An aircraft icing characteristic and shape reconstruction method based on ultrasonic full waveform inversion

Abstract

The application discloses an aircraft icing characteristic and shape reconstruction method based on ultrasonic full waveform inversion, and belongs to the field of ultrasonic imaging detection. The method depends on comprehensive numerical simulation of ultrasonic wave propagation to extract material characteristics and geometric shapes of ice layers on aircraft skin. The application iteratively updates the velocity characteristics and shapes of the ice layers by using recorded waveform data, adopts a nonlinear thin layer model, uses a two-dimensional acoustic equation and a finite differential method in Python software to model wave propagation, generates synthetic data, and matches observed waveforms with simulated waveforms until the observed waveforms match the simulated waveforms. The application combines the advantages of ultrasonic wave application in nondestructive testing and full waveform inversion, extracts all information of the ice layers on the top of the aircraft skin by using entire ultrasonic wave field data, thereby can accurately reconstruct the shapes of the model and internal properties of the medium, and can provide an accurate and reliable solution of aircraft body icing characteristic parameters, and can reduce the limitations of existing icing detection methods.

Description

Technical Field

[0001] This invention belongs to the field of ultrasonic imaging detection, specifically involving the detection of aircraft icing by reconstructing the internal properties and geometry of aircraft icing using ultrasonic full waveform inversion theory. Background Technology

[0002] Icing is considered a difficult-to-prevent and potentially dangerous phenomenon during aircraft flight. Icing on aircraft skin can be seen as a structural modification, affecting aerodynamic performance and potentially causing incalculable consequences. Three main types of ice form on aircraft fuselages during icing: frost, hoarfrost, and mixed ice. Frost is similar to the ice that forms on a refrigerator coil due to condensation. Hoarfrost is a transparent and dense (denser than frost) glassy ice coating. Mixed ice consists of irregularly shaped, rough-surfaced white particles formed by the simultaneous presence of supercooled droplets of varying sizes.

[0003] Active detection of icing can promptly alert the crew to activate the de-icing system, thereby minimizing or preventing the adverse effects of icing on flight safety and performance. In the past, pilots typically used visual inspection to assess icing levels and determine whether to activate the heating system for de-icing. However, due to limited cockpit visibility, pilots generally could only assess icing conditions during flight by observing the windshield wipers, the windshield, and a portion of the wing's upper edge.

[0004] To more comprehensively detect icing conditions, various icing detectors have been developed. For example, many airlines initially used icing detectors installed on the exterior of the aircraft skin, which had a destructive impact on the aircraft's aerodynamic performance. Furthermore, these detectors could only preliminarily detect the presence of icing, leading pilots to blindly activate anti-icing systems without knowing the thickness or type of icing. However, since de-icing systems often consume large amounts of fuel, the lack of real-time monitoring of icing thickness and de-icing progress resulted in fuel waste, which was unacceptable for airlines. Aerospace vibration probe icing detectors have been used on the Boeing 767-300F. However, this method only provides suggestive signals and cannot identify the thickness or type of ice. Moreover, the installation of this system is a destructive process, requiring cutting into the aircraft skin to accommodate its outer shell. To detect icing thickness non-destructively, airlines have developed fiber optic icing sensors that act as light intensity modulation sensors, indirectly measuring icing thickness based on light intensity. However, due to the complexity of this detection system, precise installation methods or procedures are often required. Furthermore, fiber optic sensors struggle to detect subtle changes in optical signals during the freezing process of thin water films, which limits their ability to characterize the freezing properties of water films.

[0005] Recently, ultrasonic nondestructive testing technology has been used to detect aircraft icing. For example, ultrasonic pulse-echo technology has been used to study and calculate the thickness of ice buildup on the top of aircraft. Pulse-echo ultrasonic testing uses the same principle as bats, whose sounds have ultrasonic frequencies. These sounds are transmitted, and the bat receives its echo. From the time interval between the emitted sound and the received echo, and the angle of the received echo, the bat can locate the distance to the target. Similarly, by using a point source to send a signal, we can receive the reflected wave field at the receiver. Most pulse-echo ultrasonic methods use the flight time through the surface to determine its thickness. However, as the film thins, the reflected echoes overlap. Therefore, under the influence of thin ice layers and variable ice properties, ultrasonic pulse-echo technology cannot accurately measure ice thickness.

[0006] Despite significant progress in understanding the characteristics of airframe icing and developing detection methods, considerable challenges and limitations remain. A prominent challenge is accurately measuring the thickness and properties of ice. Ice is an anisotropic and layered material, making its properties and structure highly complex. Therefore, traditional methods struggle to meet the requirements in terms of measurement accuracy and detection efficiency, especially when dealing with complex ice geometries. Furthermore, some detection methods may adversely affect the aircraft's aerodynamics.

[0007] Therefore, a new technical solution is needed to solve the above problems. Summary of the Invention

[0008] Objective of the Invention: To address the aforementioned technical problems, this invention provides an application of ultrasonic full-waveform inversion to image the velocity and density characteristics and geometry of aircraft icing. This invention revolves around utilizing ultrasonic full-waveform inversion as a model-based ultrasonic nondestructive testing technique. This method relies on comprehensive numerical simulations of ultrasonic wave propagation. Ultrasonic full-waveform inversion involves the inversion of the wave equation, which describes the propagation of ultrasonic waves in a medium. By iteratively adjusting the medium properties (including its velocity and density) to maintain consistency with the observed ultrasonic wave shape, full-waveform inversion achieves its parameter reconstruction purpose.

[0009] Technical Solution: The method for aircraft icing characteristics and shape reconstruction based on ultrasonic full waveform inversion disclosed in this invention can adopt the following technical solution.

[0010] A method for aircraft icing characteristics and shape reconstruction based on ultrasonic full-waveform inversion includes the following steps:

[0011] Step 1, Forward Modeling: Using an initial model with sound speed and density as attributes and a known sound source, calculate the synthetic wave field data and residual data of the entire initial model using acoustic equations;

[0012] Step 2, calculate the objective function: This objective function represents the degree of misfit between the synthetic data and the receiver's actual observation data, and is obtained from the residual data between the receiver's actual observation data and the synthetic data described in Step 1;

[0013] Step 3: Using the receiver as a virtual source, time reversal is performed using acoustic equations, and the residual data described in Step 2 is backpropagated to the model described in Step 1 to calculate the associated wave field of the residual data.

[0014] Step 4: Using the wavefield from Step 2 and Step 3, calculate the gradient of the objective function in terms of density and velocity.

[0015] Step 5: Estimate the step size using the straight-line search method, and update the velocity and density parameters of the initial model described in Step 1;

[0016] Step 6: Using the new model, iteratively repeat the process starting from Step 1 until the objective function described in Step 2 is small enough or the available memory is exhausted.

[0017] Furthermore, in step 1, the initial model where sound speed and density are attributes and the sound source is known is specifically as follows:

[0018] A computer program model is used to create a five-layer model, each with different material properties. The first layer represents the aircraft skin and is made of aluminum with a thickness of several millimeters. On top of the aluminum layer, two layers of ice with different densities but the same thickness are placed. Then, a water layer with the same thickness as the ice layer is placed on top of the outermost ice layer. Finally, the remaining model is filled with an infinitely thick layer of air. A signal source and a linear receiver array made of ultrasonic transducers are placed at the center of the aluminum layer. The signal source emits ultrasonic waves inside the model and collects the backscattered signals at the receivers.

[0019] Furthermore, in step 1, the synthesized wavefield data and its residual data include:

[0020] Step 1.1: After summing all wavefields for each receiver, the synthesized wavefield data is obtained using the following formula:

[0021]

[0022] Where N represents the total number of receivers, This represents the composite wave field at position x at time t. This represents the wave field acquired by the i-th receiver;

[0023] Step 1.2: Use finite differentials to derive the acoustic equations considering velocity, density, material properties, and source terms: Consider a two-dimensional spatial domain. and time Time is at the initial time and final time Between these intervals, the first-order parameterized equation of the sound wave varies with the compression rate of each layer. and density As a variable, it is obtained from the following formula.

[0024]

[0025]

[0026] In the formula, It is the source term function Pressure wave field, It's the compression speed. It's density. It is the gradient (Laplace operator) , These are medium coordinates;

[0027] Step 1.3: Calculate the residual data: The residual data ∆d represents the observed pressure wave field. and synthetic pressure wave field The difference, i.e., the difference between the synthetic data and the observed data, is calculated by the following formula.

[0028]

[0029] in, That is, in step 1.2 .

[0030] Furthermore, the objective function in step 2 Calculate using the residual data ∆d obtained in step 1.3 and the following formula.

[0031] .

[0032] Furthermore, in step 3, the calculation of the associated wave field of the residual data is as follows: using the receiver as a virtual source, time reversal is performed using acoustic equations, and the residual data ∆d described in step 1.3 is backpropagated to the model described in step 1.1 to calculate the associated wave field of the residual data ∆d, calculated according to the following formula.

[0033]

[0034] in, The associated wave field of the residual data ∆d described in step 1.3.

[0035] Furthermore, in step 4, the gradient of the objective function is calculated based on density and velocity using the following formula.

[0036] =

[0037] in, It is the backwave field of the residual data ∆d mentioned in step 1.3. It is the source term function described in step 1.2 Forward wave field.

[0038] Furthermore, step 5 specifically includes the following steps:

[0039] Step 5.1: Estimate the step size using the steepest descent method;

[0040] The steepest descent direction is given by the following formula.

[0041] ;

[0042] Use conjugate gradient direction To control the model's update speed, the conjugate gradient direction By step size The calculation yields the parameter. According to Hestenes-Stiefel Step size and Dai-Yuan Step size is calculated:

[0043]

[0044]

[0045]

[0046]

[0047] Calculate the step size using the linear search method. The formula is:

[0048]

[0049] in, Represents the Jacobian matrix. and These represent the forward wavefields of the synthetic data and the experimental data, respectively. Indicates the inner product;

[0050] Step 5.2, Update Model Parameters: In the (k+1)th iteration, the conjugate gradient method from Step 2 is used to iteratively update the model. The update process can be expressed by the following formula.

[0051]

[0052] In the formula, This represents the model parameter vector for the k-th iteration; This represents the model parameter vector at the (k+1)th iteration. Indicates the current update step size; This indicates the current update direction of the model parameter vector.

[0053] Furthermore, in step 6, the process is iteratively repeated from step 1 using the new model until the objective function described in step 2 is small enough or the available memory is exhausted.

[0054] The present invention also provides a computer device, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the steps described in the above method.

[0055] The present invention also provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps of the above-described method.

[0056] Beneficial Effects: Compared to existing technologies, the aircraft icing characteristics and shape reconstruction method based on ultrasonic full-waveform inversion proposed in this invention integrates the advantages of ultrasonic non-destructive testing, extracting all information about the ice layer on top of the aircraft skin using the entire ultrasonic field data. This method allows for the simultaneous and high-precision reconstruction of the model's shape and the internal properties of the medium. In the aerospace field, both the source and receiver are placed inside the aircraft skin, allowing for the observation of velocity changes within the model without affecting the aircraft's structural aerodynamics.

[0057] Ultrasonic full-waveform inversion holds great potential as a solution to overcome the limitations of traditional icing measurement methods. Its non-invasive nature, high-resolution imaging capabilities, real-time monitoring potential, and versatility make it a promising method for accurately characterizing airframe icing properties. Attached Figure Description

[0058] Figure 1 This is the simulation model setup described in the specific embodiments of the present invention;

[0059] Figure 2 The diagram shows the iteration results at different stages in a specific embodiment of the present invention.

[0060] Figure 3 This is a comparison diagram of the signals of the observation model, the initial model, and the reconstructed model described in a specific embodiment of the present invention;

[0061] Figure 4 This is the setup for the multilayer media simulation model described in a specific embodiment of the present invention;

[0062] Figure 5 This is a comparison diagram of the signal results between the multilayer medium observation model and the initial model in a specific embodiment of the present invention;

[0063] Figure 6 This is a diagram showing the reconstruction result of the multilayer media according to a specific embodiment of the present invention. Detailed Implementation

[0064] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be noted that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.

[0065] This invention provides a method for constructing aircraft icing characteristics and geometry using ultrasonic full-waveform inversion, comprising the following steps:

[0066] Step 1, forward modeling: using an initial model with sound speed and sound density as attributes and the sound source known, calculate the forward wave field (also known as synthetic data) and its residual data of the entire model using acoustic equations;

[0067] Step 2: Calculate the objective function, which represents the degree of misfit between the synthetic (computational) data and the experimental (observational) data. This function can be obtained from the residual between the receiver's actual observation data and the synthetic data described in Step 1.

[0068] Step 3: Using the receiver as a virtual source, time reversal is performed using acoustic equations, and the residual data described in Step 2 is backpropagated to the model described in Step 1 to calculate the associated wave field of the residual data.

[0069] Step 4: Using the wavefield from Step 2 and Step 3, calculate the gradient of the objective function in terms of density and velocity.

[0070] Step 5: Estimate the step size using the linear search method, and update the velocity and density parameters of the model described in Step 1;

[0071] Step 6: Using the new model, iteratively repeat the process starting from Step 1 until the objective function described in Step 2 is small enough or the available memory is exhausted.

[0072] Step 1 specifically includes the following steps:

[0073] Step 1.1: Design an initial computer model with 5 layers using Python, each layer having different material properties.

[0074] Table 1 provides detailed parameters for the initial model. The first layer represents the aircraft skin and is made of aluminum with a thickness of 3 mm. On top of the aluminum layer, two layers of ice with different densities and a thickness of 1 mm are placed. Subsequently, a water layer with a thickness of 1 mm is placed on top of the ice layers. Finally, the remaining model is filled with an infinitely thick layer of air. The total depth of the model is 100 mm. A 5 MHz Gaussian source is placed at the center of the aluminum layers, and a linear receiver consisting of 52 transducers is placed at each grid point.

[0075] Aluminum (-5ºC) 6480 2690 First ice layer (-5ºC) 3980 917.6 Second ice layer (0ºC) 3713 916.8 Water (0ºC) 1403 1000 Air 343 1200

[0076] Table 1 Parameters of each layer in the initial model

[0077] Table 2 provides detailed parameters for the transducers used. The signal source emits ultrasonic waves inside the model and collects the backscattered signals at the receiver.

[0078] Center frequency 5MHz Pulse duration 3 seconds Source wavelet A Gaussian distribution (at the center of the model). receiver (5:15:90) One receiver every 15 millimeters Model size 100mm x 50mm

[0079] Table 2 Detailed parameters of the transducer

[0080] After summing all wavefields for each receiver, the synthesized wavefield data is obtained using the following formula:

[0081]

[0082] Where N represents the total number of receivers, This represents the composite wave field at position x at time t. This represents the wave field acquired by the i-th receiver.

[0083] Step 1.2: Use finite differential calculus to derive the acoustic equations considering velocity, density, material properties, and source terms. Assume a two-dimensional spatial domain. and time Time is at the initial time and final time The interval between. Combining the equations of conservation of momentum and mass, the first-order parameterized equation of sound waves is expressed with respect to the compression velocity varying with each layer. and density As a variable, it is obtained from the following formula.

[0084]

[0085]

[0086] In the formula, It is the source term function Pressure wave field, It's the compression speed. It's density. It is the gradient (Laplace operator) , These are medium coordinates.

[0087] Step 1.3: Calculate the residual data. The residual data ∆d represents the observed pressure wave field. and synthetic pressure wave field The difference, i.e., the difference between the synthetic data and the observed data, is calculated by the following formula.

[0088]

[0089] in, That is, in step 1.2 .

[0090] Step 2 involves calculating the objective function. The objective function, also known as the misfit function, represents the degree of misfit between the synthesized (computational) data and the observed (experimental) data. It is calculated using the residual data ∆d obtained in step 1.3 and the following formula.

[0091]

[0092] In the formula, It is the objective function calculated based on the pressure wave field data from the synthetic model and the observation model.

[0093] Step 3 involves calculating the associated wave field of the residual data. Using the receiver as a virtual source, time reversal is performed using acoustic equations. The residual data ∆d described in step 1.3 is backpropagated into the model described in step 1.1 to calculate the associated wave field of the residual data ∆d, calculated using the following formula.

[0094]

[0095] in, The associated wave field of the residual data ∆d described in step 1.3.

[0096] Step 4 involves gradient modeling. Using the wavefield from steps 2.1 and 3.1, the gradient of the objective function is calculated based on the density and velocity, obtained using the following formula.

[0097] =

[0098] in, It is the backwave field of the residual data ∆d mentioned in step 1.3. It is the source item to which step 1.2 belongs. Forward wave field.

[0099] Step 5 specifically includes the following steps:

[0100] Step 5.1: Estimate the step size using the steepest descent method. The direction of the steepest descent is given by the following formula.

[0101]

[0102] In addition, to improve the fit between the synthetic model and the observed data through iteration, conjugate gradient directions are used. To control the model's update speed. Conjugate gradient direction. It can be done by step size The calculation yields the parameter. According to Hestenes-Stiefel Step size and Dai-Yuan Step size is calculated:

[0103]

[0104]

[0105]

[0106]

[0107] Calculate the step size using the linear search method. The formula is:

[0108]

[0109] in, Represents the Jacobian matrix. and These represent the forward wavefields of the synthetic data and the experimental data, respectively. This indicates the inner product.

[0110] Step 5.2, update the model parameters. Since the initial model is set to be uniform, and the initial velocity... It is an estimate, therefore it is used in step 1.2. To calculate the pressure wave field at the k-th iteration. If the wave field data calculated from step 1.2 fits the observed wave field data, it means that the residual in step 1.3 is very small or negligible, and further illustrates the calculated... The speed is the same as the real model. Otherwise, in the (k+1)th iteration, the model is iteratively updated using the conjugate gradient method in step 2, and the update process is expressed by the following equation.

[0111]

[0112] In the formula, This represents the model parameter vector for the k-th iteration; This represents the model parameter vector at the (k+1)th iteration. Indicates the current update step size; This indicates the current update direction of the model parameter vector.

[0113] In step 6, the process is iteratively repeated from step 1 using the new model until the objective function described in step 2 is small enough or the available memory is exhausted.

[0114] This invention proposes a method for reconstructing aircraft icing characteristics and geometry using ultrasonic full-waveform inversion. It integrates the advantages of ultrasonic non-destructive testing, extracting all information about the ice layer on top of the aircraft skin using data from the entire ultrasonic field. This method allows for the simultaneous and high-precision reconstruction of the model's shape and the internal properties of the medium. In the aerospace field, both the source and receiver are placed within the aircraft skin, enabling the observation of velocity changes within the model without affecting the aircraft's structural aerodynamics.

[0115] like Figure 1 As shown in (a), the background size of the observation model is 500*500mm, verified by numerical simulation in Python. A grid spacing of 10mm and 51 grid points in each direction were used, specifying a P-wave velocity of 3.0km / s and a density of 3.298kg / m³ for the background. 3 The goal of this verification experiment is to reconstruct a square perturbation, 100*100mm in size, located at the center of the model. This perturbation is characterized by a P-wave velocity of 4.5 km / s and a density of 3.576 kg / m³. 3 For data acquisition, 51 receivers were linearly placed at the top of the model, each receiver located at a grid point. Additionally, a Ricker source wavelet with a peak frequency of 20 Hz was introduced at the top center of the model.

[0116] from Figure 1 (b) begins with the initial model, which has the same dimensions and material properties as the background model observed. The primary objective is to reconstruct the shape and properties of the square perturbation within the model.

[0117] Figure 2 This provides a process of how the model evolves during iterations. Clearly, with each iteration, the shape of the model becomes increasingly apparent, even though the model's speed changes simultaneously. By the 100th iteration, a recognizable model shape is obtained, such as... Figure 2 As shown in (e). However, the model properties still deviated significantly from the observed velocity model. Subsequently, simulations were continued until 500 iterations were reached, successfully reconstructing the properties of the square model, as shown in (e). Figure 2 As shown in (f).

[0118] To verify the accuracy of the reconstructed model, forward modeling was performed using it, and the signals from the observed model, the initial model, and the reconstructed model were compared, such as... Figure 3 As shown. A tiny difference was found in the reconstructed background, but it was considered negligible.

[0119] A model was constructed using the parameters listed in Table 1, as follows: Figure 4 The model shown in (a) consists of an aluminum layer, followed by a nonlinear ice layer and another ice layer, topped with an air layer. A single-point source located at the top of the model emits a 5MHz Gaussian pulse. To improve the model's imaging performance, a receiver is placed every two imaging grid points, for a total of 51 receivers. The signal propagation time within the model is sufficient to reach 3 seconds. The received data, i.e., the sum of all data from each receiver, is as follows: Figure 5 As shown.

[0120] By using the method of this invention, it is possible to reconstruct the observed model from the initial model after 1000 iterations, such as... Figure 6 As shown in (a). Figure 6 As shown in (c), when comparing the wavefield data obtained from the initial model with the wavefield data from the reconstructed model, a significant change can be observed; the reconstructed wavefield is very similar to the observed wavefield data. Furthermore, Figure 6 (a) shows that the shape could not be initially discerned from the wavefield data of the reconstructed model. However, by updating the initial model, the shape of the ice layer was able to be extracted from the final model.

[0121] The observed velocity distribution and Figure 6 Comparing the reconstructed velocity distribution in (b) clearly demonstrates the effectiveness of the method of the present invention. Furthermore, all small errors in the model are easier to identify than those in the velocity curve. Through this model reconstruction process, the shape of the ice layer can be visualized based on its velocity.

[0122] In summary, the method of constructing aircraft icing characteristics and geometry based on the full waveform inversion described in this invention is effective in both numerical simulations and multi-layered media model simulations of aircraft skin, ice, and air.

[0123] The above are merely preferred embodiments of the present invention. The scope of protection of the present invention is not limited to the above embodiments. All technical solutions falling within the scope of the present invention's concept are within the scope of protection of the present invention. It should be noted that for those skilled in the art, any improvements and modifications made without departing from the principles of the present invention should be considered within the scope of protection of the present invention.

Claims

1. A method for aircraft icing characteristics and shape reconstruction based on ultrasonic full-waveform inversion, characterized in that, Includes the following steps: Step 1, Forward Modeling: Using an initial model with sound speed and density as attributes and a known sound source, calculate the synthetic wave field data and residual data of the entire initial model using acoustic equations; The initial model, with sound speed and density as attributes and the sound source known, is as follows: A computer program model with 5 layers, each with different material properties; the first layer represents the aircraft skin and is made of aluminum with a thickness of several millimeters; on top of the aluminum layer, two layers of ice with different densities and the same thickness are set; then, a water layer with the same thickness as the ice layer is set on top of the outermost ice layer; finally, the remaining model is filled with an infinitely thick air layer; a signal source and a linear receiver array made of ultrasonic transducers are placed at the center of the aluminum layer. The signal source emits ultrasonic waves inside the model and collects the backscattered signals at the receiver; Synthetic wavefield data and its residual data include: Step 1.1: After summing all wavefields for each receiver, the synthesized wavefield data is obtained using the following formula: ， Where N represents the total number of receivers, This represents the composite wave field at position x at time t. This represents the wave field acquired by the i-th receiver; Step 1.2: Use finite differentials to derive the acoustic equations considering velocity, density, material properties, and source terms: Consider a two-dimensional spatial domain. and time Time is at the initial time and final time Between these intervals, the first-order parameterized equation of the sound wave varies with the compression rate of each layer. and density As a variable, it is obtained from the following formula. ， ， In the formula, It is the source term function Pressure wave field, It's the compression speed. It's density. It's the gradient, the Laplace operator. , These are medium coordinates; Step 1.3: Calculate the residual data: The residual data ∆d represents the observed pressure wave field. and synthetic pressure wave field The difference, i.e., the difference between the synthetic data and the observed data, is calculated by the following formula. ， in, That is, in step 1.2 ; Step 2, calculate the objective function: This objective function represents the degree of misfit between the synthetic data and the receiver's actual observation data, and is obtained from the residual data between the receiver's actual observation data and the synthetic data described in Step 1; Step 3: Using the receiver as a virtual source, time reversal is performed using acoustic equations, and the residual data described in Step 2 is backpropagated to the model described in Step 1 to calculate the associated wave field of the residual data. Step 4: Using the wavefield from Step 2 and Step 3, calculate the gradient of the objective function in terms of density and velocity. Step 5: Estimate the step size using the straight-line search method, and update the velocity and density parameters of the initial model described in Step 1; Step 6: Using the new model, iteratively repeat the process starting from Step 1 until the objective function described in Step 2 is small enough or the available memory is exhausted.

2. The method for reconstructing aircraft icing characteristics and shape according to claim 1, characterized in that, The objective function in step 2 Calculate using the residual data ∆d obtained in step 1.3 and the following formula. 。 3. The method for reconstructing aircraft icing characteristics and shape according to claim 1, characterized in that, In step 3, the calculation of the associated wave field of the residual data is as follows: using the receiver as a virtual source, time reversal is performed using acoustic equations, and the residual data ∆d described in step 1.3 is backpropagated to the model described in step 1.1 to calculate the associated wave field of the residual data ∆d, using the following formula. ， in, The associated wave field of the residual data ∆d described in step 1.

3.

4. The method for reconstructing aircraft icing characteristics and shape according to claim 1, characterized in that, In step 4, the gradient of the objective function is calculated based on density and velocity using the following formula. = ， in, It is the backwave field of the residual data ∆d mentioned in step 1.

3. It is the source term function described in step 1.2 Forward wave field.

5. The method for reconstructing aircraft icing characteristics and shape according to claim 1, characterized in that, Step 5 specifically includes the following steps: Step 5.1: Estimate the step size using the steepest descent method; The steepest descent direction is given by the following formula. ； Use conjugate gradient direction To control the model's update speed, the conjugate gradient direction By step size The calculation yields the parameter. According to Hestenes-Stiefel Step size and Dai-Yuan Step size is calculated: ， ， ， ， Calculate the step size using the linear search method. The formula is: ， in, Represents the Jacobian matrix. and These represent the forward wavefields of the synthetic data and the experimental data, respectively. Indicates the inner product; Step 5.2, Update Model Parameters: In the (k+1)th iteration, the conjugate gradient method from Step 2 is used to iteratively update the model. The update process can be expressed by the following formula. ， In the formula, This represents the model parameter vector for the k-th iteration; This represents the model parameter vector at the (k+1)th iteration. Indicates the current update step size; This indicates the current update direction of the model parameter vector.

6. A computer device, characterized in that, It includes a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor, when executing the computer program, implements the steps of any one of claims 1 to 5.

7. A computer-readable storage medium, characterized in that, It stores a computer program thereon, which, when executed by a processor, implements the steps of the method described in any one of claims 1 to 5.

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