The ocean sound speed variation in depth is much greater than that in the horizontal direction, so the SSP is usually modeled as a function of depth. In order to correct the time-varying sound speed error, we have modeled the sound speed as a function of depth and time c(z,t)=c0(z)+A(t), where c0(z) denotes the background SSP, and the time-varying term A(t) is expressed as
海洋声速深度变化远大于水平方向变化，因此 SSP 通常被建模为深度的函数。为了纠正时变声速误差，我们将声速建模为深度和时间 c(z,t)=c0(z)+A(t) 的函数，其中 c0(z) 表示背景 SSP，时变项 A(t) 表示为：

View Source  查看源码 \begin{equation*}\mathrm{A}(t)=\sum\limits_{k} (a_{1k}\Phi_{1k}(t)+\cdots+a_{sk}\Phi_{sk}(t))\tag{1}\end{equation*}
\begin{方程*}\mathrm{A}（t）=\sum\limits_{k} （a_{1k}\Phi_{1k}（t）+\cdots+a_{sk}\Phi_{sk}（t））\tag{1}\end{方程*}

Compared with the GNSS-A observation configuration mentioned in [1], we have introduced the constraint of acoustic ranging observations between transponders to stable the solution and reduce the transponder positioning error. The positioning model is expressed as
与[1]中提到的 GNSS-A 观测配置相比，我们引入了应答器之间声学测距观测的约束，以稳定解并减少应答器定位误差。定位模型表示为

View Source  查看源码 \begin{equation*}\mathbf{T}=f(\mathbf{X}_{+},\mathbf{X}\_,\mathbf{m},c_{0}(z)) \tag{2}\end{equation*}

Based on the constant gradient ray tracing model [8], the jth round-trip travel time between transponder and transducer is expressed as
基于恒定梯度光线追踪模型[8]，应答器和换能器之间的 j 第次往返旅行时间表示为

View Source  查看源码 \begin{equation*}\tau_{j}=\int\limits_{Z_{j+}}^{z_{i}}\frac{1}{c(z)\sqrt{1-n_{j+,i}^{2}c^{2}(z)}}dz+\int\limits_{Z_{j-}}^{z_{i}}\frac{1}{c(z)\sqrt{1-n_{j-,i}^{2}c^{2}(z)}}dz\tag{3}\end{equation*}
\begin{方程式*}\tau_{j}=\int\limits_{Z_{j+}}^{z_{i}}\frac{1}{c（z）\sqrt{1-n_{j+，i}^{2}c^{2}（z）}}dz+\int\limits_{Z_{j-}}^{z_{i}}\frac{1}{c（z）\sqrt{1-n_{j-，i}^{2}c^{2}（z）}}dz\tag{3}\end{方程*}

The linearization approximation of the nonlinear model f() at the initial value of parameter m0 is expressed as
非线性模型 f() 在参数初始值处的线性化近似 m0 表示为

View Source  查看源码 \begin{equation*}f(\mathbf{m})\approx f(\mathbf{m}_{0})+\mathbf{J}(\mathbf{m}-\mathbf{m}_{0})\tag{4}\end{equation*}

∂τj∂ask=∫Zj+zi−Φsk(t)c2(z)[1−n2j+,ic2(z)]1/2dz+∫Zj−zi−Φsk(t)c2(z)[1−n2j−,ic2(z)]1/2dzi=1,…,I, j=1,…,J, k=1,…,K.(5)

View Source  查看源码 \begin{align*} &\frac{\partial \tau_{j}}{\partial a_{sk}}=\int\limits_{Z_{j+}}^{z_{i}}\frac{- \Phi_{sk}(t)}{c^{2}(z)\left[1- n_{j+,i}^{2}c^{2}(z)\right]^{1/2}}dz+\int\limits_{Z_{j-}}^{z_{i}}\frac{- \Phi_{sk}(t)}{c^{2}(z)\left[1- n_{j-,i}^{2}c^{2}(z)\right]^{1/2}}dz\\ &\quad i=1, \ldots, I,\ j=1, \ldots, J,\ k=1, \ldots, K.\tag{5} \end{align*}

In order to obtain a stable solution for the positioning model in (2), we treat the model parameters m as unknowns constrained with prior uncertainties and analyze the statistical properties of the parameters and observations based on the empirical Bayes approach. In our method, m contains M model parameters, the dimension of T is N, f(m) contains N predicted data, and J is N×M Jacobian matrix. The posterior probability density (PPD) P(m|T) is given by Bayes theorem [8]
为了获得（2）中定位模型的稳定解，我们将模型参数 m 视为受先验不确定性约束的未知数，并基于经验贝叶斯方法分析参数和观测值的统计性质。在我们的方法中， m 包含 M 模型参数，的 T 维度包含 N, f(m) N 预测数据， J 并且是 N×M 雅可比矩阵。后验概率密度（PPD） P(m|T) 由贝叶斯定理给出[8]

View Source  查看源码 \begin{equation*}P(\mathbf{m}\vert \mathbf{T})=\frac{P(\mathbf{T}\vert \mathbf{m})\times P(\mathbf{m})}{P(\mathbf{T})}\tag{6}\end{equation*}

Fig. 1.   图 1.

Experimental configuration: (a) Observation configuration of moving transducer and seafloor transponders; (b) Transponder mooring system; (c) Observed SSPs; (d) Ranging observations between transponders (B1-B4 indicates the ranging observations when transponder B1 acted as a master transducer).
实验配置：（a）移动换能器和海底应答器的观测配置;（b） 应答器系泊系统;（c） 观测到的 SSP;（d） 应答器之间的测距观测（B1-B4 表示应答器 B1 充当主换能器时的测距观测）。

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The observation errors are approximately Gaussian with covariance matrix CT, and the prior information is the Gaussian distribution with covariance matrix Cm0⋅ CT and Cm0 are expressed as follows using the hyperparameters σ2,ξ,λ, and μ:
观测误差近似为高斯与协方差矩阵 CT ，先验信息为高斯分布与协方差矩阵 Cm0⋅ CT Cm0 ，使用超参数 σ2,ξ,λ 表示如下 μ ：

View Source  查看源码 \begin{align*} &\mathbf{C}_{\mathbf{T}}=\sigma^{2}\left(\mathbf{C}_{\mathbf{T}^{1}}^{\prime}+\frac{\mathbf{C}_{\mathbf{T}^{2}}^{\prime}}{\lambda}\right)\tag{7}\\ &\mathbf{C}_{\mathbf{m}^{0}}=\frac{\sigma^{2}}{\xi}\left(\mathbf{C}_{\mathbf{m}^{0,1}}^{\prime}+\frac{\mathbf{C}_{\mathbf{m}^{0,2}}^{\prime}}{\mu}\right)\tag{8} \end{align*}

ml+1=ml+(JT(C′T1+C′T2λ)−1J+ξ(C′m0,1+C′m0,2μ)−1)−1×JT(C′T1+C′T2λ)−1(T−f(ml))(9)

View Source  查看源码 \begin{align*}\mathbf{m}^{l+1}=&\mathbf{m}^{l}+\left(\mathbf{J}^{T}\left({\mathbf{C}^{\prime}}_{\mathbf{T}^{1}}+\frac{{\mathbf{C}^{\prime}}_{\mathbf{T}^{2}}}{\lambda}\right)^{-1}\mathbf{J}+\xi\left({\mathbf{C}^{\prime}}_{\mathbf{m}^{0,1}}+\frac{{\mathbf{C}^{\prime}}_{\mathbf{m}^{0,2}}}{\mu}\right)^{-1}\right)^{-1}\\ &\times \mathbf{J}^{T}\left({\mathbf{C}^{\prime}}_{\mathbf{T}^{1}}+\frac{{\mathbf{C}^{\prime}}_{\mathbf{T}^{2}}}{\lambda}\right)^{-1}\left(\mathbf{T}-f\left(\mathbf{m}^{l}\right)\right)\tag{9}\end{align*}

The ABIC based on maximum entropy principle is adopted to determine the hyperparameters [9] and [10]
采用基于最大熵原理的 ABIC 确定超参数[9]和[10]

View Source  查看源码 \begin{equation*}\text{ABIC} =-2\log_{e}P\left(\mathbf{T}\vert \sigma^{2},\xi,, \lambda,\mu\right)+2K_{H}\tag{10}\end{equation*}

ABIC=Nloge(2πϕ(mMAP)N)−loge∣∣∣C′T1+C′T2λ∣∣∣−1−loge∣∣∣C′m0,1+C′m0,2μ∣∣∣−1−Mlogeξ+loge∣∣∣∣JT(C′T1+C′T2λ)−1J+ξ(C′m0,1+C′m0,2μ)−1∣∣∣+const(11)

View Source  查看源码 \begin{align*} \text{ABIC}=&N\!\log_{e}\left(\frac{2\pi\phi(\mathbf{m}_{\text{MAP}})}{N}\right)- \log_{e}\left\vert {\mathbf{C}^{\prime}}_{\mathbf{T}^{1}}+\frac{{\mathbf{C}^{\prime}}_{\mathbf{T}^{2}}}{\lambda}\right\vert ^{-1}\\ &- \log_{e}\left\vert {\mathbf{C}^{\prime}}_{\mathbf{m}^{0,1}}+\frac{{\mathbf{C}^{\prime}}_{\mathbf{m}^{0,2}}}{\mu}\right\vert ^{-1}-M \log_{e}\xi\\ &+ \log_{e}\left\vert \mathbf{J}^{T}\left({\mathbf{C}^{\prime}}_{\mathbf{T}^{1}}+\frac{{\mathbf{C}^{\prime}}_{\mathbf{T}^{2}}}{\lambda}\right)^{-1}\right.\mathbf{J}\\ &+\left. \xi\left({\mathbf{C}^{\prime}}_{\mathbf{m}^{0,1}}+\frac{{\mathbf{C}^{\prime}}_{\mathbf{m}^{0,2}}}{\mu}\right)^{-1}\right\vert +\text{const}\tag{11} \end{align*}

ϕ(mMAP)=(T−f(mMAP))T(C′T1+C′T2λ)−1(T−f(mMAP))+ξ(mMAP−m0)T(C′m0,1+C′m0,2μ)−1(mMAP−m0)(12)

View Source  查看源码 \begin{align*} \phi(\mathbf{m}_{\text{MAP}})=&(\mathbf{T}-f(\mathbf{m}_{\text{MAP}}))^{T}\left({\mathbf{C}^{\prime}}_{\mathbf{T}^{1}}+\frac{{\mathbf{C}^{\prime}}_{\mathbf{T}^{2}}}{\lambda}\right)^{-1}(\mathbf{T}-f(\mathbf{m}_{\text{MAP}}))\\ &+\xi\left(\mathbf{m}_{\text{MAP}}-\mathbf{m}^{0}\right)^{T}\left({\mathbf{C}^{\prime}}_{\mathbf{m}^{0,1}}+\frac{{\mathbf{C}^{\prime}}_{\mathbf{m}^{0,2}}}{\mu}\right)^{-1}\left(\mathbf{m}_{\text{MAP}}-\mathbf{m}^{0}\right)\tag{12} \end{align*}

σ2=ϕ(mMAP)N.(13)

View Source  查看源码 \begin{equation*}\sigma^{2}=\frac{\phi(\mathbf{m}_{\text{MAP}})}{N}.\tag{13}\end{equation*}

Seafloor positioning experiments were conducted in the South China Sea from May 1 to May 2, 2021. Five transponders were deployed on the 3km deep seafloor, as shown in Fig. 1(a). The transponders B1, B2, and B3 were installed on a rigid equilateral-triangle base, which was deployed on a seafloor station. The rigid geometric distance between the transponders is 800 mm. The transponders B4 and B5 were placed on the seafloor using “float-cord-sinker” moorings, as shown in Fig. 1(b). The observed SSPs are presented in Fig. 1(c). The surface ship carried out acoustic travel time observations according to the predetermined track shown in Fig. 1(a). In our positioning method based on the estimation of the temporal variation of sound speed profile, called ETV-SSP method, the acoustic ranging observations between seafloor transponders were introduced in this experiment. Each transponder acted as the master transducer and transmitted acoustic signals of ranging observations, and the other transponders received the acoustic signals. The ranging observations (as shown in Fig. 1(d)) were obtained using the travel time between transponders and the average sound speed.
2021 年 5 月 1 日至 5 月 2 日在南海开展海底定位实验。在 3 km 深的海底部署了 5 个应答器，如图 1（a）所示。应答器 B1、B2 和 B3 安装在刚性等边三角形底座上，该底座部署在海底站上。应答器之间的刚性几何距离为 800 毫米。应答器 B4 和 B5 使用“浮索-绳索-坠落器”系泊装置放置在海底，如图 1（b）所示。观测到的 SSP 如图 1（c）所示。水面舰艇按照图 1（a）所示的预定航迹进行了声学旅行时间观测。本实验基于估计声速剖面时间变化的定位方法（称为 ETV-SSP 方法）引入了海底应答器之间的声学测距观测。每个应答器充当主换能器，传输测距观测的声学信号，其他应答器接收声学信号。测距观测值（如图 1（d）所示）是使用应答器之间的传播时间和平均声速获得的。

In the proposed method, we first performed the estimation of the model parameters for transponders B1, B2 and B3, and then estimated the model parameters for transponders B4 and B5 using the acoustic travel time and ranging observations. And the hyperparameters ξ,λ and μ are determined by minimizing the value of ABIC using the grid search method.
在所提方法中，首先对应答器 B1、B2 和 B3 的模型参数进行估计，然后利用声学行程时间和测距观测估计了应答器 B4 和 B5 的模型参数。超参数 ξ,λ 和 μ 是利用网格搜索方法对 ABIC 的值进行最小化来确定的。

In order to verify the effectiveness of the ETV-SSP method, we used the GARPOS method as a comparison. The common method in this letter is the method that only estimates the transponder position and ignores the sound speed error. We used two groups of initial transponder position values to test the estimation performance of the methods. The position results of the common method were taken as the first group of initial values, called init1. And the position results of the ETV-SSP method but without the ranging observation constraint were set to the second group of initial values, called init2. The initial values of all sound speed parameters were set to zero.
为了验证 ETV-SSP 方法的有效性，我们使用 GARPOS 方法作为比较。这封信中常用的方法是只估计应答器位置而忽略声速误差的方法。我们使用两组初始应答器位置值来测试方法的估计性能。将常用方法的仓位结果作为第一组初始值，称为 init1。将 ETV-SSP 方法的位置结果设置为第二组初始值，称为 init2。所有声速参数的初始值都设置为零。

The comparison results of the transponder position are listed in Table 1. Since the maximum difference between the positioning results of the two groups of initial values for the proposed method is less than 10⁻⁵ m, we only listed the positioning results of the proposed method once. The two groups of results for the GARPOS method are called GARPOS of initl and GARPOS of init2 respectively. It can be seen from Table 1 that the maximum difference between the positioning results of GARPOS of init1 and GARPOS of init2 exceeds 1 m. Therefore, the proposed method has a higher level of robustness than the GARPOS method in terms of the initial values of the seafloor positioning. The rigid distance errors calculated using the estimated position results of transponders B1, B2, and B3 are as shown in Table 2. The average rigid distance error of the common method is 297.2 mm, and the other two methods improve the relative positioning accuracy. The average rigid distance error of the proposed method is less than 5 mm of the potting process error, and the error is reduced by at least 6.8 mm compared with the GARPOS method.
应答器位置的比较结果如表 1 所示。由于所提方法两组初始值的定位结果之间的最大差异小于 10 ⁻⁵ m，因此我们只列出了一次所提方法的定位结果。GARPOS 方法的两组结果分别称为 initl 的 GARPOS 和 init2 的 GARPOS。从表 1 可以看出，init1 的 GARPOS 和 init2 的 GARPOS 的定位结果最大差异超过 1 m。因此，所提方法在海底定位初始值方面比 GARPOS 方法具有更高的鲁棒性。使用应答器 B1、B2 和 B3 的估计位置结果计算的刚性距离误差如表 2 所示。常用方法的平均刚性距离误差为 297.2 mm，其他两种方法提高了相对定位精度。所提方法的平均刚距误差小于灌封工艺误差的 5 mm，与 GARPOS 方法相比，误差至少减少了 6.8 mm。

Table 1. Comparison results of seafloor transponder positioning for common method, GARPOS init1, GARPOS init2 and ETV-SSP method
表 1.通用方法 GARPOS init1、GARPOS init2 和 ETV-SSP 方法的海底应答器定位对比结果

Table 2. Rigid distance errors of positioning results
表 2.定位结果的刚性距离误差

We also tested the positioning accuracy of the seafloor transponders in terms of the long baseline positioning accuracy. As shown in Fig. 2(a), the positioning results of the transponders were used as the reference position for the onboard transducer positioning, and the long baseline positioning results were obtained using the nonlinear least square method. The positioning errors relative to the GNSS observations are shown in Fig. 2(b). The average positioning errors of GARPOS of init1 and GARPOS of init2 are 7.11 m and 7.15 m, respectively. The average positioning error of the proposed method is 5.58 m, which is at least 1.53 m better than the GARPOS method. Compared with the GARPOS method, the proposed method solves the ill-posed problem and reduces the positioning errors by introducing the constraint of ranging observations between transponders based on ABIC.
我们还测试了海底应答器的长基线定位精度方面的定位精度。如图 2（a）所示，以应答器的定位结果作为机载换能器定位的参考位置，采用非线性最小二乘法得到长基线定位结果。相对于 GNSS 观测的定位误差如图 2（b）所示。init1 的 GARPOS 和 init2 的 GARPOS 的平均定位误差分别为 7.11 m 和 7.15 m。所提方法的平均定位误差为 5.58 m，比 GARPOS 方法至少好 1.53 m。与 GARPOS 方法相比，所提方法通过引入基于 ABIC 的应答器间测距观测约束，解决了误题问题，减少了定位误差。

A new method has been proposed to estimate seafloor transponder position and SSP temporal variation. The proposed method solves the ill-posed problem caused by the strongly correlated parameters by introducing the constraint of ranging observations and adjusting the weight factors on prior information, travel time data variance, and ranging data variance based on Akaike's Bayesian information criterion. The results from the sea trial carried out in the South China Sea show that the proposed method exhibits better positioning performance of seafloor transponders than the GARPOS method in terms of rigid distance error and long baseline positioning error. This method reduces the average rigid distance error by 6.8 mm and improves the positioning accuracy of the long baseline by 1.53 m (relative to GNSS observations) when taking the positioned transponders as the reference position.
提出了一种估计海底应答器位置和 SSP 时间变化的新方法。该方法通过引入测距观测约束，并基于赤池贝叶斯信息准则调整先验信息、旅行时间数据方差和测距数据方差的权重因子，解决了强相关参数引起的误题问题。在南海进行的海试结果表明，所提方法在刚距误差和长基线定位误差方面表现出优于 GARPOS 方法的海底应答器定位性能。该方法以定位应答器为参考位置时，平均刚度距离误差降低了 6.8 mm，长基线的定位精度提高了 1.53 m（相对于 GNSS 观测值）。

Fig. 2.   图 2.

Long baseline positioning: (a) Configuration of transducer and seafloor transponders; (b) Positioning errors for common method, GARPOS init1, GARPOS init2, and ETV-SSP method.
长基线定位：（a） 传感器和海底应答器的配置;（b）常用方法、GARPOS init1、GARPOS init2 和 ETV-SSP 方法的定位误差。

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