A sequential Bayesian inversion method for geoacoustic parameters based on the time difference of arrival of shallow sea dual-node modes

Abstract

The present invention relates to a sequential Bayesian inversion method for geoacoustic parameters based on the modal arrival time difference of a shallow - sea two - node. By using two - node sensors to synchronously measure the broadband pulse signals emitted by a fixed sound source on one side (the outer side) of the two - node connection line, the modal arrival time difference data between the two received signals is extracted. The two - node sensors are moved, always ensuring that the sound source is on one side of the two - node connection line, and the measurement is repeated. Combining the measurement data and the Kraken sound field calculation model, the state equation and the measurement equation required in the sequential Bayesian inversion are established, the sequential posterior probability density distribution function of the geoacoustic parameters is inferred, and the posterior estimation mean value of the distance - related geoacoustic parameters is obtained. Compared with the prior art, its superiority lies in that: by using two - node sensors, the normal - mode observation data containing local distance - independent environmental characteristics can be obtained, avoiding the complex calculation of the coupled normal - mode model during the inversion of geoacoustic parameters, realizing the rapid inversion of geoacoustic parameters in a shallow - sea slowly - varying environment. This method has good robustness and high efficiency, and the two - node unit can be deployed flexibly. Through mobile or distributed deployment observations, multi - dimensional inversion of geoacoustic parameters from point to line and then to surface can be achieved.

Description

Technical Field:

[0001] The present invention belongs to the fields of underwater acoustic engineering, ocean engineering, sonar technology, etc., and relates to a sequential Bayesian geoacoustic parameter inversion method based on the arrival time difference of shallow sea dual-node modes, which uses the dual-node mode dispersion characteristic data to track the geoacoustic parameters varying with distance, and is applicable to the shallow sea waveguide environment with weak distance correlation. Technical Background:

[0002] The waveguide environment parameters mainly include the acoustic characteristics parameters of water body and bottom sediment, such as: sea depth, sound speed, density, attenuation coefficient, etc. These parameters are of great significance for sound field prediction, sonar system design and performance evaluation, etc. Among them, the acoustic characteristics parameters of water body can be quickly and accurately obtained by direct measurement means. However, for the acoustic characteristics parameters of bottom sediment, also known as geoacoustic parameters, direct measurement usually consumes a large amount of manpower and material resources, and only the high-frequency (>10kHz), shallow surface layer (<3m) results near the station can be obtained. Therefore, it is particularly important to use underwater acoustic remote sensing means to perform parameter inversion based on theoretical models. Geoacoustic parameter inversion has many advantages such as low cost, fast speed and wide range. The relatively mature inversion techniques include matching inversion based on optimization algorithms, linear perturbation inversion, Bayesian inversion, deep learning, etc. These methods model and solve the inverse problem from different perspectives.

[0003] In addition to using the directly observed complex sound pressure field and vector field data, the existing inversion techniques also use one or more indirect observation data including mode dispersion, seabed reflection loss, noise correlation and reverberation, etc. Different types of data contain different seabed environment information. By comprehensively using various data, reliable estimation of geoacoustic parameters can be achieved. However, the real shallow sea environment is complex and diverse, and the geoacoustic parameters have certain non-uniformity in the vertical and horizontal directions. The existing inversion techniques usually assume that the large and medium scale waveguide environment (greater than 5km) between the sound source and the receiver is horizontally uniform, and its inversion result usually represents the average characteristics within the entire experimental area. Therefore, when the horizontal non-uniformity of the waveguide environment cannot be ignored, the traditional techniques will no longer be applicable. We need to find an underwater acoustic observable that is only sensitive to the local waveguide environment, first complete the rapid inversion of the local distance-independent environment parameters, and then complete the sequential inversion of the large scale distance-dependent environment parameters.

[0004] The methods for obtaining local underwater acoustic observables are generally divided into two categories. One is the direct measurement method, that is, using a compact observation system, such as a monostatic towed system, to directly measure the near-field acoustic data and obtain local acoustic observables within a small range. The other is the indirect measurement method, usually using a large-aperture moored synchronous or asynchronous multi-sensor system, such as a vertical array and a horizontal array, combined with a moving / fixed sound source for far-field large-range acoustic observation. After obtaining the far-field data, the local acoustic observables within a small range are extracted using relevant underwater acoustic theories. Currently, in traditional acoustic survey experiments, the large-aperture moored transmitting or receiving system is more widely used than the compact towed system. Therefore, the research on obtaining local underwater acoustic observation data by indirect means has more practical application value.

[0005] In the shallow sea environment, the propagation of acoustic signals can usually be described as the propagation of different-order normal mode modes. The propagation speeds of different modes are different at different frequencies. Therefore, the received signal at a certain distance will exhibit intra-mode dispersion and inter-mode dispersion phenomena in the time-frequency domain. By extracting the dispersion characteristics and using the shape of the dispersion curve as an observable for geoacoustic parameter inversion. However, most inversion methods based on modal dispersion characteristics use a single sensor node and assume that the waveguide environment between the transmitter and the receiver is horizontally homogeneous. Therefore, they cannot be directly applied to the distance-related waveguide environment. Based on the adiabatic normal mode theory, the present invention adopts the means of indirectly obtaining local underwater acoustic observables, uses a mobile / distributed dual-node sensor to obtain multi-snapshot modal arrival time difference data that is only related to local geoacoustic parameters, and realizes the sequential inversion of distance-weakly related geoacoustic parameters in a horizontally inhomogeneous environment. Summary of the Invention:

[0006] Technical Problems to be Solved

[0007] In order to solve the practical problems that the existing shallow sea geoacoustic parameter inversion methods are difficult to apply in horizontally inhomogeneous environments and large-aperture far-field measurements, the present invention proposes a sequential Bayesian geoacoustic parameter inversion method based on the modal arrival time difference of shallow sea dual nodes.

[0008] The object of the present invention is achieved by the following technical solutions:

[0009] A sequential Bayesian geoacoustic parameter inversion method based on the modal arrival time difference of shallow sea dual nodes, which includes the following steps:

[0010] Step 1: Use a dual-node sensor to synchronously measure the broadband pulse sound source signal emitted from a fixed location on one side (outer side) of the line connecting the two nodes. When the k-th signal is emitted, the horizontal distance between the sound source and node 1 is r1(k), and the horizontal distance between the sound source and node 2 is r2(k). Denote this as the k-th snapshot, which is the k-th geophysical acoustic parameter inversion station position. According to the requirements of different actual inversion stations, change the position of the dual-node sensor to ensure that the sound source is always on one side of the line connecting the two nodes, and repeat the measurement.

[0011] Step 2: Use the dispersion cancellation transform to extract the arrival times of the dual-node modes for the k-th snapshot. Assume that the extraction results for the m-th mode are t (m) [f, r1(k)] and t (m) [f, r2(k)]. Then the observed value of the arrival time difference of the m-th dual-node mode is:

[0012] Δt (m) (f, k) = t (m) [f, r1(k)] - t (m) [f, r2(k)],

[0013] where f represents the signal frequency. According to the adiabatic normal mode theory, the time difference Δt (m) (f, k) contains the local environmental information between the two nodes. In practical applications, use the midpoint distance [r1(k) + r2(k)] / 2 between the two nodes as the distance for the k-th snapshot.

[0014] Step 3: Assume that the seabed model between the two nodes is a distance-independent sedimentary layer - basement double-layer model. In the sedimentary layer, the compressional wave speed varies linearly with depth. The unknown geophysical acoustic parameters include thickness h1, upper sound speed c 1U , lower sound speed c 1L and density ρ1. In the basement layer, the unknown geophysical acoustic parameters include compressional wave speed c2 and density ρ2. The geophysical acoustic parameters are different for different snapshots and can be represented in vector form as x k = [h1(k), c 1U (k), c 1L (k), ρ1(k), c2(k), ρ2(k)]·, k = 1, …, K, where K represents the number of snapshots, which is the number of stations for geophysical acoustic parameter inversion. Use the Kraken acoustic field model to calculate the group velocity value of the m-th mode Combined with the distance parameters r1(k) and r2(k), calculate the theoretical value of the arrival time difference of the m-th dual-node mode as:

[0015]

[0016] Step 4: Combine the observed value of the arrival time difference of the dual-node mode Δt given in Step 2(m) (f, k) and the theoretical value of the two-node modal arrival time difference ΔT given in step 3 (m) (f, x k ), construct the measurement equation and the state equation, then the sequential posterior probability density distribution function expression of the geoacoustic parameter x k is:

[0017]

[0018] where y k represents Δt (m) (f, k), which satisfies the observation independence assumption in the general sequential Bayesian theory. Y k-1 and Y k respectively represent the sets of modal arrival time difference observation vectors of the first k - 1 and the first k snapshots, that is, Y k-1 ={y1, y2, …, y k-1} and Y k ={y1, y2, ..., y k}. Solve the sequential posterior probability density distribution function to obtain the posterior estimation mean x′ k at different snapshots, that is, the inversion value of the distance-related geoacoustic parameter. On the premise that the two-node modal dispersion characteristics are distinguishable, the smaller the distance between the nodes, the higher the distance resolution of the geoacoustic parameter inversion.

[0019] The two-node sensor unit in step 1 can be part of a towed array or a distributed synchronous seafloor observation network.

[0020] The m-th order two-node modal arrival time difference observation value Δt (m) (f, k) in step 2 can be the water wave or bottom wave part of the normal mode.

[0021] The sequential posterior probability density distribution function p(x k ∣Y k ) in step 4 is solved using the particle filter algorithm.

[0022] The beneficial effects of the present invention are as follows: The present invention relates to a sequential Bayesian geoacoustic parameter inversion method based on the two-node modal arrival time difference in shallow water. Use two-node sensors to synchronously measure the broadband pulse signals emitted by a fixed sound source on one side (outer side) of the two-node connection line, and extract the modal arrival time difference data between the two received signals. Move the two-node sensors, always ensuring that the sound source is on one side of the two-node connection line, repeat the measurement, and combine the measurement data and the Kraken sound field calculation model to establish the state equation and measurement equation required in the sequential Bayesian inversion, infer the sequential posterior probability density distribution function of the geoacoustic parameter, and obtain the posterior estimation mean of the distance-related geoacoustic parameter.

[0023] The beneficial effects are as follows: In step one of this method, a mobile / distributed dual-node sensor unit is used to synchronously measure broadband pulse signals emitted by a fixed sound source at different / same moments, obtaining measurement data at multiple inversion stations. In step two, the measurement data is processed to obtain multiple groups of observed values of the dual-node mode arrival time difference. In step three, a local distance-independent seabed model and a theoretical calculation model of the dual-node mode arrival time difference are established. In step four, multiple groups of observed values and theoretical values of the dual-node mode arrival time difference are used to construct a measurement equation and a state equation, establish a sequential posterior probability density distribution function of the geoacoustic parameters, and use the particle filter algorithm for inference. The sequential posterior mean obtained is the inversion result of the distance-dependent geoacoustic parameters. This method tracks the distance-dependent geoacoustic parameters using multiple groups of observed data of the dual-node mode arrival time difference and the particle filter algorithm in steps one to four, filling the technical gap in segmentally inverting the distance-varying geoacoustic parameters using far-field large-scale acoustic observation data in a shallow sea horizontally inhomogeneous environment. This method has the following advantages:

[0024] 1) Divide the large-scale environment related to distance into several small and medium-scale environments independent of distance, realizing sequential segmented inversion of geoacoustic parameters in a complex inhomogeneous environment.

[0025] 2) The forward model is independent of distance, improving the inversion speed of geoacoustic parameters.

[0026] 3) The inversion of geoacoustic parameters combining the characteristics of normal mode water waves and bottom wave dispersion can achieve reliable estimation of shallow and deep geoacoustic parameters.

[0027] 4) The dual-node sensor unit has a simple structure and flexible deployment. Through mobile or distributed deployment, multi-dimensional inversion of geoacoustic parameters from point to line and then to surface can be realized.

[0028] 5) On the premise that the dual-node mode dispersion characteristics are distinguishable, the smaller the distance between nodes, the higher the distance resolution of the inversion of geoacoustic parameters. The refined inversion results of geoacoustic parameters can be used to correct the existing macroscopic bottom sediment acoustic database, thus better serving the sonar system. Description of the Drawings:

[0029] Figure 1 is the sound speed profile used in the simulation study of the method of the present invention.

[0030] Figure 2 is a schematic diagram of a test scenario and a seabed model of the method of the present invention, where the unknown geoacoustic parameters include: sediment layer thickness h1, compressional wave sound speed c 1U in the upper part of the sediment layer, compressional wave sound speed c 1L in the lower part of the sediment layer, sediment layer density ρ1, compressional wave sound speed c2 and density ρ2 of the basement layer, and the unknown water body parameter is water depth d w .

[0031] Figure 3 Schematic diagram of the modal arrival time extraction process of the method of the present invention: (a) Original signal; (b) Dispersion-removed transform signal; (c) 5th-order modal filtering signal; (d) 5th-order modal extraction signal.

[0032] Figure 4 Observation results and posterior prediction results of the modal arrival time difference between two nodes at different distances of the method of the present invention, where the distance parameter is corrected with the distance of the first snapshot as the zero point, and after correction, it is (a) 1 km; (b) 4 km; (c) 7 km; (d) 10 km.

[0033] Figure 5 Schematic diagram of the particle filter algorithm adopted by the method of the present invention, and the main steps include: prediction, update, and resampling.

[0034] Figure 6 Sequential inversion results of the geoacoustic parameters of the method of the present invention, where "+" is the true value and the solid line is the mean of the posterior estimate.

[0035] Figure 7 Schematic diagram of the distributed deployment of the two-node sensor unit of the method of the present invention. Specific implementation manner:

[0036] The present invention will be further described in combination with the embodiments and the accompanying drawings:

[0037] (1) Simulation calculation of data:

[0038] Figure 1 The measured sound speed profile in a certain shallow sea acoustic survey experiment is given. The seawater medium is evenly mixed, the sound speed near the sea surface is about 1516.5 m / s, and the sound speed near the seabed is about 1517.8 m / s. The subsequent simulation experiments will be based on this sound speed profile.

[0039] Figure 2 A simulation experiment environment of a shallow sea continental slope is given. As the distance increases, the sea depth slowly increases while the sediment layer thickness slowly decreases. In the simulation experiment, the sound source position is fixed, and the deployment depth is z s= 10 m. During this period, broadband pulse signals are repeatedly transmitted. The two-node hydrophone unit is deployed horizontally, with distances from the pulse sound source being r1 and r2 respectively, and r2 - r1 = 200 m is fixed. To ensure a stable and clear broadband dispersion structure is obtained in the experiment, it is usually required that r1 > 5 km. Therefore, the distances between the two nodes for the first snapshot are selected as 10 km and 10.2 km. After completing one measurement task, the two-node hydrophone unit is moved horizontally away from the sound source, with each movement distance being 200 m. This is repeated multiple times until the distances between the two nodes for the last snapshot are 20 km and 20.2 km. According to the movement trajectory of the two-node hydrophone unit, the distance-related simulation experimental environment is evenly divided into 51 distance-independent segments ( Figure 2 The shaded area in it shows one of the segments). In each segment, it is assumed that the seabed is a two-layer fluid medium of sedimentary layer - basement layer. Among them, the compressional wave sound speed of the sedimentary layer increases linearly with the increase of sediment thickness, while the density has nothing to do with the thickness. The compressional wave sound speed and density of the basement half-space layer are both constants, and the water sound speed profile does not change with distance, and all are taken from Figure 1 the sound speed profile shown. In short, the (unknown) parameters to be inverted in each segment include 6 geoacoustic parameters and the sea depth d w , and their names, units, and prior intervals for inversion are summarized in Table 1 as follows:

[0040] Table 1 Parameters to be inverted and prior intervals

[0041]

[0042]

[0043] Assume that the sound source signal is a Dirichlet function, with its frequency range being 50 Hz - 500 Hz, and the signal sampling rate f s = 2 kHz. Based on the above shallow sea simulation environment and the true values of distance-related parameters (as shown in Figure 6 ), the broadband pulse signals received by the two-node hydrophone at different distances are calculated using the parabolic equation (RAM) sound field model.

[0044] (2) Inversion of geoacoustic parameters varying with distance:

[0045] a) Data preprocessing: Extract the modal arrival times of the signals received by the two-node hydrophone. The extraction method is as shown in Figure 3As shown in the figure, it mainly includes four steps: from (a) to (b), the dispersion - free transform is used to resample the time - domain arrival signal, and the reference sound speed used in the transform is the average sound speed of the water body; from (b) to (c), the transformed signal is filtered in the time - frequency domain to separate the fifth - order modal signal after dispersion removal; from (c) to (d), the inverse dispersion - free transform is used to process the separated fifth - order modal signal to obtain the modal signal in the original time domain; from (d) to (a), the energy ridge detection is performed on the separated modal signal to obtain the fifth - order modal dispersion curve. The extraction processes of other orders of modes are the same. It should be noted that when extracting the modal dispersion curve, the reference time point needs to be accurately recorded to ensure that the arrival times of the double - node modes extracted are synchronized. After obtaining the arrival times of the double - node modes on different snapshots, calculate the arrival - time difference of the double - node modes for each snapshot. In this study, the third, fourth, fifth, and sixth - order modes are selected for the inversion study of geo - acoustic parameters, as Figure 4 shown. The calculation results of the 6th, 21st, 36th, and 51st snapshots are given respectively. Taking the distance of the first snapshot as the zero reference point, the distances of the four groups of data shown in the figure can be expressed as: 1 km, 4 km, 7 km, and 10 km. It can be seen that there are significant differences in the arrival - time difference data of the modes in different snapshots, which contain the information of unknown distance - related geo - acoustic parameters.

[0046] b) Parameter inversion: The problem of inverting the geo - acoustic parameters varying with distance is transformed into a sequential Bayesian parameter tracking problem. Based on the homogeneous Markov assumption and the observation independence assumption, the state equation (parameter evolution model) and the measurement equation (modal arrival - time difference calculation model) of the sequential Bayesian model are established, and their expressions are respectively:[[]]

[0047] x k =x k-1 +v k

[0048] y k =ΔT (m) (f,x k )+w k

[0049] where x k is the geo - acoustic parameter of the k - th snapshot (the parameter types are shown in Table 1), y k is the observation vector Δt (m) (f,k) of the arrival - time difference of the double - node modes at the k - th snapshot, ΔT (m) (f,x k ) represents the theoretical calculation model of the arrival - time difference of the double - node modes, and its core is the Kraken sound - field model. v k and w k are the state - transfer noise and the observation noise respectively, usually assumed to be zero - mean Gaussian - distributed noise.

[0050] The solution of the inverse problem is modeled in the form of a sequential posterior probability density distribution function, and its expression is:

[0051]

[0052] where Y k represents the set of modal time difference observation vectors of the first k snapshots, that is, Y k ={y1, y2,..., y k}. For the non - linear measurement equation, the particle filter algorithm is used to approximately infer p(x k |Y k ). The single - time filtering process is as Figure 5 shown, including three steps: prediction, update, and resampling. Among them, the resampling operation can effectively solve the problem of sample depletion during filtering. However, excessive resampling operations will greatly reduce the diversity of particles and produce inaccurate estimation results. Therefore, after the update step, by judging the number of effective particles, it is decided whether to perform resampling.

[0053] c) Result discussion: Figure 6 The sequential inversion results of the distance - related geo - acoustic parameters are given. Among them, the "+" symbol represents the true value of the parameter, the solid line represents the mean of the posterior estimate, and the shaded area describes the estimation uncertainty of each parameter. The lighter the color, the greater the value of the probability density function. It can be seen from the results that: the method of the present invention effectively estimates the distance - change characteristics of the parameters to be inverted. In particular, the estimation result of the sea depth d w is almost the same as the true value result, and the estimation uncertainty is the smallest; secondly, the estimation errors of the sediment layer thickness h1, the compressional wave sound speed c 1U in the sediment layer, the compressional wave sound speed c 1L under the sediment layer, and the sediment layer density ρ1 are relatively small, the estimation results are basically the same as the true value results, and they are all within the 95% confidence interval; however, the estimation errors of the basement sound speed c2 and the basement density ρ2 are the largest, which proves that the modal data used in this embodiment is not sensitive to the basement parameters. In the next step, it can be considered to jointly use the low - frequency or low - order bottom - wave dispersion characteristic data for geo - acoustic parameter inversion.

[0054] The present invention has achieved obvious implementation effects in typical simulation embodiments. Compared with the prior art, its advantages are:

[0055] (1) The double - node modal arrival time difference data is only related to the local geo - acoustic parameters between the two nodes, which conforms to the observation independence hypothesis in the general sequential Bayesian inversion theory. Through the distance - independent approximation of the local seabed model, the complex coupled normal - mode model calculation in the inversion process is effectively avoided.

[0056] (2) The sequential Bayesian inversion strategy effectively tracks the characteristics of the geoacoustic parameters varying with distance, and the inversion method has good robustness and high efficiency.

[0057] (3) The dual-node sensor unit can be deployed flexibly. As Figure 7 shown, through distributed deployment, multi-dimensional geoacoustic parameter inversion from point to line and then to surface can be realized.

Claims

1. A sequential Bayesian inversion method for geoacoustic parameters based on the time difference of arrival of modes between two nodes in shallow water, characterized in that The steps are as follows: Step 1: Use a dual-node sensor to synchronously measure the broadband pulse sound source signal emitted fixedly on one side of the dual-node connection line. When the k-th signal is emitted, the horizontal distance between the sound source and Node 1 is r1(k), and the horizontal distance between the sound source and Node 2 is r2(k), which is recorded as the k-th snapshot, that is, the k-th geosound parameter inversion station position; according to the requirements of different actual inversion station positions, change the position of the dual-node sensor to ensure that the sound source is always on one side of the dual-node connection line, and repeat the measurement; Step 2: Use the dispersion cancellation transform to extract the double-node modal arrival times of the k-th snapshot. Assume that the extraction results of the m-th mode are t (m) [f, r1(k)] and t (m) [f, r2(k)]. Then the observed value of the double-node modal arrival time difference of the m-th mode is: Δt (m) (f, k) = t (m) [f, r1(k)] - t (m) [f, r2(k)], where f represents the signal frequency. According to the adiabatic normal mode theory, the time difference Δt (m) (f,k) contains the local environmental information between two nodes; in practical applications, the midpoint distance [r1(k) + r2(k)] / 2 between the two nodes is used as the distance of the k-th snapshot; Step 3: Assume that the seabed model between two nodes is a distance-independent sediment-bedrock bilayer model. In the sediment layer, the compressional wave speed varies linearly with depth. The unknown geoacoustic parameters include the thickness h1, the upper sound speed c 1U , the lower sound speed c 1L and the density ρ1. In the bedrock layer, the unknown geoacoustic parameters include the compressional wave speed c2 and the density ρ2. The geoacoustic parameters are different for different snapshots and are represented in vector form as x k = [h1(k), c 1U (k), c 1L (k), ρ1(k), c2(k), ρ2(k)], k = 1, …, K, where K represents the number of snapshots, i.e., the number of stations for geoacoustic parameter inversion. Use the Kraken acoustic field model to calculate the group velocity value of the m-th mode Combined with the distance parameters r1(k) and r2(k), the theoretical value of the arrival time difference of the m-th two-node mode is calculated as follows: Step 4: Combine the double-node modal time difference of arrival observation value Δt (m) (f,k) given in Step 2 and the double-node modal time difference of arrival theoretical value ΔT (m) (f,x k ), construct the measurement equation and the state equation, then the expression of the sequential posterior probability density distribution function of the geoacoustic parameter x k is: Among them, y k represents Δt (m) (f, k), which satisfies the observation independence assumption in the general sequential Bayesian theory. Y k-1 and Y k respectively represent the sets of modal arrival time difference observation vectors of the first k - 1 and the first k snapshots, that is, Y k-1 ={y1, y2, …, y k-1} and Y k ={y1, y2, …, y k}. By solving the sequential posterior probability density distribution function, the posterior estimation mean x′ k is obtained, that is, the inversion value of the geoacoustic parameter related to the distance; on the premise that the modal dispersion characteristics of the two nodes can be resolved, the smaller the distance between the nodes, the higher the distance resolution of the geoacoustic parameter inversion.

2. The sequential Bayesian geoacoustic parameter inversion method based on the mode arrival time difference of two nodes in shallow water according to claim 1, wherein: The dual-node sensor unit in Step 1 is part of a towed array or a distributed synchronous seafloor observation network.

3. A sequential Bayesian inversion method for seafloor acoustic parameters based on the time difference of arrival of shallow sea dual-node modes according to claim 1, characterized in that: The arrival time difference observation value Δt of the m-th order two-node mode in the step 2 (m) (f,k) is the water wave or bottom wave part of the normal wave.

4. A sequential Bayesian inversion method for seafloor acoustic parameters based on the time difference of arrival of shallow sea dual-node modes, characterized in that: The sequential posterior probability density distribution function p(x k |Y k ) in step 4 is solved using the particle filter algorithm.

Images