window, generating a local impact frequency field represented in the form of a two-dimensional raster matrix. In addition, the system can perform real-world monitoring tasks in parallel, using edge detection and contour analysis algorithms to identify locations within the target area where aspect ratios and compactness abruptly change, and classify them into lists of point-like, linear, or sheet-like damage and fracture events, providing a factual benchmark reference for the prediction model.

[0027] Step 103: Extract shock cooling features based on infrared temperature field time series and shock event set, and track the thermophysical response of each shock event in the shock event set based on infrared temperature field time series to calculate the thermophysical response degradation features of the target area.

[0028] In this embodiment, the specific process of extracting the impact cooling characteristics includes determining the peak temperature triggered by the target impact event within its heat-affected zone in the infrared temperature field time series for the target impact event in the impact event set. Further, the system extracts the temperature cooling curve from the moment corresponding to the peak temperature from the infrared temperature field time series, and fits the temperature cooling curve using a preset exponential decay model to obtain a cooling time constant characterizing the local heat dissipation rate.

[0029] Specifically, the exponential decay model is expressed by the following formula: ΔT(t) = ΔT peak *e^(-t / τ cool );

[0030] In the formula, ΔT peak For the corresponding peak temperature, the constant e is the base of the natural logarithm, ^ is the superscript sign, t is the time variable with the peak time as zero, and τ cool This represents the cooling time constant, which characterizes the local heat dissipation rate. Under the constraints of a limited observation time window and spatial resolution, the single exponential decay model can stably characterize the phenomenological cooling scale with minimal parameters, avoiding the risk of overfitting in the solution process of complex heat transfer error functions. Furthermore, the system uses the peak temperature and cooling time constant as characteristics of shock cooling.

[0031] Accordingly, for the same spatial grid point, the system collects the cooling time constant τ extracted from each impact event. cool A historical time series of cooling time constants was constructed based on the chronological order of the impact events. The system uses the impact event number as the independent variable and the corresponding cooling time constant as the dependent variable, and employs weighted linear regression to obtain the slope of the series, i.e., the cooling time constant drift rate γ. τ Similarly, the system responds to the peak temperature ΔT of each impact event. peak Construct historical time series data and obtain the peak temperature degradation rate γ using the same weighted linear regression method. peak .

[0032] Based on this, the system continuously tracks the evolution trajectory of impact cooling characteristics under various impact events at the same spatial location as the service process progresses, thereby calculating the thermophysical response degradation characteristics that characterize the accumulation of microscopic damage and the decline in heat transfer capacity within the material. Specifically, the system extracts the cooling time constant from the same spatial grid point in each impact event, constructing a historical evolution time series of the cooling time constant with the impact event number or timestamp as the horizontal axis and the cooling time constant value as the vertical axis. The system uses a weighted linear regression fitting method to analyze the trend of this time series, where data points closer to the current time are assigned higher weights. The slope of the fitted line is the cooling time constant drift rate, denoted as γ. τ A positive drift rate indicates that the heat transfer path inside the material is blocked and the heat dissipation capacity continues to decline. Similarly, the system constructs a historical evolution time series of the peak temperature based on the peak temperatures extracted from each impact event, and extracts its slope using the same weighted linear regression fitting method, which is the peak temperature degradation rate, denoted as γ. peak A negative degradation rate indicates a declining trend in the material's surface heat flux absorption capacity. The cooling time constant drift rate and the peak temperature degradation rate together constitute a biaxial degradation characteristic that characterizes the direction of material degradation evolution, and can be considered as a form of thermophysical response degradation characteristic.

[0033] Step 104: Using the infrared temperature field time series and the set of impact events, extract the precursory anomaly indicators of the target area during the quiet period of adjacent impacts.

[0034] In actual working conditions, the phase transformation process of moisture inside prestressed steel wires or the frictional opening and closing of microcracks are often accompanied by weak endothermic or exothermic effects. These minute heat flow fluctuations are easily masked by the temperature jumps caused by the main impact of the molten droplets. Therefore, this embodiment defines the quiet period as the time interval between two adjacent determined impact events, unaffected by the main heat flow tail wave. During the quiet period, the system extracts high-frequency temperature residual fluctuation information purely caused by the evolution mechanism of microscopic damage within the material by constructing and subtracting the macroscopic residual thermal conductivity attenuation trend baseline, and calculates it as a precursor anomaly index using statistical methods. This index can serve as an early warning signal for microstructural instability, compensating for the physical lag in macroscopic temperature amplitude monitoring.

[0035] Step 105: Construct single-event damage contribution based on impact cooling characteristics, dynamically update the pre-configured damage fracture probability distribution parameters by combining thermophysical response degradation characteristics and precursor anomaly indicators, and calculate the spatial damage fracture probability distribution of the target area based on the cumulative results of single-event damage contribution and the updated damage fracture probability distribution parameters.

[0036] As an alternative implementation, in order to convert the physical quantity of thermal shock into a damage metric in the sense of reliability engineering, the system constructs the single-event damage contribution by calculating three dimensionless product factors.

[0037] Specifically, the system calculates a dimensionless temperature factor characterizing the degree of local overheating based on the peak temperature of the target impact event and a preset damage initiation temperature threshold. The formula for calculating this factor is:

[0038] f T =max(0,(ΔT peak -ΔT th ) / ΔT th ) m ;

[0039] In the formula, ΔT peak ΔT represents the peak temperature caused by the target impact event. th The damage initiation temperature threshold is set as a preset value, and m is the temperature nonlinearity exponent. Specifically, the damage initiation temperature threshold can be determined through thermodynamic experiments or by consulting material handbooks, based on material properties such as the thermal expansion coefficient and fracture toughness of the target prestressed steel wire. In this embodiment, the damage initiation temperature threshold is set to 50℃. When the peak temperature does not exceed the damage initiation temperature threshold, this factor is zero, indicating that the impact does not constitute effective damage. The exponent m reflects the nonlinear physical law of the accelerated degradation of the material's residual compressive strength with increasing temperature amplitude.

[0040] Accordingly, the system calculates a dimensionless time factor characterizing the thermal energy deposition retention effect based on the cooling time constant of the target impact event and a pre-configured reference time constant. The formula for calculating this factor is:

[0041] f τ =τ cool / τ ref ;

[0042] In the formula, τ cool τ is the cooling time constant obtained by fitting the current impact. ref This is a pre-configured reference time constant. The reference time constant can be taken as the median value of the overall cooling time constant obtained during the initial system calibration phase. A larger cooling time constant indicates that heat remains on the material surface for a longer period, resulting in greater thermal fatigue damage.

[0043] Furthermore, the system uses the cooling time constant and the preset heat-affected zone expansion rate to determine the spatial decay radius of the Gaussian kernel function, and accordingly calculates the spatial decay factor characterizing the degree of damage spatial diffusion. The formula for calculating the spatial decay radius is R. dam =v th ×τ cool In the formula, v thThis is the preset heat-affected zone expansion rate. The formula for calculating the spatial attenuation factor is f. s (r)=exp(-r 2 / (2×R dam 2 )), where r is the Euclidean distance from the target space grid point to the impact centroid.

[0044] Accordingly, multiplying the dimensionless temperature factor, the dimensionless time factor, and the spatial attenuation factor yields the single-event damage contribution characterizing the intensity of a single impact failure, expressed as dk(x,y)=f T ×f τ ×f s (r).

[0045] Furthermore, during the continuous acquisition of single-event damage contributions, the system spatially accumulates these contributions over time. Simultaneously, utilizing the acquired thermophysical response degradation characteristics and precursor anomaly indicators, the system applies targeted dynamic corrections to the pre-calibrated baseline Weibull scale parameters and baseline Weibull shape parameters based on historical data, obtaining damage fracture probability distribution parameters updated in real-time with the material's service status. Finally, the system substitutes the accumulated damage results into a Weibull cumulative distribution model constructed using the updated parameters, deducing the probability value of macroscopic damage fracture at each spatial grid point within the target area, forming a two-dimensional spatial damage fracture probability distribution that fully covers the observation field.

[0046] Step 106: Output closed-loop control commands based on the spatial damage fracture probability distribution and precursor anomaly indicators.

[0047] In this embodiment, the system extracts the probability values ​​of each spatial grid point within the target area in the spatial damage fracture probability distribution. To avoid malfunctions of production equipment caused by errors in a single statistical model or accidental surface thermodynamic fluctuations, this embodiment designs a dual-channel false alarm judgment logic. Grid points whose probability values ​​exceed the pre-configured risk probability threshold and whose precursor anomaly indicators at the same spatial location simultaneously meet the preset anomaly judgment conditions are marked as high-risk areas, forming a dual-channel early warning condition of physical and probabilistic factors. When the dual-channel early warning condition is triggered, the impact point offset coordinates are determined based on the spatial location information of the high-risk area, and the frequency adjustment factor is determined based on the degree of exceedance of the probability value. A closed-loop control command containing the impact point offset coordinates and the frequency adjustment factor is generated to control the droplet release device to avoid the high-risk area and reduce the impact frequency.

[0048] Specifically, the system determines whether a grid point is dangerous not only by requiring its probability value to exceed a pre-configured risk probability threshold, but also by requiring that precursory anomalies at the same spatial location simultaneously meet preset anomaly judgment conditions. The risk probability threshold can be configured according to the safety margin requirements of the actual working conditions, and can be determined by those skilled in the art by combining historical damage and fracture statistics and process safety standards. Only when both the theoretical calculation channel and the physical observation channel are satisfied simultaneously will the grid point be marked as a high-risk area, thus triggering the dual-channel early warning condition of physical and probabilistic factors. Once this early warning condition is triggered, the system immediately generates a closed-loop control command, specifically manifested as a control structure including the landing point offset coordinates and frequency adjustment factor. This command is sent to the lower-level machine of the equipment to control the robotic arm and throttle valve of the droplet release device, forcing subsequent dripping actions to avoid the high-risk area and reducing the overall impact frequency, thus suppressing the malignant accumulation of local thermal stress.

[0049] Example 2 describes a method for tracking apparent thermal diffusivity based on the inversion of the partial differential equation of thermal conduction (PDE).

[0050] Preferably, this embodiment describes abandoning the superficial statistics based on macroscopic temperature amplitude, focusing on the underlying physical laws of transient heat conduction partial differential equations, and using numerical inversion methods to deeply explore the intrinsic heat transfer hindrance mechanism inside the material, so as to achieve high-dimensional and high-precision extraction of the degradation characteristics of thermophysical response.

[0051] In one possible implementation, the thermophysical response degradation characteristics of the target region are calculated, such as... Figure 2 As shown, it includes the following steps:

[0052] Step 201: During the cooling phase of the target impact event, based on the range corresponding to the preset proportion of the temperature increment decreasing to the peak temperature in the infrared temperature field time series, the corresponding time segment is extracted as the analysis window.

[0053] Accordingly, after acquiring the droplet impact response in the target region, not all temperature drop periods are suitable for extracting intrinsic thermal conductivity parameters. In the initial stage immediately following the impact, the thermodynamic process is often interspersed with latent heat of phase change released during droplet solidification and surface convection heat transfer due to temperature rise, at which point the partial differential equations of thermal conductivity are not applicable. Furthermore, in the final, extremely late stage of the cooling process, minute temperature changes are easily masked by the background white noise of the infrared detector. Therefore, the system needs to extract an ideal time period during the cooling phase of the target impact event, where pure three-dimensional thermal conduction within the solid dominates.

[0054] Specifically, the system uses the highest temperature increment corresponding to the peak temperature as a benchmark and extracts data where the temperature drops to a predetermined proportion. This predetermined proportion range can be set to 20% to 80% of the highest temperature increment. The set of continuous sampling frames within this dimensionless proportion range constitutes the analysis window. Through the time-series extraction mechanism, the system physically eliminates the effects of initial nonlinear phase transition interference and final signal-to-noise ratio degradation.

[0055] Step 202: Within the analysis window, calculate the central difference time derivative and the five-point difference space Laplace operator based on the infrared temperature field time series.

[0056] Accordingly, after establishing the analysis window, the system performs numerical discretization calculations based on the transient heat conduction partial differential equations. According to the fundamental laws of thermodynamics, the spatiotemporal evolution of the temperature field within an isotropic medium without an internal heat source follows physical constraints, namely, the spatial Laplace operator multiplied by the apparent thermal diffusivity equals the time derivative of temperature. To transform the continuous equations into a computer-processable discrete form, the system extracts multidimensional grid data based on the infrared temperature field time series. For a target grid point with spatial coordinates (x, y) and time t, the system calculates the central difference time derivative using data from adjacent frames. The specific calculation formula is dT. dt =(T(x,y,t+1)-T(x,y,t-1)) / (2*dt), where dt is the sampling time interval.

[0057] Furthermore, the system extracts the temperature values ​​of the target grid point and its four orthogonally adjacent grid points (up, down, left, and right) at the same time, and calculates the five-point difference space Laplacian operator. The formula for this operator is Lap... T =(T(x+1,y,t)+T(x-1,y,t)+T(x,y+1,t)+T(x,y-1,t)-4*T(x,y,t)) / dx 2 In the formula, dx represents the resolution size of the spatial grid. In some optional implementations, to further suppress the sensitivity of the time derivative calculation to high-frequency noise, the system can pre-smooth the time series data within the analysis window using a polynomial smoothing filter algorithm before performing central difference.

[0058] Step 203: Construct an overdetermined system of equations using the central difference time derivative and the five-point difference space Laplace operator, and obtain the local apparent thermal diffusivity of the target region by least squares estimation.

[0059] As an optional implementation, the system continuously performs the aforementioned difference calculations at multiple time sampling points covered by the analysis window, collecting a series of time derivative values ​​and corresponding spatial Laplace operator values. The system aggregates the data over multiple time points to construct an overdetermined system of equations in the form of matrix-vector multiplication. Specifically, the system arranges all central difference time derivative values ​​as column vectors b, and the corresponding five-point difference spatial Laplace operator values ​​as a single-column matrix A. The parameter to be solved is the local apparent thermal diffusivity α. app The equation is in the form A*α app =b. Since the number of time sampling points is much greater than the number of unknowns, this system of equations constitutes a standard overdetermined problem. The system uses the least squares estimation method to find its optimal analytical solution, and its analytical expression is α. app =(A T *A) (-1) *A T b, where A T Let represent the transpose of matrix A. Through a multi-point joint solution framework, the system can suppress random errors caused by single-frame computation to the greatest extent possible, extracting the intrinsic material parameters implicit in the spatiotemporal evolution.

[0060] In one possible implementation, during the process of obtaining the local apparent thermal diffusivity of the target region through least squares estimation, such as Figure 3 As shown, it also includes the following steps:

[0061] Step 203a: Calculate the Laplace signal-to-noise ratio based on the signal variance of the five-point difference space Laplace operator and the pre-stored background noise variance, and calculate the fitting determination coefficient based on the fitting residuals of the least squares estimation.

[0062] Furthermore, in high-temperature industrial environments, intense high-temperature molten droplet impacts and airflow disturbances, or background radiation reflections, may cause local pixels to deviate from standard thermal conductivity laws. To ensure the engineering robustness of the inversion results, the system introduces a data quality assessment mechanism.

[0063] Specifically, the system calculates the signal variance of the five-point difference spatial Laplacian operator sequence extracted within the analysis window, and divides it by the variance of the background floor noise pre-collected under cold conditions to obtain the Laplacian signal-to-noise ratio. This parameter is used to measure whether the current pixel has a sufficiently strong spatial temperature gradient driving force. After solving the overdetermined equations, the system calculates the sum of squared residuals between the predicted value of the time derivative reconstructed from the inverted local apparent thermal diffusivity and the actual extracted central difference time derivative. Based on this, it calculates the least squares estimation fitting coefficient of determination, which characterizes the degree of conformity between the current local thermodynamic decay process and the assumptions of the internal heat conduction model.

[0064] Step 203b: Remove unreliable pixels whose Laplacian signal-to-noise ratio is lower than a preset signal-to-noise ratio threshold or whose fitted coefficient of determination is lower than a preset coefficient of determination threshold.

[0065] Optionally, the system applies the aforementioned two evaluation metrics to perform a filtering logic for each pixel involved in the calculation. The system pre-sets threshold standards in the parameter configuration table. For example, the preset signal-to-noise ratio (SNR) threshold can be set to 4.0, and the preset coefficient of determination (COD) threshold can be set to 0.85. When the calculated Laplace SNR of a target pixel is lower than the aforementioned threshold, it indicates that its spatial temperature distribution is relatively flat, and solving overdetermined equations is prone to matrix ill-conditioning and numerical explosion. The system determines that this point is excessively affected by convective radiation interference. Similarly, if the fitted COD is lower than the set COD threshold, it indicates that the heat dissipation process at this point deviates from the diffusion physics model. For pixels that touch any of the threshold red lines, the system marks them as unreliable pixels and removes them from the subsequent spatial feature extraction domain, filtering out abnormal data.

[0066] Step 203c: Extract the spatial median of the local apparent thermal diffusivity of the remaining reliable pixels in the thermally affected area of ​​the target impact event, and use it as the local apparent thermal diffusivity of the grid points where the target area is located.

[0067] Accordingly, after cleaning up the poor-quality data, for the effective thermally affected area covered by the same target impact event, the system obtains a set of local apparent thermal diffusivity composed of multiple spatially adjacent remaining reliable pixels. To further eliminate statistical bias caused by individual extreme points, the system abandons the arithmetic mean method, which is easily influenced by outliers, and instead sorts the data within this set, extracting the value in the middle position, i.e., the spatial median. The system uses this spatial median as an indicator representing the comprehensive heat dissipation capacity of the entire target area's macroscopic grid points under the current impact event.

[0068] Furthermore, the following demonstrates the inversion process of the aforementioned physical laws, providing a simplified normalization example to illustrate its numerical logic. Assume that a grid point within the target region, after smoothing and spatiotemporal difference discretization, yields a normalized central difference time derivative of approximately -0.4 at a given moment, while the corresponding five-point difference space Laplacian operator value is -2.0. Simultaneously, the system detects that the Laplacian signal-to-noise ratio at this point reaches 10, far exceeding the preset signal-to-noise ratio threshold of 4.0, and the fitting determination coefficient meets the standard, thus classifying it as a reliable pixel. Substituting this into the equation system A*α... app =b is represented as -2.0*α app =-0.4. The system obtains algebraic value of 0.2 for the local apparent thermal diffusivity at the current moment through least squares estimation.

[0069] Step 204: Construct a time series based on the local apparent thermal diffusivity under each impact event in the impact event set and perform weighted linear regression fitting to obtain the thermal diffusivity degradation rate, and use the thermal diffusivity degradation rate as the characteristic of thermophysical response degradation.

[0070] As an alternative implementation, with the passage of service, the gradual increase in the micro-porosity inside the prestressed steel wire and the continuous initiation of delamination cracks will block the heat conduction path, which is macroscopically manifested as a continuous decrease in the equivalent thermal conductivity. The system collects multiple local apparent thermal diffusivity calculations for the same spatial grid point over a long period of time and constructs a structured time series according to the chronological order of the occurrence of each impact event.

[0071] Furthermore, the system employs a time-weighted linear regression fitting algorithm similar to the aforementioned alternative to process the time series. Specifically, the system fits an evolution line with time as the horizontal axis and the obtained thermal diffusivity value as the vertical axis; the slope of this line represents the thermal diffusivity degradation rate. Following the aforementioned normalization example, after tracking multiple impacts, if the system observes that the local apparent thermal diffusivity at a grid point gradually decreases from an initial 0.2 to a smaller absolute value of 0.1 or even 0, the system determines that intrinsic thermal diffusivity degradation has occurred in that region. Finally, the system encapsulates the calculated thermal diffusivity degradation rate and inputs it as a high-quality thermophysical response degradation feature into the subsequent failure mode identification and probability correction module. This embodiment avoids the superficial interference caused by changes in surface emissivity or fluctuations in ambient temperature, achieving penetrating tracking of deep structural aging of materials.

[0072] According to one aspect of this application, when simultaneously obtaining the cooling time constant drift rate and the thermal diffusivity degradation rate, the method further includes:

[0073] The estimation uncertainty of the cooling time constant drift rate and the estimation uncertainty of the thermal diffusivity degradation rate are obtained respectively, and each estimation uncertainty is characterized by the degree of dispersion of the corresponding weighted linear regression fitting residual.

[0074] Normalization is performed based on the reciprocals of the estimated uncertainties to determine the first fusion weight corresponding to the cooling time constant drift rate and the second fusion weight corresponding to the thermal diffusivity degradation rate;

[0075] Using the first fusion weight and the second fusion weight, the cooling time constant drift rate and the thermal diffusivity degradation rate are weighted and fused into a composite degradation rate index;

[0076] The composite degradation rate index is used instead of the cooling time constant drift rate as the degradation driving input for online correction of Weibull parameters.

[0077] In some embodiments, when the second uncertainty index exceeds a preset upper uncertainty threshold, the second fusion weight approaches zero, and the composite degradation rate index automatically reverts to the cooling time constant drift rate. The random components from their respective independent noise sources in the composite index are suppressed due to the weighted averaging effect, increasing the contrast of the degradation trend signal relative to the noise floor. This makes the subsequent update of the Weibull scale parameter in the first-layer exponential reduction correction more stable, reducing scale parameter jumps caused by occasional outliers in a single index.

[0078] Example 3 describes the process of biaxial failure mode identification and target parameter anti-false alarm correction.

[0079] As an optional implementation, this embodiment performs physical dimensionality reduction and matrix classification on the extracted degradation features. By constructing a biaxial degradation space, the system can identify different failure mechanisms such as deep delamination and surface effects, and apply targeted updates and anti-false alarm correction mechanisms to the core parameters of the dynamic Weibull cumulative damage prediction model.

[0080] Optionally, the cooling time constant drift rate and peak temperature degradation rate can be combined to construct a two-dimensional degradation feature space.

[0081] Specifically, the cooling time constant drift rate characterizes the degree of obstruction of the internal conduction path of the material, while the peak temperature degradation rate reflects the decline in the material's surface heat flux absorption capacity. The system combines these two orthogonal physical evolution dimensions, using the peak temperature degradation rate as the x-axis and the cooling time constant drift rate as the y-axis, to construct a mathematical expression: (γ... peak ,γ τ The continuous coordinate system is the two-dimensional degenerate feature space. Through biaxial mapping, the originally isolated one-dimensional scalar time series is transformed into a spatial state vector that can indicate the trajectories of crack initiation and propagation in different types of structures, providing a basic analytical domain for subsequent matrix-based qualitative classification.

[0082] Optionally, in the two-dimensional degradation feature space, the failure mode categories of the target region are classified using pre-configured drift rate significance thresholds and degradation significance thresholds.

[0083] Accordingly, after constructing the dual-axis space, the system uses threshold segmentation logic to discretize the continuous two-dimensional plane into four quadrants or regions with independent physical meaning. The system pre-configures a drift rate significance threshold for the vertical axis and a degradation significance threshold for the horizontal axis in the expert knowledge base. These two feature thresholds are orthogonal to each other, forming a cross-shaped segmentation boundary in space. Based on the current state vector (γ) of the target region... peak ,γ τBased on the specific quadrant position of the cross-shaped dividing boundary, the system classifies macroscopic material damage into four failure mode categories. For example, when both characteristic values ​​exceed the corresponding thresholds, the system determines it to be a surface macroscopic cracking mode, in which both surface heat absorption and deep heat conduction are severely deteriorated; while when both characteristic values ​​are below the corresponding thresholds, the system determines that the target area is in a normal stable service mode.

[0084] Optionally, when the peak temperature degradation rate does not exceed the degradation significance threshold and the cooling time constant drift rate exceeds the drift rate significance threshold, the failure mode category is determined to be deep delamination mode.

[0085] In this embodiment, when the impact heat flow acts on the target area, if the material surface remains relatively intact, the maximum temperature rise observed by the infrared thermal imager remains relatively stable, meaning the peak temperature degradation rate remains at a low level and does not exceed the degradation significance threshold. However, if a penetrating delamination air gap has formed deep within the parallel surface of the material, this gap will constitute a large contact thermal resistance, preventing heat from being smoothly transferred to the cold end of the material. Heat energy is forced to accumulate on the surface, prolonging the local cooling process, manifested as a sharp increase in the cooling time constant drift rate exceeding the drift rate significance threshold. After the system captures the characteristic combination of biaxial divergence, it immediately locks its failure mode category as a deep delamination mode, providing a physical basis for subsequent targeted high-risk probability correction.

[0086] Optionally, when the peak temperature degradation rate exceeds the degradation significance threshold and the cooling time constant drift rate does not exceed the drift rate significance threshold, the failure mode category is determined to be the surface effect mode.

[0087] As an alternative implementation, in contrast to the aforementioned deep-seated defects, during long-term operation of the industrial high-temperature molten droplet impact test platform, the surface of the prestressed steel wire is prone to weld slag adhesion or shallow oxidation reactions. This type of surface oxidation or change in emissivity leads to a systematic decrease in the radiant energy received by the detector, manifested as the calculated peak temperature degradation rate exceeding the degradation significance threshold. However, since the internal lattice heat transfer structure of the material does not undergo substantial fracture, the timescale law of heat dissipation remains unchanged, thus the cooling time constant drift rate remains stable and does not exceed the drift rate significance threshold. After identifying this combination of characteristics in biaxial space, the system accurately classifies its failure mode as a surface effect mode. This classification mechanism eliminates false degradation signals and is the foundation of the early warning system's anti-false alarm capability.

[0088] Optionally, the pre-calibrated Weibull datum scale parameters and Weibull datum shape parameters can be obtained offline.

[0089] Furthermore, to transform the aforementioned qualitative failure classification into quantitative inputs for damage and fracture probability calculations, the system needs to invoke the basic reliability model. Before the system was put into online real-time monitoring, initial physical indices representing the service life distribution of this batch of prestressed steel wires under ideal, non-degradation conditions were obtained by analyzing a large amount of historical damage and fracture experimental data and using statistical methods such as maximum likelihood estimation. Specifically, the system reads the Weibull baseline scale parameters and Weibull baseline shape parameters from the preloaded configuration memory. Among them, the scale parameters mainly determine the length of the material's characteristic life, while the shape parameters control the steepness and predictability of the failure rate curve's rise over time.

[0090] Optionally, a first-level exponential reduction correction is applied to the Weibull baseline scale parameters based on the peak temperature degradation rate and the cooling time constant drift rate.

[0091] After obtaining the ideal baseline parameters, the system introduces real-time extracted biaxial degradation features for dynamic degradation. Specifically, the system constructs a linear combination weighted term containing two feature variables, which is obtained by multiplying a preset degradation sensitivity coefficient by the peak temperature degradation rate and the cooling time constant drift rate, and then summing the results. Further, the system uses the base of the natural logarithm as the base and the negative of this weighted term as the exponent to calculate an exponential decay multiplier between 0 and 1. The system multiplies the Weibull baseline scale parameter with this exponential decay multiplier to complete the first layer of exponential reduction correction. This expression is set as λ1=λ0*exp(-(w peak *γ peak +w τ *γ τ ), where λ0 is the reference scale parameter, w peak and w τ These are degradation sensitivity coefficients for temperature and time constants, respectively, used to normalize the weighting term to a dimensionless quantity. This type of exponential correction structure mathematically guarantees that the characteristic lifetime parameter is positive and maps the linearly accumulated degradation rate to a nonlinear approximation of the lifetime limit.

[0092] In some embodiments, when the failure mode category is determined to be a deep delamination mode, in the first layer exponential reduction correction, the cooling time constant drift rate or composite degradation rate index is used as the only degradation driving quantity, and the correction contribution of the peak temperature degradation rate is suppressed or set to zero; wherein, the determination condition of the deep delamination mode is that the absolute value of the peak temperature degradation rate is lower than a preset peak significance threshold, and the cooling time constant drift rate exceeds a preset drift significance threshold.

[0093] As an optional implementation, the grid points (x) g ,y gWhen identified as Q2 mode, i.e., during deep delamination / debonding, its physical characteristic is that the peak temperature response amplitude does not change significantly (|γ) peak |≤γ th_class The cooling time constant showed an increasing trend (γ). τ >γ th_class In this mode, γ peak It does not carry meaningful degradation information, and its value fluctuates within the range of statistical noise. Including it in the correction formula cannot provide incremental information, but may instead introduce noise-driven spurious corrections.

[0094] Therefore, for grid points in the Q2 model, the first-level correction formula for the Weibull scale parameter is adjusted as follows:

[0095] λ 1_Q2 =λ0×exp(−μ Q2 ×max(0,γ τ (x g ,y g )));

[0096] Where, λ 1_Q2 Here, λ0 represents the first-layer corrected Weibull scale parameter in Q2 mode, and μ represents the baseline scale parameter calibrated offline. Q2 γ is the degradation sensitivity coefficient for deep delamination modes. τ This represents the drift rate of the cooling time constant at that grid point. This formula does not include γ. peak Contribution item, γ peak In Q2 mode, it is judged to be insignificant and therefore excluded from the correction driving quantity to avoid the interference of noise components on the scale parameter correction.

[0097] For grid points identified as being in a deep delamination mode, the peak temperature degradation rate contribution term in the above correction formula is set to zero, and only the cooling time constant drift rate or composite degradation rate index is retained as the driving force for correction.

[0098] Optionally, when the failure mode category is surface effect mode, the correction amount of the first layer of exponential reduction correction applied to the Weibull reference scale parameter is multiplied by a preset anti-false alarm attenuation coefficient to eliminate the probability overestimation caused by changes in surface emissivity.

[0099] To further demonstrate the anti-false alarm logic under the surface effect mode, a simplified normalized example is provided. Assume that, after pre-calibration, the Weibull baseline scale parameter for a certain grid point is initially set to 1.0, and the degradation sensitivity coefficient for temperature characteristics is also 1.0. In a certain monitoring cycle, the system detects that only temperature amplitude degradation occurred at this point, with a normalized peak temperature degradation rate of 0.2 and a drift rate of 0, thus confirming it as a surface effect mode. If the conventional full-load correction logic is followed, the new scale parameter should be corrected to the original parameter multiplied by exp(-0.2), resulting in approximately 0.82. This would cause the system to overestimate the risk of damage and fracture, potentially triggering a false alarm and premature pipeline shutdown.

[0100] However, under the intervention of the anti-false alarm mechanism in this embodiment, the system determines that the current feature change may only be caused by surface oxidation. At this time, the system calls the preset anti-false alarm attenuation coefficient, which can be set to 0.5. The system multiplies the feature value of 0.2 in the original exponent by the anti-false alarm attenuation coefficient of 0.5, so that the correction exponent is halved to 0.1. Then, the correction calculation formula of the scale parameter is updated to 1.0 multiplied by exp(-0.1), and the final updated value is 0.90. By applying an artificial subtraction reduction to the correction amount, the system reduces the dimensionality of non-fatal surface interference factors in the probabilistic calculation model, thereby enhancing the prediction robustness in complex industrial environments.

[0101] Optionally, when the current mega-anomaly index exceeds a preset threshold, a second layer of correction is applied to the reduced and corrected Weibull baseline scale parameter.

[0102] Accordingly, after completing the first layer of correction for the macroscopic biaxial degradation characteristics, the system further introduces high-frequency microscopic fluctuation information derived from the quiescent period. When the system detects that the precursor anomaly index, calculated by combining physical baseline detrending and residual variance testing, exceeds the preset statistical test threshold, it indicates that the microcracks inside the material are already in an active tribothermic heating or propagation stage. At this point, the system applies an additional exponential decay multiplier based on the magnitude of the anomaly index to the Weibull scale parameters, which have already undergone the first layer of reduction correction, to perform the second layer of correction. This mechanism enables the system to rapidly shorten the theoretical predicted lifetime of the model with high sensitivity when encountering sudden microstructural instability, reserving a sufficient early warning time window for impending severe damage and fracture.

[0103] Optionally, when the failure mode category is deep delamination, the Weibull baseline shape parameters are adjusted downward to obtain updated damage fracture probability distribution parameters.

[0104] In this embodiment, the system modifies the underlying distribution pattern to address a specific deep delamination mechanism. During conventional material aging, the Weibull shape parameter is typically greater than 1, indicating a regular wear-out phase where the failure risk monotonically increases over time, exhibiting relatively good predictability. However, when the target region is determined to be in a deep delamination mode, the deep, parallel air gaps prevent the surface from transmitting sufficient precursor deformation information, potentially causing brittle fracture under any ordinary droplet impact. This mechanism leads to a degradation of the local material's failure characteristics towards a random failure mode.

[0105] Accordingly, the system initiates the shape parameter reduction logic, subtracting a preset reduction step size component from the Weibull baseline shape parameter. Reducing the shape parameter, from a probabilistic perspective, makes the cumulative distribution function curve flatter and more long-tailed on the time axis, effectively widening the time window for high-risk events. Through the double-layer reduction of the scale parameter and the targeted reduction of the shape parameter, the system finally completes a comprehensive parameter reconstruction, encapsulating these two highly customized feature values ​​into updated damage and fracture probability distribution parameters, and then outputting the spatial damage and fracture probability distribution deduction module. Through this targeted mechanism, this embodiment overcomes the limitation of rigid, constant parameters in traditional reliability models.

[0106] Example 4 describes the detection of precursor fluctuations based on the physical baseline detrending during the silent period.

[0107] This embodiment, as a specific implementation scheme for extracting precursor anomaly indicators, describes how, under the cover of temperature jump caused by droplet impact, spatial constraints and baseline subtraction techniques are used to extract weak thermal flow fluctuation signals caused by the evolution of microscopic damage inside the material (such as water phase change and microcrack friction), thereby achieving early warning of microscopic precursors.

[0108] In one possible implementation, extracting precursory anomaly indicators of the target region during the quiet period of an adjacent impact includes the following steps:

[0109] Step 401: Determine the effective silence period that is located between two adjacent target impact events and meets the preset duration condition.

[0110] As an alternative implementation, in the actual operation of an industrial high-temperature droplet impact testing platform, the droplet falling is a random and discrete process. Each impact triggers intense heat deposition and dissipation in a localized area. If another impact occurs immediately after the cooling phase of one impact, the thermal effects of the two impacts will severely overlap. In this case, any minute temperature fluctuation may be interpreted as the superposition of residual heat or the start of a new round of heating, losing its independence as an indicator of damage precursors. Therefore, in order to extract microscopic fluctuations purely caused by changes in the material's own physical state, the system needs to be filtered and isolated in the time dimension.

[0111] Specifically, the system continuously monitors the infrared temperature field time series. For the same spatial location, the system records the starting point of the temperature rise caused by the previous target impact event and monitors its subsequent cooling process. The system calculates the point at which the temperature increment drops to 5% of the peak value and defines this as the safe boundary point where the heat from the previous impact has been largely dissipated. Only when the next target impact event occurs later than this safe boundary point, and the time interval between the two is greater than twice the cooling time constant, does the system consider that this period is not disturbed by the macroscopic main heat flow. The system marks this time window as the effective quiet period. Through dynamic duration constraints based on exponential decay characteristics, the system can adaptively avoid thermal tail interference, defining a reliable time domain range for subsequent high-frequency residual analysis.

[0112] Step 402: During the effective silence period, calculate the expected residual temperature decay baseline using the cooling time constant of the preceding target impact event.

[0113] Accordingly, after determining the time window, the system needs to further strip away the masking effect of macroscopic heat transfer on microscopic fluctuations. During the effective quiescent period, although the temperature change on the material surface is slow, it is essentially still a long-tailed macroscopic cooling process dominated by the continued diffusion of heat deposited from the previous impact into the deeper parts of the material and radiative convection into the environment. This long-tailed process constitutes the background of a slowly decreasing trend. If this is not subtracted, calculating the temperature variance reveals that the low-frequency, large-amplitude changes caused by this macroscopic trend will overwhelm the high-frequency, small-amplitude fluctuations caused by microstructural instability.

[0114] To calculate the trend background, the system calls the cooling time constant obtained by fitting the preceding target impact event. Further, based on a single exponential cooling model, the system uses the start of the effective quiescent period as the time zero point and the extrapolated residual temperature at that moment as the initial amplitude to generate a smooth theoretical curve that decreases exponentially with time. This smooth curve, derived from macroscopic heat transfer physics, is the expected residual temperature decay baseline. This baseline is expressed as T. base (t)=T residual *exp(-t / τcool ), where T residual τ represents the residual temperature rise at the start of the quiescent period. cool This refers to the cooling time constant of the preceding event. By constructing a physical baseline instead of polynomial smoothing, the system ensures that detrending operations do not filter out anomalous signals caused by abrupt changes in material properties.

[0115] In one possible implementation, before calculating the variance of the detrended temperature residual within the target region, optionally, the thermally affected zone expansion rate is determined based on the thermally affected zone isotherm expansion information of the target impact event in the infrared temperature field time series, and the precursor analysis spatial radius is calculated using the thermally affected zone expansion rate and the cooling time constant.

[0116] Furthermore, before extracting spatial features, the system introduces an engineering-robust spatial constraint mechanism. Because industrial prestressed steel wire specimens are not perfectly homogeneous and their surfaces may exhibit geometric micro-undulations, extrapolating the residual temperature decay baseline over a large area using a single point—typically the impact centroid—will introduce model extrapolation errors as the spatial distance increases. If the analysis range is too large, these spatial extrapolation errors will be treated as high-frequency fluctuating residuals, triggering false alarms.

[0117] To this end, the system pre-calculates the isotherm propagation rate, i.e., the heat-affected zone (HAZ) expansion rate, after the target impact event. The HAZ expansion rate can be determined by tracking the outward propagation speed of the isotherm after a single impact in a calibration experiment. Those skilled in the art can determine a suitable value based on the thermophysical properties of the actual material and the impact conditions. The system multiplies this expansion rate by the cooling time constant and introduces a geometric adjustment coefficient between 0.5 and 1.0 to dynamically calculate the physical distance threshold, i.e., the precursor analysis spatial radius. This formula can be expressed as R... precursor =eta*v th *τ cool Where eta is the geometric adjustment coefficient, v th This represents the expansion rate of the heat-affected zone. By introducing this physical boundary constraint, the system can adaptively define an analysis circle with smaller model errors based on the energy of a single impact and the local thermal conductivity of the material.

[0118] Step 403: Subtract the expected residual temperature decay baseline from the infrared temperature field time series to obtain the detrended temperature residual.

[0119] Specifically, after establishing the baseline model and analysis scope, the system performs a detrending operation. During the effective quiet period, the system traverses all spatial grid points within the local neighborhood defined by the aforementioned precursor analysis spatial radius. For each sampling point, the system subtracts the theoretical temperature value corresponding to the expected residual temperature decay baseline, calculated by the aforementioned exponential decay equation, from the infrared temperature field time series data actually acquired by the infrared thermal imager. After point-by-point subtraction, the remaining portion purely represents a high-frequency weak signal that deviates from the macroscopic heat transfer law and contains microscopic physical phase transition or crack opening and closing heating information; this is called the detrended temperature residual.

[0120] Optionally, the spatial range of the variance of the calculated detrended temperature residual can be limited to a local neighborhood centered on the target impact event and with the radius of the precursor analysis space, thereby controlling the spatial expansion error of the expected residual temperature decay baseline.

[0121] Furthermore, the system confines subsequent statistical feature extraction operations to a physically reliable spatial range. Instead of blindly calculating variance across the global grid within the field of view, the system defines a circular local neighborhood centered on the centroid coordinates of the target impact event and using the radius of the precursor analysis space as the physical boundary. All detrended temperature residual data outside this neighborhood are ignored. This embodiment physically severs the path of divergence errors generated by the macroscopic heat conduction model at distant boundaries to microscopic precursor anomaly indicators, improving the spatial calibration accuracy of the early warning system in complex, large-area scenarios.

[0122] Step 404: Calculate the variance of the detrended temperature residual within the target area, and calculate the ratio of the variance to the pre-stored background variance to obtain the precursor anomaly index as a precursor anomaly indicator.

[0123] Accordingly, under the dual constraints of spatiotemporal constraints of local neighborhood and effective silence period, the system performs statistical analysis on the extracted detrended temperature residual sequence to calculate its dispersion, i.e., the fluctuation variance. To determine whether this fluctuation variance originates from normal detector white noise or whether an abnormal evolution has indeed occurred within the material, the system introduces a variance ratio test mechanism from statistics. During the system debugging and preprocessing stage before the equipment is put into operation, the system has long recorded the inherent thermal noise generated by the infrared detector itself under stable operating conditions of constant temperature and no droplet impact on the high-temperature droplet impact experimental platform, and extracted and calculated the stable pre-stored background fluctuation variance.

[0124] Simultaneously, the system uses the currently calculated real-time variance as the numerator and the pre-stored background variance as the denominator, performing a division ratio calculation. According to statistical principles, if no anomalies occur within the material, the detrended residual should primarily consist of background white noise, and its ratio should fluctuate around 1.0. Once the ratio deviates from 1.0, it indicates a physical anomaly exceeding the detector's inherent error. The system defines this ratio calculation result as the precursor anomaly index and inputs it as the final output precursor anomaly indicator into the macroscopic probability prediction model.

[0125] Furthermore, to illustrate the statistical verification process of this micro-signal amplification, a simplified normalization example is provided here. Assume that after spatiotemporal filtering and baseline subtraction, the system calculates the normalized variance of the detrended residuals within the local effective neighborhood to be 0.05. The system retrieves the pre-loaded background white noise variance, which is 0.01. The system substitutes these two values ​​into the ratio formula to calculate the anomaly index I = 0.05 / 0.01 = 5.0. The system's preset chi-square test threshold is 4.0. Since the calculated value of 5.0 exceeds the threshold of 4.0, the system immediately marks the target region as potentially experiencing a microscopic precursor of microcrack tip closure frictional heating and immediately triggers an early microscopic warning signal to the upper-level model.

[0126] Example 5 describes a transient snapshot probability inference method based on independent grid points.

[0127] This embodiment, as a basic implementation and parallel alternative for calculating the probability distribution of spatial damage fracture, describes how, in complex systems where model parameters dynamically change with service time, transient snapshot logic and spatially independent grid point inference mechanism are used to achieve spatial failure probability calculation with high sensitivity and self-correction characteristics while saving online computing resources.

[0128] In one possible implementation, the spatial damage fracture probability distribution of the target region is calculated based on the cumulative result of single-event damage contribution and the updated damage fracture probability distribution parameters, such as... Figure 5 As shown, it includes the following steps:

[0129] Step 501: Spatially linearly superimpose the damage contributions of single events to construct the current cumulative damage field.

[0130] As an optional implementation, after acquiring the damage metric derived from the physical characteristics extracted from previous impacts, the system tracks and records the fatigue load borne by each spatial grid point along a macroscopic time elapsed axis. For any predetermined coordinate position within the target area, the system algebraically sums the dimensionless single-event damage contributions corresponding to all historical droplet impacts that could reach that position. On a two-dimensional spatial plane, the system integrates the accumulated values ​​of all discrete grid points into a continuous two-dimensional matrix structure. This matrix represents the current cumulative damage field. This damage field reflects the total thermomechanical fatigue load state borne by the surface and shallow materials of the entire target area at the current moment and serves as the core independent variable data base input into the subsequent statistical inference model.

[0131] Step 502: Use the updated damage fracture probability distribution parameters as the scale and shape parameters of the Weibull distribution model.

[0132] The system retrieves the latest dynamic parameter set from the pre-installed failure mode identification and targeted update module. Specifically, the system loads scale parameters, which have undergone qualitative identification in a dual-axis feature space and may have already been augmented with multi-layered exponential reduction corrections and false alarm suppression, along with shape parameters that have triggered downsampling correction logic under severe deep delamination conditions, into the basic Weibull lifetime statistical physics model. Within this model architecture, scale parameters determine the tensile span and expected characteristic lifetime of the prediction curve along the time or cumulative damage axis, while shape parameters control the steepness and random distribution characteristics of the material failure rate as the load increases. Through a real-time hot-plug mechanism for parameters, this probabilistic statistical model does not rely on static empirical curves and can adapt in real time to the evolving physical state of material aging within the current target region.

[0133] Step 503: Using the current cumulative damage field as the independent variable, independently calculate the cumulative distribution function value of each spatial grid point in the target area under the Weibull distribution model, and use it as the spatial damage fracture probability distribution.

[0134] Optionally, after establishing the model's independent variable matrix and the latest parameter vector, the system performs parallel failure probability extrapolation calculations across the entire field of view. In the specific calculation implementation, the system uses the standard Weibull cumulative distribution function, expressed as P = 1 - exp(-(D / λ)). kIn this equation, P represents the cumulative macroscopic damage fracture probability calculated from the target spatial grid points, D represents the absolute cumulative damage value extracted from the corresponding coordinate position in the current cumulative damage field, the constant exp is the base of the natural logarithm, λ is the updated scale parameter loaded in real time, and k is the corresponding updated shape parameter. The system performs numerical solutions for hundreds of thousands or even millions of pixels in the target area, individually and spatially independently, by substituting them into this formula. Finally, the solution results are combined and reconstructed into a two-dimensional pseudo-color probability mapping layer, which is the spatial damage fracture probability distribution.

[0135] In the above inference process, this embodiment employs an engineering approximation strategy that satisfies the ultimate industrial response speed. In high-order reliability mathematics, when the lifetime distribution parameters of a complex system, such as scale and shape parameters, dynamically evolve with the service process, calculating the absolute cumulative failure probability at a predetermined cross-section typically requires performing extremely computationally intensive multiple integration operations along the historical evolution path on the instantaneous hazard rate function. However, in industrial settings, high-frame-rate infrared video stream monitoring scenarios are common. Continuous-time integration of the historical degradation states of millions of high-resolution pixels increases the system's online computing power requirements and memory overhead, affecting the engineering effectiveness of millisecond-level closed-loop reverse control.

[0136] Optionally, as a further extension and supplement to the aforementioned independent probability inference scheme, the system can also output, in parallel, the instantaneous damage fracture hazard rate, representing the immediate damage fracture risk, during the background calculation cycle of calculating the spatial damage fracture probability distribution. The expression for this index is h=(k / λ)*(D / λ). (k-1) This indicator, possessing the significance of a physical derivative, reveals the instantaneous conditional probability density of macroscopic damage that occurs at physical grid points that have not yet undergone substantial fracture when subjected to the thermal stress of the next newly falling molten droplet impact. This instantaneous hazard rate field can serve as a supplementary defense to the absolute probability value distribution, enabling highly sensitive detection of the acceleration trend of a rapid shift in the center of gravity of risk within the target area.

[0137] Furthermore, to enhance the forward-looking decision-making capabilities of industrial production scheduling and preventative maintenance, the system can automatically calculate the first derivative of the evolution of damage values ​​in the current cumulative damage field over time based on recent sliding time monitoring windows, extracting the recent growth rate of local damage. Using this damage growth rate, the system linearly extrapolates and pre-funds the expected damage increment within a preset future time observation span. Further, the system substitutes the future expected total cumulative damage data, including the extrapolated increment, into the aforementioned Weibull cumulative distribution model equation, calculating the predicted conditional probability for a future time period across the current time and space. This derivative-based extrapolation-assisted prediction output allows on-site process experts and control equipment to predict the optimal downtime maintenance window half an hour or even longer in advance, or automatically plan changes to material delivery strategies, making the entire dynamic probability prediction model a decision-making unit in modern construction sites.

[0138] Example 6 describes a phased parameter calibration and long-term system verification method.

[0139] This embodiment describes how to calibrate the hyperparameters of a dynamic Weibull cumulative damage prediction model using offline historical data before the system is put into high frame rate online real-time prediction.

[0140] Optionally, a training set containing multiple known historical cases of macroscopic damage and fracture can be obtained in advance, and feature data of each impact and the corresponding actual damage and fracture time can be extracted from the training set.

[0141] As an optional implementation, the system needs sufficient prior physical data as statistical support before making any parameter inferences. Specifically, engineers collect high-definition infrared video recordings over several maintenance cycles, along with corresponding equipment damage and repair records, on the same research equipment. The system backtracks and analyzes the historical recordings, extracting complete lifecycle data for each eventual material failure, and constructs a training set. For each independent sample in this training set, the system extracts characteristic data such as the cooling time constant and peak temperature corresponding to each droplet impact from its pristine state until macroscopic delamination or fracture occurs, and records the absolute time point of the final devastating fracture, i.e., the actual fracture moment. This historical fatigue archive constitutes the benchmark for all subsequent calibration algorithms.

[0142] Optionally, in the first-stage parameter calibration, the temperature nonlinearity index on which the calculation of single-event damage contribution depends is locked to a constant empirical value to eliminate the mathematical compensation effect in the multidimensional parameter space.

[0143] Correspondingly, in the process of transforming discrete thermophysical characteristics into dimensionless damage, there exists a sensitive exponential hyperparameter, namely the temperature factor exponent represented by the variable m. If this exponent is jointly searched in the same high-dimensional space as the scale and shape parameters of the Weibull distribution during the initial calibration phase, the algorithm is prone to compensation effects.

[0144] Specifically, the optimization algorithm may output a large temperature factor exponent, coupled with a small Weibull shape parameter. While this set of parameters can still fit the historical failure curve algebraically, its physical meaning is lost, leading to an over-amplification of minor temperature perturbations during online prediction.

[0145] Furthermore, to avoid physical distortion in parameter fitting, the system introduces a staged control approach. At the start of the first stage of parameter calibration, the system closes the multidimensional search space and locks the temperature factor exponent to a constant empirical value consistent with heat transfer principles. For example, this constant empirical value can be set to 4.0. Through this locking operation, the system reduces the original three-dimensional non-convex optimization problem to a two-dimensional optimization problem, providing a physical basis for subsequent extraction of the lifetime benchmark.

[0146] Optionally, the log-likelihood of the training set can be maximized using the maximum likelihood estimation method to obtain the initial calibration values ​​of the Weibull baseline scale parameters and the Weibull baseline shape parameters.

[0147] Specifically, within the reduced-dimensional parameter space after eliminating the high-dimensional compensation effect, the system employs a maximum likelihood estimation algorithm for numerical solution. The system substitutes all extracted historical impact damage features into the standard Weibull probability density function to construct a joint probability distribution equation for the entire training sample set. To facilitate floating-point optimization by the computer, the system takes the natural logarithm of this joint probability equation, transforming it into a log-likelihood function. Furthermore, the system uses conventional numerical optimization methods to find the extremum of this log-likelihood function. Since the temperature index is already locked, the optimization process exhibits high convergence stability and can quickly output the two core parameters that maximize the overall probability of historical damage fracture events. These two outputs are recorded as the initial calibration values ​​of the Weibull baseline scale parameter and the Weibull baseline shape parameter, respectively.

[0148] Optionally, in the second-stage parameter calibration, the temperature factor exponent is released, and the initial calibration value is used as the starting point for the joint optimization iteration.

[0149] Accordingly, after obtaining two-dimensional parameter anchor points with a solid physical foundation, the system smoothly transitions to a higher-precision global fine-tuning stage. In the second-stage parameter calibration, the system removes the locking constraint on variable m, allowing it to float freely within reasonable physical constraint boundaries. Simultaneously, the system abandons the inefficient and easily divergent random initialization strategy, instead injecting the initial calibration values ​​obtained in the previous stage into the input of the multidimensional optimizer, using them as the starting point for global search iterations. This type of relay-like optimization strategy accelerates the convergence process of the high-dimensional algorithm.

[0150] Optionally, a leave-one-out cross-validation strategy is used to calculate the root mean square error between the predicted damage fracture time and the actual damage fracture time, and the final solidified model hyperparameters are output by minimizing the root mean square error.

[0151] Furthermore, to find the optimal parameter combination that can generalize to unknown working conditions to the greatest extent in the multidimensional parameter space, the system adopts a leave-one-out cross-validation strategy. Specifically, it is assumed that the training set contains N independent historical damage fracture samples. In each iteration of evaluation, the system deliberately extracts one sample from the dataset as a hidden test blind box, and uses the remaining N-1 samples to refit the theoretical life curve under the current parameter combination. Further, the system uses this curve to deduce the expected failure time of the hidden sample, i.e., the predicted damage fracture time.

[0152] Simultaneously, the system compares the predicted damage fracture time with the actual recorded damage fracture time, calculating the residual. The system iterates through this extraction and extrapolation process N times until each sample has served as a test blind box. Finally, the system sums and averages the square roots of the squared residuals from all N iterations to obtain the evaluation index, namely the root mean square error (RMSE). The RMSE expression is: RMSE = sqrt(sum((T)) pred -T true ) 2 ) / N), where T pred T is the predicted damage fracture time for a predetermined sample. true It refers to the actual moment of damage and fracture.

[0153] In this embodiment, the system uses a gradient-free optimization algorithm or a grid search strategy to continuously fine-tune the released temperature factor exponent and its associated scale and shape parameters, repeatedly performing the aforementioned cross-validation loop until a parameter combination is found that reduces the root mean square error to its global minimum. This parameter combination, having undergone blind testing, is the final, fixed model hyperparameter output by the system.

[0154] Optionally, online updates of hyperparameters can be calibrated using independent long-term validation experimental data.

[0155] As an optional implementation, after the baseline Weibull parameters, namely the temperature nonlinearity exponent, the initial baseline Weibull scale parameters, and the initial baseline Weibull shape parameters, are solidified, the system enters the calibration phase of online hyperparameter update. The system acquires one or more sets of pre-stored long-term continuous verification experimental data, which are independent of the training set and include historical impact records and the actual macroscopic damage fracture frequencies occurring within a known time period.

[0156] Accordingly, the system uses the actual damage fracture frequency as the calibration target. Within a pre-defined parameter space, it performs a grid search or gradient-free optimization scan on a set of online update hyperparameters required for the online operation phase, including the degradation sensitivity coefficient, false alarm attenuation coefficient, precursor anomaly correction coefficient, and shape parameter downsizing step size. For each candidate hyperparameter combination in the parameter space, the system uses the fixed benchmark parameters to execute an online prediction simulation process on long-term validation experimental data, statistically analyzes the predicted damage fracture frequency, and calculates the root mean square deviation (RMS) between the predicted and actual damage fracture frequencies. The system extracts the hyperparameter combination that minimizes the global RMS deviation and uses it as the final online update hyperparameter.

[0157] Furthermore, the system pre-configures and burns the temperature nonlinearity exponent, initial baseline Weibull scale parameters, initial baseline Weibull shape parameters, and online update hyperparameters into the online monitoring host for real-time prediction.

[0158] Those skilled in the art will understand that the numerical examples in the above embodiments are used to illustrate the calculation logic of each step. Specific performance indicators in actual applications may vary depending on the type of high-temperature molten droplet impact test platform, the grade of prestressed steel wire, the performance of the infrared detector, and the operating conditions. This solution, by introducing innovative mechanisms such as deep degradation feature extraction based on the inversion of the partial differential equation of heat conduction, microscopic precursor detection based on physical baseline detrending, and targeted updating of Weibull parameters based on failure mode identification, can achieve a systematic improvement over traditional static macroscopic temperature amplitude assessment methods at the principle level.

[0159] This scheme decouples the propagation of microcracks inside the steel wire from surface oxidation by constructing a biaxial degradation feature space that integrates the drift rate of the cooling time constant and the peak temperature degradation rate. For physical modes characterized as surface effects, the system introduces an anti-false alarm attenuation coefficient when updating the probability distribution parameters, artificially reducing the probability correction caused by non-fatal disturbances. This improves the problem of false warnings easily triggered by a single apparent temperature amplitude, thus achieving state identification.

[0160] Meanwhile, by introducing a Gaussian kernel function with a spatial decay radius dynamically calibrated using the cooling time constant to construct a damage spatial diffusion model, and by dynamically scaling the spatial influence scale of the damage using the local apparent thermal diffusivity, this scheme enables the spatial sweep range of the damage to adaptively adjust as the intrinsic heat dissipation capacity of the material degrades. Furthermore, the system abandons static parameters and directly uses biaxial degradation characteristics to dynamically and targetedly update the scale and shape parameters of the statistical lifetime model. This approach avoids the limitations of traditional constant parameters, enabling the model to adapt to the unstable fracture evolution of materials.

[0161] Furthermore, by constructing and subtracting the expected residual temperature decay baseline using the preceding cooling time constant within the selected effective quiescent period of adjacent impacts, this scheme removes the masking effect of macroscopic long-tail heat transfer laws. Combining the spatial radius constraint of precursor analysis and the local variance ratio test, the system extracts high-frequency weak signals of microstructural frictional heating from the detrended temperature residuals. This method improves upon the problem of early weak physical precursor signals being lost due to conventional macroscopic main heat flow feature extraction methods, endowing the system with highly sensitive microscopic early warning and interception capabilities. For PCCP pipeline construction and repair scenarios, this method, by identifying the abnormal drift of the cooling time constant caused by molten droplet impact, can output targeted closed-loop control commands before the steel wire undergoes brittle fracture, thereby blocking the microscopic thermal damage evolution path of the prestressed steel wire and reducing the long-term risk of PCCP pipeline rupture.

[0162] It should be noted that the various specific technical features described in the above embodiments can be combined in any suitable manner without contradiction. To avoid unnecessary repetition, the present invention will not describe the various possible combinations separately.

Claims

1. A dynamic probability prediction method for physical precursors of high-temperature molten droplet damage fracture in prestressed steel wires of water transmission pipelines, characterized in that, include: Obtain a continuous time series of infrared temperature field in the target area; By detecting droplet impact events based on infrared temperature field time series, a set of impact events in the target region is obtained. Impact cooling features are extracted based on infrared temperature field time series and impact event set, and the thermophysical response of each impact event in the impact event set is tracked based on infrared temperature field time series to calculate the thermophysical response degradation features of the target region. By utilizing the infrared temperature field time series and the set of impact events, precursory anomaly indicators of the target area are extracted during the quiescent period of adjacent impacts. Based on the characteristics of shock cooling, a single-event damage contribution is constructed. Combined with the characteristics of thermophysical response degradation and precursor anomaly indicators, the pre-configured damage fracture probability distribution parameters are dynamically updated in a targeted manner. Based on the cumulative results of the single-event damage contribution and the updated damage fracture probability distribution parameters, the spatial damage fracture probability distribution of the target area is calculated. Output closed-loop control commands based on spatial damage fracture probability distribution and precursor anomaly indicators; For a target impact event in a set of impact events, the peak temperature induced by the target impact event in its thermally affected zone is determined in the infrared temperature field time series. The temperature cooling curve from the moment corresponding to the peak temperature is extracted from the infrared temperature field time series. The temperature cooling curve is fitted using a preset exponential decay model to obtain the cooling time constant characterizing the local heat dissipation rate. Peak temperature and cooling time constant are used as characteristics of shock cooling; Based on the peak temperature of the target impact event and the preset damage initiation temperature threshold, a dimensionless temperature factor characterizing the degree of local overheating is calculated. Based on the cooling time constant of the target impact event and the pre-configured reference time constant, a dimensionless time factor characterizing the thermal energy deposition residence effect is calculated. The spatial decay radius of the Gaussian kernel function is determined by using the cooling time constant and the preset heat-affected zone expansion rate, and the spatial decay factor characterizing the degree of damage spatial diffusion is calculated. The single-event damage contribution is obtained by multiplying the dimensionless temperature factor, the dimensionless time factor, and the spatial decay factor. The damage contributions from single events are linearly superimposed in space to construct the current cumulative damage field. The updated damage fracture probability distribution parameters are used as the scale and shape parameters of the Weibull distribution model; Using the current cumulative damage field as the independent variable, the cumulative distribution function value of each spatial grid point in the target area under the Weibull distribution model is independently calculated as the spatial damage fracture probability distribution.

2. The method according to claim 1, characterized in that, The thermophysical response degradation characteristics of the target region are calculated in the following way: During the cooling phase of the target impact event, based on the range corresponding to the preset proportion of the temperature increment decreasing to the peak temperature in the infrared temperature field time series, the corresponding time segment is extracted as the analysis window. Within the analysis window, the central difference time derivative and the five-point difference space Laplace operator are calculated based on the infrared temperature field time series. An overdetermined system of equations was constructed using the central difference time derivative and the five-point difference space Laplace operator, and the local apparent thermal diffusivity of the target region was obtained by least squares estimation. A time series of local apparent thermal diffusivity under each impact event in the impact event set is constructed and a weighted linear regression is performed to obtain the thermal diffusivity degradation rate, which is then used as a characteristic of thermophysical response degradation.

3. The method according to claim 1, characterized in that, During the quiet period between adjacent shocks, precursory anomalies in the target region are extracted, including: Determine the effective silence period that is located between two adjacent target impact events and meets the preset duration conditions; During the effective silence period, the expected residual temperature decay baseline is calculated using the cooling time constant of the preceding target impact event; The detrended temperature residual is obtained by subtracting the expected residual temperature decay baseline from the infrared temperature field time series. Calculate the variance of the detrended temperature residual within the target area, and then calculate the ratio of the variance to the pre-stored background variance to obtain the precursor anomaly index as a precursor anomaly indicator.

4. The method according to claim 1, characterized in that, Based on the spatial damage fracture probability distribution and precursor anomaly indicators, closed-loop control commands are output, including: Extract the probability values ​​of each spatial grid point in the spatial damage and fracture probability distribution within the target area; Grid points whose probability values ​​exceed the pre-configured risk probability threshold and whose precursor anomaly indicators at the same spatial location simultaneously meet the preset anomaly judgment conditions are marked as high-risk areas, forming a dual-channel early warning condition of physical and probabilistic characteristics. When the dual-channel warning condition is triggered, the landing point offset coordinates are determined based on the spatial location information of the high-risk area, and the frequency adjustment ratio is determined based on the degree of exceeding the probability value. A closed-loop control command containing the landing point offset coordinates and the frequency adjustment ratio is generated to control the droplet release device to avoid the high-risk area and reduce the impact frequency.

5. The method according to claim 1, characterized in that, Before acquiring the continuous infrared temperature field time series of the target area, an offline parameter calibration stage is also included. This stage is used to predetermine the baseline values ​​required for calculating the damage contribution of a single event and updating the damage fracture probability distribution parameters. The offline parameter calibration stage includes: Acquire pre-stored historical damage fracture experimental data, and fix the temperature factor exponent based on the physical law of material residual compressive strength degradation, and use it as a physical prior parameter; Under the condition of fixed physical prior parameters, a likelihood function is constructed using historical damage fracture experimental data, and the initial benchmark Weibull scale parameter and the initial benchmark Weibull shape parameter are solved by joint maximum likelihood estimation. A damage fracture prediction model is constructed based on the initial benchmark Weibull scale parameters and the initial benchmark Weibull shape parameters. The actual damage fracture frequency in the pre-acquired long-term validation experimental data is used as the calibration target. The parameter combination that minimizes the root mean square error between the predicted damage fracture frequency output by the damage fracture prediction model and the actual damage fracture frequency is scanned and extracted within the set parameter space. This combination is used as the online update hyperparameter for updating the damage fracture probability distribution parameters. The physical prior parameters, initial benchmark Weibull scale parameters, initial benchmark Weibull shape parameters, and online update hyperparameters are pre-configured as components of the pre-configured damage fracture probability distribution parameters for real-time prediction.

6. The method according to claim 2, characterized in that, The process of obtaining the local apparent thermal diffusivity of the target region through least squares estimation also includes: The Laplace signal-to-noise ratio is calculated based on the signal variance of the five-point difference space Laplace operator and the pre-stored background noise variance, and the fitting determination coefficient is calculated based on the fitting residuals of the least squares estimation. Unreliable pixels with a Laplacian signal-to-noise ratio lower than a preset signal-to-noise ratio threshold or a fitting coefficient of determination lower than a preset coefficient of determination threshold are removed. The spatial median of the local apparent thermal diffusivity of the remaining reliable pixels within the thermally affected zone of the target impact event is extracted and used as the characterization of the local apparent thermal diffusivity of the grid points where the target region is located.

7. The method according to claim 3, characterized in that, Before calculating the variance of the detrended temperature residuals within the target region, the following steps are also included: Based on the isotherm expansion information of the thermally affected zone of a target impact event in the infrared temperature field time series, the expansion rate of the thermally affected zone is determined, and the spatial radius of the precursor analysis is calculated using the expansion rate of the thermally affected zone and the cooling time constant. The spatial range of the variance of the calculated detrended temperature residual is limited to a local neighborhood centered on the target impact event and with the radius of the precursor analysis space, thereby controlling the spatial expansion error of the expected residual temperature decay baseline.

Images