GNSS-Acoustic positioning error in the vertical component considering the uncertainty of a reference sound speed profile

doi:10.21203/rs.3.rs-4711386/v1

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GNSS-Acoustic positioning error in the vertical component considering the uncertainty of a reference sound speed profile

https://doi.org/10.21203/rs.3.rs-4711386/v1

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~2 and ~ 6 cm in the former and latter cases, respectively. Moreover, we optimized the vertical GNSS-A positioning errors by considering the uncertainty of a reference SSP and other error sources, and the optimization of the vertical GNSS-A positioning errors was more reasonable than the conventional error evaluation. Although this study focused on the uncertainty of a reference SSP, it is important to carefully evaluate various other systematic error sources to precisely detect vertical seafloor motion.

GNSS-Acoustic positioning

seafloor geodesy

underwater sound speed

vertical crustal deformation

Global Navigation Satellite System–Acoustic (GNSS-A) positioning system is a seafloor geodetic observation system combining GNSS positioning on a sea surface platform (e.g., research vessel [Sato et al. 2013], buoy [Imano et al. 2019], Wave Glider (WG) [Iinuma et al. 2021]) and acoustic ranging between the sea surface platform and seafloor transponders (Fig. 1a), which was firstly contrived by Spiess (1985). Multiple (three to six) seafloor transponders were deployed at each GNSS-A site, forming a triangular or square-shaped array. The array size (diameter of the circumcircle of the array) typically corresponds to the water depth. The seafloor transponders adopt a mirror-type acoustic ranging system. The transponders temporally recorded the acoustic signal transmitted from the sea-surface platform and then retransmitted the recorded signal to the sea-surface platform. Since the GNSS-A observations require a sea surface platform, the GNSS-A observation has been typically conducted as a campaign-style survey with a survey period of ~ 0.5–1 day except while employing a moored buoy (e.g., Imano et al. 2019). In GNSS-A positioning, individual transponder positions or array displacement (a translation of the entire transponder array from the initial transponder positions) for each campaign are estimated by minimizing the residuals between the observed travel times and modeled travel times (e.g., Tomita & Kido 2024).

Since the late 1990s, GNSS-A observations have revealed seafloor crustal deformation associated various important tectonic phenomena: inter-seismic deformation due to interplate locking (e.g., Gagnon et al. 2005; Yokota et al. 2016; Tadokoro et al. 2018), coseismic deformation (e.g., Sato et al. 2011; Kido et al. 2011), postseismic deformation (e.g., Tomita et al. 2017; Watanabe et al. 2021; Brooks et al. 2023), slow-slip events (e.g., Honsho et al. 2019; Yokota & Ishikawa 2020), incoming plate motion in a subduction zone (e.g., Tomita et al. 2015), back-arc spreading (e.g., Chen et al. 2021), and plate motion near a ridge-transform boundary (Chadwell & Spiess 2008). Most GNSS-A studies only employed horizontal components because the vertical component has larger positioning errors than the horizontal components (e.g., Sato et al. 2013) and because some studies did not estimate the vertical component considering the theoretical difficulty in estimating the vertical component (e.g., Spiess et al. 1998). However, several studies have successfully reported significant vertical motion in cases where the crustal deformation was quite large (e.g., coseismic and postseismic deformation associated with the 2011 Tohoku-oki earthquake [e.g., Sato et al. 2011; Kido et al. 2011; Watanabe et al. 2021]). Although ocean-bottom pressure gauges can detect seafloor vertical motions and have high precision in detecting instantaneous deformation, they are poor at detecting long-term deformations because of sensor drifts (Bürgmann & Chadwell 2014). Thus, it is important to appropriately detect seafloor vertical deformation using GNSS-A observations and evaluate the positioning error.

Various factors affect the GNSS-A positioning errors in the vertical component. First, various acoustic incident angles in a campaign dataset are essential for detecting the vertical component. The vertical component tends to be a trade-off with the underwater sound speed variation, and a variety of acoustic incident angles are required to solve this trade-off (e.g., Sato et al. 2013). Classic GNSS-A observation studies have located a sea surface platform at a fixed point above the center of the seafloor transponder array (a fixed-point survey: e.g., Spiess et al. 1998), and in principle, it is difficult to detect the vertical component. In recent studies, especially those performed by Japanese research groups, surveys have been conducted in which a sea surface platform moves around with elaborate survey tracks (moving survey: e.g., Sato et al. 2013) and successfully detected the vertical component (e.g., Watanabe et al. 2021). Tomita et al. (2019) proposed another technique for collecting a variety of acoustic incident angles by elaborating the transponder distribution for a site (called multiangled transponders). Second, modelling errors in the underwater sound speed structure (SSS) degrade the accuracy of GNSS-A positioning. In acoustic ranging, we typically calculate synthetic travel times assuming a horizontally stratified SSS depending on a reference sound speed profile (SSP) and model travel time delays due to spatiotemporal disturbances of the SSS (e.g., Tomita & Kido 2024). The reference SSP with a large bias often provides a synthetic bias in the vertical component but does not significantly affect the horizontal components (Ishikawa, 2019). Moreover, mismodelling of the travel time delays also generates synthetic errors in both the horizontal and vertical components. Third, systematic errors in the antenna-transducer (ATD) offset of a sea-surface platform directly cause positioning bias. In the GNSS-A positioning technique, the GNSS antenna position is transformed into the transducer position attached to the sea surface platform. The ATD offset is measured by optical ranging when the sea surface platform is small (e.g., a WG) or is simultaneously estimated by the acoustic ranging with the seafloor transponder positions when the sea surface platform is large (e.g., a research vessel). In the latter case, the ATD offset is optimized using multiple campaign datasets employing the same sea-surface platform and is fixed (e.g., Honsho & Kido 2017; Watanabe et al. 2020). In addition to the above factors, various other factors could provide systematic errors in the vertical component, such as GNSS positioning errors and misdetection of observed travel times (e.g., Honsho et al. 2021). Most error sources appeared to be biased. Thus, it is typically difficult to assess the uncertainty of each error source.

In the present study, we examined the GNSS-A positioning error in the vertical component caused by the uncertainty of the reference SSP. The reference SSP is typically obtained by oceanographical measurements such as the Expendable Bathy Thermograph (XBT), Expendable Conductivity, Temperature, and Depth (XCTD), and a Conductivity Temperature Depth profiler (CTD). However, the SSS could be spatiotemporally perturbed during the observational period (~ 0.5–1 day), so the reference SSP has uncertainty owing to the spatiotemporal variation in SSS. Moreover, actual SSPs obtained through oceanographical measurements are not necessarily available in recent GNSS-A observations because unmanned sea surface platforms (e.g., WG, moored buoys) have often been employed. In this case, statistical models (e.g., WOA2018: Garcia et al. 2019) or numerical data assimilation models (e.g., JCOPE2M: Miyazawa et al 2017; Miyazawa et al. 2019) were employed to generate a reference SSP. As these unmanned sea surface platforms are the mainstream of GNSS-A observations (e.g., Iinuma et al. 2021; DeSanto et al. 2023), investigation of the uncertainty due to reference SSP biases is important. Furthermore, we propose a new technique to assess the vertical positioning error for each campaign dataset and the vertical displacement rate, considering various systematic positioning errors.

2.1 Principle of the GNSS-A positioning

The individual seafloor transponder positions or array displacements of the seafloor transponders were estimated by minimizing the residuals between the observed and synthetic travel times in the GNSS-A positioning. In this study, we focused on estimating array displacement. Various observation equations have been developed to perform GNSS-A positioning (e.g., Spiess et al. 1998; Ikuta et al. 2008; Watanabe et al. 2020; Tomita & Kido 2024). One of the basic observation equations for the $\:n$th acoustic shot to the $\:k$th transponder provided by Tomita and Kido (2024) is written as

$$\:{T}_{n,k}^{\text{o}\text{b}\text{s}}={T}^{\text{c}\text{a}\text{l}}\left({\mathbf{p}}_{k};\mathbf{u}\left({t}_{n,k}^{+},{\mathbf{b}}_{0}\right);\mathbf{u}\left({t}_{n,k}^{-},{\mathbf{b}}_{0}\right);{v}_{0}\left(z\right)\right)+\varDelta\:{T}_{n,k}$$

where $\:{T}_{n,k}^{\text{o}\text{b}\text{s}}$ and $\:{T}^{\text{c}\text{a}\text{l}}$are the observed and synthetic round-trip travel times, respectively, and $\:\varDelta\:{T}_{n,k}$ is the travel time delay. The seafloor transponder position is $\:{\mathbf{p}}_{k}$, and the sea surface transducer position is $\:\mathbf{u}$. $\:{t}_{n,k}^{+}$ and $\:{t}_{n,k}^{-}$ indicate the times when the acoustic signal is transmitted from the sea surface platform and received at the sea surface platform, respectively. The sea surface platform position was originally provided as the sea surface GNSS antenna position by kinematic GNSS positioning, and the GNSS antenna position was then transformed into the sea surface transducer position $\:\mathbf{u}$ using the ATD offset $\:{\mathbf{b}}_{0}$ and the attitudes of the sea surface platform. Although it is possible to calculate travel time in a 4-dimensional (spatiotemporally-variated) velocity field, it incurs high calculation costs and presents difficulties in modelling the 4-dimensional velocity field. Thus, the synthetic travel time is typically calculated assuming a horizontally stratified SSS based on a reference SSP $\:{v}_{0}\left(z\right)$ ($\:z$ indicates the depth from the sea surface), and the travel time delay due to the spatiotemporal disturbance of the actual SSS is separately modeled.

To model travel time delay, only temporal variations have often been considered. For instance, travel time delay was modeled as a temporal variation of the average sound speed (e.g., Fujita et al. 2006) or temporal variation of nadir total delay (NTD: Kido et al. 2008; Honsho & Kido 2017). The NTD is analogous to the Zenith Total Delay in the GNSS positioning (e.g., Marini 1972); a travel time delay integrated into the nadir direction from the sea surface platform appears equally among seafloor transponders, assuming that the disturbance of the actual SSS is caused by temporal variation in the shallow portion of a horizontally stratified SSS (Honsho & Kido 2017; Tomita & Kido 2024). Recent studies have often included a horizontal gradient in the SSS in addition to the above temporal variations (e.g., Yokota et al. 2018; Honsho et al. 2019; Watanabe et al. 2020; Kinugasa et al. 2020; Tomita & Kido 2022).

In the present study, to investigate the GNSS-A positioning error in vertical components due to the uncertainty of the reference SSP, we employed the GNSS-A positioning method of the “static_array” function in the GNSS-A positioning software “SeaGap” version 1.0.1 (Tomita & Kido 2024) assuming that the travel time delay is modeled by temporal variation of NTD. As shown by Tomita and Kido (2022), the horizontal gradient in the SSS does not affect the vertical components of the GNSS-A positioning; thus, we can sufficiently investigate the vertical GNSS-A positioning error under this assumption. The observation equation of “static_array” function of SeaGap is as following:

$$\:{\frac{1}{{M}_{n,k}}T}_{n,k}^{\text{o}\text{b}\text{s}}=\frac{1}{{M}_{n,k}}{T}^{\text{c}\text{a}\text{l}}\left({\mathbf{p}}_{k}^{\text{i}\text{n}\text{i}}+\delta\:\mathbf{p};\mathbf{u}\left({t}_{n,k}^{+},{\mathbf{b}}_{0}\right);\mathbf{u}\left({t}_{n,k}^{-},{\mathbf{b}}_{0}\right);{v}_{0}\left(z\right)\right)+C\left({t}_{n,k}\right)$$

with a mapping function of

$$\:{M}_{n,k}=\frac{1}{\text{cos}{\xi\:}_{n,k}},$$

where, $\:C\left({t}_{n,k}\right)$ expresses the temporal variation of NTD, $\:{t}_{n,k}$ corresponds to $\:\frac{{t}_{n,k}^{+}+{t}_{n,k}^{-}}{2}$, $\:{\xi\:}_{n,k}$ is the incident angle, $\:{\mathbf{p}}_{k}^{\text{i}\text{n}\text{i}}$ is the initial positions of the seafloor transponders, and $\:\delta\:\mathbf{p}$ is the array displacement. The temporal variation of the NTD was expressed by 3-dimensional B-Spline functions, and the temporal basis interval was optimized using the Bayesian information criterion (BIC: Schwartz 1978). In the “static_array” function, the array displacement and the coefficients for the 3-dimensional B-Spline basis of NTD are estimated by the Gauss-Newton method. The array displacement has three components, which are East-West (EW), North-South (NS), and up-down (Z) components, as $\:\delta\:\mathbf{p}=\left(\delta\:{p}_{\text{E}\text{W}},\delta\:{p}_{\text{N}\text{S}},\delta\:{p}_{\text{Z}}\right)$. The details of this method were described by Tomita and Kido (2024).

In the GNSS-A observational studies, repeated campaign surveys were conducted, and the displacements among the campaigns were estimated. The GNSS-A positioning errors were evaluated using the following three approaches. The first approach employs the uncertainty that appears in the model covariance matrix or in the posterior probability distribution; hereafter, we refer to this error as the inversion modelling error. In GNSS-A positioning, the array displacement is estimated by a nonlinear least squares method, such as the Gauss-Newton method or the Markov Chain Monte Carlo (MCMC) method (e.g., Watanabe et al. 2023; Tomita & Kido 2024), and these techniques provide an inversion modelling error. As discussed by Tomita and Kido (2024), the inversion modelling error does not necessarily include systematic biases such as those due to the ATD offset and tends to be smaller than the actual positioning error. However, the inversion modelling error shows the geometrical resolution of seafloor transponders (e.g., Imano et al. 2019); that is, the vertical component of the inversion modelling error reflects the positioning error depending on the variation in the acoustic incident angles in a campaign dataset. In practice, some studies have employed multiples of inversion modelling errors as the positioning error for an individual campaign. The second approach employs deviations from the fitting curve of multiple campaign solutions in a time series. When significant short-term steps (e.g., coseismic displacements) are not caused or corrected, we often assume that the array displacements show steady motion or simple decay as inter- or postseismic deformation, respectively, and calculate a fitting curve for the array displacements. The deviation from the fitting curve shows positioning errors, including systematic biases. The root mean square (RMS) from the fitting curve in the vertical component is ~ 4 cm when the campaign datasets include sufficient variation in acoustic shot angles (e.g., Sato et al. 2013; Tomita & Kido 2022). However, the RMS tends to be large for campaign datasets with insufficient variation in acoustic shot angles (Sato et al. 2013). Moreover, it is difficult to evaluate the positioning error for an individual campaign dataset and assess the positioning error sources. The third approach involves the dispersion of solutions estimated from data subsets. Some studies have divided each campaign dataset into multiple data subsets and evaluated the positioning error as a dispersion of the subset solutions (e.g., Fujita et al. 2006). This approach can easily assess the positioning precision of each campaign; however, it does not necessarily consider systematic biases.

2–2. Approach

In the present study, we first investigated the uncertainty of a reference SSP. When a reference SSP is directly measured by oceanographical measurements (XBT, XCTD, and CTD), the reference SSP has a spatiotemporal disturbance in the SSS during the observational period for each survey (~ 0.5–1 day). Therefore, we examined the deviation among multiple SSPs measured during a survey to assess the uncertainty of the reference SSP. When an SSP was not directly measured, the reference SSP was obtained from the data assimilated model, JCOPE-2M (Miyazawa et al. 2017; 2019). In this case, we examined the reproducibility of JCOPE-2M relative to the measured SSPs; thus, we calculated the modeled SSPs from JCOPE-2M at the oceanographic measurement points and compared the modeled and measured SSPs. We then investigated the GNSS-A positioning error in the vertical component due to the above uncertainties in the reference SSPs (hereafter referred to as the reference SSP positioning error). In this step, we generated many SSPs from a reference SSP, which included perturbations based on uncertainties, and the reference SSP positioning error was examined using the Monte Carlo method with the generated SSPs.

Next, we optimized the GNSS-A positioning error in the vertical component due to various error sources. As noted in Section 1, there are many sources of vertical positioning errors other than the uncertainty of the reference SSP. However, it is difficult to quantitatively assess each positioning error because most are caused by systematic errors. Here, we assumed that the vertical positioning error was modeled as a combination of multiples of the inversion modelling and reference positioning errors. The vertical positioning error defined in this study $\:{\widehat{\sigma\:}}_{\text{Z}}$ is written as:

$$\:{\widehat{\sigma\:}}_{\text{Z}}\equiv\:\sqrt{{\alpha\:}^{2}\bullet\:{\sigma\:}_{\text{Z}}^{2}+{\sigma\:}_{\text{S}\text{S}\text{P}}^{2}},$$

where, $\:{\sigma\:}_{\text{Z}}$ is the inversion modelling error in 1$\:\sigma\:$ calculated from the model variance-covariance matrix, $\:{\sigma\:}_{\text{S}\text{S}\text{P}}$ is the reference SSP positioning error calculated as the standard deviation of the Monte Carlo samples, and $\:\alpha\:$ is a coefficient that multiplies the inversion modelling error. The formulation of Eq. 4 assumes that the error sources $\:{\sigma\:}_{\text{Z}}$ and $\:{\sigma\:}_{\text{S}\text{S}\text{P}}$ are independent of each other. Evidence of their independence is presented in Section 4. We optimized $\:\alpha\:$ by a MCMC method so that the vertical positioning error is equivalent to the deviation from a fitting curve of multiple campaign solutions in a time-series. Because the deviation from the fitting curve contains various systematic errors, this optimization enables us to express the actual positioning error using Eq. 4.

To investigate the uncertainty of a reference SSP, we employed the direct oceanographical measurement (XBT, XCTD, and CTD) data obtained at the GNSS-A sites off East Japan along the Japan and Kuril trenches (G01–G13 and G15–G25: Fig. 1b) and calculated their SSPs: hereafter, the observed SSPs. A total of 317 observed SSPs were employed. The XBT and XCTD types used were XBT-5 (depth limit of measurement: 1850 m) and CTD-2 (depth limit of measurement: 1950 m), respectively. CTD measurements were conducted from the sea surface to the seafloor. Measurements were typically conducted twice for each survey at each site. Using the conversion formula to calculate the underwater sound speed from water temperature and salinity (Chen & Millero 1977), we calculated the observed SSP from each measurement. As each XBT measurement did not include information on salinity, we employed a salinity profile obtained by XCTD or CTD measurements around the same period in the vicinity of the XBT measurement. It has been known that the depth component of the XBT measurements has a systematic bias, so we corrected it by the correction parameters of Kizu et al. (2005). We then extrapolated the SSPs at a depth from 1800 m to the seafloor, using the WOA2018 climatological mean model estimated from oceanographic measurements since 1955 (Garcia et al. 2018). As the underwater sound speed structure in the deep portion was stable, we employed an annual climatological mean model for extrapolation. All observed SSPs were converted into profiles with the 1-meter increments.

JCOPE2M distributes daily model values of various oceanographic parameters (Miyazawa et al. 2017; 2019). Using the parameters of JCOPE2M on potential water temperature and salinity, we calculated the SSP for the date when each observed SSP was collected. A campaign survey was conducted at each site using the conversion formula of Chen and Miller (1977). Although JCOPE2M provides these parameters even in the deep portion, we employed parameters shallower than 1800 m and extrapolated the SSPs using the WOA2018 model to maintain consistency with the above direct measurement cases. All JCOPE2M SSPs were also interpolated as profiles with the 1-meter increments.

We employed the GNSS-A observational datasets at G02 during 2012–2023 (Fig. 1b) to examine the vertical positioning errors using actual observational data. This site is located far from the rupture area of the 2011 Tohoku-oki earthquake (e.g., Tomita et al. 2020), and its movement is considered to be small. Actually, Fig. 2 shows the horizontal and vertical array displacement at G02 solved by the “static_array” function of SeaGap (Equations 2 and 3), and the horizontal and vertical displacement rate at G02 is ~ 1 cm/yr. If the actual movement is large, it is difficult to distinguish between systematic errors and actual movements. Therefore, it is effective to employ observational datasets at sites with small movements to assess positioning errors. Moreover, we collected GNSS-A datasets obtained using both research vessels with direct SSP measurements (circles in Fig. 2) and a WG (Iinuma et al. 2021) (cyan color solutions in Fig. 2) without direct SSP measurements (squares in Fig. 2) at G02. Considering these factors, the GNSS-A observational datasets at G02 are suitable for examining the vertical positioning errors due to the uncertainty of the reference SSP.

4-1. Uncertainty of a reference sound speed profile

We investigated the deviations among multiple SSPs measured during a campaign survey to assess the spatiotemporal disturbance of the SSS within the observation period (~0.5–1 day). We extracted direct oceanographic measurements for which multiple SSPs were available during the survey campaign at each site. Then, we randomly selected one of the SSPs for each campaign survey and calculated the difference between the other SSPs and the selected SSP. From the 317 observed SSPs, we obtained 145 difference SSPs. In Figure 3a, the gray curves show the difference SSPs and the red curves show the standard deviation of the absolute value of the difference SSPs (this value is positive, but its y-axis symmetric curve is shown in this figure). Figure 3a shows that the spatiotemporal instability in the SSS during each survey period appeared in the portion shallower than 600 m and provided a sound speed perturbation of typically ~5 m/s (maximum ~20 m/s). This instability corresponded to the uncertainty of the reference SSP when the observed SSP was employed as the reference SSP.

To assess the uncertainty of each JCOPE2M SSP, we calculated the difference between the JCOPE2M SSPs and observed SSPs, as shown in Figure 4a. The gray and red curves in Figure 4a show the difference in SSPs and the standard deviation of the absolute value of the difference in SSPs, respectively. Figure 4a shows that the deviation of the JCOPE2M SSPs relative to the observed SSPs appeared in the portion shallower than 600 m and provided a sound speed perturbation typically over 10 m/s (maximum ~40 m/s). Figures 3a and 4a indicate that the uncertainty of the reference SSP based on JCOPE2M was much larger than that based on direct oceanographic measurements.

4−2. Positioning error estimation using a Monte Carlo method

In this study, we examined the reference SSP positioning errors using actual GNSS-A observational datasets at G02. To analyze each campaign dataset, we first generated several SSPs based on a reference SSP, where each generated SSP $\:v\left(z\right)$ is composed of the reference SSP and a perturbation profile as follows:

$$\:v\left(z\right)={v}_{0}\left(z\right)+\delta\:v\left(z\right),$$

where, $\:{v}_{0}\left(z\right)$ and $\:\delta\:v\left(z\right)$ are the reference SSP of each campaign and the perturbation profile, respectively. We generated many perturbation profiles using random numbers based on the difference in SSPs, as shown in Figs. 3a and 4a. Each perturbation profile is generated by combining randomly selected two difference SSPs $\:{\delta\:v}_{1}$ and $\:{\delta\:v}_{2}$ from one of the observed and JCOPE2M cases (Figs. 3a and 4a) as:

$$\:\delta\:v\left(z\right)={\tau\:}_{1}\bullet\:{\delta\:v}_{1}\left(z\right)+{\tau\:}_{2}\bullet\:{\delta\:v}_{2}\left(z\right),$$

where, $\:{\tau\:}_{1}$ and $\:{\tau\:}_{2}$ are random numbers generated from a uniform distribution with the range from − 1 to 1. Examples of the generated perturbation profiles are shown as gray curves in Figs. 3b and 4b. In both the observed and JCOPE2M cases, the generated perturbation profiles successfully reproduced the original differences in the SSPs (Figs. 3a and 4a). This generation system enables us to create a large number of SSPs that can be used in the Monte Carlo method beyond the total number of original difference SSPs.

In this study, 300 SSPs were generated for each reference SSP by using the Monte Carlo method. Then, using these 300 generated SSPs, we collected 300 samples of the array displacement for each campaign dataset by the “static_array” function of SeaGap. Figure 5 shows the sampled vertical array displacements for the campaign datasets conducted on August 10, 2013 (Fig. 5a–c) and September 20, 2023 (Fig. 5d–f). In the 2013 campaign survey, a research vessel was employed as the sea surface platform, and direct oceanographic measurements were available; hence, we performed the Monte Carlo method based on the difference SSPs in the observed reference SSP case (Fig. 3), and the 300 vertical array displacements $\:\delta\:{p}_{\text{Z}}$ were sampled as shown in Fig. 5b. The standard deviation of 300 vertical array displacements was 2.62 cm. In the 2023 campaign survey, a WG was employed as the sea surface platform, and no direct oceanographic measurement was available; hence, we performed the Monte Carlo method based on the difference SSPs in the JCOPE2M reference SSP case (Fig. 4) and the 300 vertical array displacements $\:\delta\:{p}_{\text{Z}}$ were sampled as shown in Fig. 5e. The standard deviation of the 300 vertical array displacements was 7.09 cm. The estimation results for all the campaign datasets of G02 are listed in Table 1. From the results, $\:{\sigma\:}_{\text{S}\text{S}\text{P}}$ is ~ 2 cm for the observed reference SSP and is ~ 6 cm for the JCOPE2M reference SSP. Meanwhile, we also obtained the inversion modelling error $\:{\sigma\:}_{\text{Z}}$ for each sample and the sampled inversion modelling errors are shown in Figs. 5c and 5f. The standard deviation of the sampled inversion modelling errors is very small (~ 0.2 mm); thus, the inversion modelling error $\:{\sigma\:}_{\text{Z}}$ is independent of the reference SSP positioning error $\:{\sigma\:}_{\text{S}\text{S}\text{P}}$.

Table 1

The estimation results of the vertical array displacements at G02. The platform number indicates the types of the employed sea surface platform, which corresponds to the colors in Fig. 2. The SSP pattern A indicates that the direct SSP measurement data were available, which corresponds to the circle shapes in Fig. 2; the SSP pattern B indicates that the direct SSP measurements were unavailable, which corresponds to the square shape in Fig. 2.
Campaign Number	Year	$\:\delta\:{p}_{\text{Z}}$ [cm]	$\:{\sigma\:}_{\text{S}\text{S}\text{P}}\:$[cm]	$\:{\sigma\:}_{\text{Z}}\:$[cm]	$\:{\widehat{\sigma\:}}_{\text{Z}}\:$[cm]	Platform Number	SSP Pattern	Number of shots	Moving survey
1	2012.723	2.56	1.62	0.49	4.13	1	A	6943	Yes
2	2013.608	1.63	2.62	0.45	4.36	2	A	8128	Yes
3	2013.838	0.80	2.43	0.38	3.85	2	A	5886	Yes
4	2015.178	0.95	2.84	0.36	3.97	3	A	7444	Yes
5	2015.838	-13.07	2.14	0.47	4.25	3	A	4239	Yes
6	2016.367	-4.49	2.05	0.99	8.02	4	A	3281	Yes
7	2019.504	-1.81	2.53	0.64	5.60	5	A	1518	Yes
8	2019.509	5.79	1.89	1.72	13.55	6	A	2312	No
9	2020.493	4.23	4.25	1.05	9.22	6	B	10663	No
10	2020.783	-18.13	5.36	2.60	20.97	6	B	2460	No
11	2020.925	-9.02	6.41	0.88	9.38	6	B	2750	Yes
12	2021.287	3.74	8.96	0.47	9.68	6	B	5924	Yes
13	2022.381	6.37	7.12	0.45	7.94	6	B	8942	Yes
14	2022.772	-9.14	6.33	0.48	7.34	6	B	6100	Yes
15	2023.290	14.10	7.79	1.09	11.55	6	B	7412	No
16	2023.646	4.26	2.13	0.83	6.83	5	A	2029	Yes
17	2023.720	-4.58	7.09	0.38	7.69	6	B	8444	Yes

5 − 1. Optimization method

We optimized the hyper-parameter $\:\alpha\:$, which balances the inversion modelling error and the reference SSP positioning error, by the MCMC method to obtain the vertical positioning error $\:{\widehat{\sigma\:}}_{\text{Z}}$ defined in Eq. 4. In the MCMC method, $\:\alpha\:$ is optimized to match the vertical positioning error $\:{\widehat{\sigma\:}}_{\text{Z}}$ with the deviation from a fitting curve of multiple campaign solutions in a time-series. Assuming that the displacement rate at G02 was constant during the observational period, we applied weighted linear line fitting to the vertical array displacements for the multiple campaign datasets at G02: By defining the vertical array displacement, $\:\delta\:{p}_{\text{Z}},$ for the $\:i$th campaign dataset as $\:{d}_{i}$, the fitting line can be written as follows:

$$\:{d}_{i}^{\text{c}\text{a}\text{l}}={a}_{\text{Z}}{\bullet\:t}_{i}+{b}_{\text{Z}},$$

where, $\:{d}_{i}^{\text{c}\text{a}\text{l}}$ is the predicted vertical array displacement; $\:{t}_{i}$ is the time of the $\:i$th survey campaign; $\:{a}_{\text{Z}}$ is the vertical displacement rate; and $\:{b}_{\text{Z}}$ is the intercept of the fitted line.

Assuming that misfits between $\:{d}_{i}$ and $\:{d}_{i}^{\text{c}\text{a}\text{l}}$ follow the normal distribution with the mean of 0 and the standard deviation of $\:{\widehat{\sigma\:}}_{\text{Z}}$, we approximated the posterior density functions of the hyper-parameter $\:\alpha\:$ by the random walk Metropolis algorithm (Metropolis et al. 1953) which is one of the basic MCMC algorithms. Following the Bayes’ theorem, the posterior probability density $\:p\left(\alpha\:|\mathbf{d}\right)$ is calculated as

$$\:p\left(\alpha\:|\mathbf{d}\right)\propto\:p\left(\mathbf{d}|\alpha\:\right)p\left(\alpha\:\right).$$

To calculate the marginal likelihood $\:p\left(\mathbf{d}|\alpha\:\right)$, we need to estimate the linear parameters, $\:{a}_{\text{Z}}$ and $\:{b}_{\text{Z}}$, for the weighted line fitting shown in Eq. 7. Defining the linear parameter vector $\:\mathbf{x}$ as $\:\mathbf{x}=\left({a}_{\text{Z}},{b}_{\text{Z}}\right)$, we rewrite the observation equation (Eq. 7) as $\:\mathbf{d}=\mathbf{H}\mathbf{x}$ in matrix form. According to Fukuda & Johnson (2010), the marginal likelihood $\:p\left(\mathbf{d}|\alpha\:\right)$ is extended using the linear parameters when the observation equation includes linear parameters:

$$\:p\left(\mathbf{d}|\alpha\:\right)=\int\:p\left(\mathbf{d},\mathbf{x}|\alpha\:\right)\:d\mathbf{x}=\int\:p\left(\mathbf{d}|\alpha\:,\mathbf{x}\right)p\left(\mathbf{x}|\alpha\:\right)\:d\mathbf{x}.$$

The likelihood $\:p\left(\mathbf{d}|\alpha\:,\mathbf{x}\right)$ is defined based on the probability density of the normal distribution as:

$$\:p\left(\mathbf{d}|\alpha\:,\mathbf{x}\right)\propto\:{\left|\mathbf{E}\left(\alpha\:\right)\right|}^{-\frac{1}{2}}\bullet\:\text{e}\text{x}\text{p}\left(-\frac{1}{2}{\left(\mathbf{d}-\mathbf{H}\mathbf{x}\right)}^{\text{T}}{\mathbf{E}}^{-1}\left(\alpha\:\right)\left(\mathbf{d}-\mathbf{H}\mathbf{x}\right)\right),$$

where, $\:\mathbf{E}\left(\alpha\:\right)$ is the data variance-covariance matrix and is defined as $\:\mathbf{E}\left(\alpha\:\right)=\text{d}\text{i}\text{a}\text{g}\left({\widehat{\sigma\:}}_{Z,1}^{2},\cdots\:,{\widehat{\sigma\:}}_{Z,I}^{2}\right)$. $\:I$ is the total number of campaigns. Note that the vertical positioning error of the $\:i$th campaign survey is denoted as $\:{\widehat{\sigma\:}}_{\text{Z},i}=\sqrt{{\alpha\:}^{2}\bullet\:{\sigma\:}_{\text{Z},i}^{2}+{\sigma\:}_{\text{S}\text{S}\text{P},i}^{2}}$. In this study, we provided no prior information on $\:p\left(\mathbf{x}|\alpha\:\right)$; thus, $\:p\left(\mathbf{x}|\alpha\:\right)=1$ and Eq. 9 is rewritten as:

$$\:p\left(\mathbf{d}|\alpha\:\right)\propto\:{\left|\mathbf{E}\left(\alpha\:\right)\right|}^{-\frac{1}{2}}\int\:\text{e}\text{x}\text{p}\left(-\frac{1}{2}{\left(\mathbf{d}-\mathbf{H}\mathbf{x}\right)}^{\text{T}}{\mathbf{E}}^{-1}\left(\alpha\:\right)\left(\mathbf{d}-\mathbf{H}\mathbf{x}\right)\right)\:d\mathbf{x}.$$

Letting $\:{\left(\mathbf{d}-\mathbf{H}\mathbf{x}\right)}^{\text{T}}{\mathbf{E}}^{-1}\left(\alpha\:\right)\left(\mathbf{d}-\mathbf{H}\mathbf{x}\right)$ as $\:f\left(\mathbf{x}\right)$, $\:f\left(\mathbf{x}\right)$ is rewritten as:

$$\:f\left(\mathbf{x}\right)=f\left({\mathbf{x}}^{\varvec{*}}\right)+{\left(\mathbf{x}-{\mathbf{x}}^{\varvec{*}}\right)}^{\text{T}}{\mathbf{H}}^{\text{T}}{\mathbf{E}}^{-1}\left(\alpha\:\right)\mathbf{H}\left(\mathbf{x}-{\mathbf{x}}^{\varvec{*}}\right),$$

where, $\:{\mathbf{x}}^{\varvec{*}}$ is the least squared solution minimizing $\:f\left(\mathbf{x}\right)$ as:

$$\:{\mathbf{x}}^{\varvec{*}}={\left({\mathbf{H}}^{\text{T}}{\mathbf{E}}^{-1}\left(\alpha\:\right)\mathbf{H}\right)}^{-1}{\mathbf{H}}^{\text{T}}{\mathbf{E}}^{-1}\left(\alpha\:\right)\mathbf{d}.$$

According to Fukuda and Johnson (2010), the integral in Eq. 11 is analytically computed as

$$\:\int\:\text{e}\text{x}\text{p}\left(-\frac{1}{2}{\left(\mathbf{d}-\mathbf{H}\mathbf{x}\right)}^{\text{T}}{\mathbf{E}}^{-1}\left(\alpha\:\right)\left(\mathbf{d}-\mathbf{H}\mathbf{x}\right)\right)\:d\mathbf{x}=\text{e}\text{x}\text{p}\left(-\frac{1}{2}f\left({\mathbf{x}}^{\varvec{*}}\right)\right)\times\:2\pi\:{\left|{\mathbf{H}}^{\text{T}}{\mathbf{E}}^{-1}\left(\alpha\:\right)\mathbf{H}\right|}^{-\frac{1}{2}}.$$

In the present study, we provided a prior information on $\:\alpha\:$ which restricts $\:\alpha\:>0$ as:

$$\:p\left(\alpha\:\right)=\left\{\begin{array}{cc}0&\:\text{i}\text{f}\:\alpha\:\le\:0\\\:1&\:\text{i}\text{f}\:\alpha\:>0\end{array}\right.$$

Thus, when $\:\alpha\:>0$, the posterior probability density $\:p\left(\alpha\:|\mathbf{d}\right)$ is rewritten using Equations 8, 11–15 as:

$$\:p\left(\mathbf{d}|\alpha\:\right)\propto\:{\left|\mathbf{E}\left(\alpha\:\right)\right|}^{-\frac{1}{2}}{\left|{\mathbf{H}}^{\text{T}}{\mathbf{E}}^{-1}\left(\alpha\:\right)\mathbf{H}\right|}^{-\frac{1}{2}}\bullet\:\text{e}\text{x}\text{p}\left(-\frac{1}{2}{\left(\mathbf{d}-\mathbf{H}\mathbf{x}\right)}^{\text{T}}{\mathbf{E}}^{-1}\left(\alpha\:\right)\left(\mathbf{d}-\mathbf{H}\mathbf{x}\right)\right)$$

In the random-walk Metropolis algorithm, an unknown parameter is iteratively updated in the MCMC chain. Letting $\:\alpha\:$ in the $\:j$th iteration is $\:{\alpha\:}^{\left(j\right)}$, the unknown parameter in the next iteration $\:{\alpha\:}^{{\prime\:}}$ is generated as:

$$\:{\alpha\:}^{{\prime\:}}={\alpha\:}^{\left(j\right)}+\delta\:\alpha\:$$

where, $\:\delta\:\alpha\:$ is a random value and is generated from a uniform distribution $\:U\left(-\text{0.5,0.5}\right)$ in this study. The generated unknown parameter is then evaluated using the acceptance probability

$$\:{P}_{\text{a}\text{c}\text{c}\text{e}\text{p}\text{t}}=\text{m}\text{i}\text{n}\left[1,\frac{p\left({\alpha\:}^{{\prime\:}}|\mathbf{d}\right)}{p\left({\alpha\:}^{\left(j\right)}|\mathbf{d}\right)}\right]$$

If the acceptance probability exceeds a random number generated from a uniform distribution $\:U\left(\text{0,1}\right)$, the generated unknown parameter is accepted as $\:{\alpha\:}^{\left(j+1\right)}$. We performed $\:1.1\times\:{10}^{6}$ iterations and discarded the samples of the first $\:1\times\:{10}^{5}$ iterations as the burn-in period. We adopted mode of the samples as the optimal value of $\:\alpha\:$. We then calculated the displacement rate, $\:{a}_{\text{Z}}$, and its error using a weighted linear least-squares method, considering the optimized positioning errors. Although we can simultaneously estimate posterior distributions of the linear parameters, $\:{a}_{\text{Z}}$ and $\:{b}_{\text{Z}}$, as with $\:\alpha\:$ by the method of Fukuda & Johnson (2010), we employed the displacement rate and its error based on the optimal value of $\:\alpha\:$ to compare with the solutions from the conventional positioning errors.

5 − 2. Optimized positioning errors

Applying the optimization method addressed in Section 5.1 to the positioning results at G02 including $\:{\sigma\:}_{\text{S}\text{S}\text{P}}$ shown in Table 1. Figure 6 shows a histogram of the sampled $\:\alpha\:$, and the mode of the samples is 7.8.

Figure 7 shows time-series of the vertical array displacements with three types of the positioning errors: the conventional positioning errors (10$\:{\sigma\:}_{\text{Z}}$: multiples of the inversion modelling error) in Fig. 7a, the reference SSP positioning errors ($\:{\sigma\:}_{\text{S}\text{S}\text{P}}$ in Table 1) in Fig. 7b, and the optimized positioning errors ($\:{\widehat{\sigma\:}}_{\text{Z}}=\sqrt{{7.8}^{2}\bullet\:{\sigma\:}_{\text{Z}}^{2}+{\sigma\:}_{\text{S}\text{S}\text{P}}^{2}}$) in Fig. 7c. The conventional positioning errors reflected the observation conditions, especially the insufficiency of the moving survey; for example, $\:{\sigma\:}_{\text{Z}}$ at the campaign number 10 is large (Table 1) because we did not conduct a moving survey at the campaign (Figure S1). However, the deviation from the regression line was occasionally larger than the conventional positioning error when a direct oceanographical measurement of SSP was not conducted (mainly when using WG; squares in Fig. 7a). The reference SSP positioning errors provided large errors to the campaigns without direct oceanographical measurements of the SSP. However, the overall positioning errors were evaluated to be too small to compensate for the deviation of the estimates from the regression line. The optimized positioning errors successfully express the deviation of the estimates from the regression line by including the influence of reference SSP positioning errors, unlike conventional positioning errors.

The present study addressed the uncertainty of a reference SSP employed in GNSS-A positioning and examined the GNSS-A positioning error owing to the uncertainty of a reference SSP. When a reference SSP measured using oceanographic measurements was available, the uncertainty of the reference SSP provided a GNSS-A positioning error of approximately 2 cm for each campaign dataset. Meanwhile, when a reference SSP was obtained from the oceanographic data assimilation model, the uncertainty of the reference SSP provided a GNSS-A positioning error of ~ 6 cm in the case that we employed JCOPE2M to calculate the reference SSP. Then, we optimized the GNSS-A positioning error by unifying both the conventional inversion modelling error and reference SSP positioning error.

We optimized the GNSS-A positioning errors for campaign datasets at G02 site. Regarding to the G02 case, the multiplying factor $\:\alpha\:$ balancing the inversion modelling error and the reference SSP positioning error (Eq. 4) was determined to be 7.8. To estimate the multiplying factor in a stable manner, we must employ a sufficient number of campaign datasets and optimize the multiplying factor using campaign datasets from multiple observational sites. We then calculated the displacement rate of the vertical component at G02 based on three types of GNSS-A positioning errors (inversion modelling error, reference SSP positioning error, and optimized positioning error; shown in Fig. 7a, 7b, and 7c, respectively). The optimized positioning errors successfully express the deviation from the fitting line, while the displacement rates estimated from three types of the positioning errors were not significantly different from each other. However, if we use a small number of campaign datasets without a directly measured reference SSP, these campaigns will work as outliers because of the reference SSP positioning error and cause mismodelling in the calculation of the displacement rate. Therefore, it is important to optimize the GNSS-A positioning error by considering the reference SSP positioning error.

In the optimization of the GNSS-A positioning error, we assume that the GNSS-A positioning error caused by error sources other than the reference SSP positioning error is expressed as a multiple of the inversion modelling error (Eq. 4). However, this assumption is not necessarily valid when other positioning error sources are independent of the inversion modelling error, such as systematic positioning errors in the ATD offset and kinematic GNSS positioning. Thus, future studies must identify systematic error sources and quantify their influence. To understand the long-term vertical deformation field in an offshore region, it is important to evaluate the GNSS-A positioning error sources in detail.

ATD

Antenna-transducer

BIC

Bayesian information criterion

GNSS

Global Navigation Satellite System

GNSS-A

GNSS-acoustic

MCMC

Markov Chain Monte Carlo

NTD

nadir total delay

RMS

root means square

SSP

sound speed profile

SSS

sound speed structure

Wave Glider

Ethics approval and consent to participate

Not applicable.

Consent for publication

Not applicable.

Competing interests

The authors declare no conflicts of interest.

Authors' information

Not applicable.

Funding

A part of this study was supported by JSPS KAKENHI Grant Number JP19H05596.

Author Contribution

F.T. contrived the approach and method in this study, collected the datasets, performed the analysis, and wrote the manuscript. M.K., T.I. and Y.O. contributed to collect the datasets and contributed to the discussion of the results. M.K. improved the approach and the manuscript. All the authors have read and approved the final version of this manuscript.

Acknowledgement

The observation and operation using the WG were supported by Japan Agency for Marine-Earth Science and Technology (JAMSTEC) and Tohoku University. We thank the captains and crews of the R/V Shinsei-maru (KS-14-17, KS-15-03, KS-16-14, KS-17-13, KS-18-10, KS-19-12, KS-20-16, KS-21-05, KS-21-25, KS-23-04, KS-23-14), R/V Kaimei (KM17-05, KM18-06, KM21-04, KM22-07, KM23-07), R/V Hakuho (KH15-02), R/V Yokosuka (YK16-02), R/V Mirai (MR20-E02), and the other charter ships for supporting to collect GNSS-A observational data and the SSP data. We thank JAMSTEC Application Laboratory for distributing the JCOPE2M modeling results. We also thank Editage (www.editage.jp) for English language editing. Some figures were generated using Generic Mapping Tools software (Wessel et al. 2019).

Data Availability

The original data of this study (XBT, XCTD, GNSS-Acoustic observational datasets) are available from the corresponding author upon a reasonable request.

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GNSS-Acoustic positioning error in the vertical component considering the uncertainty of a reference sound speed profile

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Abstract

method

Figures

1. Introduction

2. Method

2.1 Principle of the GNSS-A positioning

2–2. Approach

3. Data and pre-processing

4. Positioning error due to uncertainty of a reference sound speed profile

4−2. Positioning error estimation using a Monte Carlo method

5. Optimization of the positioning error

5 − 1. Optimization method

5 − 2. Optimized positioning errors

6. Discussion and conclusion

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Authors' information

Funding

Author Contribution

Acknowledgement

Data Availability

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