A Virtual-Beacon-Based Calibration Method for Precise Acoustic Positioning of Deep-Sea Sensing Networks

Abstract

The rapid expansion of deep-sea sensing networks underscores the critical need for accurate underwater positioning of observation base stations. However, achieving precise acoustic localization, particularly at depths exceeding 4 km, remains a significant challenge due to systematic ranging errors, clock drift, and inaccuracies in sound speed modeling. This study proposes and validates a three-tier calibration framework consisting of a Dynamic Single-Difference (DSD) solver, a geometrically optimized reference buoy selection algorithm, and a Virtual Beacon (VB) depth inversion method based on sound speed profiles. Through simulations under varying noise conditions, the DSD method effectively mitigates common ranging and clock errors. The geometric reference optimization algorithm enhances the selection of optimal buoy layouts and reference points. At a depth of 4 km, the VB method improves vertical positioning accuracy by 15% compared to the DSD method alone, and nearly doubles vertical accuracy compared to traditional non-differential approaches. This research demonstrates that deep-sea underwater target calibration can be achieved without high-precision time synchronization and in the presence of fixed ranging errors. The proposed framework has the potential to lower technological barriers for large-scale deep-sea network deployments and provides a robust foundation for autonomous deep-sea exploration.

1. Introduction

The global ocean harbors vast living, mineral, and energy resources, and the scale and strategic importance of marine operations continue to grow worldwide [1,2]. Accurate underwater positioning supports a broad spectrum of marine activities, ranging from Autonomous Underwater Vehicle (AUV) navigation and sub-sea infrastructure inspection to seafloor geodesy and plate-boundary monitoring [3,4,5]. Because radio-frequency signals such as the Global Navigation Satellite System (GNSS) do not propagate effectively in water, underwater vehicles lose direct GNSS access once submerged [6]. Consequently, sub-sea navigation must rely on alternative techniques: inertial navigation systems, whose errors grow unbounded with time [7], and acoustic positioning systems, which provide absolute fixes but require careful calibration and are sensitive to environmental disturbances [8,9].

In practice, the accuracy of acoustic positioning is limited by systematic measurement and modelling errors [10]. One dominant source is the uncertainty in the three-dimensional sound-speed structure [11,12]. Even with modern profile observation methods, temporal, and spatial fluctuations (e.g., thermocline, salinity gradients, internal waves) cause the actual acoustic velocity along a path to deviate from the assumed model [13,14,15]. A mean bias of only 1 m/${\mathrm{s}}^{-1}$ over a 5000 m two-way path introduces range errors of several meters. A second critical source is the transducer timing/lever-arm bias, which includes fixed electronic delays and spatial offset between acoustic transducers; if not calibrated, such biases impose constant errors on all ranges [16]. Although classical acoustic processing treats these biases as additional least-squares parameters [17,18], the resulting solutions are often strongly correlated. For example, an over-estimated sound speed can be partially compensated by shifting the estimated depth. Mitigation strategies such as circular or radial survey patterns, multi-epoch observations, or multi-platform campaigns reduce but do not eliminate residual biases [19,20]. Vertical positioning is particularly vulnerable, as any unmodelled propagation delay maps directly into a depth offset [21].

To avoid permanent seafloor infrastructure, we focus on absolute position calibration of temporarily deployed seafloor beacons using a surface buoy array [22,23]. After deployment, a seafloor station drifts before settling; its final coordinates must be determined. Multiple buoys form a surface array, each periodically transmitting an acoustic pulse and recording the travel time to the target on the sea floor. Combining the measured travel times with the precisely known GNSS positions yields a set of range constraints that can be inverted for the three-dimensional position of the seafloor station [24,25].

To address these shortcomings, we propose a three-layer calibration framework that combines and extends existing ideas in a novel way. The architecture consists of: (i) a Dynamic Single-Difference (DSD) solver, (ii) a geometry-aware reference-buoy selection algorithm, and (iii) a Virtual-Beacon (VB) depth inversion scheme. The virtual beacon concept, in essence, introduces an additional depth-sensitive constraint by leveraging the time-of-flight vs. depth relationship. This concept has been explored in prior works–for example, using virtual reference points or ‘virtual long baselines’ to aid underwater positioning [26,27], but here we integrate it specifically to improve vertical accuracy in a multi-buoy acoustic array. Our novelty lies in the synergistic integration of the VB depth constraint with a single-difference ranging strategy and real-time optimal reference selection. This combined approach effectively cancels common-mode errors while anchoring the vertical solution, resulting in further improved calibration accuracy for seafloor beacons, especially in the vertical direction. The architecture consists of three parts.

Dynamic Single-Difference (DSD) solver. Acoustically measured ranges are different between multiple transponders within the same epoch or between successive pings to the same transponder. Many common-mode errors, for example, constant-range bias or timing bias, are thereby canceled or greatly suppressed, enhancing the solution accuracy and robustness.

Geometry-aware reference-buoy selection. Because single differencing eliminates an observation, the choice of reference directly affects the noise amplification matrix. A real-time algorithm based on data-driven selects the buoy that minimizes the single-difference Position-Dilution of Precision (PDOP), thus preserving the favorable geometry.

Virtual-beacon (VB) depth inversion. To overcome the poor vertical accuracy inherent to single-difference positioning, which is primarily caused by the unfavorable geometry of having all buoys confined to the sea surface, we introduce a virtual-beacon depth-inversion scheme. By combining the virtual beacon depth with the curvature information contained in the time-of-flight versus depth relationship derived from the sound speed profile, the method synthesizes an additional, depth-sensitive constraint that effectively anchors the vertical solution.

The DSD–VB combination jointly suppresses systematic errors resulting from sound speed mismodelling, clock drift, and transducer offsets, thus improving robustness to environmental disturbances. Figure 1 summarises the observation geometry considered in this study and illustrates the central idea behind the Virtual–Beacon (VB) depth inversion. Surface buoys provide slant ranges to a seafloor target; by scanning candidate depth slices and examining the spatial compactness of the resulting range-intersection clouds, we recover the depth that best matches the acoustic data and thereby strengthen the vertical component of the 3-D solution. Taken together, the integrated DSD–geometry–VB framework provides high horizontal positioning accuracy while significantly enhancing vertical localization performance, thereby reducing the reliance on ultra-precise ranging hardware and lowering the deployment barrier of large-scale sub-sea Internet-of-Things systems [28]. This offers a robust navigation foundation for future autonomous deep-ocean operations.

The rest of this paper is organized as follows. Section 2 details the principles and mathematical models of the three components. Section 3 presents Monte-Carlo simulations under progressively complex error scenarios—range bias, clock drift and sound-speed mismodelling—to quantify performance gains. Section 4 analyses the results in terms of error propagation and proposes a calibration workflow for deep-ocean deployments. Finally, Section 5 summaries the main findings and their significance for next-generation high-precision underwater positioning.

2. Methods

This section formalizes the three pillars of the proposed calibration framework and provides sufficient algorithmic detail for independent replication. Section 2.1 develops the DSD observation model, its linearized least squares solution, and the associated error-suppression mechanism. Section 2.2 derives closed-form expressions for PDOP in both non-differential and single-difference modes, and introduces an on-line criterion for optimal reference-buoy selection. Section 2.3 describes the VB depth-fixing strategy: from sound-speed profile (SSP) pre-processing and time-of-flight (ToF) database construction, through exhaustive depth slice evaluation, to the compactness metric that yields the final vertical solution.

2.1. Dynamic Single-Difference Positioning

In GNSS-A underwater positioning systems, systematic errors, including inaccuracies in acoustic velocity and sensor calibration biases, pose significant challenges to accurate positioning. To effectively mitigate these systematic influences, this study introduces a single-difference dynamic positioning methodology between underwater acoustic reference stations.

In each observation epoch the one–way acoustic range inferred for surface (or buoy) beacon j to the underwater target at state $X_t=(x_t,y_t,z_t)$ is modeled as

$$R_j=G_j\left(X_t\right)+{\tau }_j+{\beta }_j+{\gamma }_j+{\eta }_j,$$

(1)

where all additive terms are expressed in meters of range. The physical interpretation of each term is as follows:

$G_j\left(X_t\right)$

Geometric path length (m) between the target $X_t$ and beacon j at known coordinates $(x_j,y_j,z_j)$, computed under the assumed propagation model (Section 3.1).

${\tau }_j$

Transducer/electronics fixed range bias (m): equivalent path length representing fixed Tx/Rx electronics delays and signal-processing latencies particular to beacon j.

${\beta }_j$

Clock-synchronization range error (m): residual range error arising from epoch-constant transmit/receive timing misalignment between beacon j and the timing reference. If the relative timing offset is $\Delta t_j$ (s), then ${\beta }_j\approx c_{\mathrm{est}}\Delta t_j$.

${\gamma }_j$

Sound-speed modeling contribution (m): bias introduced when the assumed propagation speed differs from the true path-integrated water-column sound speed. Because the induced error scales with distance, ${\gamma }_j$ is typically approximately proportional to $G_j\left(X_t\right)$ under small fractional speed errors ($\Delta c/c\ll 1$), i.e., ${\gamma }_j\simeq {\displaystyle \frac{\Delta c}{c_{\mathrm{est}}}}{G}_j\left(X_t\right)$ to first order.

${\eta }_j$

Random measurement noise (m): zero-mean stochastic residual capturing timing jitter, unresolved multipath, and unmodeled effects after quality control.

Selecting beacon 0 as the reference, we form the dynamic single-difference (SD) observable

$$\delta R_{j|0}=R_j-R_0={\Psi }_{j|0}\left(X_t\right)+\delta {\tau }_{j|0}+\delta {\beta }_{j|0}+\delta {\gamma }_{j|0}+\delta {\eta }_{j|0},$$

(2)

with component definitions

$$\begin{array}{cc}\hfill {\Psi }_{j|0}\left(X_t\right)& =G_j\left(X_t\right)-G_0\left(X_t\right),\hfill \\ \hfill \delta {\tau }_{j|0}& ={\tau }_j-{\tau }_0,\hfill \\ \hfill \delta {\beta }_{j|0}& ={\beta }_j-{\beta }_0,\hfill \\ \hfill \delta {\gamma }_{j|0}& ={\gamma }_j-{\gamma }_0,\hfill \\ \hfill \delta {\eta }_{j|0}& ={\eta }_j-{\eta }_0.\hfill \end{array}$$

Under homogeneous instrumentation and within a short observation epoch, the additive hardware and clock terms are effectively common mode so that $\delta {\tau }_{j|0}\approx 0$ and $\delta {\beta }_{j|0}\approx 0$. However, the sound-speed (scale) contribution $\delta {\gamma }_{j|0}$ does not cancel in general, because a fractional sound-speed error $\Delta c/c$ maps proportionally to path length and therefore differs across beacons. Retaining this term yields the working SD observation used in our solver,

$$\delta R_{j|0}\approx {\Psi }_{j|0}\left(X_t\right)+\delta {\gamma }_{j|0}+\delta {\eta }_{j|0}.$$

(3)

Introducing intermediate auxiliary variables, the functional form of ${\Psi }_{j|0}\left(X_t\right)$ can be detailed as:

$${\Psi }_{j|0}\left(X_t\right)=\parallel X_t-X_j\parallel -\parallel X_t-X_0\parallel$$

(4)

where the distance norm $\parallel ·\parallel$ between two points $X_a=(x_a,y_a,z_a)$ and $X_b=(x_b,y_b,z_b)$ is given by:

$$\parallel X_a-X_b\parallel =\sqrt{{(x_a-x_b)}^2+{(y_a-y_b)}^2+{(z_a-z_b)}^2}$$

(5)

To obtain the positioning solution, the non-linear equation can be linearized around an initial approximate position $X_t^{\left(0\right)}$, resulting in a linear system:

$$\mathbf{A}\Delta X_t=\mathbf{l}$$

(6)

where the incremental parameter vector $\Delta X_t={[\Delta x,\Delta y,\Delta z]}^T$, the observation vector $\mathbf{l}$, and the Jacobian design matrix $\mathbf{A}$ are defined as:

$$\mathbf{l}={[\delta R_{1|0}-{\Psi }_{1|0}\left(X_t^{\left(0\right)}\right),\delta R_{2|0}-{\Psi }_{2|0}\left(X_t^{\left(0\right)}\right),\dots ,\delta R_{n|0}-{\Psi }_{n|0}\left(X_t^{\left(0\right)}\right)]}^T$$

(7)

$$\mathbf{A}={\left[\begin{array}{ccc}{\displaystyle \frac{\partial {\Psi }_{1|0}}{\partial x}}& {\displaystyle \frac{\partial {\Psi }_{1|0}}{\partial y}}& {\displaystyle \frac{\partial {\Psi }_{1|0}}{\partial z}}\\ {\displaystyle \frac{\partial {\Psi }_{2|0}}{\partial x}}& {\displaystyle \frac{\partial {\Psi }_{2|0}}{\partial y}}& {\displaystyle \frac{\partial {\Psi }_{2|0}}{\partial z}}\\ ⋮& ⋮& ⋮\\ {\displaystyle \frac{\partial {\Psi }_{n|0}}{\partial x}}& {\displaystyle \frac{\partial {\Psi }_{n|0}}{\partial y}}& {\displaystyle \frac{\partial {\Psi }_{n|0}}{\partial z}}\end{array}\right]}_{X_t=X_t^{\left(0\right)}}$$

(8)

Before linearizing the non-linear range equations (Equation (12)), a starting estimate of the target position $X_t^{\left(0\right)}=(x_t^{\left(0\right)},y_t^{\left(0\right)},z_t^{\left(0\right)})$ must be specified for the iterative single-difference least-squares solution. In practical deployments such an estimate is always available at modest accuracy. Useful sources include: (i) the recorded deployment/drop coordinates of the seafloor station; (ii) a coarse multilateration computed from the first few one-way travel-time measurements under a nominal constant sound speed; and/or (iii) the centroid of the initial set of range-circle intersections, which provides a quick horizontal fix. The depth component may be initialized from an onboard pressure/depth sensor reading or from local bathymetry if available. This initial guess does not need to be highly accurate—it only needs to fall within a reasonable vicinity (typically tens to a few hundreds of meters) of the true location for the iterative least-squares adjustment to converge robustly. Linearizing about $X_t^{\left(0\right)}$ yields the system summarized in Equation (12).

The optimal least-squares solution for incremental position update $\Delta X_t$ is:

$$\Delta X_t={\left({\mathbf{A}}^T\mathbf{A}\right)}^{-1}{\mathbf{A}}^T\mathbf{l}$$

(9)

This Dynamic Single-Difference methodology significantly reduces systematic biases and ensures enhanced robustness and accuracy of underwater dynamic target positioning. Strategic selection and arrangement of reference points are crucial for optimizing overall positioning precision.

2.2. Spatial Position Dilution of Precision

The PDOP expresses how the geometry of a sensor constellation amplifies measurement noise into positioning error. For clarity, we first restate the classical non-differential definition using a compact matrix notation, then derive its single-difference counterpart.

2.2.1. Linearized Observation Model

Let ${\mathbf{p}}_i={(x_i,y_i,z_i)}^{\mathsf{T}}$$(i=1,\dots ,N)$ be the known coordinates of the N acoustic beacons and $\mathbf{x}={(x,y,z)}^{\mathsf{T}}$ the unknown position of the target. Given the provisional estimate ${\mathbf{x}}_0$, the range residual to beacon i is

$$\delta {\rho }_i=\underset{{\displaystyle {\mathbf{u}}_i^{\mathsf{T}}}}{\underbrace{{\displaystyle \frac{{\mathbf{x}}_0-{\mathbf{p}}_i}{\parallel {\mathbf{x}}_0-{\mathbf{p}}_i\parallel }}}}\delta \mathbf{x}+{\nu }_i,$$

(10)

where ${\mathbf{u}}_i$ is the unit line-of-sight (LOS) vector and ${\nu }_i\sim \mathcal{N}(0,{\sigma }_{\rho }^2)$ the range noise. Stacking (10) for all beacons yields

$$\delta \rho =\mathbf{A}\delta \mathbf{x}+\nu ,\mathbf{A}=\left[\begin{array}{c}{\mathbf{u}}_1^{\mathsf{T}}\\ {\mathbf{u}}_2^{\mathsf{T}}\\ ⋮\\ {\mathbf{u}}_N^{\mathsf{T}}\end{array}\right]\in {\mathbb{R}}^{N\times 3}.$$

(11)

2.2.2. PDOP for Non-Differential Measurements

With unit weights, the least squares covariance of $\widehat{\mathbf{x}}$ is $\mathbf{Q}={\left({\mathbf{A}}^{\mathsf{T}}\mathbf{A}\right)}^{-1}{\sigma }_{\rho }^2$. Following GPS convention, the scalar PDOP is defined as the root-sum square of the position variances.

$$\mathrm{PDOP}=\sqrt{tr\left[{\left({\mathbf{A}}^{\mathsf{T}}\mathbf{A}\right)}^{-1}\right]}=\sqrt{{\lambda }_1^{-1}+{\lambda }_2^{-1}+{\lambda }_3^{-1}},$$

(12)

where ${\lambda }_1\ge {\lambda }_2\ge {\lambda }_3>0$ are the eigenvalues of ${\mathbf{A}}^{\mathsf{T}}\mathbf{A}$. Equation (12) shows that a constellation is geometrically robust when its three principal LOS directions are nearly orthogonal, i.e., all ${\lambda }_i$ are of comparable magnitude.

2.2.3. PDOP for Single-Difference Observations

To remove common timing or sound speed biases, one beacon r is selected as a reference, and single difference (SD) ranges are formed via the differencing operator ${\mathbf{D}}_r=[-{\mathbf{1}}_{N-1}{\mathbf{I}}_{N-1}]$, here $-{\mathbf{1}}_{N-1}$ is a column vector of length $(N-1)$ containing all $-1$ entries (used to subtract the reference observation), and ${\mathbf{I}}_{N-1}$ is the $(N-1)\times (N-1)$ identity matrix. Applying ${\mathbf{D}}_r$ to the raw range vector $\rho$ gives

$${\mathbf{D}}_r\delta \rho =\left({\mathbf{D}}_r\mathbf{A}\right)\delta \mathbf{x}+{\mathbf{D}}_r\nu ={\mathbf{A}}_r\delta \mathbf{x}+\tilde{\nu },$$

(13)

where $\mathrm{cov}\left(\tilde{\nu }\right)=2{\sigma }_{\rho }^2{\mathbf{I}}_{N-1}$. The SD covariance matrix is therefore $2{\left({\mathbf{A}}_r^{\mathsf{T}}{\mathbf{A}}_r\right)}^{-1}{\sigma }_{\rho }^2$ and the single-difference PDOP is

$${\mathrm{PDOP}}_r=\sqrt{2tr\left[{\left({\mathbf{A}}_r^{\mathsf{T}}{\mathbf{A}}_r\right)}^{-1}\right]}.$$

(14)

Because (i) one equation is lost and (ii) the noise variance doubles, ${\mathrm{PDOP}}_r$ is generally larger than its non-differential counterpart. However, it is immune to any bias common to all ranges, making it indispensable when high-precision synchronization is unavailable.

2.2.4. Optimal Choice of Reference Beacon

For each candidate r the geometry matrix ${\mathbf{A}}_r$ and hence ${\mathrm{PDOP}}_r$ vary. The bias-free geometry best suited for single-difference positioning at a given location is obtained by

$$r^{★}=\underset{r\in \{1,\dots ,N\}}{argmin}{\mathrm{PDOP}}_r.$$

(15)

Equation (15) is evaluated online within our simulation to guarantee that the SD solution uses the most favorable beacon as a reference at every point on the grid.

2.3. Aided Depth Fixing Method Based on Virtual Beacon Construction

The program initially reads the input SSP and performs initialization procedures, such as extrapolation at the beginning and end. Subsequently, it calculates the ranges of possible emission angles and propagation times. Within these calculated ranges, propagation times are iterated in steps of 1 m/s (0.001 s), while emission angles are iterated in steps of 0.01 degrees. Using the layered constant-gradient acoustic ray tracing model, the program computes the endpoint positions, thereby generating an initial database containing propagation time, emission angle, and endpoint position information. Specifically, this model approximates the ocean sound-speed profile by dividing the water column into multiple layers, each with a constant linear gradient of sound speed versus depth. Within each layer, analytic ray paths can be computed efficiently using Snell’s law.

Based on this database, the horizontal distance associated with each depth and propagation-time pair is retrieved within a reasonable interval. The original dataset is then refined by filtering out infeasible ray paths, including those experiencing multipath effects, surface/bottom reflections, and mid-water ray turning. The term mid-water ray turning refers to the physical phenomenon whereby refracted acoustic rays change their vertical direction after passing through the minimum sound-speed channel axis, potentially reversing direction without encountering any boundary. In the database construction, only direct-path first arrivals are retained, while rays that exhibit multiple turnings or any boundary interactions are systematically excluded. Thus, a refined database consisting of depth, propagation time, and horizontal range information suitable for depth inversion is established. As shown in Figure 2, the program’s primary workflow proceeds as follows.

For realistic application scenarios, initial depth search values for underwater targets are defined, and potential depth intervals are systematically explored.Specifically, for i reference buoys, there are $\left({\displaystyle \genfrac{}{}{0pt}{}{i}{2}}\right)$ unique buoy pairs, each potentially yielding up to two intersection points, resulting in at most $2\left({\displaystyle \genfrac{}{}{0pt}{}{i}{2}}\right)$ candidate horizontal solution points. Many of these candidate solutions are geometrically redundant “mirror” intersections or points that do not simultaneously satisfy all range constraints. By analyzing the consistency and proximity of these candidate points, erroneous or virtual positions are systematically identified and eliminated. True positions typically exhibit significant consistency and compact clustering, whereas virtual positions show considerable dispersion.

When the assumed depth closely aligns with the actual target depth, the candidate intersection points cluster tightly, indicating a high consistency with the observed data. Conversely, incorrect hypothesized depths yield widely scattered candidate points. Consequently, the reliability and accuracy of the depth estimation directly correlate with the degree of compactness among candidate positions. The iterative depth evaluation proceeds until the minimal dispersion threshold is identified, accurately determining both the depth and horizontal position of the target. The algorithm logic is summarized in Algorithm 1.

Algorithm 1: Virtual-Beacon-Aided Depth Inversion

For each tested depth slice $z_k$ the Virtual–Beacon (VB) routine produces a set of candidate horizontal points ${\mathcal{P}}_k$ by pairwise intersecting the range circles implied by the i active surface buoys: circle p has radius $R_p\left(z_k\right)$ about buoy coordinates ${\mathbf{b}}_p$, and similarly for q. Solving

$$\parallel \mathbf{x}-{\mathbf{b}}_p\parallel =R_p\left(z_k\right),\parallel \mathbf{x}-{\mathbf{b}}_q\parallel =R_q\left(z_k\right),$$

(16)

for every unordered buoy pair $(p,q)$ yields 0, 1 (tangent), or up to 2 intersection points. Because there are $\left({\displaystyle \genfrac{}{}{0pt}{}{i}{2}}\right)=i(i-1)/2$ unique buoy pairs, at most $2\left({\displaystyle \genfrac{}{}{0pt}{}{i}{2}}\right)$ candidate horizontal points can arise at depth $z_k$. Many of these are merely geometric “mirror” solutions that cannot satisfy the ranges to all buoys simultaneously; we refer to such inconsistent solutions as virtual points.

To isolate the physically admissible solution we perform a simple cluster analysis. Let

$${\mathbf{c}}_k=\mathrm{centroid}\left({\mathcal{P}}_k\right)={\displaystyle \frac{1}{|{\mathcal{P}}_k|}}\sum _{\mathbf{p}\in {\mathcal{P}}_k}\mathbf{p}$$

(17)

be the centroid of all candidates, and define the cluster compactness (root–mean–square scatter)

$$C_k=\sqrt{{\displaystyle \frac{1}{|{\mathcal{P}}_k|}}\sum _{\mathbf{p}\in {\mathcal{P}}_k}{∥\mathbf{p}-{\mathbf{c}}_k∥}^2}.$$

(18)

When $z_k$ is close to the true target depth the independent range circles from all buoys intersect near a common point; the candidates therefore form a tight cluster and $C_k$ is small. At incorrect depth slices the circles do not share a common intersection, producing a widely dispersed candidate cloud and a correspondingly large $C_k$.

The VB algorithm selects the depth estimate

$$\widehat{z}=arg\underset{z_k}{min}{C}_k,$$

(19)

i.e., the depth slice whose candidate set is most compact. Having identified $\widehat{z}$ we remove outliers by retaining only those candidates that lie within a distance threshold ${\tau }_{\widehat{z}}$ of the centroid ${\mathbf{c}}_{\widehat{z}}$:

$${\mathcal{P}}_{\widehat{z}}^{*}=\left\{\mathbf{p}\in {\mathcal{P}}_{\widehat{z}}|\parallel \mathbf{p}-{\mathbf{c}}_{\widehat{z}}\parallel \le {\tau }_{\widehat{z}}\right\}.$$

(20)

The threshold may be set as a user–defined multiple of the intrinsic scatter (e.g., ${\tau }_{\widehat{z}}=\kappa C_{\widehat{z}}$ with a modest $\kappa$); points outside this bound are flagged as virtual and discarded. The final horizontal fix at depth $\widehat{z}$ is then taken as the centroid of the retained cluster,

$$\widehat{\mathbf{p}}=\mathrm{centroid}\left({\mathcal{P}}_{\widehat{z}}^{*}\right).$$

(21)

This procedure operationalizes the distinction between the geometrically consistent (true) solution and spurious virtual intersections.

The validity of selecting the depth slice that minimizes the compactness statistic $C_k$ rests on a simple geometric principle. In the absence of measurement noise, the range circles associated with the correct depth all intersect at a single common point (the true target projection), whereas circles evaluated at any incorrect depth fail to meet at one location and instead intersect in mutually inconsistent pairs. Consequently, the candidate intersection set at the correct depth collapses to a single point (zero scatter), while incorrect depths yield a dispersed set with nonzero scatter.

With realistic noise the situation becomes probabilistic: the true intersection is blurred into a compact cluster whose size reflects the measurement errors, but a pronounced contrast in dispersion persists between the correct depth slice and slices that are far from the truth. Hence the depth slice at which $C_k$ attains a clear minimum provides a robust internal indicator of the target depth.

3. Simulation Experiments and Results

This section presents a comprehensive suite of simulation experiments that evaluate the proposed Single-Difference methodology, the PDOP-optimised buoy-array design, and the Virtual Beacon–aided depth fixing strategy.

3.1. Simulation Environment, Propagation Model, and Error Metrics

All simulation and test data used in this study were generated under a controlled deep-ocean environment so that the performance of the calibration framework could be evaluated reproducibly.

Water depth and bottom: We model a uniform open-ocean water column of total depth $H=4000\mathrm{m}$. Because only reliable direct-path (first-arrival) travel times are used in the positioning algorithms—all multipath arrivals are excluded during data preparation—the seafloor boundary is treated as flat and perfectly absorptive. Under this assumption, bottom-reflected energy does not contribute to the travel-time data set.

Sound-speed profile (SSP): The depth variation of true sound speed follows the canonical deep-ocean Munk profile. With reference speed $c_0=1500\mathrm{m}{\mathrm{s}}^{-1}$, amplitude $\epsilon =0.00737$, thermocline scale depth $B=1300\mathrm{m}$, and sound-speed minimum at $z_0=1300\mathrm{m}$, the profile over $0\le z\le H$ is

$$c\left(z\right)=c_0\left[1+\epsilon \left(e^{-(z-z_0)/B}+(z-z_0)/B-1\right)\right].$$

(22)

To investigate the impact of sound-speed measurement error on positioning performance, we introduce an equivalent mean bias relative to the above Munk field during processing, implemented as an effective constant sound speed.

Sound-propagation model: Acoustic travel times are computed with a layered constant-gradient ray-tracing approach: the water column is discretized into depth layers, and within each layer the vertical gradient $\partial c/\partial z$ is assumed constant, permitting analytic Snell-law eigenray solutions and efficient tabulation of time–distance relationships. A lookup table of travel time versus depth and horizontal range is generated at 1 ms time resolution and ${0.01}^{\circ }$ launch-angle resolution. The direct-path database so obtained is used consistently in both the Single-Difference (DSD) and Virtual-Beacon (VB) components of the calibration workflow.

Error metrics and statistical procedures: All performance statistics are computed relative to the known ground–truth trajectory available in simulation. Let $(x_t,y_t,z_t)$ be the estimated target coordinates at epoch $t=1,\dots ,N$ and $(x_t^{\mathrm{true}},y_t^{\mathrm{true}},z_t^{\mathrm{true}})$ the truth. Define coordinate offsets $\Delta x_t=x_t-x_t^{\mathrm{true}}$, $\Delta y_t=y_t-y_t^{\mathrm{true}}$, $\Delta z_t=z_t-z_t^{\mathrm{true}}$, horizontal error magnitude $e_{H,t}=\sqrt{\Delta x_t^2+\Delta y_t^2}$, vertical error magnitude $e_{V,t}=|\Delta z_t|$, and 3-D error magnitude $e_{3\mathrm{D},t}=\sqrt{\Delta x_t^2+\Delta y_t^2+\Delta z_t^2}$. Tables list “mean ± std” values for each error component, where the mean is the sample mean of $e_{H,t}$, $e_{V,t}$, or $e_{3\mathrm{D},t}$ over all epochs and “std” is the corresponding sample standard deviation (solution stability). For selected cases we also plot error histograms (with kernel density estimates) to visualize spread and bias, confirming that RMSE summaries are representative. Geometric sensitivity is characterized by PDOP, HDOP, and VDOP, computed from the appropriate design matrices for undifferenced and single–difference geometries. These unitless factors indicate how range noise maps into position error and are used to interpret the RMSE results, especially in connection with the reference–buoy selection strategy.

3.2. Dynamic Single-Difference Positioning

This subsection validates the proposed dynamic single-difference (DSD) strategy under three error scenarios of increasing complexity.

The simulation uses five surface buoys arranged in a cross-shaped radial array. One buoy is located at the array center and four buoys are placed at the cardinal directions (north, south, east, west), each at a horizontal distance $L=1500\mathrm{m}$ from the center. (This configuration is equivalent to a “radiation” pattern consisting of a central buoy and four peripheral buoys on a circle of radius $1500\mathrm{m}$). All buoys are assumed to float at the sea surface ($z=0$). The underwater target is at a true depth of $z_{\mathrm{true}}=-4000\mathrm{m}$ and moves along a horizontal circular trajectory of radius $1000\mathrm{m}$ centered on the array origin, over 50 epochs. Three solvers are compared, including (1) non-difference least-squares (ND), (2) single-difference, center reference (DSD-C), and (3) single-difference, edge reference (DSD-E).

3.2.1. Case I—Random Noise + Constant Range Bias

Zero-mean Gaussian range noise $\mathcal{N}(0,{\sigma }_r^2)$ with ${\sigma }_r=0.05$ m added. Each range is biased by a fixed $+1.5$ m offset (for example, transducer delay). The positioning-error results for Case I are illustrated in Figure 3 and

Table 1. Positioning Error Statistics under Case I.

Method	Horizontal (m)	Vertical (m)	3-D (m)
ND	0.351 ± 0.094	1.537 ± 0.033	1.580 ± 0.024
DSD-C	0.207 ± 0.121	0.013 ± 0.909	0.781 ± 0.523
DSD-E	0.211 ± 0.123	0.023 ± 0.913	0.787 ± 0.523

.

3.2.2. Case II—Random Noise + Range Bias + Clock Offset

Besides the random noise and 1.5 m bias, a 1 m/s transmit-clock error is added. The positioning-error results for Case II are illustrated in Figure 4 and

Table 2. Positioning Error Statistics under Case II.

Method	Horizontal (m)	Vertical (m)	3-D (m)
ND	0.678 ± 0.097	$3.075\pm 0.033$	3.150 ± 0.023
DSD-C	0.207 ± 0.121	$0.013\pm 0.909$	0.781 ± 0.523
DSD-E	0.211 ± 0.123	$0.023\pm 0.913$	0.787 ± 0.523

.

3.2.3. Case III—Random Noise + Range Bias + Clock Offset + Sound-Speed Measurement Error

Building on the preceding error scenario, we introduce an additional sound-speed bias of $1\mathrm{m}{\mathrm{s}}^{-1}$.The positioning-error results for Case III are illustrated in Figure 5 and

Table 3. Positioning Error Statistics under Case III.

Method	Horizontal (m)	Vertical (m)	3-D (m)
ND	1.969 ± 0.115	$5.879\pm 0.040$	6.201 ± 0.024
DSD-C	0.200 ± 0.116	$3.062\pm 0.776$	3.072 ± 0.770
DSD-E	0.204 ± 0.116	$3.072\pm 0.782$	3.083 ± 0.774

.

3.3. Full–Depth PDOP, HDOP, VDOP Distributions

A cubic grid ($x,y\in [-1500,1500]$, $z\in [0,-4000]$) was used to evaluate the precision factor of the Decagon,+,Center matrix. The array consists of ten buoys uniformly distributed along a radius circle R in the horizontal plane ($z=0$), positioned at angular intervals of 36° to form a regular decagon. An additional buoy is located at the center of the array, resulting in a total of eleven elements. For every node within the grid, we computed the undifferenced DOP triplet $\left\{\mathrm{PDOP},\mathrm{HDOP},\mathrm{VDOP}\right\}$, and the single-difference DOPs for two reference-selection strategies: center buoy and edge buoy.

The complete grid rendered as a semitransparent point cloud (x–y–z), color-mapped to the DOP magnitude.

(1) Undifferenced solution.

The undifferenced scenario provides baseline DOP distributions, illustrating the geometric precision achievable without differencing. The undifferenced DOP distributions over the full 3-D grid are presented in Figure 6.

(2) Center-Referenced Single Difference.

Employing the center buoy as the differencing reference eliminates common-mode biases such as clock drift or constant range errors but introduces geometric rank deficiency directly below the array center. Figure 7 illustrates the corresponding DOP distributions obtained with single-difference processing that uses the centre buoy as reference.

(3) Edge-Referenced Single Difference.

Select an edge buoy as the differencing reference.Figure 8 illustrates the corresponding DOP distributions obtained with single-difference processing that uses an edge buoy as reference.

3.4. Virtual-Beacon-Aided Depth Fixing

To assess the validity of the virtual-beacon (VB) inversion, we repeated the simulations of Section 2.3 under the following settings:

(1) Five surface buoys arranged in a cross pattern $\left\{[0,0],[\pm 1500,0],[0,\pm 1500]\right\}$ m;

(2) Gaussian range noise ${\sigma }_{\rho }=0.05$ m;

(3) A 1.5 m range bias, a 1 m/s sound–speed modeling error, and a 1 m/s clock offset (the same as in Case III of Section 3.2).

Two complementary experiments were carried out to validate the virtual-beacon (VB) inversion.

(1) Static convergence test: The VB routine was first applied to two fixed target depths, $z=-3000$ m and $z=-4000$ m. In each slice, the 30 candidate horizontal points obtained from the intersections of the circles were plotted. Figure 9 and Figure 10 visualize how candidates collapse into a tight two-point cluster precisely at the true depth, confirming the depth focusing ability of the method.

(2) Dynamic depth fix under Case III errors: Adopting the exact error budget of Section 3.2 Case III. We simulated 50 consecutive epochs for an AUV following $\mathtt{trajFun}\left(t\right)=\left[1000cos\left(0.12t\right),1000sin\left(0.12t\right),-4000\right]$ m. The VB inversion (minimum-$C\left(z\right)$ slice) was executed at every epoch; the resulting depth time series and error histogram are shown in Figure 11.

Table 4. Vertical positioning error statistics under Case III.

Method	Vertical (m)	Method	Vertical (m)
ND	5.879 ± 0.040	DSD-C	3.072 ± 0.782
DSD-E	3.062 ± 0.776	VB	2.770 ± 0.140

sums the mean and $1\sigma$ depth errors, and juxtaposes them with the corresponding statistics of the non-difference (ND) and single-difference solutions referenced to the center buoy (DSD_C) and an edge buoy (DSD_E).

4. Discussion

This section interprets the numerical results presented in Section 3, relates them to the underlying error mechanisms, and derives a recommended calibration workflow for deep-ocean position operations. The final subsection outlines the main limitations of the present study and the direction for future work.

4.1. Error-Cancellation Capability of the Single-Difference Solver

Additive (range-bias) errors: Under Case I conditions—random noise plus a 1.5 m range bias, the single difference (DSD) solution exhibits a markedly smaller mean error than the non-difference (ND) solution (Figure 3, Table 1), which confirms its effectiveness in suppressing common mode biases. However, the standard deviation of DSD is slightly larger than that of ND. This is a direct consequence of the statistical properties of the model.

(1) Observation redundancy loss—one row is removed from the design matrix, inflating ${\left({\mathbf{A}}_r^{\mathsf{T}}{\mathbf{A}}_r\right)}^{-1}$;

(2) Geometry degradation—the deleted line-of-sight (LOS) vector weakens column orthogonality, especially beneath the reference buoy;

(3) Correlated observation noise—the resulting negative covariance between different observations is often ignored in a unit-weight adjustment, causing a further, though modest, variance inflation.

Together these effects lead to the covariance relationship

$${\mathbf{Q}}_{\mathrm{DSD}}=2{\left({\mathbf{A}}_r^{\mathsf{T}}{\mathbf{A}}_r\right)}^{-1}{\sigma }_{\rho }^2>{\mathbf{Q}}_{\mathrm{ND}}={\left({\mathbf{A}}^{\mathsf{T}}\mathbf{A}\right)}^{-1}{\sigma }_{\rho }^2,$$

(23)

which explains the larger $1\sigma$ dispersion observed for DSD-C and DSD-E in Table 1. In practice, this penalty is acceptable because eliminating the deviation of the system at a micrometer is more important than suppressing the random noise at a centimeter.

Epoch-constant clock offsets. Case II introduces a 1 ms transmit clock error on top of Case I. The ND solution now suffers a depth bias of $\simeq 3$ m, whereas the DSD statistics and time histories remain virtually unchanged (Figure 4, Table 2). Immunity comes from the fact that the clock term $c\delta {\tau }^{\left(t\right)}$ is identical for all beacons at a given epoch and is therefore eliminated exactly by the single difference operator. This property renders DSD particularly attractive for buoy networks with loose synchronization tolerances.

Scale (sound-speed) errors. In Case III the equivalent sound speed is underestimated by 1 m ${\mathrm{s}}^{-1}$, introducing a multiplicative error. Because the bias is proportional to the residual range $G_j-G_0$, it survives single differencing (Section 3.2). The resulting vertical bias is about 2.7 m at 4 km depth. Accurate environmental compensation, such as profile-based sound-speed correction or in-situ velocity sensing, therefore remains indispensable for multikilometer operations.

4.2. Geometry Strength Versus Differencing Strategy

The full-depth PDOP maps in Figure 6 and Figure 7 show that ND produces uniformly smaller PDOP, HDOP, and VDOP than either single-difference variant. This purely geometric superiority stems from (i) the loss of one observation and (ii) the doubled noise variance in DSD. However, DOP is a bias-free noise amplification metric; once common errors are present at the meter level, ND suffers systematic biases two orders of magnitude higher than its stochastic noise, while DSD eliminates them at the cost of increasing random dispersion (Section 4.1). Numerically, DSD therefore retains an overall accuracy advantage in realistic error environments.

4.3. Reference-Buoy Selection in a Radiation Array

Comparing DSD-C and DSD-E, the solution referenced in the center exhibits lower DOP over most of the workspace. Deleting the LOS of the center buoy removes little horizontal information, whereas deleting an edge buoy eliminates one of the longest baselines and inflates HDOP markedly. Analytically, the center buoy minimizes

$${\mathrm{PDOP}}_r=\sqrt{2tr\left[{\left({\mathbf{A}}_r^{\mathsf{T}}{\mathbf{A}}_r\right)}^{-1}\right]},$$

and is therefore the preferred reference for any star-shaped surface array.

4.4. Virtual-Beacon Depth Fixing Under Case III Errors

The static convergence plots (Figure 9 and Figure 10) show that the circle-intersection candidates collapse into a tight two-point cluster exactly at the true depth slice, confirming the depth-focusing nature of the VB inversion. With the full Case III error budget applied, the 50-epoch VB run achieves a mean vertical error of $2.77\pm 0.14$ m (Table 4), a 15% improvement over DSD-C/DSD-E and more than a two-fold improvement over ND. The gain stems from exploiting the curvature of the time-of-flight–depth relationship.

4.5. Recommended Calibration Workflow

Based on the above findings, the following three-step workflow is recommended for deep-ocean position calibration.

(1) Horizontal positioning: Solve target coordinates using the differential formulation to suppress range, clock, and other systematic errors;

(2) Geometry screening: Analyze the spatial distribution of the DOP in differencing to determine the optimal configuration of the array and the selection of reference buoys based on mission-specific requirements;

(3) Depth correction: Apply virtual-beacon-based inversion to the screened data to refine the target depth estimate.

4.6. Limitations and Future Work

Model idealizations: Simulations assumed stationary buoys, horizontally stratified sound speed, and Gaussian noise. Buoy drift, anisotropic sound-speed anomalies, and multipath interference merit further study.

Array topology: Only the Decagon Center and cross-layouts were examined. Denser or asymmetric constellations under real-world deployment constraints remain to be evaluated.

Real-time sound-speed estimation: VB currently relies on a prebuilt lookup table; assimilating real-time CTD data is expected to further improve depth accuracy.

Addressing these issues will help translate the present simulation study into routine field practice, enabling high-precision navigation for long-range autonomous underwater missions.

5. Conclusions

This study proposes and rigorously validates a three-tier calibration architecture to fix the deep-ocean position, comprising (i) a solver dynamic single-difference (DSD), (ii) a geometry-aware reference-buoy selection algorithm, and (iii) a depth-inversion technique virtual-beacon (VB).

First, 50-epoch dynamic simulations were performed under three escalating error scenarios: random noise with range bias, additional clock bias, and additional sound-speed scale error to assess additive bias suppression and stochastic noise inflation of DSD relative to a conventional non-difference (ND) solution. Second, for a radiation array, full-depth PDOP/HDOP/VDOP maps for center referenced (DSD-C) and edge referenced (DSD-E) cases were generated, enabling a quantitative comparison of reference choice on geometric dilution and motivating a real-time minimum-PDOP selector. Finally, in the complete Case III error budget, the VB depth inversion method was validated through static slice and 50-epoch dynamic tests: candidate convergence at multiple depth slices was visualized, vertical error statistics were obtained, and performance was bench marked against ND and DSD-C/E solutions.

The DSD solver effectively suppresses such fixed systematic errors; for multiplicative errors such as sound-speed bias, DSD still outperforms ND; (2) the geometry-aware reference selector quantitatively evaluates the impact of the array layout and the reference buoy choice on positioning accuracy and allows optimal design for user-defined regions and dimensions of interest; (3) at a depth of 4 km, the VB inversion reduces vertical error by approximately 15% relative to DSD and by more than a factor of two relative to ND.

Taken together, the integrated DSD–geometry–VB framework provides high horizontal positioning accuracy while significantly enhancing vertical localization performance, thereby relaxing the engineering demands on ultra-precision range, lowering the deployment barrier of large-scale sub-sea Internet-of-Things systems, and delivering a robust navigation foundation for future autonomous deep-ocean operations.