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Development of a kinematic GNSSAcoustic positioning method based on a statespace model
Earth, Planets and Space volume 71, Article number: 102 (2019)
Abstract
GNSSA (combination of Global Navigation Satellite System and Acoustic ranging) observations have provided important geophysical results, typically based on static GNSSAcoustic positioning methods. Recently, continuous GNSSAcoustic observations using a moored buoy have been attempted. Precise kinematic GNSSAcoustic positioning is essential for these approaches. In this study, we developed a new kinematic GNSSA positioning method using the extended Kalman filter (EKF). As for the observation model, parameters expressing underwater sound speed structure [nadir total delay (NTD) and underwater delay gradients] are defined in a similar manner to the satellite geodetic positioning. We then investigated the performance of the new method using both the synthetic and observational data. We also investigated the utility of a GNSSAcoustic array geometry composed of multiangled transponders for detection of vertical displacements. The synthetic tests successfully demonstrated that (1) the EKFbased GNSSAcoustic positioning method can resolve the GNSSAcoustic array displacements, as well as NTDs and underwater delay gradients, more precisely than those estimated by the conventional kinematic positioning methods and (2) precise detection of vertical displacements can be achieved using multiangled transponders and EKFbased GNSSAcoustic positioning. Analyses of the observational data also demonstrated superior performance of the EKFbased GNSSAcoustic positioning method, when assuming a laterally stratified sound speed structure. Further, we found three superior aspects to the EKFbased array positioning method when using observational data: (1) robustness of the solutions when some transponders fail to respond, (2) precise detection for an abrupt vertical displacement, and (3) applicability to realtime positioning when sampling interval of the acoustic ranging is shorter than 30 min. The precision of the detection of abrupt steps, such as those caused by coseismic slips, is ~ 5 cm (1σ) using this method, an improvement on the precision of ~ 10 cm of conventional methods. Using the observational data, the underwater delay gradients and the horizontal array displacements could not be accurately solved even using the new method. This suggests that shortwavelength spatial heterogeneity exists in the actual ocean sound speed structure, which cannot be approximated using a simple horizontally graded sound speed structure.
Introduction
The GNSSAcoustic (GNSSA) positioning method, which was contrived by the Scripps Institute of Oceanography (Spiess 1985; Spiess et al. 1998), is a combination technique of kinematic GNSS positioning on a sea surface platform and underwater acoustic ranging between the seasurface platform and a seafloor acoustic benchmark (Fig. 1). The GNSSA observations enable measurement of seafloor displacements in a global geodetic coordinate system. The system has provided important geodetic observation results including detection of interseismic (e.g., Gagnon et al. 2005; Tadokoro et al. 2012; Yokota et al. 2016), coseismic (e.g., Tadokoro et al. 2006; Sato et al. 2011; Kido et al. 2011), and postseismic (e.g., Watanabe et al. 2014; Tomita et al. 2015, 2017; Honsho et al. 2019) deformation associated with earthquake cycles in subduction zones, and plate motions near ridgetransform boundaries (Chadwell and Spiess 2008).
Although GNSSA observations are typically collected by campaignstyle surveys using a research vessel as the sea surface platform, continuous GNSSA observations have recently been developed using moored buoys (e.g., Imano et al. 2015; Kido et al. 2018; Kato et al. 2018; Imano et al. 2019) for an early warning system through instant offshore geodetic positioning. To support these efforts, we investigate a precise “kinematic” GNSSA positioning method. Most recently developed GNSSA positioning methods were designed as a “static” positioning method, estimating a single positioning solution using longterm (over several hours) campaign survey data (e.g., Fujita et al. 2006; Ikuta et al. 2008; Honsho and Kido 2017; Yokota et al. 2018). Although the classic GNSSA positioning method (Spiess et al. 1998; Kido et al. 2006) can be used to estimate a kinematic solution from a single acoustic ping, each solution has tens of centimeters of positioning error due to spatiotemporal fluctuations in the underwater sound speed structure (SSS). However, it should be noted that it is possible to obtain a comparable solution with static GNSSA positioning by averaging the kinematic solutions from longterm observational data (e.g., Kido et al. 2006; Tomita et al. 2015). Furthermore, the kinematic positioning method is useful for the detection of temporally noisy data and for determining a final solution without using the noisy data. Thus, the classic positioning method is still an effective technique and has provided important observational results (e.g., Gagnon et al. 2005; Tomita et al. 2017).
As noted above, a major source of GNSSA positioning error is thought to be the spatiotemporal fluctuations in underwater SSS. In past studies, a laterally stratified SSS has been assumed, and this assumption is generally applicable (e.g., Kido et al. 2008). If horizontal heterogeneity in SSS is present, systematic positioning errors will arise. Additional file 1: Figure S1 shows a schematic image that a horizontally graded SSS (simple case of the horizontal heterogeneity in SSS) produces systematic bias in positioning results. As a response to this, static GNSSA positioning methods that consider the horizontally graded structure of underwater sound speeds that are persistent over the longterm (over several hours) have recently been developed (e.g., Yasuda et al. 2017; Yokota et al. 2018; Honsho et al. 2019), and have demonstrated some reduction in biased positioning errors. However, it has been found that shortterm fluctuations in the horizontal heterogeneity of the SSS can be caused by internal gravitational waves (e.g., Spiess et al. 1998; Kido et al. 2006; Tomita et al. 2015), and these shortterm fluctuations have degraded the precision of the kinematic GNSSA positioning method.
In this study, we attempt to develop a kinematic GNSSA positioning method based on a statespace model using the extended Kalman filter (EKF) (Kalman 1960) to improve the precision of the kinematic GNSSA positioning method. The Kalman filter (KF) has frequently been used in GNSS positioning systems to precisely estimate GNSS antenna position, tropospheric delay, ionospheric delay and other parameters (e.g., Lichten and Border 1987). Since the KF can constrain the behavior of timedependent unknown parameters based on a given stochastic processes, it is a useful tool for kinematic inversion techniques. In this study, we first formulate the spatiotemporal fluctuations in the SSS for the GNSSA positioning, based on the formulation for the tropospheric delay in the satellite geodetic measurements. Subsequently, we apply the above formulation to the EKF framework. Then we investigate the performance of this EKFbased approach using synthetic tests, and finally we apply the approach to observational data from the offTohoku region (northeast Japan).
Method
Principles of the GNSSA positioning method
A conceptual model of the GNSSA observation system is shown in Fig. 1. The seafloor benchmark for each GNSSA observation site is composed of several (~ 3–6) transponders forming a triangular or squareshaped array. In the surveys, first, the position of a GNSS antenna attached to the seasurface platform was measured using a kinematic GNSS positioning technique. We then determined the position of an acoustic transducer attached to the seasurface platform from the GNSS antenna positions, the relative position between the GNSS antenna and the acoustic transducer, and the attitude of the seasurface platform. As well as positioning the acoustic transducer, the acoustic transducer transmits an acoustic signal and receives the returned acoustic signal from the seafloor transponders; thus, roundtrip travel times between the seasurface acoustic transducer and the seafloor transponders were obtained. We finally estimated the positions of the seafloor transponders by an iterative nonlinear least means square technique minimizing residuals between the observed travel times and the calculated travel times (e.g., Spiess et al. 1998; Kido et al. 2006; Fujita et al. 2006; Ikuta et al. 2008).
One of the most important assumptions of the GNSSA analysis is the rigid movement of the seafloor transponder array. Since an individual seafloor transponder position can be estimated using the acoustic traveltime data collected by a “moving survey” (the acoustic pings are transmitted from various seasurface points) (e.g., Ikuta et al. 2008; Honsho and Kido 2017; Additional file 1: Figure S2a), as similar to the determination of the seismological hypocenter (e.g., Hirata and Matsu’ura 1987), we can determine array geometry composed of the transponders. Then, constraining the array geometry, array displacements relative to the predetermined array position can be precisely estimated (e.g., Spiess et al. 1998; Kido et al. 2006; Matsumoto et al. 2008; Honsho and Kido 2017; Chen et al. 2018; Tadokoro et al. 2018b). If the array geometry is well determined, an array displacement can be estimated using only a single acoustic ping which simultaneously transmits to all seafloor transponders (e.g., Spiess et al. 1998; Kido et al. 2006; Additional file 1: Figure S2b). This type of the GNSSA positioning, using individual pings, previously introduced as the classic GNSSA positioning method in “Introduction” and often called “array positioning”, can in principle be achieved using the kinematic GNSSA positioning method. Note that observational data collected from “point surveys” (where the acoustic pings are transmitted from the center of the seafloor transponder array) are required for kinematic GNSSA positioning because accuracy of the array displacements degrades away from the array center (e.g., Kido 2007; Imano et al. 2015, 2019). In the following sections, we introduce a kinematic GNSSA positioning method based on array positioning where the array geometry is already determined.
The observation equation analogy to satellite measurements
In this section, observation equations for the kinematic array positioning method are introduced. This method assumes either a laterally stratified SSS or a horizontally graded SSS. For the method assuming a laterally stratified SSS, we utilize a concept of nadir total delay (NTD) for expressing the temporal fluctuation in the average sound speed, following Kido et al. (2008) and Honsho and Kido (2017). The NTD corresponds to the zenith total delay (ZTD) that expresses tropospheric delay in the satellite geodetic measurements (e.g., Marini 1972). For the method assuming a horizontally graded SSS, we formulated the underwater delay gradient of sound speed based on the azimuthal dependence, as similar to the expression of the tropospheric delay gradient in the satellite geodetic measurements (e.g., MacMillan 1995).
For the assumption of a laterally stratified SSS, the observation equation expressing a roundtrip travel time for the seafloor transponder \(k\) at time \(t_{n}\) is written as follows:
with
where \(T_{k,n}^{\text{obs}}\) and \(T_{k,n}^{\text{cal}}\) represent an observed and calculated travel time, respectively. The travel time is calculated from the initial transponder position, \({\mathbf{p}}_{k}\), and the array displacement, \(\delta {\mathbf{p}}_{n}\). \({\mathbf{d}}_{n}\) represents the position of a seasurface acoustic transducer at time \(t_{n}\) and \(v_{0}\) represents the initial sound speed profile. \({\text{NTD}}_{n}\) represents the NTD for time \(t_{n}\) and \(M\left( {\epsilon_{k,n} } \right)\) represents the mapping function for NTD, which depends on an inclination angle \(\epsilon_{k,n}\) of the acoustic ray path (Fig. 1). In this study, we adopted a simple mapping function using the sin function (Marini 1972):
This mapping function can also be represented using a shot angle of the acoustic ray path, \(\xi_{k,n}\), which is the same formulation of the array positioning introduced by Kido et al. (2006, 2008). In Eq. (1), the unknown parameters for each time step are the threedimensional array displacement, \(\delta {\mathbf{p}}_{n}\), and NTD. This observation equation is similar to that presented in Kido et al. (2006, 2008), but we newly introduced a vertical component for the array displacement, \(\delta p_{n}^{\text{Z}}\), as an unknown parameter. The vertical array displacement cannot be determined from kinematic array positioning using the point survey data at conventional GNSSA sites because of the tradeoff between vertical array displacements and the changes in sound speed (NTDs) (Additional file 1: Figure S3a). Thus, the conventional kinematic array positioning method fixes the vertical motion and estimates only the horizontal motions. Meanwhile, it is known that the moving survey data are essential to detect displacements in the vertical component (e.g., Sato et al. 2013) because the variation in shot angle of the seafloor transponders obtained from the moving survey is required to solve the tradeoff relationship. At some recent GNSSA observation sites, six transponders have been used to form a combination of small and large triangular arrays (Kido et al. 2015), providing variation in the shot angles even for a point survey (Additional file 1: Figure S3b). Thus, we can detect the vertical array displacement even when using kinematic array positioning for a site with “multiangled transponders”. The effects of using the multiangled transponders are investigated in “Synthetic test” and “Application”.
For the assumption of a horizontally graded sound speed structure, the observation equation expressing a roundtrip travel time for the seafloor transponder, \(k\), at time, \(t_{n}\), is written as follows:
where \(G_{n}^{\text{EW}}\) and \(G_{n}^{\text{NS}}\) represent the underwater delay gradients for the east–west and north–south components, respectively. \(\phi_{k,n}\) is the azimuth of the seafloor transponder from the seasurface platform at time, \(t_{n}\). This formulation for the underwater delay gradient is the same as that for the tropospheric delay gradient in the satellite geodetic measurements (MacMillan 1995). Although Kido (2007) proposed a formulation for describing the contributions to the sound speed gradient similar to our own formulation, the formulation of Kido (2007) did not account for the effect related to each seafloor transponder depth, represented by \(\cot \epsilon_{k,n}\) in Eq. (5). Thus, our formulation is an updated version of Kido (2007). Yasuda et al. (2017) and Yokota et al. (2018) present alternative formulations for the sound speed gradient. However, their formulations necessitate that the acoustic ranging data must be collected from various seasurface shot points by moving the seasurface platform; therefore, they are unsuitable for use in kinematic positioning. Our formulation is suitable for use in kinematic positioning, but it should be noted that the seasurface platform should be kept in position above the array center to obtain good resolution as pointed out by Kido (2007).
Application to the extended Kalman filter
We apply the above observation equation for the kinematic GNSSA positioning to EKF. Since an inversion problem in the GNSSA positioning is nonlinear, EKF is utilized in this study. This nonlinear state space model is governed by a system model (state transition equation) defined as
and the observation model (observation equation) is defined as
where \({\mathbf{y}}_{n}\) and \({\mathbf{x}}_{n}\) are the observation vector and the state vector at time, \(t_{n}\), respectively. \({\mathbf{u}}_{n}\) is the control input vector, \({\mathbf{v}}_{n}\) is the process noise vector and \({\mathbf{w}}_{n}\) is the measurement error vector. Since the EKF assumes that the processing noise and measurement errors follow the Gaussian distribution, their probability density functions are denoted as follows:
and
The process noise vector, \({\mathbf{v}}_{n}\), is under the Gaussian noise with average zero and the covariance matrix (the process noise matrix), \({\mathbf{Q}}_{n}\). The measurement error, \({\mathbf{w}}_{n}\), is under the Gaussian noise with average zero and the covariance matrix, \({\mathbf{R}}_{n}\). \({\mathbf{F}}_{n}\) and \({\mathbf{H}}_{n}\) in Eqs. (6) and (7) are the nonlinear functions for calculating the state transition and the synthetic observations from the state parameters, respectively. When a linear state space model is assumed for the system model, Eq. (6) is reformulated using a linear transition matrix, \({\bar{\mathbf{F}}}_{n}\), as follows:
In the EKF, \({\mathbf{H}}_{n}\) is linearized based on a firstorder Taylor expansion in the neighborhood of the state \({\bar{\mathbf{x}}}_{n}\) predicted from the system model:
with the linearized Jacobian matrix
Using the linearized matrix, the EKF can be applied in the same way as the linear KF; an update of the state parameters from the actual observation data based on the observation equation and a prediction of the state parameters based on the system model are performed alternately and iteratively at each time step. The predicted state parameter \({\mathbf{x}}_{n + 1}\) is obtained using the estimated state parameter of the previous time step \({\hat{\mathbf{x}}}_{n}\) by the system model as shown in the following equation:
The predicted covariance matrix, \({\bar{\mathbf{P}}}_{n + 1}\), is obtained using the estimated covariance matrix at the previous time step, \({\hat{\mathbf{P}}}_{n}\), using the following equation:
The estimated state parameter \({\hat{\mathbf{x}}}_{n}\) is obtained using the predicted state parameter \({\bar{\mathbf{x}}}_{n}\) using the following equation:
where \({\mathbf{K}}_{n}\) is the Kalman gain, with
The updated covariance matrix is then obtained using the following equation:
For the assumption of a laterally stratified sound speed structure, the state vector is defined as
We assume the white noise in the stochastic process for the estimation of the array displacement, while we assume the random walk in the stochastic process for estimating NTD. The assumption of the white noise process for estimation of the array displacement is introduced to estimate these parameters in a kinematic manner, independent of time. The assumption of a random walk process for estimation of the NTD is often used for modeling the tropospheric delay in satellite measurements (e.g., Herring et al. 1990). The assumptions are implemented by defining the linear transition matrix, \({\mathbf{F}}\), as
and by defining the process noise matrix as
with
The variance used in the process noise matrix should be assigned in advance. As explained later in this section, the variance for the NTD, \(\sigma_{\text{NTD}}^{2}\), is determined by fitting to the observational data. The variance of the array displacements is assumed to be 1.0 m^{2} since the array displacements can be estimated even for large abrupt displacements by assigning large values to the variance in the array displacements. Note that, to adjust the average levels of the array displacements using initial array displacements, \(\delta {\mathbf{p}}_{0}\), we define the control input vector as follows:
Unifying Eqs. (18)–(22), the system model [Eqs. (8) and (10)] for the assumption of a laterally stratified sound speed structure is defined. As for the observation model, the observation vector is obtained as
where \(k_{n}\) is the total number of the replied seafloor transponders at time, \(t_{n}\). The nonlinear Jacobian matrix is defined as
and the linearized Jacobian matrix is defined via Eq. (12) as
We assume the same weight among the observed roundtrip travel times regardless of the time step as follows:
where \(\sigma^{2}\) corresponds to variance of the measurement errors for the roundtrip travel times. Considering that the precision of the acoustic ranging is < ~1 cm (e.g., Fujimoto 2014) and that the precision of the kinematic GNSS positioning on the seasurface platform is roughly a few centimeters (e.g., Sugimoto et al. 2009), \(\sigma^{2}\) is assumed to be 1.0 × 10^{−9} s^{2}, which corresponds to an ~ 5 cm measurement error in the lineofsight direction.
For the assumption of a horizontally graded sound speed structure, we define the state vectors as
Similar to the assumptions for a laterally stratified sound speed structure, the array displacements are assumed to be a white noise process, while the NTD is assumed to be a random walk process. Further, the underwater delay gradient is assumed to be a random walk processes, like the tropospheric delay gradients in the satellite measurements (e.g., BarServer et al. 1998). Then, as for the system model, the transition matrix and the process noise matrix are defined as follows:
The variance in the array displacements and the NTD is the same as for the case of the laterally stratified sound speed structure. The variance in the underwater delay gradients are determined to fit the observational data, as for that of the NTD. To adjust the average levels of the array displacements using the initial array displacements, \(\delta {\mathbf{p}}_{0}\), we also define the control input vector as
As for the observation model, the observation vector is obtained using Eq. (23). The kth component of the nonlinear Jacobian matrix H_{n}(x_{n}) is defined as
and the linearized Jacobian matrix is defined as
We also assume the same covariance matrix for the measurement errors using Eq. (26).
The optimal variance for the process noise of the NTD is determined by maximizing likelihood. Given the variance parameter vector as
for the case of the horizontally graded SSS (for the case of the laterally stratified SSS, the variance parameter is \(\sigma_{\text{NTD}}^{2}\), not forming a vector), the loglikelihood is written as follows (e.g., Kitagawa 2005; Segall and Matthews 1997):
with
Synthetic test
Model setting
To evaluate the performance of the introduced EKFbased array positioning method, we conducted synthetic tests. We analyzed synthetic observational data assuming a horizontally graded SSS with a temporal fluctuation by both the conventional array positioning (e.g., Kido et al. 2006; Kido 2007) method and the EKFbased array positioning method. Note that we performed the above analysis assuming both a laterally stratified SSS and the horizontally graded SSS to investigate how the temporally fluctuating gradient structure affects estimations compared with the case of a laterally stratified SSS. Synthetic roundtrip traveltime data were produced based on the observation equation [Eq. (5)] by temporally varying the unknown parameters \(\left( {\delta {\mathbf{p}}_{n} , NTD_{n} , G_{n}^{\text{EW}} , G_{n}^{\text{NS}} } \right)\) as follows:
To calculate the synthetic traveltime data, we specified the initial positions of the transponders, \({\mathbf{p}}_{k}\), synthetic positions of an acoustic transducer at a seasurface platform, \({\mathbf{d}}_{n}\), and an initial sound speed profile, \(v_{0}\), as synthetic observational data.
We generated synthetic observational data based on three different geometries for the seafloor transponder array (i.e., the initial positions of the transponders, \({\mathbf{p}}_{k}\)): four transponders forming a square array geometry (Fig. 2a); six transponders forming a combined large and small triangular array (Fig. 2b); four transponders forming a combined triangular array with an additional transponder at its center (Fig. 2c). The squareformed array geometry is the most popular pattern among the recently established GNSSA sites in Japan, and the doubletriangular array geometry has been adopted at five sites on a trial basis, which provides a multiangled transponder array (e.g., Kido et al. 2015). The assumed geometries for the squareformed array (Fig. 2a) and for the doubletriangular array (Fig. 2b) are same with actual observation sites, the G08 (water depth: 3473 m) and G19 (water depth: 5725 m), located in the offTohoku region (Tomita et al. 2017), respectively. The triangularcentered array geometry (Fig. 2c) has not yet been trialed. However, it can be treated as a multiangled transponder site using only four transponders as illustrated in Fig. 2c; we investigate its performance. Note that we assumed 4000 m of water depth for the synthetic triangularcentered array geometry.
Synthetic positions for an acoustic transducer on a seasurface platform, \({\mathbf{d}}_{n}\), were generated within a radius of 50 m from the array center, imitating a point survey. Note that the actual positions of the seasurface acoustic transducers were used to calculate the synthetic traveltime data; however, when estimating the unknown parameters in the synthetic tests, we used positions for the seasurface acoustic transducers contaminated by GNSS positioning errors, with standard deviation of 3 cm for each component. These values where chosen as the precision of the kinematic GNSS positioning on the seasurface platform is roughly a few centimeters (e.g., Sugimoto et al. 2009).
The initial sound speed profile, \(v_{0}\), for each site (G08 and G19) was calculated from temperature and salinity data obtained from the World Ocean Atlas 2013 (WOA13; Locarnini et al. 2013; Zweng et al. 2013). WOA13 distributed average underwater structure models for temperature, salinity and other indexes, complied with observational data over the past 60 years. We calculated a temperature and salinity profile for each site and converted them into a sound speed profile following Chen and Millero (1977). The initial sound speed profile for the triangularcentered array geometry is same as that for G17 site also located in the offTohoku region, at water depth of 4232 m.
Then we assigned synthetic temporal fluctuations in the unknown parameters. We assumed temporally fixed array displacements for the synthetic tests (all components of the array displacement were fixed to 0.5 m). Note that we provided an initial array displacement of 0.3 m for all components as the control input for Eq. (30) when using the EKFbased array positioning method. Temporal fluctuations in the NTD were expressed as a sine wave with an amplitude of 0.1 ms and a period of 4 h. The east–west component of the temporal fluctuation in the underwater delay gradient, \(G_{n}^{\text{EW}}\), was also expressed as a sine wave with an amplitude of 0.1 ms and a period of 2 h. The north–south component of the underwater delay gradient, \(G_{n}^{\text{NS}}\), was fixed in time.
Based on the above model setup, we calculated the synthetic traveltime data using Eq. (36). In this calculation, a total of 300 shots of the synthetic acoustic pings were sampled at intervals of 60 s. Finally, we added measurement errors to the traveltime data assuming Gaussian noise with a standard deviation of 1.0 × 10^{−5} s (corresponding to ~ 0.75 cm in the oneway slant path) since precision of the acoustic ranging is < ~ 1 cm (e.g., Fujimoto 2014).
Results
Figure 3 shows the kinematic array positioning results for the synthetic data assuming the squareformed array geometry. The solutions estimated by the conventional array positioning method, which fixed the vertical array displacements to be zero (e.g., Kido et al. 2006, 2008) (black triangles), demonstrated a timedependent systematic error in the east–west component of the array displacements (Fig. 3a), caused by the prescribed gradient in SSS (Fig. 3e). Since the vertical array displacements are fixed to be zero (Fig. 3c), the estimated NTDs are biased from the true solutions; however, the estimated NTDs could explain the true temporal evolution (Fig. 3d). The solutions obtained by the conventional array positioning method, estimating both vertical and horizontal array displacements, are shown by orange crosses. The estimated vertical array displacements are out of plotting range in Fig. 3c, with displacements of up to ± ~ 10 m. As the estimated NTDs are also out of plotting range in Fig. 3d, the tradeoff relationship between the vertical array displacements and the NTDs cannot be solved in the squareformed array geometry when only conducting a point survey. Further, the unsolved NTDs might degrade the precision of the horizontal array displacements (Fig. 3a, b). The EKFbased array positioning, assuming a laterally stratified SSS (blue squares), prevents the vertical array displacements and the NTDs from diverging. However, the tradeoff cannot be solved even when using the EKFbased array positioning, meaning that the vertical array displacements and the NTDs deviated from the true values over time. The horizontal components of the array displacements are almost the same as the solutions obtained from the conventional array positioning method with fixed vertical array displacements (Fig. 3a, b) since the NTDs did not diverge through the EKFbased array positioning in this time window. Estimation results of the EKFbased array positioning assuming a horizontally graded SSS (purple circles) showed similar vertical array displacements and NTDs to predictions assuming a laterally stratified SSS (Fig. 3c, d). In the EKFbased array positioning, the horizontal components of the array displacements also deviated from the true values with time because of the tradeoff relationship between the array displacements and the underwater delay gradients (Fig. 3a, b).
Figure 4 shows the kinematic array positioning results for the synthetic data assuming the doubletriangle array geometry. The conventional array positioning method of fixed zero vertical array displacements showed similar results to the case of the squareformed array geometry (black triangles in Fig. 4a–l), but for this new geometry, the conventional array positioning method was also able to resolve the vertical array displacements and the NTDs (orange crosses in Fig. 4a–f). This improvement is a result of the multiangled transponders solving the vertical array displacements with the NTDs as explained in “The observation equation analogy to satellite measurements”. Furthermore, the EKFbased array positioning method, assuming the laterally stratified SSS, further demonstrates the superior performance for the vertical array displacements (blue squares in Fig. 4c) since the EKF provides a temporal constraint on NTDs that avoids numerical instability when solving the vertical array displacements and NTDs. The conventional array positioning method [an updated method from Kido (2007) based on Eq. (5)] and the EKFbased array positioning method for the horizontally graded SSS are shown in Fig. 4g–l as green diamonds and purple circles, respectively. The conventional method approximately solves the gradient parameters (Fig. 4k, l), but the solutions of the gradient parameters and of the horizontal array displacements have large dispersions, probably because of the potential tradeoff relationship among them. Meanwhile, the EKFbased array positioning method stably solves those parameters by constraining the temporal evolution of the gradient parameters and greatly reduces the dispersion in the horizontal array displacements.
Figure 5 shows the kinematic array positioning results for the synthetic data assuming the triangularcentered array geometry. The conventional array positioning method can resolve the vertical array displacements and the NTDs unlike for the case of the squareformed array geometry (orange crosses in Fig. 5c, d). However, note that the dispersion of the vertical array displacements is a little larger that the case of the doubletriangularformed array geometry probably because of insufficient variation in the shot angles. The EKFbased array positioning method can improve the precision of the vertical array displacements (blue squares in Fig. 5c) by temporally constraining NTDs similar to the doubletriangularformed array geometry. Although the underwater delay gradients cannot be solved in the case of the triangularcentered array geometry (Fig. 5e, f), this geometry enables us to detect kinematic vertical array displacements accurately, using only four transponders (the doubletriangularformed array geometry requires six transponders). Furthermore, the precision of the vertical array displacements can be improved using the EKFbased array positioning method, assuming a laterally stratified SSS.
Through the synthetic test, we confirmed (1) the utility of the multiangled transponders for detecting the kinematic vertical array displacements, (2) the accurate performance of the EKFbased array positioning method for precisely detecting the vertical array displacements at the multiangled transponder sites, and (3) the accurate kinematic solutions to the underwater delay gradients for the doubletriangularformed array geometry site via the EKFbased array positioning method.
Point (1) shows the importance of acoustic data with various shot angles during a point survey. Effects of a temporal sound speed fluctuation can be expressed by NTDs that are independent of shot angles, whereas the vertical array displacement is sensitive to variation of shot angles. We validated this fact through a simple numerical test. We assumed a twodimensional field as shown by Fig. 6a, and five transponders are vertically located. When perturbation is given in the average sound speed, the oneway traveltime residual can be expressed as follows:
Using the mapping function, contribution of the sound speed perturbation to the traveltime residual is common with each transponder. In other words, the traveltime residual normalized by the mapping function is independent of the horizontal distance of the transducer (i.e., shot angle) as follows:
Figure 6b shows the normalized traveltime residuals when various values of the sound speed perturbation are given. Figure 6b clearly demonstrates that the normalized traveltime residuals are same among the transducers. Meanwhile, when perturbation in depth is given, the traveltime residual can be written as follows:
Then the normalized travel time can be also written as follows:
Figure 6c shows the normalized traveltime residuals when various values of the perturbation in depth are given. Figure 6c demonstrates that the normalized traveltime residuals vary depending on the absolute horizontal distance from the array center (i.e., absolute shot angles). This indicates that if the absolute shot angles for the transponders from the array center are same (such as the squareformed array geometry), we cannot distinguish contributions of the sound speed perturbation from those of the depth perturbation. Thus, variation in absolute shot angles is essential to precisely estimate vertical displacements in the kinematic GNSSA positioning.
In the next section, we apply the EKFbased array positioning methods to actual observational data.
Application
Actual observation data
We used the actual GNSSA observational data collected by Tomita et al. (2017) in the offTohoku region. We used the observational data collected on March 14, 2015, at the G08 site (longitude: 143.647°E, latitude: 38.721°N, depth: 3473 m) as an example of the squareformed array geometry (Fig. 7). Moreover, we used the observational data collected on February 28, 2015, at the G19 site (longitude: 142.671°E, latitude: 36.496°N, depth: 5725 m) as an example of the doubletriangularformed array geometry (Fig. 8). The details of the observational data are shown in Tomita et al. (2017).
For GNSS positioning using the KF, the process noises are often assigned as fixed values (e.g., BarServer et al. 1998; Hirata and Ohta 2016) because there is high computational cost to determine the optimal process noise values. Therefore, it is worthwhile to search for “general” process noise values to use with the EKFbased GNSSA array positioning method. From the synthetic tests, we have found that the doubletriangularformed array geometry is suitable for the EKFbased array positioning method, regardless of the assumed SSS. Therefore, we investigated the optimal process noise values by maximizing the likelihood of the actual observational data, at sites with the doubletriangularformed array geometry. Thus, we also analyzed all the observational data at the sites forming the doubletriangularformed array geometry collected by Tomita et al. (2017): G04 (longitude: 143.897°E, latitude: 39.566°N, depth: 4587 m), G10 (longitude: 143.483°E, latitude: 38.302°N, depth: 3271 m), G15 (longitude: 143.521°E, latitude: 37.677°N, depth: 5264 m) and G19 sites. Then we compared the determined process noise values.
Results
Figure 7 shows the kinematic array positioning results for the actual observational data at the G08 site (the squareformed array geometry). We analyzed the observational data using the conventional array positioning method of fixed vertical array displacements (black triangles), the conventional array positioning method estimating the vertical array displacements as well as the horizontal array displacements (orange crosses), and the EKFbased array positioning method assuming a laterally stratified SSS (blue squares). The general features of the positioning results are similar to the synthetic test results, assuming the squareformed array geometry (Fig. 3); the vertical array displacements could not be solved due to the tradeoff relationship between the vertical array displacements and the NTDs, even by the EKFbased array positioning method, and the horizontal array displacements estimated by the EKFbased array positioning method were almost the same with those estimated by the conventional array positioning method of fixed vertical array displacements.
Figure 8a–f shows the kinematic array positioning results for the actual observational data at the G19 site (the doubletriangularformed array geometry) analyzed using the positioning method assuming a laterally stratified SSS. The general features of these positioning results are similar to those of the synthetic test for the doubletriangularformed array geometry (Fig. 4a–f); the horizontal components of the solutions estimated by this method are roughly in accordance. The vertical array displacements can be determined using the conventional array positioning method of estimated vertical array displacements (orange crosses) and the EKFbased array positioning method (blue squares). Moreover, the EKFbased array positioning method can detect the vertical array displacements more precisely than the conventional array positioning method. Although the horizontal array displacements estimated by the conventional array positioning method of fixed vertical displacements agree well with those estimated using the EKFbased array positioning method in the synthetic test (Fig. 4a, b), they sometimes disagree for the actual observational data (black triangles in Fig. 8a, b). The causes of this are discussed later in “Robustness in the case of an unresponsive transponder”. Unlike the synthetic test, the vertical array displacements estimated using the EKFbased array positioning method fluctuate with time, and the fluctuation is up to a few tens of centimeters, although the vertical array displacements should be constant during the cruise; the causes of this are discussed later in “Improvements to detection of vertical array displacements”.
Features of the positioning results for the actual observational data at the G19 site, analyzed using the positioning methods assuming a horizontally graded SSS (Fig. 8g–l), are quite different from the results of the synthetic test (Fig. 4g–l). The horizontal components of the solutions showed large fluctuations with time in both cases of the conventional array positioning method (green diamonds) and of the EKFbased array positioning method (purple circles). The variations are larger than those observed from the solutions obtained assuming a laterally stratified SSS. These variations show shortterm (~ a few hours) periodic fluctuations (nonrandom variations); the degraded positioning results are attributed to systematic modeling errors in the underwater delay gradients. These modeling errors are discussed later in “Sound speed structure in actual ocean”. Meanwhile, the estimated vertical array displacements and NTDs are almost the same as the solutions obtained assuming a laterally stratified SSS. Thus, the NTDs can be solved independently of the crucial modeling errors in the underwater delay gradients. Additional file 1: Figures S4–S8 show positioning results at the G19 site for the actual observational data collected from the other cruises, and the features of the positioning results mentioned above are also found in the results of other cruises.
The process noise values determined from the actual observational data, for the doubletriangularformed array geometry, are summarized in Table 1. The optimal process noise values (\(\sigma_{\text{NTD}}\)) are determined to be ~ 2.0 × 10^{−3} (s/s^{1/2}) regardless of the assumed SSS, the observation site and the observation period. Thus, this value would be generally applicable. Although the process noise values of the underwater delay gradient were determined to be ~ 1.9 × 10^{−3} (s/s^{1/2}) in both the east–west and north–south components (\(\sigma_{{G^{\text{EW}} }} ,\;\sigma_{{G^{\text{NS}} }}\)), these values are not reliable because the estimated horizontal array displacements assuming a horizontally graded SSS were not accurately determined, as explained above.
Discussion
Performance of the EKFbased array positioning method
Here, we discuss the utility of our newly developed EKFbased array positioning method. As shown in “Results”, the EKFbased array positioning method, assuming a laterally stratified SSS, shows superior performance when compared to the conventional array positioning methods. The advantages of the EKFbased array positioning method are (1) robust positioning accuracy even when some seafloor transponders are unresponsive; (2) precise detection of the vertical array displacements; (3) applicability to continuous GNSSA positioning. These advantages are discussed in this section. However, in instances of a horizontally graded SSS, the EKFbased array positioning method failed to improve on the precision of the array positioning results, when compared to other methods. The causes of this failure are discussed in “Sound speed structure in actual ocean”.
Robustness in the case of an unresponsive transponder
The horizontal array displacements for the actual observational data estimated using the conventional array positioning method with fixed vertical array displacements (black triangles in Fig. 8a, b) show little scatter when compared with those estimated using the conventional and the EKFbased array positioning methods. Figure 9a, b also shows the horizontal array displacements for the same data estimated using the conventional array positioning method of fixed vertical array displacements, and the plotted color represents the number of the responding transponders for each ping. This illustrates that the scattered array displacements appear when some transponders are not responding. Such response failures can arise for various reasons including, for example, significant background mechanical noise, bad weather and overlaps in the signal response. For examples, in the observational data obtained in March 2015 at G19 (Fig. 8), three transponders configuring the inner triangle well responded (data acquisition ratio is ~ 97 percent), while the other three transponders configuring the outer triangle occasionally failed to responded (data acquisition ratio is ~ 83 percent). Yet in the observational data obtained in November 2015 at G19 (Additional file 1: Figure S7), all transponders well responded (data acquisition ratio is ~ 97 percent).
To investigate the influence of the unresponsive transponders, we performed additional synthetic test with intermittent transponder failures (Fig. 10). The conditions of these synthetic test are similar to those assuming the doubletriangularformed array geometry, but the temporal fluctuations in the underwater delay gradients are omitted (thus, the horizontal array displacements should be constant with time). In the test, 40% of the responses from the transponders are randomly excluded. Figure 10a–c shows the array displacements estimated using the kinematic array positioning method (true array displacements are 0.5 m for all components). As seen for the actual observational data (Fig. 8), the conventional (orange crosses) and the EKFbased (colored dots) methods estimating the vertical array displacements generally detect true horizontal array displacements, whereas the conventional array positioning method of fixed vertical array displacements (black triangles) provides scattered solutions. However, assuming that the true vertical array displacements are zero, the conventional array positioning method with fixed vertical array displacements successfully provides comparable horizontal array displacements (black triangles in Fig. 10d–e; note that they are totally overlapped with the colored dots). These synthetic tests suggest that deviation of the vertical array displacements from true values would produce crucial modeling errors in the horizontal array displacements when some transponders fail to respond. We considered that this modeling error is related to the apparent shift in the point of the array center when some transponders fail to respond. Since the accuracy of the array positioning degrades away from the array center (e.g., Kido 2007; Imano et al. 2015, 2019), deviation of the vertical array displacements results in serious modeling errors due to the apparent shift in the point of the array center. This problem can be avoided by estimating the vertical array positions precisely. Since the precision of the vertical array displacements estimated using the EKFbased array positioning method is better than that of the conventional array positioning method, the EKFbased array positioning method can stably perform the array positioning not only for the vertical array displacements but also for the horizontal array displacements, as shown from the analyses of the actual observational data (e.g., Fig. 8a–c; Additional file 1: Figure S8a–c).
Figure 10 further illustrates the robustness of the EKFbased array positioning method in the instance of unresponsive transponders. In principle, the EKFbased array positioning method can estimate the unknown parameters even when the number of observations is smaller than the number of unknown parameters. The colors of the dots (the solutions from the EKFbased array positioning method) in Fig. 10 represent the number of unresponsive transponders; the EKFbased array positioning method can provide accurate solutions when more than three responsive transponders are available. When less than two responsive transponders are available, the array positions are estimated to be 0.3 m for all components, which corresponds to the initial array displacements for the control input in Eq. (30), as explained in “Model setting”. Therefore, the array displacements cannot be constrained using the observational data in this case. However, since the conventional array positioning method requires more than four responsive transponders to estimate the three components of the array position and NTDs (orange crosses in Fig. 10), the EKFbased array positioning method can increase positioning opportunities when only three responsive transponders are available.
Improvements in the detection of vertical array displacements
As shown both in the synthetic tests (Figs. 4, 5) and in the actual observational data analysis (Fig. 8; Additional file 1: Figures S4–S8), the EKFbased array positioning method successfully improves the precision of the array displacements, especially for the vertical component when using multiangled transponders. However, unlike the synthetic tests, the positioning results using the actual observational data showed temporal fluctuations in the vertical array displacements of up to a few tens of centimeters (Fig. 8c; Additional file 1: Figures S4c–S8c). The primary reasons for this temporal fluctuation are (1) GNSS positioning errors on the seasurface platform, and (2) deviation from a laterally stratified SSS in the actual ocean. Note that Earth tide effects were eliminated in the shown vertical positioning results in advance although the Earth tide potentially produces longterm fluctuation in the vertical component.
The GNSS positioning errors directly propagate to the positions of the acoustic transducers on the seasurface platform and then propagate to the array displacements. Although it is difficult to evaluate positioning errors for this moving body, we can consider that the vertical positions of the acoustic transducer on the seasurface platform are inherently bounded to the sea surface. Therefore, eliminating oceanic tidal effects and geoid heights from the acoustic transducer positions, we can roughly evaluate the relative temporal fluctuation of the vertical GNSS positioning errors, although the absolute GNSS positioning errors cannot be evaluated (Fujita and Yabuki 2003). The vertical positions of the acoustic transducers are shown as bold gray curves in Fig. 8c and Additional file 1: Figures S4c–S8c, eliminating the oceanic tidal effects using the NAO.99Jb model (Matsumoto 2000) and the geoid heights calculated from Fukuda (1990). Note that the acoustic transducer positions are smoothed by a 5min moving average filter because the positions are shaken by sea waves with short timescale. Since the longterm fluctuation in the vertical array positions is consistent with that of the acoustic transducer positions, the GNSS positioning errors are considered to be the major source of error in the vertical array displacements. Table 2 shows 1σ standard deviations of the vertical array positions for the actual observational data estimated by the conventional array positioning method (columns 3–5) and by the EKFbased array positioning method (columns 6–8). The average standard deviation of 11.14 cm (column 6: std.) is reduced to 9.18 cm (column 7: corrected std.) by subtracting the acoustic transducer positions in the case of the EKFbased array positioning method [the reduction can be up to ~ 5 cm in the observational data for G04 (March, 2015)]. Although one of the factors causing the longterm GNSS positioning errors is the very long baseline for kinematic differential GNSS positioning (e.g., Colombo et al. 2000), kinematic precise point positioning (PPP) (Zumberge et al. 1997) techniques, which do not require a terrestrial reference station, have improved and would be an alternative way of determining the position of the seasurface platform. Watanabe et al. (2017) reported that kinematic PPP provided more stable solutions than longbaseline differential positioning. These developments in GNSS positioning would enable us to further improve the precision of the vertical array displacements using the EKFbased array positioning to less than ~ 10 cm. It should be noted that the precisions discussed here are relative precisions in the halfday observation data, and are not the absolute accuracy of the vertical positions. To discuss the absolute accuracy, much longer observational data are required.
In addition to longterm GNSS positioning errors, the vertical array displacements also have shortterm variations with periods of ~ tens of minutes (e.g., Figure 8c; Additional file 1: Figures S4c–S8c). These shortterm variations may result from deviation from a laterally stratified SSS in the actual ocean. Although the horizontally graded SSS does not strongly affect positioning of the vertical array displacements as shown by the synthetic tests (Figs. 4, 5), more complicated spatial heterogeneity may affect the positioning of the vertical array displacements. In fact, the horizontal array displacements often fluctuate over similar time periods (Fig. 8; Additional file 1: Figures S4–S8). The GNSS positioning errors of the seasurface platform possibly produced the shortterm fluctuations in the vertical array displacements, but it is hard to evaluate their contributions. Since the positioning errors of the vertical array displacements appear to be systematic with time, detection of an abrupt step in the array position, such as a coseismic slip event, would be more precisely determined than a stable array displacement. To investigate the precision of the array displacements relative to the previous time step for detection of an abrupt step, we calculated differential sequences of the vertical array displacements and then calculated the standard deviation of the differential sequences (columns 5 and 8 in Table 2: difference std.). As a result, the standard deviation for the EKFbased array positioning method is on average 4.50 cm, while that of the conventional array positioning method is on average 9.13 cm. Since the EKFbased array positioning method constrains the temporal evolution of the NTDs, we can precisely detect an abrupt step in the vertical component eliminating the potential tradeoff relationship between the vertical array displacements and the NTDs.
Applicability to realtime GNSSA positioning
Our main finding, accurate detection of vertical displacements, comes from appropriate estimation of NTDs constrained by the EKF. Although the constraint of temporal variations in NTDs can be achieved using conventional inversion techniques using a batch of acoustic ranging data (e.g., Honsho and Kido 2017), the EKFbased GNSSA positioning method has an additional utility: instant processing suitable for realtime positioning.
The batchtype positioning requires high computational cost because a significant amount of data should be simultaneously processed; however, the EKFbased GNSSA positioning method can determine a position using only acoustic ranging data for each ping, when the process noise values are fixed, as shown in “Results”. In these circumstances, our method requires less computational cost, comparable to the simplest kinematic GNSSA positioning methods which do not provide temporal constraints of NTDs (e.g., Spiess et al. 1998; Kido et al. 2006). Furthermore, the EKFbased positioning method can provide kinematic solutions instantly, just after collection of the acoustic ranging data for each ping, because this technique does not need “future” acoustic ranging data to constrain the temporal variation in NTDs. Recently, some trials of realtime and continuous GNSSA observations have been conducted using a moored buoy (e.g., Imano et al. 2015; Kido et al. 2018; Kato et al. 2018; Tadokoro et al. 2018a). In this application, kinematic GNSSA positions are estimated using a small computer attached to the buoy, which are then transmitted to an onshore station via satellite relay. Thus, the computational cost of GNSSA positioning using a moored buoy is as low as possible. Our proposed method has a low computational cost and can provide a kinematic position with constraints on the temporal variation of NTDs, using realtime processing, making it suitable for realtime and continuous GNSSA observations.
To estimate vertical positions using the EKFbased GNSSA positioning method, frequency of acoustic ranging is an important factor to constrain the temporal variation NTDs. The recent yearly trials of GNSSA observations using a moored buoy have performed acoustic ranging less frequency than campaigns using a research vessel, to conserve the battery life of the buoy and sensors. The interval of acoustic pings in the observations obtained using a research vessel are generally 30–60 s, whereas moored buoy systems provided a set of 11 acoustic pings with an interval of 65 s in a week (e.g., Imano et al. 2015; Kido et al. 2018), or acoustic ranging data with intervals of 180 s (e.g., Kato et al. 2018; Tadokoro et al. 2018a). In this study, we investigated the influence of sampling frequency on the vertical positioning using actual campaign observational data at site G19. We resampled the observational data for each campaign with various sampling intervals from 1 to 180 min, and then we estimated the kinematic positions using the EKFbased GNSSA positioning using the optimal process noise for NTD (2.0 × 10^{−3} s/s^{1/2}). Figure 11a demonstrates standard deviations of the vertical positions for the different sampling intervals relative to those for the sampling interval of 60 s. Note that we consider that the most frequent sampling interval (here, 60 s) could provide the best solutions. Since the degree of improvement in the positioning is different from the cruise data, Fig. 11b shows the standard deviations normalized by those estimated using the classical GNSSA positioning method without temporal smoothing for NTD. In most campaigns, the standard deviations for the sampling intervals longer than 30 min do not show clear difference, and they are close to those calculated using the classical kinematic positioning method (the normalized standard deviation is ~ 0.9). However, the standard deviations for the sampling intervals shorter than 30 min are significantly improved. Thus, a sampling interval shorter than 30 min should be used for continuous GNSSA observations to obtain the most benefit from introduction of the EKFbased positioning method. Therefore, our method would provide usable constraints for acoustic ranging data with sampling intervals of a few minutes, such as those obtained using a moored buoy, while it would not be useful for the weekly interval of the acoustic ranging data (Imano et al. 2015; Kido et al. 2018). Since continuous GNSSA observations using a moored buoy are still at the testing stage, specification of the system, such as sampling interval, can change. As long as the sampling interval is shorter than 30 min, it is worthwhile implementing our proposed method to continuous GNSSA observation systems to obtain precise kinematic vertical positions.
Another issue that should be overcome for achieving the precise realtime GNSSA positioning is how to keep the seasurface platform position at the array center. Regardless of implementation of the EKF, the kinematic array positioning basically requires the point survey data because accuracy of the array displacements degrades away from the array center as explained in “Principles of the GNSSA positioning method” (e.g., Kido 2007; Imano et al. 2015, 2019). In the past trials of the GNSSA measurement using the moored buoy, the buoy occasionally moved out of the transponder array, and the accuracy of the positioning was degraded (Imano et al. 2019). Thus, future technical improvement in operating the moored buoy is an important factor for the precise realtime GNSSA positioning. This issue is beyond the scope of this study, but we emphasize that it is important to develop the both tangible (such as the moored buoy itself) and intangible (such as the EKFbased positioning method) factors.
Sound speed structure in actual ocean
Although the underwater delay gradients and the horizontal array displacements are well resolved in the synthetic test using the EKFbased array positioning method (Fig. 4), the horizontal array positions for the actual observational data fluctuate with time and deteriorate when compared to the solutions obtained assuming a laterally stratified SSS (Fig. 8). Although it is inherently difficult to resolve the underwater delay gradients and the horizontal array displacements due to their tradeoff relationship, the numerical instability due to this tradeoff should be random (as shown by the conventional array positioning method in the synthetic test: green diamonds in Fig. 5) rather than have a temporal fluctuation (as found from the actual observational data: purple circles in Fig. 8). Thus, we consider that the temporal fluctuations in the horizontal array displacements may be caused by systematic errors in modeling SSS not by numerical instability; SSS in the actual ocean might not be well expressed using horizontal grading. As Kido (2007) pointed out, spatial heterogeneity in SSS with long wavelengths can be expressed as a horizontally graded SSS even if multiple gradients exist across different water depths. Note that the long wavelength implies that the wavelength is longer than distance among acoustic path lines at the depth where the graded structure exists. However, spatial heterogeneity in SSS with short wavelengths cannot be expressed using a horizontally graded SSS. Thus, in our results, we think that the underwater delay gradients were overestimated for such temporally fluctuating shortwavelength spatial heterogeneities, and that the horizontal array displacements also fluctuated due to the overestimated underwater delay gradients.
We found that the SSS might have shortwavelength heterogeneities (shorter than the distance among acoustic path line), with short time periods (~ tens of minutes), whereas some previous studies have already successfully improved accuracy of the GNSSA positioning assuming the longlived horizontally graded SSS (e.g., Yasuda et al. 2017; Yokota et al. 2018). However, these previous studies performed static GNSSA positioning using longterm (over hours) observational data collected from moving surveys. Therefore, we consider that the actual ocean has mixed spatiotemporal heterogeneities in SSS, comprising both shortterm and shortwavelength heterogeneities, and longterm and longwavelength heterogeneities. Accordingly, the static GNSSA positioning has successfully modeled the longterm and longwavelength heterogeneity of the SSS via moving surveys, while the kinematic GNSSA positioning method has faced serious modeling errors due to the shortterm and shortwavelength heterogeneities in the SSS, especially for the detection of precise horizontal array displacements. Since precise modeling of the shortterm and shortwavelength heterogeneity of the SSS is important not only to improve accuracy of the kinematic GNSSA positioning but also to further improve precision of the static GNSSA positioning, further developments of SSS modeling techniques are required. The simplest way to model such shortwavelength heterogeneities in the SSS is to increase the number of seafloor transponders and the number of seasurface platforms although the financial costs would be high. Otherwise, it would be helpful for precise GNSSA positioning to understand the spatiotemporal behavior of the internal gravity wave. The internal wave has been considered to be a source of the shortterm and shortwavelength heterogeneity in the SSS (e.g., Spiess et al. 1998). However, the detailed characteristics of the internal waves (such as spatial extents of the internal waves) have not been well investigated. Such oceanographic information would improve the design of the observation model and/or the system model for the EKF formulation, and would reduce the systematic modeling errors.
Conclusion
In this study, we developed the EKFbased kinematic GNSSA array positioning method, and we investigated its performance using both the synthetic data and actual observational data. The synthetic tests demonstrated the superiority of the EKFbased array positioning methods when compared with the conventional array positioning methods, for both cases assuming a laterally stratified SSS and a horizontally graded SSS. Through analyses of the observational data, we found that using the EKFbased array positioning method and assuming a laterally stratified SSS significantly improved the precision of the array displacements, especially for the vertical component. The precision of the vertical array displacements is ~ 10 cm; the longterm GNSS positioning errors on a seasurface platform and the shortterm deviation of SSS from a laterally stratified SSS might reduce the precision of the vertical array displacements and contribute to temporal fluctuations. Thus, detection of an abrupt step, such as due to a coseismic slip event, would be carried out much more precisely via the EKFbased array positioning, with precision of ~ 5 cm. Furthermore, the EKFbased array positioning method also demonstrated robust performance in the instance of unresponsive transponders. However, as for the actual observational data, the EKFbased array positioning method, assuming a horizontally graded SSS, produced apparent fluctuations in the horizontal array displacements that made the precision worse than the results obtained assuming a laterally stratified SSS. These apparent fluctuations may be caused by shortwavelength spatial heterogeneities in the SSS; therefore, modeling such heterogeneities would be important to enable precise kinematic GNSSA array positioning.
Availability of data and materials
The original data and findings of this study are available from the corresponding author upon a reasonable request.
Abbreviations
 EKF:

extended Kalman filter
 GNSS:

Global Navigation Satellite System
 GNSSA:

combination of Global Navigation Satellite System and Acoustic ranging
 KF:

Kalman filter
 NTD:

Nadir total delay
 PPP:

precise point positioning
 SSS:

sound speed structure
 WOA13:

World Ocean Atlas 2013
 ZTD:

zenith total delay
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Acknowledgements
We acknowledge Dr. Y. Ohta for his helpful comment on designing the study. We thank two anonymous reviewers and the editor Takuya Nishimura for providing us valuable comments to improve the manuscript. The figures were generated using Generic Mapping Tools software (Wessel and Smith 1998). We thank Editage (www.editage.jp) for English language editing.
Funding
This research was supported by Ministry of Education, Culture, Sports, Science and Technology (MEXT; Grant Number: 1222), Japan in the Project for “Development of GPS/Acoustic Technique” and by Japan Science and Technology Agency (JST) with Council for Science, Technology and Innovation (CSTI), Crossministerial Strategic Innovation Promotion Program (SIP) “Enhancement of societal resiliency against natural disasters”, and by Japan Society for the Promotion of Science (JSPS) KAKENHI (Grant Number: 17J026523).
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FT analyzed all the calculations shown in this study and drafted the manuscript including figure. MK, CH, and RM contributed to discussion and interpretation, and they improved the developed technique and the manuscript. All authors read and approved the final manuscript.
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Correspondence to Fumiaki Tomita.
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Keywords
 GNSSA positioning
 Seafloor geodesy
 Extended Kalman filter
 Realtime positioning