Simulation of Non-Stationary Mobile Underwater Acoustic Communication Channels Based on a Multi-Scale Time-Varying Multipath Model

Abstract

Traditional Underwater Acoustic Communication (UAC) typically assumes static or slowly varying channels over short observation periods and models multipath amplitude fluctuations with single-state statistical distributions. However, field measurements in shallow-water high-speed mobile scenarios reveal that the combined effects of rapid platform motion and dynamic environments induce multi-scale time-varying amplitude characteristics. These include distance-dependent attenuation, fluctuations in average energy, and rapid random variations. This observation directly challenges traditional single-state models and wide-sense stationary assumptions. To address this, we propose a multi-scale time-varying multipath amplitude model. Using singular spectrum analysis, we decompose amplitude sequences into hierarchical components: large-scale components modeled via acoustic propagation physics; medium-scale components characterized by Hidden Markov Models; and small-scale components described by zero-mean Gaussian distributions. Building on this model, we further develop a time-varying impulse response simulation framework validated with experimental data. The results demonstrate superior performance over conventional single-state distribution and autoregressive models in statistical distribution matching, temporal dynamics representation, and communication performance testing. The model effectively characterizes non-stationary time-varying channels, supporting high-precision modeling and simulation for mobile UAC systems.

1. Introduction

Underwater Acoustic Communication (UAC), as a core technology for ocean exploration and development, relies fundamentally on channel modeling accuracy to ensure reliable information transmission links. The UAC channel exhibits inherent characteristics, such as high propagation loss, narrow available bandwidth, and strong background noise. Due to the low sound speed in water (approximately 1500 m/s), multipath propagation phenomena caused by surface/bottom reflections, water refraction, and object scattering are particularly pronounced [1]. This multipath effect can cause signal waveform distortion, leading to severe inter-symbol interference and deep fading, constituting one of the major bottlenecks restricting the performance of UAC. Therefore, multipath delay–amplitude models are widely used in UAC modulation schemes and network simulations. For instance, adaptive UAC systems generally require real-time amplitude parameters from channel state feedback [2,3,4].

In recent years, Unmanned Underwater Vehicle (UUV) technologies represented by autonomous underwater vehicles, remotely operated vehicles, and underwater gliders have experienced rapid development [5], leading to increasing demand for mobile UAC. In mobile UAC scenarios, the coupling effects of propagation geometry alterations caused by vehicle motion and the spatiotemporal heterogeneity of marine environments result in complex time-varying channel characteristics [6,7]. Such spatiotemporal coupled variations in multipath structures invalidate traditional static channel models. This degradation affects the adaptive prediction and tracking capabilities of UAC systems. Research [8] indicates that, even under slight relative transmitter–receiver drift, the average fading prediction error of static models increases twofold. Therefore, establishing an accurate time-varying multipath model is a critical theoretical foundation for enhancing mobile UAC system performance.

The early studies concentrated on statistical probabilistic models for path amplitudes. Chitre [9] modeled non-line-of-sight path amplitudes as independent Rayleigh distributions. Radosevic et al. [10] compared Rician, Nakagami-m, and lognormal distributions, finding Rician fits for shallow-water measurements. Kim et al. [11] observed the presence of Rayleigh and K-distribution amplitudes in shallow-water environments ranging from 6 to 12 kHz. Borowski [12] noted that shallow-water channel amplitudes transition from approximate Rician to Nakagami-m distributions with increasing distance. Li et al. [13] proved that inverse Gaussian distributions effectively characterize multipath clusters in deep-sea sound axis propagation. These models treat multipath amplitudes as single-state stochastic processes, neglecting platform motion-induced property variations.

With the advancement of computational power, ray tracing-based time-varying channel simulation methods have evolved to model multipath amplitude variations as discrete-time sequences. The approximation of motion-induced changes is achieved through the sampling of receiver/transmitter trajectories at signal rates, followed by the computation of arrival path parameters at each position using Bellhop [14]. The VIRtual Time series EXperiment (VirTEX) [15] platform motion algorithm applies to two-dimensional single-input–single-output scenarios with uniform motion, while the Time-Varying Acoustic Propagation Model (TV-APM) [16] supports UAC network simulations in three-dimensional single-input–multiple-output (SIMO) scenarios with uniform motion. The simulation accuracy of temporal variability in ray tracing-based methods depends on the number of sampling positions, and Ref. [17] reduced computational complexity through local spline interpolation. Although these models can reflect the acoustic propagation loss caused by vehicle motion, they remain inadequate in characterizing small-scale disturbances such as water turbulence and scattering objects.

Recent studies have begun to focus on dual-scale time-varying modeling. Tong et al. [18] observed dual-scale amplitude fluctuations through sea trials, attributing these to fast-varying paths caused by dynamic surfaces and relatively stable or slow-varying line-of-sight paths. Socheleau et al. [19] proposed a quasi-stationary model that uses Empirical Mode Decomposition (EMD) to separate multipath amplitudes into slow-varying trend components and fast-varying random components; the latter can be well approximated by a zero-mean Gaussian. Qarabaqi et al. [20] categorized channel time-variation into two scales: large-scale variations caused by displacements spanning multiple wavelengths, and small-scale variations caused by displacements of a few wavelengths. The former employ acoustic field calculations to obtain the propagation loss of dominant multipath components, while the latter use statistically equivalent functions derived from stochastic geometric models to characterize the superposition effect of scattered components. Combining both, path amplitude is equivalent to a Rician distribution with a slowly time-varying mean. Building upon this, Tu et al. [21] further incorporated the Hall–Novarini model to simulate wind-driven surface wave-induced small-scale variations and validated its accuracy using sea trial data collected in the Gulf of Mexico. Dual-scale time-variability has gained wide acceptance [22,23,24]. However, most existing models assume that, over short observation periods, the impact of vehicle motion manifests only as a Doppler time-scaling effect on the received signal, neglecting the temporally correlated fading trend arising from the continuously changing transmitter–receiver distance. Consequently, these models still fall short in adequately adapting to rapidly mobile scenarios.

We introduce the results from a field experiment to illustrate the complex time-varying characteristics of multipath amplitudes in rapidly mobile scenarios, as shown in Figure 1. The data originates from a shallow-water channel measurement with an average depth of 130 m, and the relative motion speed is about 15 knots (7.7 m/s). The path amplitude exhibits three distinct timescale variations: (1) over brief observation periods of 55 s, the amplitude attenuation trend correlates clearly with movement distance, distinguishing the paths; (2) within a sliding moving average window on the order of seconds, the average amplitude varies slowly between high and low levels; (3) alongside this, there are rapidly fluctuating random components with small energy. These phenomena directly reveal the limitations of traditional wide-sense stationary assumptions and dual-scale time-varying models, highlighting the need to incorporate motion-induced amplitude variation trends in UAC channel modeling.

In response to these experimental findings, we propose a multi-scale time-varying model to achieve more realistic multipath simulations that better align with real channel dynamics. The amplitude time-varying process is decomposed into three scale-specific components, modeling each, respectively: a large-scale propagation loss model based on acoustic propagation physics, a medium-scale shadow fading model established by a Hidden Markov Model (HMM), and a small-scale scattering fading model derived from measurement statistics. Ultimately, the overall time-varying amplitude sequence is generated through superimposing these components. Supported by shallow-water mobile channel measurements, the singular spectrum analysis (SSA) method is employed to decompose the measured data into three-scale components. We then fully validate the fitting goodness of the proposed model. Driven by the measured parameters, we further conduct a time-varying multipath channel simulation. These are compared against analyses of both classical single-state statistical distribution models and the first-order autoregressive (AR-1) model from [20] to illustrate the simulation fidelity of the proposed model. The evaluation metrics include the cumulative distribution function (CDF), temporal autocorrelation function (ACF), and communication system simulation Bit Error Rate (BER).

The remainder of this paper is organized as follows: Section 2 introduces the multi-scale time-varying multipath amplitude model and the SSA-based multi-scale decomposition method. Section 3 details the separate modeling of each component and proposes a measurement-driven time-varying impulse response (TVIR) simulation framework. Section 4 analyzes the measured channel characteristics and evaluates the goodness of fit of the proposed model. Section 5 conducts channel simulation and comparative performance evaluations. The conclusions and discussion are provided in Section 6.

2. Time-Varying Underwater Acoustic Channel Model

2.1. Multi-Scale Time-Varying Path Amplitude Model

The UAC channel is modeled as a linear random time-varying system so that the input $x\left(t\right)$ and the output $y\left(t\right)$ of this system satisfy

$$y\left(t\right)={\int }_{-\infty }^{\infty }h\left(t,\tau \right)x\left(t-\tau \right)d\tau +n\left(t\right)$$

(1)

where $h\left(t,\tau \right)$ is channel impulse response, t is the observation time, and $\tau$ is the relative delay. $n\left(t\right)$ represents additive noise. The TVIR reflects the multiplicative fading caused by the signal passing through the UAC channel.

The TVIR can be represented by a time-domain multipath amplitude-delay model as

$$h\left(t,\tau \right)=\sum _{l=1}^L{a}_l\left(t\right)\delta \left(t-{\tau }_l\left(t\right)\right)$$

(2)

where L is the number of propagation paths. $a_l\left(t\right)=A_l\left(t\right)e^{j{\theta }_l\left(t\right)}$ is a complex weight with the magnitude $A_l$ of the l-th path arriving with a delay ${\tau }_l\left(t\right)$. Considering micro-path superposition effects and propagation delay wandering, ${\theta }_l\left(t\right)$ is often assumed to follow a uniform distribution over $\left[-\pi ,\pi \right)$ [25].

Based on the observation phenomenon shown in Figure 1, we model the path amplitude $A_l\left(t\right)$ as the linear superposition of three components:

$$A_l\left(t\right)=A_T\left(t\right)+A_M\left(t\right)+A_S\left(t\right)$$

(3)

where $A_T\left(t\right)$ is the attenuation trend related to the distance moved, defined as the large-scale time-varying component; $A_M\left(t\right)$ is the average value within a short observation window length, defined as the medium-scale time-varying component; and $A_S\left(t\right)$ is the residual rapidly random fluctuation, defined as the small-scale time-varying component.

2.2. Amplitude Decomposition Based on Singular Spectrum Analysis

The path amplitude $A_l\left(t\right)$ can be regarded as a non-stationary stochastic process. To accurately establish and validate the proposed multi-scale model, we need a suitable operator to separate the three scale components from $A_l$. The difficulty lies in the fact that these three components lack fixed fluctuation periods, and their frequency spectra overlap. This indicates that classical linear filtering operators commonly used in time series analysis (e.g., Fourier transform and wavelet decomposition) may be unsuitable for addressing our problem. Ref. [19] employed EMD to decompose the slowly time-varying amplitude, but this method is susceptible to endpoint effects and mode mixing issues, making it difficult to obtain a monotonic trend component in the time-domain waveform. Therefore, we adopt the SSA method for multi-scale decomposition of the multipath amplitude. SSA offers the advantage of adaptively capturing nonlinear and non-stationary features without predefined basis functions or prior models, making it particularly suitable for multi-scale separation of complex non-stationary signals.

Within a finite observation window length, the time series of each path amplitude $A_l=[{x_1}^l,{x_2}^l,\cdots ,{x_N}^l]$ is constructed into a Hankel matrix:

$$\left.{\mathbf{A}}^l=\left[\begin{array}{cccc}{x_1}^l& {x_2}^l& \cdots & {x_K}^l\\ x_2& x_3& \cdots & {x_{K+1}}^l\\ ⋮& \cdots & \ddots & ⋮\\ {x_M}^l& {x_{M+1}}^l& \cdots & {x_N}^l\end{array}\right.\right]$$

(4)

where M is the window length, N is the number of observation samples, $1<M<N/\phantom{N2}2$, and each row vector has length $K=N-M+1$. Then, perform Singular Value Decomposition (SVD) on ${\mathbf{A}}^l$:

$${\mathbf{A}}^l=\sum _{i=1}^q{\mathbf{U}}_i{\lambda }_i{V_i}^T$$

(5)

where $U_i$ and $V_i$ are eigenvector matrices, ${\lambda }_i$ represents the singular values, q is the number of singular values, and ${\mathbf{A}}_i={\mathbf{U}}_i{\lambda }_i{V_i}^T$ is the i-th elementary matrix, corresponding to an “oscillation mode” of the time series. The relative value $E_i={\lambda }_i^2/\phantom{{\lambda }_i^2{\sum }_{j=1}^q{\lambda }_j^2}{\sum }_{j=1}^q{\lambda }_j^2$ of the singular value ${\lambda }_i$ represents the energy proportion of this mode.

The key to multi-scale decomposition lies in grouping these elementary matrices within the singular spectrum domain. Subsequently, calculate the weighted correlation function for mode grouping [26]:

$${\mathcal{R}}_{m,i}=1-{\displaystyle \frac{\sum \gamma S_m{S}_i}{\sqrt{\sum \gamma {\left(S_m\right)}^2}\sqrt{\sum \gamma {\left(S_i\right)}^2}}},\gamma =\left\{\begin{array}{c}t,1\le t\le M\\ M,M<t\le K-1\\ N-t+K,K\le t\le N\end{array}\right.$$

(6)

where $\gamma$ is the weight vector, $S_m$ is the time series reconstructed by diagonal averaging of the elementary matrix ${\mathbf{A}}_m$ corresponding to the largest singular value ${\lambda }_m$, and $S_i,i\ne m$ is the time series reconstructed from the remaining elementary matrices. Set a threshold $\chi$ and group all elementary matrices whose index i satisfies ${\mathcal{R}}_{m,i}<\chi$ into one group, obtaining the matrix ${\mathbf{Y}}_1={\displaystyle \sum _i}{\mathbf{A}}_i+{\mathbf{A}}_m$. Repeat the above grouping operation on the remaining elementary matrices to obtain ${\mathbf{Y}}_2$ and ${\mathbf{Y}}_3$.

Perform diagonal averaging on the grouped matrices $\mathbf{Y}$ to reconstruct the time-domain signal sequences $\tilde{\mathbf{x}}=\left[{\tilde{x}}_1,{\tilde{x}}_2,\cdots ,{\tilde{x}}_N\right]$:

$${\tilde{x}}_n={\displaystyle \frac{1}{n_t}}\sum _{\left(i,j\right)\in \omega }{\mathbf{Y}}_{i,j},t=1,\cdots ,N$$

(7)

where $\omega =\left\{\left(i,j\right)\left|i+j-1=t\right.\right\}$, and $n_t$ is the number of elements in $\omega$. Then, check whether monotonic trends exist in ${\tilde{\mathbf{x}}}_1$, ${\tilde{\mathbf{x}}}_2$, and ${\tilde{\mathbf{x}}}_3$. If so, $A_T={\tilde{\mathbf{x}}}_1,A_M={\tilde{\mathbf{x}}}_2,A_S={\tilde{\mathbf{x}}}_3$; otherwise, $A_T=0,A_M={\tilde{\mathbf{x}}}_1+{\tilde{\mathbf{x}}}_2,A_S={\tilde{\mathbf{x}}}_3$. The entire decomposition process adaptively extracts the time-frequency coupled structures of the multipath amplitude without requiring preset basis functions. It is worth noting that the choice of window length M and threshold $\chi$ directly affects the separation results. In practical operation, iterative adjustment is used to select the optimal values.

3. Multi-Scale Time-Varying Multipath Amplitude Model

3.1. Large-Scale Propagation Loss Model

In mobile UAC scenarios, the average intensity of acoustic signals decreases with increasing propagation distance. The large-scale propagation loss in underwater acoustic signals is attributable to two predominant phenomena: geometric spreading loss and medium absorption loss, calculated as [27]

$$TL\left(r,f\right)=L_0{r}^{K_s}\alpha {\left(f\right)}^r$$

(8)

where r is the propagation distance in meters, f is signal center frequency in Hz, $L_0$ is a proportionality constant, and $K_s$ is the spreading factor related to the propagation environment, typically ranging $K_s\in \left[1, 3\right]$ [27]. $\alpha \left(f\right)$ is the frequency-dependent absorption coefficient. The commonly used frequency band for UAC is between 1 and 40 kHz. The Thorp empirical formula [28] is generally applied to calculate the absorption coefficient for frequencies below 50 kHz

$${\alpha }_{\mathrm{dB}}\left(f\right)=0.11{\displaystyle \frac{f^2}{1+f^2}}+{\displaystyle \frac{44f^2}{4100+f^2}}+2.75\times {10}^{-4}{f}^2+0.003.$$

(9)

yielding absorption coefficients in dB/km, requiring unit conversion to match the energy representation in Equation (8). Notably, alternative absorption coefficient formulations exist for different frequency bands and hydrological conditions, including the Francois–Garrison model comprehensively reviewed in [29].

Consider a two-dimensional isospeed geometric propagation model [25], as shown in Figure 2a. The horizontal relative distance d and vertical immersion depth z are known. Assuming the vehicle moves from an initial horizontal distance $d_0$ to $d_1$, the propagation distance variation $\Delta r$ for the first stable arriving path is approximately

$$\Delta r=\sqrt{{\left(z_1-z_t\right)}^2+d_1^2}-\sqrt{{\left(z_0-z_t\right)}^2+d_0^2}.$$

(10)

Substituting Equation (8), the ratio of large-scale propagation loss $\beta$ is

$$\beta ={\displaystyle \frac{TL\left(r_1\right)}{TL\left(r_0\right)}}={\left(1+{\displaystyle \frac{\Delta r}{r_0}}\right)}^{K_s}\alpha {\left(f\right)}^{\Delta r}$$

(11)

where $r_0$ is the propagation distance at the initial position. Assuming the initial amplitude of this path is $A_0$, the amplitude variation caused by large-scale propagation loss can be quantified as

$$A\left(t\right)={\displaystyle \frac{A_0}{\sqrt{\beta }}}={\displaystyle \frac{A_0}{\sqrt{{\left(1+\frac{\Delta r}{r_0}\right)}^{K_s}{10}^{\frac{{\alpha }_{dB}\times \Delta r}{10000}}}}}.$$

(12)

3.2. Medium-Scale Shadow Fading Model

When moving over a large range, an underwater vehicle may enter the sound shadow zone generated by the reflective or refractive effects of various rough interfaces, leading to a sudden drop in received instantaneous acoustic intensity. Furthermore, the presence of submarine mountains, artificial structures, and reef clusters within underwater environments can occlude sound rays, as shown in Figure 2b, thereby inducing slow variations in the average signal intensity over the course of navigation. This phenomenon is termed the shadow effect, and the resulting slow fluctuation is called shadow fading, which reflects the variation trend of signal envelope mean over medium-scale ranges.

Given the impracticality of precise physical propagation calculations for such shadow effects, the HMM emerges as a viable alternative. HMM offers sufficient flexibility and accuracy to simulate various stochastic processes with fewer parameters and relatively simple reconstruction.

3.2.1. Discretization Modeling

As HMM is a discrete stochastic process, the medium-scale fading $A_M\left(t\right)$ requires discretization. The amplitude range of $A_M$ is divided into M non-overlapping intervals based on thresholds ${\Gamma }_m\left(m=1,2,\cdots ,M\right)$, where $-\infty <{\Gamma }_1<min\left(A_M\right)$, ${\Gamma }_M<max\left(A_M\right)$, yielding discrete observations as

$$o_k=arg\underset{m}{min}\left[\left(A_M^k-{\Gamma }_m\right)\ge 0\right]$$

(13)

where $A_M^k$ is the amplitude value at k-th sampling instant of $A_M\left(t\right)$, and $o_k$ is the corresponding discrete observation, forming a symbolic observation set $V=\left\{v_1,v_2,\cdots ,v_M\right\}$, $1\le v_m\le M$. Following Equation (13), the observation sequence $O=\left\{o_1,o_2,\cdots ,o_K\right\}$ is obtained.

According to HMM theory, it is assumed that the random sequence $A_M$ has N hidden states. To reduce model training complexity, the K-means clustering algorithm is applied for unsupervised state partitioning. The state mapping follows the nearest centroid principle

$$s_k=arg\underset{n}{min}\left[d\left(o_k,c_n\right)\right]$$

(14)

where $c_n$ is the centroid of the corresponding cluster, and the distance operator d typically adopts Euclidean distance. The indexes of N cluster centroids form the state set $Q=\left\{q_1,q_2,\cdots ,q_N\right\}$. Using the observation sequence O and Equation (14), the state sequence $S=\left\{s_1,s_2,\cdots ,s_N\right\}$ is mapped point by point.

HMM modeling relies on two fundamental assumptions:

(1)

The homogeneous Markov chain assumption:

$$a_{i,j}=P\left(s_{k+1}\left|s_k,s_{k-1},\cdots ,s_1\right.\right)=P\left(s_{k+1}=q_j\left|s_k=q_i\right.\right)$$

(15)

where $a_{i,j}$ is the transition probability from state i to state j, forming the state transition matrix $\mathbf{A}={\left[a_{i,j}\right]}_{N\times N}$.

(2)

The observation independence assumption:

$$b_j\left(m\right)=P\left(o_k\left|s_k,o_{k-1},\cdots ,o_1\right.\right)=P\left(o_k=v_m\left|s_k=q_j\right.\right)$$

(16)

where $b_j\left(m\right)$ is the probability of generating observation $s_k=q_j$ under a specific state $o_k=v_m$, forming the observation probability matrix $\mathbf{B}={\left[b_j\left(m\right)\right]}_{N\times M}$.

Additionally, an initial hidden state probability distribution $\Pi ={\left[{\pi }_i\right]}_{1\times N}$ is required, directly determined from the state sequence

$${\pi }_i=P\left(s_1=q_i\right),\sum _{i=1}^N{\pi }_i=1.$$

(17)

With this, the modeling of medium-scale fading $A_M\left(t\right)$ is complete, and the entire HMM parameter set is

$$\lambda =\left(V,Q,\mathbf{A},\mathbf{B},\Pi \right).$$

(18)

3.2.2. Model Parameter Estimation

Using the discretized results of measured $A_M\left(t\right)$, the HMM parameters are estimated via maximum likelihood estimation given the state sequence S and observation sequence O. The state transition matrix $\mathbf{A}$ is estimated by counting transition frequency between states

$$a_{i,j}={\displaystyle \frac{{\displaystyle \sum _{k=1}^{K-1}}\xi \left(s_k=q_i\&s_{k+1}=q_j\right)}{{\displaystyle \sum _{j=1}^N}{\displaystyle \sum _{k=1}^{K-1}}\xi \left(s_k=q_i\&s_{k+1}=q_j\right)}}$$

(19)

where $\xi$ is an indicator operator equal to 1 if the condition is satisfied and 0 otherwise. The observation probability matrix $\mathbf{B}$ is estimated by counting generation frequency of the observation symbol under each state

$$b_j\left(m\right)={\displaystyle \frac{{\displaystyle \sum _{k=1}^K}\xi \left(s_k=q_j\&o_k=v_m\right)}{{\displaystyle \sum _{m=1}^M}{\displaystyle \sum _{k=1}^K}\xi \left(s_k=q_j\&o_k=v_m\right)}}.$$

(20)

3.2.3. Stochastic Process Reconstruction

To simulate the dynamic behavior of $A_M\left(t\right)$, the Viterbi algorithm is employed for stochastic process reconstruction. The Viterbi algorithm is a dynamic programming method, whose core idea is to recursively compute the optimal path probability for each state at each time step given an observation sequence

$${\delta }_k\left(j\right)=\underset{1\le i\le N}{max}\left[{\delta }_{k-1}\left(i\right)·a_{i,j}\right]·b_j\left(o_k\right).$$

(21)

The initial path probability is set as ${\delta }_1\left(j\right)={\pi }_j·b_j\left(o_1\right)$, and the backtracking pointer for the previous state i that maximizes ${\delta }_k\left(j\right)$ is stored

$${\phi }_k\left(j\right)=arg\underset{i}{max}\left[{\delta }_{k-1}\left(i\right)·a_{i,j}\right].$$

(22)

At the final time step K, the optimal path endpoint corresponding to the final maximum probability is determined as

$$s_K^{*}=arg\underset{i}{max}\left[{\delta }_K\left(i\right)\right].$$

(23)

Backtracking is then performed stepwise as follows:

$$s_k^{*}={\phi }_{k+1}\left(s_{k+1}^{*}=q_j\right).$$

(24)

The optimal path $S^{*}=\left\{s_1^{*},s_2^{*},\cdots ,s_K^{*}\right\}$ is obtained through the backtracking pointer of each step. Finally, the cluster centroids are used to convert $S^{*}$ into actual amplitude sequences $A_M^{*}$, reconstructing the medium-scale fading process

$$A_M^{*}=\mathbf{F}\left(s_k^{*}=q_n,c_n\right)$$

(25)

where $\mathbf{F}$ is the inverse clustering operator.

3.3. Small-Scale Scattering Fading Model

During underwater acoustic propagation, sound rays undergo scattering effects at rough interfaces, such as undulating sea surfaces and microscopic particles, decomposing into numerous micro-paths characterized by distinct phases and amplitudes. These micro-paths experience destructive or constructive interference, resulting in rapid fluctuations in transient path energy, a phenomenon termed scattering fading. A critical assumption in modeling this scattering effect is that micro-paths arriving at the receiver after multiple scattering and reflection events are statistically independent and identically distributed. Under such conditions, the central limit theorem confirms that the superposition of a sufficiently large number of micro-paths converges to a complex Gaussian distribution. Empirical studies [20] demonstrate that, even with as few as ten micro-paths, the statistical behavior aligns well with Gaussian characteristics.

The essence of multi-scale process separation lies in removing the slowly time-varying mean of path amplitude. Consequently, the residual fading caused by small-scale scattering can be modeled as a zero-mean Gaussian process

$$A_S\left(t\right)\sim \mathcal{N}\left(0,{\sigma }^2\right).$$

(26)

Compared to alternative methods such as Rician or Rayleigh fading models, the Gaussian assumption proves particularly advantageous in highly scattering environments, such as shallow-water regions dominated by scattered multipath components rather than line-of-sight paths. Subsequent validation using measured data further confirms the accuracy of the zero-mean Gaussian model.

3.4. Multi-Scale Time-Varying Channel Simulation

Based on the above analysis, we propose a multi-scale time-varying path amplitude simulation model, as illustrated in Figure 3. Algorithm 1 details the measurement-driven time-varying channel simulation process.

For each path, first, a large-scale propagation loss model is established using sound propagation physics, generating the sequence $A_T^{\mathrm{sim}}\left(t\right)$ by measuring the movement trajectory of underwater vehicles. Second, the medium-scale shadow fading model is constructed using HMM, and the sequence $A_M^{\mathrm{sim}}\left(t\right)$ is generated via the Viterbi algorithm. Third, the small-scale scattering fading model is modeled as a zero-mean Gaussian process, with the sequence $A_S^{\mathrm{sim}}\left(t\right)$ generated using the Monte Carlo simulation. After superimposing these three components to obtain the amplitude $A_l^{\mathrm{sim}}\left(t\right)$ of each path, uniform distribution $\mathcal{U}\left[-\pi ,\pi \right)$ is applied to derive the phase of each path. The time delay of each simulated path is then determined according to measured path delays. Finally, the simulated UAC channel impulse response is obtained using Equation (2).

Algorithm 1 Measurement-driven time-varying channel simulation

Input: moving trajectory $\Delta r$, spread factor K, measured TVIR ${\mathbf{H}}_{\mathrm{real}}$   Initialization: Multipath number L, zero matrix $\mathbf{H}$, sampling rate $f_s$   For 1 ← to L do      1. Path amplitude: decompose $A_l^{\mathrm{real}}$ using SSA to obtain $A_T^{\mathrm{real}}$, $A_M^{\mathrm{real}}$, and $A_S^{\mathrm{real}}$      2. Large-scale model: calculate ${\alpha }_{\mathrm{dB}}$ ← Equation (8), convert $\alpha ={10}^{\left({\alpha }_{\mathrm{dB}}/20\right)}$.       Using $\Delta r$, K and $A_T^{\mathrm{sim}}\left(0\right)$, calculate $\beta$ ← Equation (11). Generate $A_T^{\mathrm{sim}}$ ← Equation (12).      3. Medium-scale model: perform state partitioning on $A_M^{\mathrm{real}}$ to obtain O and S.       Count element categories in O and S, obtain V and Q. Calculate $\Pi$ ← Equation (17),       $\mathbf{A}$ ← Equation (19), $\mathbf{B}$ ← Equation (20), to form HMM $\lambda =\left(V,Q,A,B,\Pi \right)$.       Using the Viterbi Algorithm to backtrack the optimal path $S^{*}$ ← Equations (21)–(24).       Generate $A_M^{\mathrm{sim}}$ according to Equation (25).      4. Small-scale model: estimate the Gaussian fitting variance ${\sigma }^2$ of $A_S^{\mathrm{real}}$.       build a Gaussian random number generator $\mathcal{N}\left(0,{\sigma }^2\right)$, sample to generate $A_S^{\mathrm{sim}}$.      5. Path amplitude: calculate $A_l^{\mathrm{sim}}=A_T^{\mathrm{sim}}+A_M^{\mathrm{sim}}+A_S^{\mathrm{sim}}$ ← Equation (3).      6. Random phase: build a uniform random number generator $\mathcal{U}\left[-\pi ,\pi \right)$,       sample to generate ${\theta }_l$.   end      7. Reorganization TVIR: obtain the path delay $\left\{{\tau }_l\right\}$ from measuring TVIR ${\mathbf{H}}_{\mathrm{real}}$       insert amplitudes $\left\{A_l^{\mathrm{sim}}\right\}$ and phases $\left\{{\theta }_l^{\mathrm{sim}}\right\}$ into the zero matrix $\mathrm{H}$ corresponding       delay index ${\tau }_l\times f_s$. Generate $\mathbf{H}=\left\{h\left(t,\tau \right)\right\}$ according to Equation (2).   Output: ${\mathbf{H}}_{\mathrm{sim}}=\left\{h\left(t,\tau \right)\right\}$

4. Measured Channel Analysis

4.1. Channel Sounding and Parameter Estimation

Channel data are obtained from a single-input–single-output (SISO) shallow-water mobile UAC experiment conducted in Lake Fuxian, Yunnan Province, with an average water depth of 130 m. Two measurements, labeled FX1 and FX2, are carried out. In FX1, the transmitter and receiving hydrophone are deployed at depths of 20 m and 35 m, respectively, with a transmission distance ranging from 700 to 1200 m. In FX2, they are placed at 20 m and 50 m, over a transmission distance of 1000 to 1400 m. The channel measurement deployment and environment are shown in Figure 4. A stationary omnidirectional transmitter is used throughout. During FX1, an unmanned vehicle carrying the hydrophone moved toward the transmitter at a speed of about 15 knots, with minor speed fluctuations due to currents. During FX2, the vehicle underwent variable acceleration followed by deceleration, reaching a maximum speed of approximately 15 knots. Figure 5a shows the Sound Speed Profile (SSP) of the experimental water area, which is a typical shallow-water negative thermocline. Figure 5b presents the Bellhop-simulated channel impulse response (CIR) based on geometric deployment parameters under this SSP, used to validate subsequent channel measurements.

Linear Frequency Modulation (LFM) pulses are used for channel sounding, with a frequency band of 9–13 kHz and a cyclic sounding period of 0.12 s. Leveraging the strong autocorrelation of LFM signals, a matched filter channel sounder is employed to estimate the baseband TVIR. For mobile scenarios, average Doppler compensation [30] is applied to improve the accuracy of path amplitude estimation. Figure 6a illustrates the TVIR estimation process incorporating time-varying Doppler compensation. Since the path delay varies with time, amplitudes cannot be directly extracted at a fixed tap index. We use a threshold-based contour extraction method for multipath sparsification and parameter estimation, as shown in Figure 6b. First, an appropriate threshold is set to filter out background noise interference, and sub-threshold amplitudes are set to zero. Secondly, the search region is divided based on the delay fluctuation range, and peak points are identified, with non-peak positions set to zero for path sparsity. Transient interference or abrupt multipath structural changes may introduce false peaks, which are eliminated by analyzing delay fluctuation trends. Finally, non-zero values are retrieved to locate and extract path parameters.

Figure 7 shows an example of the measured TVIR after average Doppler compensation. The CIR reveals four stable multipath arrivals within a delay observation window of approximately 40 ms, consistent with the Bellhop simulation, validating the accuracy of the channel sounding system. Subsequent multipath contour extraction yielded the sparsified TVIR and the amplitude of each stable path, as shown in Figure 8. The measured TVIR reveals four stable arrival paths, labeled P1, P2, P3, and P4, respectively. In subsequent analysis, these path labels will be used directly for simplified expression. It should be noted that the normalized amplitude mentioned hereafter refers to the relative value compared to the maximum measured amplitude among the four paths.

4.2. Multi-Scale Model Validation

The measured TVIR in Figure 8 is used to verify the accuracy of the proposed multi-scale time-varying amplitude model. Taking Path 1 as an example, the SSA decomposition results of the measured path amplitude are shown in Figure 9.

4.2.1. Large-Scale Model Validation

The decomposed large-scale component $A_T$ is used to validate the acoustic propagation loss model, with results shown in Figure 10. The fitted curve follows Equation (12), where the propagation distance variation $\Delta r$ is derived from measured moving distance.

The fitting parameter settings are shown in

Table 1. Fitting parameter settings and goodness of fit for large-scale models.

	Constant Term ${\mathit{A}}_0$	Spreading Factor ${\mathit{K}}_{\mathit{s}}$	Absorption Factor ${\mathit{\alpha }}_{\mathbf{dB}}$	${\mathit{R}}_{\mathbf{RMSE}}\left({\mathit{A}}_{\mathit{T}}^{\mathbf{real}},{\mathit{A}}_{\mathit{T}}^{\mathbf{sim}}\right)$
FX1	0.78	2.55	2.17 dB/km	9.8 × 10−3
FX2	0.75	3.02	2.32 dB/km	1.5 × 10−2

. In shallow-water acoustic propagation, the spreading loss exponent typically falls within the range of $\left[2, 3\right]$ [27]. Given the system’s upper frequency limit of f = 13 kHz, the empirical value calculated using Equation (9) is $a_{\mathrm{dB}}=1.91$ dB/km. The fitted value in Table 1 is consistent with this empirical result, which validates the parameter configuration of the large-scale model. The overall attenuation trend of measured amplitudes, the large-scale component $A_T^{\mathrm{real}}$, and the fitting results $A_T^{\mathrm{sim}}$ exhibit strong alignment in Figure 10. To evaluate the model fitting goodness, the Root Mean Square Error (RMSE) is employed as a similarity evaluation metric between decomposition results $X^{\mathrm{real}}$ and the simulated outputs $X^{\mathrm{sim}}$:

$$R_{RMSE}\left(X^{\mathrm{real}},X^{\mathrm{sim}}\right)=\sqrt{{\displaystyle \frac{1}{T}}\sum _{i=1}^T{\left(X_i^{\mathrm{real}}-X_i^{\mathrm{sim}}\right)}^2}.$$

(27)

The calculated fitting goodness values $R_{RMSE}\left(A_T^{\mathrm{real}},A_T^{\mathrm{sim}}\right)$ for the large-scale model in FX1 and FX2 are $1.5\times {10}^{-2}$ and $9.8\times {10}^{-3}$, respectively, demonstrating that the model closely approximates the overall attenuation trend of the measured path amplitudes.

4.2.2. Medium-Scale Model Validation

The decomposed medium-scale component $A_M$ is used to validate the HMM. Figure 11 shows an example of state partitioning with N = 5 hidden states using the k-means clustering algorithm. The clustering results divide the sampled amplitudes $A_M^k$ into N non-uniform and non-overlapping regions (Figure 11b), with the corresponding state sequence S shown in Figure 11c. This indicates that the HMM states automatically partitioned by k-means clustering algorithm are physically interpretable, effectively reflecting periodic fluctuations in average power caused by medium-scale shadowing effects. The probability density function (PDF) shows the non-uniform distribution of $A_M^k$ (Figure 11a). Compared with conventional equal-width binning methods, the clustering approach better captures the inherent data distribution patterns, enabling the HMM to learn dominant transition patterns more accurately while avoiding Viterbi algorithm backtracking failures caused by excessive extremely low values in state transition matrix $\mathbf{A}$ and observation probability matrix $\mathbf{B}$.

The selection of observation symbol number M and hidden state number N in an HMM directly affects model expressiveness and computational efficiency [31,32]. To evaluate the rationality of parameter settings, the former uses $R_{RMSE}\left(A_M^{\mathrm{real}},O\right)$ and $R_{RMSE}\left(A_M^{\mathrm{real}},A_M^{\mathrm{sim}}\right)$ to measure observation quantization error and state decoding error, respectively, while the latter employs the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) to assess model overfitting

$$\begin{array}{c}\mathrm{AIC}=2G-2ln\left(O\left|\lambda \right.\right)\hfill \\ \mathrm{BIC}=Gln\left(T\right)-2ln\left(O\left|\lambda \right.\right)\hfill \end{array}$$

(28)

where G is the number of model parameters, $ln\left(O\left|\lambda \right.\right)$ is maximum likelihood function value, and T is total length of the observation sequence. Lower values of these criteria suggest simpler model structures. By iterating over $2\le N\le 10$ and $2\le M\le 20$, optimal parameter configurations are selected based on metric trends. Using P1 of FX1 as an example, Figure 12a,b show the bar charts of evaluation metrics when fixing $N=2$ and $M=12$, respectively. When N and M exceed a certain threshold, the reduction in RMSE becomes marginal, while the AIC and BIC values increase more rapidly. The final parameter configurations and evaluation metrics of the HMM for each path in both experiments are summarized in

Table 2. HMM model parameter settings and evaluation results.

	Path	$\left(\mathit{N},\mathit{M}\right)$	${\mathit{R}}_{\mathbf{RMSE}}\left({\mathit{A}}_{\mathit{M}}^{\mathbf{real}},\mathit{O}\right)$	${\mathit{R}}_{\mathbf{RMSE}}\left({\mathit{A}}_{\mathit{M}}^{\mathbf{real}},{\mathit{A}}_{\mathit{M}}^{\mathbf{sim}}\right)$	AIC	BIC
FX1	P1	(6, 13)	5.2 $\times {10}^{-3}$	1.1 $\times {10}^{-2}$	1328.7	1748.6
	P2	(8, 15)	1.5 $\times {10}^{-2}$	2.9 $\times {10}^{-2}$	1375.2	2066.6
	P3	(9, 14)	1.4 $\times {10}^{-2}$	2.1 $\times {10}^{-2}$	1457.6	2235.5
	P4	(7, 14)	5.3 $\times {10}^{-3}$	1.1 $\times {10}^{-2}$	819.8	1367.2
FX2	P1	(7, 12)	5.9 $\times {10}^{-3}$	1.0 $\times {10}^{-2}$	919.1	1408.4
	P2	(6, 14)	6.7 $\times {10}^{-3}$	1.5 $\times {10}^{-2}$	1141.8	1585.8
	P3	(7, 15)	8.5 $\times {10}^{-3}$	1.8 $\times {10}^{-2}$	1127.7	1703.3
	P4	(6, 14)	6.7 $\times {10}^{-3}$	1.5 $\times {10}^{-2}$	1185.1	1629.1

, and the corresponding HMM reconstruction results are shown in Figure 13. The reconstructions effectively capture the temporal variations of medium-scale components, with errors on the order of ${10}^{-2}$.

Furthermore, to highlight the advantages of HMM, this subsection employs the Markov chain model [32] and the Kalman filter as control groups. Using P1 as an example, Figure 14 compares the modeling performance of these three approaches for the medium-scale component. Both the Markov chain and the HMM are configured with a state number of $N=6$, while the Kalman filter is configured with process noise covariance $Q=0.01$ and measurement noise covariance $R=1$. The corresponding reconstruction errors are provided in

Table 3. Medium-scale component reconstruction errors $R_{RMSE}\left(A_M^{\mathrm{real}},A_M^{\mathrm{sim}}\right)$ of Markov chain, Kalman filter, and HMM.

	Markov Chain	Kalman Filter	HMM
FX1, P1	1.5 $\times {10}^{-2}$	3.4 $\times {10}^{-2}$	1.1 $\times {10}^{-2}$
FX2, P1	1.6 $\times {10}^{-2}$	3.0 $\times {10}^{-2}$	1.1 $\times {10}^{-2}$

. The results demonstrate that, compared to the Markov model and the Kalman filter, the HMM is more suitable for modeling the medium-scale component within the multi-scale time-varying multipath channel simulation framework proposed in this work.

4.2.3. Small-Scale Model Validation

The decomposed small-scale component $A_S$ is used to validate the Gaussian model. The PDF of $A_S$ for each path is statistically analyzed in Figure 15, with Gaussian fitting parameters provided in

Table 4. Fitting model parameter settings and JS divergence evaluation results.

	Project	P1	P2	P3	P4
FX1	$\sigma$	2.7 $\times {10}^{-2}$	3.9 $\times {10}^{-2}$	4.1 $\times {10}^{-2}$	7.5 $\times {10}^{-2}$
	$JS\left(P_{A_S}∥\mathcal{N}\left(0,{\sigma }^2\right)\right)$	1.5 $\times {10}^{-3}$	1.5 $\times {10}^{-3}$	1.4 $\times {10}^{-3}$	2.6 $\times {10}^{-3}$
	$JS\left(P_{{\theta }_l}∥\mathcal{U}\left[-\pi ,\pi \right)\right)$	3.5 $\times {10}^{-2}$	9.3 $\times {10}^{-3}$	6.7 $\times {10}^{-3}$	6.3 $\times {10}^{-3}$
	$JS\left(P_{A_l^{\mathrm{real}}}∥P_{A_l^{\mathrm{sim}}}\right)$	4.2 $\times {10}^{-4}$	9.2 $\times {10}^{-4}$	9.2 $\times {10}^{-3}$	6.0 $\times {10}^{-3}$
FX2	$\sigma$	2.8 $\times {10}^{-2}$	3.2 $\times {10}^{-2}$	5.0 $\times {10}^{-2}$	7.7 $\times {10}^{-2}$
	$JS\left(P_{A_S}∥\mathcal{N}\left(0,{\sigma }^2\right)\right)$	1.8 $\times {10}^{-3}$	2.1 $\times {10}^{-3}$	1.7 $\times {10}^{-3}$	3.6 $\times {10}^{-3}$
	$JS\left(P_{{\theta }_l}∥\mathcal{U}\left[-\pi ,\pi \right)\right)$	8.6 $\times {10}^{-3}$	7.9 $\times {10}^{-3}$	7.3 $\times {10}^{-3}$	8.7 $\times {10}^{-3}$
	$JS\left(P_{A_l^{\mathrm{real}}}∥P_{A_l^{\mathrm{sim}}}\right)$	1.2 $\times {10}^{-2}$	1.4 $\times {10}^{-3}$	1.9 $\times {10}^{-3}$	1.5 $\times {10}^{-3}$

. To quantify the similarity between measured data and the Gaussian model, the Jensen–Shannon (JS) divergence is calculated:

$$JS\left(P_{A_S}∥\mathcal{N}\right)={\displaystyle \frac{1}{2}}\left(\sum P_{A_S}^{i}log{\displaystyle \frac{P_{A_S}^i}{{\mathcal{N}}_i}}+\sum {\mathcal{N}}_{i}log{\displaystyle \frac{{\mathcal{N}}_i}{P_{A_S}^i}}\right).$$

(29)

The JS divergence is non-negative, with smaller values indicating closer distributions. Statistical fitting and JS divergence tests confirm the validity of modeling the small-scale fading process as a zero-mean Gaussian distribution.

4.2.4. Phase Model Validation

Additionally, to validate the rationality of uniform distribution assumption for multipath phase modeling, Figure 16a,c compares the CDF of measured phases with the ideal uniform distribution, with corresponding JS divergence $JS\left(P_{{\theta }_l}∥\mathcal{U}\left[-\pi ,\pi \right)\right)$ listed in Table 4. Except for the P1 in FX1, which exhibits a certain deviation, the phase distributions of the remaining paths align well with the expected uniform distribution characteristics. Furthermore, Figure 16b,d shows the temporal correlation functions of the phases for the four paths. The results indicate that the temporal correlation of phase is negligible. Therefore, the simulation method in Algorithm 1, which employs a random number generator to generate path phase ${\theta }_l$, is reasonable.

5. Simulation Testing and Model Comparison

Driven by experimental measurements, we conduct channel simulation tests and compare with the classical single-state statistical distribution model and AR-1 model from [20], thereby demonstrating the simulation advantages of the proposed multi-scale time-varying model. After statistical fitting of the measured data, we chose the Nakagami-m distribution for single-state model simulation, and its probability density function is defined as

$$p_w\left(x\right)=\left\{\begin{array}{c}{\displaystyle \frac{2m^m{x}^{2m-1}{e}^{-\left(m/\Omega \right)x^2}}{\gamma \left(m\right){\Omega }^m}},x\ge 0\hfill \\ 0,x<0\hfill \end{array}\right.$$

(30)

where $\gamma$ is the gamma function. The AR-1 model is represented as

$$x\left(t\right)=a_{1}x\left(t-1\right)+c+\epsilon \left(t\right)$$

(31)

where $a_1$ is the autoregressive coefficient, satisfying $\left|a_1\right|<1$. c is a constant term; $\epsilon$ is a noise error term and follows a zero-mean distribution with variance ${\sigma }_{AR}^2$. Optimal fitting results are used to set simulation model parameters. In the FX1 experiment, Nakagami-m parameters corresponding to the four multipath components are $m=\left[4.7,1.8,2.5,1.5\right]$ and $\Omega =\left[0.3,0.3,0.2,0.1\right]$, and AR-1 model parameters are $a_1=\left[0.94,0.96,0.89,0.78\right]$, $c=\left[3.6,2.1,4.8,5.4\right]\times {10}^{-2}$, and ${\sigma }_{AR}^2=\left[1.8,3.1,4.6,4.2\right]\times {10}^{-3}$. In the FX2 experiment, Nakagami-m parameters are $m=\left[4.2,10.0,2.3,1.0\right]$ and $\Omega =\left[0.3,0.3,0.1,0.1\right]$, and AR-1 model parameters are $a_1=\left[0.95,0.89,0.88,0.97\right]$, $c=\left[3.0,5.7,3.9,1.0\right]\times {10}^{-2}$, and ${\sigma }_{AR}^2=\left[1.6,1.4,2.6,2.1\right]\times {10}^{-3}$. The proposed multi-scale time-varying amplitude model in this work is named MSTV for simplified reference in subsequent analysis. All methods performed 100 Monte Carlo simulations to generate corresponding datasets.

5.1. Statistical Distribution Function

Figure 17 shows the CDFs of the measured amplitudes (solid lines) and MSTV model simulated results (dashed lines) for multipath components. The CDF curves of the simulated and measured amplitudes show strong agreement across all four paths, with corresponding JS divergences $JS\left(P_{A_l^{\mathrm{real}}}∥P_{A_l^{\mathrm{sim}}}\right)$ listed in Table 4, validating the statistical validity and generalization capability of the proposed MSTV model.

Taking Path 1 as an example, Figure 18 compares the CDFs of the different simulation methods. While all three methods closely match the measured results, the proposed model exhibits superior performance in specific intervals (zoomed regions). Notably, the Nakagami-m and AR-1 models exhibit CDF tails in the range $A_l<0.25$, whereas empirically measured minima exceed 0.25, indicating these models tend to generate unrealistic low amplitudes with non-negligible probability. Such extreme values could destabilize fixed-tap adaptive multipath tracking algorithms, adversely affecting communication system testing.

5.2. Temporal Dynamic Behavior

Figure 19 illustrates the time-domain waveforms of the measured and simulated amplitudes. Despite sharing identical statistical distributions (see Figure 18), the Nakagami-m model produces amplitude sequences with only random fluctuations, lacking the slow time-varying mean trends observed in the measurements. The AR-1 model captures mean slow variations but fails to align these trends with measured data, neglecting the continuous modulation effect of large-scale propagation loss. In contrast, the MSTV model accurately reproduces the temporal dynamics behavior of path amplitude.

Table 5. Average RMSE between simulations and measurements for different models.

	${\mathit{R}}_{\mathbf{RMSE}}\left({\mathit{A}}_{\mathit{l}}^{\mathbf{real}},{\mathit{A}}_{\mathit{l}}^{\mathbf{sim}}\right)$	P1	P2	P3	P4
FX1	Nakagami-m model	0.18	0.28	0.20	0.15
	AR-1 model	0.17	0.27	0.20	0.14
	MSTV model	0.04	0.06	0.06	0.10
FX2	Nakagami-m model	0.19	0.12	0.16	0.24
	AR-1 model	0.19	0.11	0.15	0.22
	MSTV model	0.05	0.05	0.07	0.01

lists the RMSE between the simulated and measured path amplitudes, averaged over the simulation dataset.

The temporal dynamic behavior can be characterized by the ACF. By analyzing the oscillatory decay pattern of ACF, one can evaluate the temporal variation rate and the implicit periodicity of stochastic process. The ACF for sequence $\left\{x_t\right\}$ with lag $\Delta t$ is

$$\rho \left(\Delta t\right)={\displaystyle \frac{{\displaystyle \sum _{t=\Delta t+1}^T}\left(x_t-\overline{x}\right)\left(x_{t-\Delta t}-\overline{x}\right)}{{\displaystyle \sum _{t=1}^T}\left(x_t-\overline{x}\right)\left(x_t-\overline{x}\right)}}.$$

(32)

Figure 20 compares the ACFs of measured and simulated path amplitudes. The ACF of Nakagami-m model decays rapidly to zero, deviating significantly from measurements. The AR-1 model fails to reproduce realistic oscillatory tails and lacks negative correlation features. In contrast, the proposed model effectively captures measured temporal correlation, validating its superiority in reproducing non-stationary time-varying characteristics.

5.3. Communication Performance Prediction

Signals experience fading during propagation due to the random superposition of different path components. Therefore, the simulation accuracy of multipath amplitudes significantly impacts the prediction of communication performance. Using the path delay indices from Figure 8 and the simulated time-varying amplitudes derived from Algorithm 1, we generated the simulated TVIR. Figure 21 shows examples of simulated TVIR using different methods, driven by the sparsified channel in Figure 8. Subsequently, we test the BER under different channel and Signal-to-Noise Ratio (SNR) conditions to highlight the benefits of high-fidelity channel modeling for communication system design and validation.

The test system employs an Orthogonal Frequency Division Multiplex (OFDM) scheme with 1/2 rate convolutional coding, a symbol rate of 4000 symbols/s matching the sounding bandwidth, and 1/4 comb-type pilot subcarriers for Least-Squares (LS) equalization. Gaussian white noise is added, with BER results being averaged over 100 Monte Carlo tests. It should be noted that the goal of the receiver setting here is not to achieve optimal communication performance but to be applicable for comparing BER performance.

Figure 22 shows the BER performance under different SNRs and channel conditions. Compared with the other two simulation methods, the proposed MSTV model yields communication prediction results with significantly higher credibility. The BER curve of the MSTV model closely aligns with measured channels, particularly in low-SNR regions (<5 dB). These results indicate that signal fading caused by time-varying multipath amplitudes can be accurately simulated by multi-scale models, demonstrating the reliability of this simulation framework for evaluating the performance of UAC systems.

6. Conclusions

This study proposes a multi-scale time-varying amplitude model for mobile UAC channels, integrating physical mechanisms and statistical modeling. Data-driven simulations validate its accuracy and fidelity in high-speed mobile scenarios. The research results demonstrate that traditional single-state statistical models (e.g., Nakagami-m distribution) and AR-1 models, while capable of reflecting statistical characteristics of multipath amplitudes, fail to capture the attenuation trends and mean fluctuation characteristics caused by rapid motion. In contrast, the proposed multi-scale model enhances dynamic characterization and accurate reproduction through hierarchical modeling and coupled superposition mechanisms.

The experimental results show that the advantages of the proposed model are mainly reflected in three aspects: Firstly, in terms of statistical distribution, the CDF of the simulated path amplitude aligns closely with the measurements, avoiding unrealistic tailing in low-amplitude regions, indicating that the MSTV model can more accurately describe the occurrence probability of deep fading events. Secondly, in terms of temporal dynamics, the large-scale component based on the propagation loss model and the medium-scale component based on the HMM reconstruct the overall trend and mean fluctuation characteristics, respectively, and ACF tail shapes consistent with the measurements. This validates the model’s ability to reproduce temporal non-stationarity. Thirdly, in terms of system-level performance, the MSTV model-generated TVIR achieves BER curves nearly identical to the measured channels, with significantly lower prediction errors than the alternatives, which proves the model’s reliability for communication system design and performance prediction.

However, the proposed model still has certain limitations. First, its computational complexity may hinder real-time deployment in resource-constrained applications. Therefore, the current framework is suitable primarily for laboratory-based UAC system simulations. Second, although modeling the small-scale scattering component as a zero-mean Gaussian process performs well in high-scattering shallow-water environments, it is not universally applicable, particularly under turbulent conditions. Therefore, the proposed multi-scale time-varying channel simulation framework allows the integration of alternative small-scale statistical models—such as Rician, Rayleigh, Lognormal, composite K, or Weibull distributions—to better match empirical measurements in different environments. This enhancement underscores the model’s extensibility and fidelity across varying channels. Through the above improvements, the multi-scale time-varying amplitude model is expected to become a core tool for supporting the simulation and prediction of non-stationary mobile UAC systems.