A data assimilation framework for datadriven flow models enabled by motion tomography
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Abstract
Autonomous underwater vehicles (AUVs) have become central to data collection for scientific and monitoring missions in the coastal and global oceans. To provide immediate navigational support for AUVs, computational datadriven flow models described as generic environmental models (GEMs) construct a map of the environment around AUVs. This paper proposes a data assimilation framework for the GEM to update the map using data collected by the AUVs. Unlike Eulerian data, Lagrangian data along the AUV trajectory carry timeintegrated flow information. To facilitate assimilation of Lagrangian data into the GEM, the motion tomography method is employed to convert Lagrangian data of AUVs into an Eulerian spatial map of a flow field. This process allows assimilation of both Eulerian and Lagrangian data into the GEM to be incorporated in a unified framework, which introduces a nonlinear filtering problem. Considering potential complementarity of Eulerian and Lagrangian data in estimating spatial and temporal characteristics of flow, we develop a filtering method for estimation of the spatial and temporal parameters in the GEM. The observability is analyzed to verify the convergence of our filtering method. The proposed data assimilation framework for the GEM is demonstrated through simulations using two flow fields with different characteristics: (i) a doublegyre flow field and (ii) a flow field constructed by using real ocean surface flow observations from highfrequency radar.
Keywords
Motion tomography Data assimilation Generic environmental models1 Introduction
Autonomous underwater vehicles (AUVs) are proven versatile instruments for ocean sampling and monitoring (Curtin et al. 1993; Fratantoni and Haddock 2009; Leonard et al. 2010). However, environmental changes such as ocean currents often hinder the survey performance of AUVs. To cope with environmental changes, an AUV can employ environmental models (e.g., Smith et al. 2010; Chang et al. 2015 for path planning and Song and Mohseni 2017 for localization). For geophysical flow, typical regional ocean models (e.g., Luettich et al. 1992; Bleck 2002; Shchepetkin and McWilliams 2005; Haidvogel et al. 2008) numerically solve partial differential equations (PDEs) for a large spatial area with longterm prediction. However, their high complexity makes solving these PDEs computationally expensive. To reduce the computational burden, regional ocean models typically solve these PDEs with large spatial and temporal resolutions that may not be suitable for AUV planning and control. Datadriven flow models (Petrich et al. 2009; Mokhasi et al. 2009; Chang et al. 2014) can provide high spatial and temporal resolutions computationally fast for a small spatial area with shortterm prediction. Hence, datadriven flow models are suitable for local guidance of AUVs.
Extending our previous work (Chang and Zhang 2016b), this paper focuses on an estimation problem for datadriven computational flow models described as generic environmental models (GEMs) (Zhang 2016). After initialized by using prior information of the environment, possibly obtained from geophysical ocean models, the GEM constructs a map of the environment to provide immediate navigational support for AUVs. When a flow model is involved in AUV navigation, the performance of AUV navigation may, to a large extent, rely on the accuracy of the model. In Szwaykowska (2014), Szwaykowska and Zhang (2018), the authors analyze the influence of the geophysical ocean model on AUV navigation and demonstrate that the error growth rate of the vehicle position depends on the accuracy and resolution of the model. This concept can also be extended to datadriven models. To improve the model accuracy, observation data can be incorporated into the estimation of the ocean state or parameters in the model, which is generally known as data assimilation (Robinson and Lermusiaux 2008).
Observation data incorporated in data assimilation can be categorized into Eulerian and Lagrangian. Eulerian data are framed with a fixed grid in space and time, and Lagrangian data are obtained while moving with the flow over time. Since most flow models are Eulerian, assimilation of Eulerian data into the model is relatively straightforward. However, Lagrangian observations such as data collected by AUVs carry timeintegrated information along the trajectory and have both direct and indirect connections with the state variables and observation noise (e.g., observation model error and sensor noise) of flow models (Kuznetsov et al. 2003). Therefore, assimilation of Lagrangian data raises difficulties and requires an additional framework in comparison to Eulerian data assimilation.
For geophysical ocean models, Lagrangian data assimilation mostly employs passive drifters or floats. In Kuznetsov et al. (2003), a method for Lagrangian data assimilation is developed by using the augmented state that combines the Eulerian ocean state with the positions of Lagrangian observations traced by using advection equations. Another method introduced in Molcard (2003) computes Lagrangian velocity as the difference between the positions of a drifter over the position update interval. Then, this velocity is interpolated to be assimilated into a flow model. However, since AUVs are not passive but ‘controlled’, these methods using the positions of passive drifters may not be easily applicable to datadriven flow models combined with observations from AUVs. In addition, data assimilation for geophysical ocean models with high complexity is computationally expensive. In Hackbarth et al. (2014), flow field estimation and data assimilation have been studied for AUV path planning using a computational fluid dynamics model and flow measurements from AUVs on the fly. However, AUVs typically have a limited capability of flow measurement. In this paper, we present a framework of computationally efficient data assimilation for datadriven flow models using AUVs with no direct flow measurement and limited underwater localization capabilities.
To facilitate assimilation of Lagrangian data collected by AUVs into the GEM, we employ the motion tomography (MT) method (Wu et al. 2013; Chang et al. 2016, 2017). Along the trajectory of a vehicle, its motion is typically perturbed by ambient flow, deviating the actual trajectory of the vehicle from its predicted trajectory. By deriving an expression of this deviation using a vehicle motion model, MT constructs a system of equations that describes the influence of flow on the vehicle trajectory. Through solving this system of equations, MT converts Lagrangian data into an Eulerian spatial map of a flow field, allowing Lagrangian data to be incorporated together with Eulerian data into a unified data assimilation framework. Note that instead of direct flow measurements, our method assimilates an underlying flow field estimated by MT from timeintegrated flow information carried in the vehicle trajectory. Extending our previous work (Chang and Zhang 2016a), MT in this paper is formulated in both space and time so that timeintegrated flow information along the trajectory can be inferred in both space and time. This formulation enables the spatial map of a flow field constructed by MT to account for the temporal variability. To the best of our knowledge, this paper is the first work on a unified data assimilation framework for datadriven flow models that incorporates both Eulerian data and Lagrangian data collected by AUVs. This framework may also be a helpful analysis tool for geophysical ocean models dealing with Lagrangian data collected by AUVs.
In this paper, the GEM is constructed by using spatial and temporal basis functions along with their corresponding parameters. Assimilation of both Eulerian and Lagrangian data into the GEM through the unified framework involves a nonlinear filtering problem to estimate the spatial and temporal parameters of the GEM. We design a filtering method in which the spatial and temporal parameters of the GEM are individually estimated through assimilation of Lagrangian and Eulerian data, respectively. We analyze the observability to check the convergence of the proposed filtering method. The GEM combined with the proposed data assimilation framework can provide effective navigational support for AUVs in real time. To demonstrate its capability for AUV guidance, we compare three GEMs assimilating (i) both Eulerian and Lagrangian data, (ii) only Eulerian data, and (iii) only Lagrangian data, respectively. The three GEMs are used to guide simulated stationkeeping vehicles in the following two flow fields with different characteristics: (i) a doublegyre flow field and (ii) a flow field constructed by using real ocean surface flow observations from highfrequency (HF) radar. The differences in the performance of the three GEMs show that the GEM combined with the proposed unified framework can favorably account for the spatial and temporal variations of flow by assimilating both Eulerian and Lagrangian data.
The rest of the paper is organized as follows. Section 2 presents datadriven flow modeling with emphasis on its application to the GEM for the guidance of AUVs. Section 3 provides an overview of MT to describe how an Eulerian spatial map is constructed from Lagrangian data collected by AUVs. Section 4 introduces a data assimilation method for the GEM in which assimilation of both Eulerian and Lagrangian data is achieved in a unified framework to estimate the spatial and temporal parameters of the GEM. Section 5 validates our method through simulations and analyzes the results. Lastly, Sect. 6 concludes the paper with future directions.
2 Datadriven flow modeling
Flow field modeling and estimation have been studied in oceanography, fluid dynamics, and marine robotics. Typical regional ocean models for geophysical flow have high computational costs and coarse spatial and temporal resolutions (e.g., on the order of kilometers and output on the order of hours to days), which are not suitable for AUV planning and control. By compromising detailed physical insights, datadriven flow models obtain flow estimates computationally faster and in higher resolution than PDEbased geophysical ocean models. This section presents a datadriven approach for flow modeling for the guidance of AUVs.
2.1 Data collection of flow
In this section, we describe Eulerian and Lagrangian representations of flow and present typical Eulerian and Lagrangian data sources that can be used for datadriven flow models.
2.1.1 Eulerian and Lagrangian representations of flow
2.1.2 Data sources of flow
This section reviews the following three typical data sources: moorings, HF radar systems, and AUVs. These data sources have different complexities and ranges, and their observation data can be complementary to understand the spatial and temporal characteristics of flow.
A mooring is a stationary oceanographic instrumentation anchored on the sea floor with a collection of sensors. A mooring with the current meter can measure ocean currents typically on an hourly basis at different depths from near the sea surface to near the sea bottom. Although a mooring provides Eulerian data on fast time scales that are useful for time series analysis and interpretation, since it is fixed at one location, its flow data provide insufficient spatial variability for AUVs unless a set of moorings are installed in a dense network or array.
HF radar systems (Paduan et al. 1995; Gurgel et al. 1999; Shay et al. 2007) are shorebased remote sensing systems that use overthehorizon technology to measure surface ocean currents, and are widely used in coastal ocean observing to provide near realtime observations. Shore stations of HF radar emit radio signals that bounce off surface waves and return to the receiver. The received radio signals are used to compute ocean surface current movement relative to ocean surface wave movement and the surface current observation is typically processed by averaging 20 min to 1 h timeseries. The spatial and temporal resolutions of the HF radar systems depend on the signal frequency and configurations of the system. Most HF radar systems provide Eulerian data hourly with a typical spatial resolution of \(3\times 3\) or \(6\times 6\ \mathrm{km}^{2}\) for a large spatial coverage. However, the spatial resolution is considered low for AUV guidance and the data from the HF radar national network data server (http://cordc.ucsd.edu/projects/mapping/) require a 3 h processing delay.
AUVs are an important tool for adaptive ocean sampling and can be used to collect transect data in horizontal and vertical spaces. AUVs typically have limited localization capabilities while underwater (Zhang et al. 2015), so their positions are only available when they are at the surface of water. Therefore, AUVs in general follow predicted trajectories generated prior to diving into the water. While traveling underwater, the motion of an AUV is perturbed by ambient flow, deviating its actual trajectory from its predicted trajectory. This deviation accumulates along the vehicle trajectory and provides important Lagrangian information of flow. For example, the deadreckoning method of underwater gliders employs flow estimates computed from this deviation.
2.2 Generic environmental models
The GEM is a class of datadriven computational models that estimate a map of the environment using the data collected by AUVs to support their navigation (Zhang 2016). The GEM is named “generic” in the sense that it is independent of boundary conditions that are required to compute the states of typical PDEbased models. This section introduces the structure of the GEM and the modeling of coastal ocean flow as an example of the GEM.
2.2.1 Model structure
We consider 2dimensional flow fields for simplicity of presentation. Let subscripts x and y denote the x and y components in \({\mathbb {R}}^{2}\) and define a flow vector as \({\mathbf{f }}=[f_{x},f_{y}]^{T}\in {\mathbb {R}}^{2}\). We assume that spatial and temporal characteristics of flow can be approximated by a series of spatial and temporal basis functions. Let us denote the time by \(t\in {\mathbb {R}}\) and the position by \({\mathbf{r }}\in {\mathbb {R}}^{2}\). For position \({\mathbf{r }}\), we define a series of spatial basis functions indexed by m as \(\phi _{m}({\mathbf{r }})\in {\mathbb {R}}\) to approximate spatial variation of flow. For time t, a series of temporal basis functions indexed by n is given by \(\psi _{n}(t)\in {\mathbb {R}}\) to approximate temporal variation of flow.
The basis functions of the GEM can be constructed by using prior information of the environment obtained from existing ocean models or historic observation data and its parameters can be trained on nowcast or forecast data from the ocean models. Ocean models are typically updated on a daily basis. However, to support AUV navigation in dynamic environments, the model parameters should be updated more frequently. During the time between the ocean model updates, one may use other observations such as data from AUVs or moorings to update the parameters of the GEM. This process can be considered as simplified data assimilation for datadriven flow models. Details of this data assimilation process for (1) is discussed in Sect. 4.
2.2.2 An example of the GEM
3 Motion tomography: Eulerian flow mapping from Lagrangian data
To facilitate assimilation of Lagrangian data collected by AUVs into the GEM, we convert the data into an Eulerian spatial map through MT so that Lagrangian data assimilation can be achieved together with Eulerian data assimilation in a unified framework. In this section, we first study how the flow field affects the horizontal motion of AUVs. Then, by exploiting multiple AUVs traveling in a domain, MT is formulated with spatiotemporal discretization, allowing to account for both spatial and temporal variability of flow. Lastly, the flow field estimation problem through MT and its parameterization are introduced.
3.1 Horizontal motion of AUVs under flow
To model the influence of flow on the vehicle trajectory, we consider the horizontal motion of AUVs in the presence of flow. The horizontal vehicle motion is modeled using a firstorder particle model with constant speed \(s_{\mathrm {h}}\) and vehicle heading \(\theta\). Let us denote the time by \(t\in {\mathbb {R}}\) and the position by \({\mathbf{r }}\in {\mathbb {R}}^{2}\). Let \({\mathcal {T}}=[t^{0},t^{f}]\) be a bounded time interval called the observation interval. Let us consider that a vehicle travels using constant flow prediction \(\hat{{\mathbf{f }}}\) during the observation interval. Suppose that the actual position \({\mathbf{r }}\) of a vehicle is available only at the beginning and ending times of its travel, \(t=t^{0}\) and \(t=t^{f}\), respectively.
The motionintegration error has been used to estimate a flow field in the vicinity of an AUV that does not measure ambient flow directly. For example, the underwater glider, a buoyancydriven AUV (Davis et al. 2002), computes a spatially and temporally averaged flow estimate from the motionintegration error along the vehicle trajectory traveled over one subsurface interval (Merckelbach et al. 2008). This method is very efficient in computation, and the glider incorporates this estimate into navigation to reduce the motionintegration error for the next subsurface interval. The work Petrich et al. (2009) presents a similar way of estimating a flow velocity to identify model parameters for a timeinvariant flow field. However, the effectiveness of this method significantly degrades in the presence of flow with strong spatial and temporal variations (Chang et al. 2015).
3.2 Formulation of MT
To construct the flow vector, let us denote the flow velocity in grid cell \(D_{j}\) for time interval \({\mathcal {T}}_{\tau }\) by \({\mathbf{f }}_{(j,\tau )} = {\mathbf{f }}({\mathbf{r }},t)\), \({\mathbf{r }}\in D_{j}\), \(t\in t^{\tau }\) and assume \({\mathbf{f }}_{(j,\tau )}\) is constant within \((D_{j},{\mathcal {T}}_{\tau })\). Let us stack flow velocities for all the spatiotemporal intervals and define vector \(\overline{{\mathbf{f }}} = [\overline{{\mathbf{f }}}_{x}^{T},\overline{{\mathbf{f }}}_{y}^{T}]^{T}\), where \(\overline{{\mathbf{f }}}_{x}=[\dots ,f_{x,(P,\tau 1)},f_{x,(1,\tau )},\dots ,f_{x,(P,\tau )},f_{x,(1,\tau +1)},\dots ]^{T}\) and \(\overline{{\mathbf{f }}}_{y}=[\dots ,f_{y,(P,\tau 1)},f_{y,(1,\tau )},\dots ,f_{y,(P,\tau )}, f_{y,(1,\tau +1)},\dots ]^{T}\) are the x and y components of \(\overline{{\mathbf{f }}}\), respectively. Assuming that vehicle heading \(\theta _{(j,\tau )}^{i}\) of the ith vehicle within \((D_{j},{\mathcal {T}}_{\tau })\) is constant, this flow setting constructs the linear trajectory in \((D_{j},{\mathcal {T}}_{\tau })\), leading to the piecewise linear trajectory over \(({\mathcal {D}},{\mathcal {T}})\). We assume that vehicle heading \(\theta _{(j,\tau )}^{i}\) can be measured by a compass or estimated with small bounded error. The influence of the vehicle heading error on MT is analyzed in our previous work (Chang et al. 2017).
3.3 Flow field mapping
To solve the equations in (19) for flow \(\overline{{\mathbf{f }}}\), we first need to determine \(\mathbf{L }(\overline{{\mathbf{f }}})\) based on the knowledge of vehicle trajectories. However, because of limited localization capabilities of AUVs, their underwater trajectories are often unknown and thus must be estimated before solving (19). Therefore, flow field mapping through MT (see Algorithm 1) is achieved through an iterative process consisting of two key steps: trajectory tracing and flow field estimation. In this section, we introduce flow field estimation and refer to Chang et al. (2017) for details about trajectory tracing.
3.3.1 Flow field estimation
To solve the underdetermined nonlinear system of equations in (19), our previous work (Wu et al. 2013; Chang et al. 2016, 2017) developed a method by extending the Kaczmarz method (Kaczmarz 1937, 1993). The Kaczmarz method, also known as the algebraic reconstruction technique (Gordon et al. 1970), iteratively solves an underdetermined linear system of equations for computerized tomography (Kak and Malcom 2001; Natterer 1986; Cierniak 2011) in medical imaging. As its extension, our method linearizes the system of equations associated with MT at each iteration and solves the linearized system of equations.
3.3.2 Parameterization
Since we discretize (15) in both space and time, the dimension of the solution space for \(\overline{{\mathbf{f }}}_{x}\) and \(\overline{{\mathbf{f }}}_{y}\) in (19) may become very high. In addition, we may not have enough trajectory information to estimate full solution, causing the solution variable to be sparse. To address these issues, instead of solving the system of equations in (22) directly, we parameterize (22) using (1) and solve the parameterized system of equations. That is, we estimate the parameters of the model first and then reconstruct a flow field. By using a sufficiently small number of basis functions associated with parameters to approximate the flow field, we can reduce the dimension of the solution space.
Based on the above two alternating optimization problems, we obtain an iterative algorithm for flow model parameter estimation (Algorithm 2). In the algorithm, the updating equations include relaxation parameter \(\lambda ^{k}\) which affects the convergence rate. To check the convergence, for k that satisfies \(\bmod (k,K) + 1 = K\), we first reconstruct \(\overline{{\mathbf{f }}}^{k}\) using the estimated parameters, \(\varvec{\rho }^{k}\) and \(\varvec{\eta }^{k}\). Then, we stop the algorithm if \(\Vert \mathbf{L }(\overline{{\mathbf{f }}}^{k})\overline{{\mathbf{f }}}_{x}^{k}  \overline{\mathbf{d }}_{x}\Vert\) and \(\Vert \mathbf{L }(\overline{{\mathbf{f }}}^{k})\overline{{\mathbf{f }}}_{y}^{k}  \overline{\mathbf{d }}_{y}\Vert\) are sufficiently small (i.e., below a threshold \(\epsilon _{\overline{{\mathbf{f }}}}\)).
In the next section, we present assimilation of Lagrangian data, converted into a spatial map of a flow field by MT, into the GEM. Note that flow field estimation (Algorithm 2) in MT is deterministic. Hence, by incorporating Lagrangian data into the data assimilation framework, we can reduce the uncertainties in Lagrangian observation data.
4 Data assimilation
Remark 1
Assimilation of Eulerian and Lagrangian data into the GEM helps to reduce uncertainties in the noisy data. For such information fusion to be achieved in a unified framework of data assimilation, we should have a clear understanding of the covariance of noise in both data. Since Eulerian data are collected at a fixed location, the covariance of noise in Eulerian data can be obtained from historic data or geophysical ocean models. However, since Lagrangian data are collected from mobile sensors, the covariance of noise in Lagrangian data is difficult to identify. We assume that the noise in the field is spatially uniform with respect to covariance and the noise in Lagrangian data is ergodic with respect to covariance. Then, the covariance of noise in Lagrangian data can be obtained by computing the covariance of noise in Eulerian data averaged over time at a fixed location. We also consider local correlation of noise covariance in the field. We assume that each grid cell in the Eulerian spatial map converted from Lagrangian data is identically correlated with its neighboring grid cells with respect to noise covariance. That is, noise covariance matrix \({\mathbf{R}}_{k}\) is constructed such that \({\mathbf{R}}_{k,(p,p)} = {\mathbb {E}}[\omega _{k,p}\omega _{k,p}^{T}] = \omega\) and \({\mathbf{R}}_{k,(p,q)} = {\mathbb {E}}[\omega _{k,p}\omega _{k,q}^{T}] = \omega\) if the pth and qth grid cells are neighboring where \(p=\{1,\dots ,P\}\), \(q=\{1,\dots ,P\}\), and \(\omega\) is constant.
Because of the nonlinear observation equation, parameter estimation for the system (29) and (31) becomes a nonlinear filtering problem. To deal with this problem, we decompose it into two linear subfiltering problems for (i) spatial parameter estimation and (ii) temporal parameter estimation. For each subfiltering problem, we fix either set of spatial or temporal parameters and estimate the other set of parameters. Because of their different spatial and temporal sampling scales, Eulerian data better account for temporal variability of a flow field while Lagrangian data better for spatial variability. Therefore, we use Eulerian data to update temporal parameters and Lagrangian data to update spatial parameters. This decomposition of the nonlinear filtering problem is valid provided that the spatial and temporal characteristics of flow can be individually estimated from Lagrangian and Eulerian data, respectively. In the following sections, we present our filtering method and analyze the observability to verify the convergence of the derived filters.
4.1 Temporal and spatial parameter estimation
As described in the previous section, Eulerian and Lagrangian data are provided on different time scales, leading to different time scales for Eulerian and Lagrangian data assimilation (i.e., every time step for Eulerian data and every \(\alpha\)th time step for Lagrangian data). Suppose Eulerian data are available at time step k and the latest estimates of the spatial parameters were computed at time step \(\lfloor k/\alpha \rfloor \alpha\). We fix all the spatial parameters \(\varvec{\eta }_{k}\) using their latest estimates \(\varvec{\eta }_{\lfloor k/\alpha \rfloor \alpha }\) and estimate the temporal parameters \(\varvec{\rho }_{k}\). Now, in state vector \(\varvec{\varTheta }\), \(\varvec{\rho }\) is the only unknown, formulating the filtering problem that is linear in \(\varvec{\rho }\).
Suppose Lagrangian data are available at the kth step. Since Lagrangian data are collected over relatively a long time period, the temporal scale of Lagrangian data does not match that of Eulerian data. To resolve this issue, we construct an Eulerian spatial map for time step k by using the estimated parameters through MT using Algorithm 2 and assimilate the constructed map into the GEM. Suppose that the latest estimates for the temporal parameters were computed at the \((k1)\)th step. In contrast to temporal parameter estimation, we fix all the temporal parameters \(\varvec{\rho }\) using their latest estimates \(\varvec{\rho }_{k1}\). Then, the subsequent filtering problem is linear in \(\varvec{\eta }\).
4.2 Observability analysis
Given a system with no control input, a Kalman filter converges if the system is uniformly completely observable (see Jazwinski 1970). We redefine uniform complete observability in Jazwinski (1970) as follows.
Definition 1
Lemma 1
Let nonzero vectors\(\mathbf{u }_{i}\in {\mathbb {R}}^{n}\), \(i\in \{1,\dots ,n\}\)be linearly independent. Then, \(\varvec{{\mathcal {M}}} = \sum _{i=1}^{n}\mathbf{u }_{i}\mathbf{u }_{i}^{T}\in {\mathbb {R}}^{n\times n}\)has full rank.
Proof
Consider nonzero vector \(\mathbf{v }_{1}\in \mathrm {span}\{\mathbf{u }_{2},\dots ,\mathbf{u }_{n}\}^{\perp }\). Then, \(\varvec{{\mathcal {M}}}\mathbf{v }_{1} = (\mathbf{u }_{1}\mathbf{u }_{1}^{T})\mathbf{v }_{1} = \mathbf{u }_{1}(\mathbf{u }_{1}^{T}\mathbf{v }_{1})\) is a nonzero scalar multiple of \(\mathbf{u }_{1}\). Similarly, for \(\mathbf{v }_{i}\in \mathrm {span}\{\mathbf{u }_{j}\}^{\perp }_{j\ne i}\), \(\varvec{{\mathcal {M}}}\mathbf{v }_{i}\) is a nonzero scalar multiple of \(\mathbf{u }_{i}\). In other words, nonzero scalar multiples of each \(\mathbf{u }_{i}\) are in the range of \(\varvec{{\mathcal {M}}}\) and the dimension of the range of \(\varvec{{\mathcal {M}}}\) is n, which is equivalent to \(\varvec{{\mathcal {M}}}\) having full rank (c.f., Horn and Johnson 1985, p. 13). \(\square\)
In the following theorem, we prove uniform complete observability for spatial parameter estimation:
Theorem 1
 (Cd1)

The matrix\({\mathbf{R}}_{j}^{\eta }\)is uniformly bounded for allj (i.e., \(\beta _{3}I_{(P+1)\times (P+1)}\preccurlyeq \mathbf{R}_{j}^{\eta }\preccurlyeq \beta _{4}I_{(P+1)\times (P+1)}\)for some constants\(\beta _{3}, \beta _{4} > 0\).
 (Cd2)

Among\(\varvec{\varPhi }(\cdot )\)’s evaluated at the position of the mooring, \({\mathbf{r }}^{E}\), and the positions of the grid cells, \({\mathbf{r }}_{i}^{L}\), \(i \in \{1,\dots ,P\}\), at leastMnumber of\(\varvec{\varPhi }(\cdot )\)’s are linearly independent.
Proof
Remark 2
Condition (Cd1) represents that the covariance of noise is uniformly bounded. Note that observation \(\mathbf{z }\) is obtained from a stationary sensor and the MT method. By assuming that the stationary sensor is reliable and the parameter estimation algorithm in MT is convergent, (Cd1) can be satisfied. Condition (Cd2) can be satisfied by choosing spatial basis functions appropriately. Consider Gaussian RBFs indexed by m, \(\phi _{m}({\mathbf{r }}) = \exp \left( \frac{\Vert {\mathbf{r }}{\mathbf{c }}_{m}\Vert }{2\sigma ^{2}}\right)\), where \({\mathbf{c }}_{m}\) is the center and \(\sigma\) is the width. If we use Gaussian RBFs as spatial basis functions for \(\varvec{\varPhi }({\mathbf{r }}) = [\dots ,\phi _{m}({\mathbf{r }}),\dots ]^{T}\), \(m\in \{1,\dots ,M\}\), (Cd2) can be satisfied by choosing M number of different centers.
In the following theorem, we prove uniform complete observability for temporal parameter estimation:
Theorem 2
Proof
Remark 3
Even though the system in (32) and (33) and the system in (34) and (35) are uniformly completely observable, the convergence rate and accuracy of the derived filters may depend on how observable the systems are. For this problem, we can incorporate observability metrics such as the local unobservability index and the local estimation condition, introduced in Krener and Ide (2009), that evaluate the degree of observability or unobservability.
5 Simulation results
This section validates the proposed method by simulating one stationary and multiple mobile sensors that collect data in (i) a doublegyre flow field and (ii) a flow field constructed by real ocean surface flow observations from HF radar. Compared to a doublegyre flow field which is relatively smooth and has a periodic spatiotemporal pattern, an ocean surface flow field observed by HF radar is nonsmooth and erratic. For the construction of the GEM and the parameterization of MT, we use the flow model in (1). The collected Eulerian and Lagrangian data are assimilated into the GEM through the proposed unified data assimilation framework. To show that the performance of the proposed method combined with the GEM, we compare three implementations of the GEM through AUV guidance: (i) GEMEL, assimilating both Eulerian and Lagrangian data, (ii) GEME, assimilating Eulerian data only, and (iii) GEML, assimilating Lagrangian data only.
5.1 GEM in simulated doublegyre flow
For the flow model in the GEM, the domain is discretized into \(6\times 6\) grid cells. Among the grid points, the centers of 4 Gaussian RBFs are determined by a Kmeans clustering algorithm to construct the spatial basis functions of the GEM. To determine the width for the RBFs, we first compute the distances between each individual center and other centers, and choose the minimum distance for each center. Then, compute an average of all the minimum distances associated with the 4 centers and divide the average by 2 (i.e., \(\sigma = \frac{\sum _{m=1}^{M}\min _{i\in m}\{\Vert {\mathbf{c }}_{i}{\mathbf{c }}_{m}\Vert \}}{2M}\), \(m\in \{1,\dots ,M\}\) where \(M=4\) in the simulation). The temporal basis functions of the GEM are initialized such that the tidal flow consists of only the tidal residual flow which is constant and the nontidal flow is approximated by using 0th to 4th Laguerre polynomials with time scaling factor \(\zeta =\frac{1}{10}\).
In data assimilation, one time step corresponds to 1 h and the spatial and temporal parameters of the GEM are initialized as \([0.01,\dots ,0.01]^{T}\) for both x and y components at the beginning. To apply the proposed filters for data assimilation, we let \(\mathbf{A }_{k}=I\), \(\mathbf{Q }_{k}^{\rho } = 10^{4}I_{(L+2N+2)\times (L+2N+2)}\), and \(\mathbf{Q }_{k}^{\eta } = 10^{4}I_{M\times M}\) for the system (29) and (31). The noise covariance is constructed such that for the pth grid cell, \({\mathbf{R}}_{k,(p,p)} = 10^{2}\) and \({\mathbf{R}}_{k,(p,q)} = 10^{2}\) if the pth and qth grid cells are neighboring. By assimilating Eulerian data, the temporal parameters of the GEM are updated at each time step.
For Lagrangian data, the two vehicles always attempt to crisscross around the mooring by heading towards the mooring, but to make a wide spatial coverage to capture the spatial variability of flow, the actual heading is adjusted by adding \(18^{\circ }\) to the heading straight towards the mooring. The Lagrangian data collected over 24 h are converted by MT in a batch into an Eulerian spatial map. For the spatial map, we select the rectangle domain that can enclose the vehicle positions observed at the surface over 24 h. Since each vehicle generates 6 trajectories over 24 h, we discretize the spatial domain into \(6\times 6\) spatial grid cells. The spatial basis functions of MT are constructed using the centers and width of RBFs chosen for the spatial basis functions of the GEM. We also discretize the 24 h temporal domain into 3 temporal grid cells, but to estimate the most uptodate temporal parameters, we construct the temporal basis functions of MT for the latest time step.
For MT, we need the initial guesses of the spatial and temporal parameters. We first compute an estimate of constant flow velocity over an individual vehicle trajectory during one observation interval of 4 h in a similar way that a glider computes its depthaveraged flow velocity (see Chang et al. 2017 for details). Then, for the grid cells that the trajectory of each vehicle passes through, we assign the flow estimate corresponding to the trajectory and construct an initial guess of the underlying flow field. Then, based on this initial guess of the flow field, we compute the initial guesses of parameters using a nonlinear least squares method using (1). Since a small number of AUVs is collecting Lagrangian data for MT, we lack information to solve the system of equations associated with MT. Therefore, we run Algorithm 2 up to 500 iterations with a convergence threshold of \(\epsilon _{\overline{{\mathbf{f }}}}=300\) m and a relaxation parameter of \(\lambda =0.01\), and choose the estimates with the lowest error. Along with Algorithm 2, we run trajectory tracing with a convergence threshold of \(\epsilon _{\gamma }=10\) m up to 5 times. By combining estimated parameters of MT with the basis functions of MT, an Eulerian spatial map is constructed. Then, this spatial map is assimilated into the GEM to update the spatial parameters of the GEM.
Remark 4
Instead of multiple vehicles, we can also use a single vehicle for MT. In this case, we may not have sufficient information of vehicle trajectories for MT. Therefore, estimation of the spatial and temporal parameters of the nonlinear flow model (1) through MT can be challenging. As a workaround, we can fix the temporal parameters of MT using the temporal parameters of the GEM and use MT to estimate the spatial parameters of MT only. Then, an Eulerian spatial map can be constructed by using the temporal parameters of the GEM and the spatial parameters of MT.
To demonstrate the proposed data assimilation framework combined with the GEM, we simulate 3 stationkeeping vehicles guided by GEMEL, GEME, and GEML, respectively. We assume that the vehicles are capable of canceling out the flow, i.e., the vehicle can always move in the opposite direction of flow at the same speed of flow. From each time step until the next, the simulated flow field for the stationkeeping vehicles is assumed constant with the flow velocity evaluated at the time between the current and next time steps. Then, starting from \((x,y)=(5000,5000)\) in the domain, the vehicles predict flow using their own GEMs and attempt to maintain their current positions.
RMS errors for stationkeeping vehicles guided by GEMs under doublegyre flow
GEM for guidance  EL  E  L 

RMSE (m)  145.3524  244.3035  217.2148 
5.2 GEM in real flow observed by HF radar
To simulate the flow field, we use historic HF radar data that are postprocessed for our convenience with \(3\times 15\) km\(^{2}\) spatial resolution and 30 min temporal resolution. For the region of the AUV and mooring deployment, we choose the region centered at \((80.0240^{\circ },31.3261^{\circ })\) being \((x,y)=(0,0)\) where we deploy a mooring. As is done in the previous section, we construct the spatial domain whose size is \([2520,2520]\times [2520,2520]\) m\(^{2}\) around the mooring. We discretize the domain into \(6\times 6\) spatial grid cells whose grid points are used to compute the centers of 4 Gaussian RBFs using a Kmeans clustering algorithm. Then, we determine the width of the RBFs in the same way as in the previous section and construct the spatial basis functions of the GEM. This region is close to the Gulf Stream current which causes strong spatial and temporal variability in the region (see Fig. 9). Since the flow in this region is strong compared to the horizontal throughwater speed 0.35 m/s of the AUV, we construct a simulated flow field by using a half magnitude of the actual flow velocity. See Fig. 2 for the actual flow velocity observed by HF radar and Fig. 10 for the reduced flow velocity at the location of the mooring.
To approximate the temporal variation of the flow, we first decompose ocean flow into tidal and nontidal components using the T_Tide MATLAB^{®} toolbox (Pawlowicz et al. 2002). Since the tidal component of ocean flow can be described by a superposition of multiple tidal constituents characterized by magnitude and phase, we use a series of sinusoidal basis functions. Using 31day historical data from January 1, 2012 00:23 to January 31, 2012 23:53 as the initialization data set, we identify the three major (M2, N2, and S2) tidal constituents for tidal flow. In contrast to the doublegyre flow in the previous section, real flow off the coast of Georgia contains very complex nontidal and nonperiodic components of flow (see Fig. 10). Therefore, nontidal flow is approximated by using 0th to 14th weighted Laguerre polynomials as temporal basis functions with time scaling factor \(\zeta =\frac{1}{20}\). Based on the above tidal and nontidal approximations, we construct the temporal basis functions of the GEM.
As is the case in the previous section, a time step for data assimilation corresponds to 1 h and the spatial and temporal parameters of the GEM are initialized as \([0.01,\dots ,0.01]^{T}\) for both x and y components before data assimilation begins. To apply the proposed filters for data assimilation, we let \(\mathbf{A }_{k}=I\), \(\mathbf{Q }_{k}^{\rho } = 10^{4}I_{(L+2N+2)\times (L+2N+2)}\), and \(\mathbf{Q }_{k}^{\eta } = 10^{4}I_{M\times M}\) for the system (29) and (31). The noise covariance is constructed such that for the pth grid cell, \({\mathbf{R}}_{k,(p,p)} = 10^{2}\) and \({\mathbf{R}}_{k,(p,q)} = 10^{2}\) if the pth and qth grid cells are neighboring. By assimilating Eulerian data, the temporal parameters of the GEM are updated at each time step.
Since the flow in this region is very dynamic and complex, we have 3 AUVs crisscrossing around the mooring to collect data. The heading of the vehicles is always towards the mooring with \(18^{\circ }\) adjustment. The flow field simulated for the GEM and MT is the segment of HF radar data next to the period of the initialization data set. However, please note that the initialization data set is used only to identify the major tidal constituents for the domain. Based on Lagrangian data collected over 24 h, we first construct the spatial and temporal basis functions of MT. The spatiotemporal domain is discretized into \(6\times 6\) spatial grid cells and 3 temporal grid cells. Then, MT estimates the spatial and temporal parameters of MT using the initial guesses computed as in the previous section and construct an Eulerian spatial map for the latest time step. By assimilating this spatial map into the GEM, we update the spatial parameters of the GEM.
RMS errors for stationkeeping vehicles guided by GEMs under real surface flow observed by HF radar
GEM for guidance  EL  E  L 

RMSE (m)  270.6909  396.5027  598.1049 
The operational conditions of AUVs used in the simulation are motivated by underwater gliders which are robust sensor platforms widely used in the oceanographic community. Typical horizontal throughwater speed of underwater gliders is 0.25–0.35 m/s. As observed in this section, real flow is very complex and dynamic, and can significantly affect the navigation of AUVs, especially slowlymoving AUVs such as gliders. Therefore, data collected by gliders are not necessarily synoptic and can thus convolve spatial and temporal representations of ocean features as they are sampled. Outside of its capability for the guidance of AUVs, the data assimilation framework presented in this paper could serve as a useful tool for analysis and interpretation of data from AUVs through rapidly evolving features in oceanographic studies.
6 Conclusions and future work
This paper has presented a unified framework for assimilating both Lagrangian and Eulerian data into datadriven computational flow models described as generic environmental models (GEMs). In the framework, Lagrangian data collected by autonomous underwater vehicles (AUVs) are converted into an Eulerian spatial map through the motion tomography (MT) method. This conversion has allowed for Lagrangian data assimilation to be achieved together with Eulerian data assimilation in the unified framework. Leveraging different spatial and temporal scales of Eulerian and Lagrangian data, Eulerian data are assimilated to update the temporal parameters of the GEM and Lagrangian data to update the spatial parameters. Assimilation of both Eulerian and Lagrangian data in the unified framework leads estimation of the spatial and temporal parameters in the GEM to a nonlinear filtering problem. To solve this nonlinear problem, two linear subfilters are derived for spatial and temporal parameter estimation, respectively and to verify the convergence of the filters, the observability is analyzed. Lastly, the paper has demonstrated that the proposed data assimilation framework combined with the GEM can improve AUV navigation through simulations using a doublegyre flow field and a flow field constructed from real ocean surface flow observed by HF radar. Since geophysical ocean models incorporate Lagrangian data collected by AUVs, the proposed framework may be a helpful analysis tool for geophysical ocean models. Future work will study the design of vehicle trajectories so that we can minimize the uncertainty in the flow field constructed from Lagrangian data and maximize the accuracy of the flow model.
Notes
References
 Bishop, C.M.: Neural Networks for Pattern Recognition. Oxford University Press, Oxford (1995)zbMATHGoogle Scholar
 Bleck, R.: An oceanic general circulation model framed in hybrid isopycnicCartesian coordinates. Ocean Model 37, 55–88 (2002)CrossRefGoogle Scholar
 Chang, D., Liang, X., Wu, W., Edwards, C.R., Zhang, F.: Realtime modeling of ocean currents for navigating underwater glider sensing networks. In: Koubâa, A., Khelil, A. (eds.) Cooperative Robots and Sensor Networks. Studies in Computational Intelligence, vol. 507, pp. 61–75. Springer, Berlin (2014)CrossRefGoogle Scholar
 Chang, D., Wu, W., Edwards, C.R., Zhang, F.: Motion tomography: mapping flow fields using autonomous underwater vehicles. Int. J, Robot. Res. 36(3), 320–336 (2017)CrossRefGoogle Scholar
 Chang, D., Wu, W., Zhang, F.: Glider CT: analysis and experimental validation. In: Chong, N.Y., Cho, Y.J. (eds.) Distributed Autonomous Robotic Systems, Springer Tracts in Advanced Robotics, vol. 112, pp. 285–298. Springer, Tokyo (2016)CrossRefGoogle Scholar
 Chang, D., Zhang, F.: Distributed motion tomography for timevarying flow fields. In: OCEANS 2016Shanghai, pp. 1–7 (2016)Google Scholar
 Chang, D., Zhang, F.: Resolving temporal variations in datadriven flow models constructed by motion tomography. In: IFACPapers OnLine, vol. 49, pp. 182–187. Elsevier, Amsterdam (2016)Google Scholar
 Chang, D., Zhang, F., Edwards, C.R.: Realtime guidance of underwater gliders assisted by predictive ocean models. J. Atmos. Ocean. Technol. 32(3), 562–578 (2015)CrossRefGoogle Scholar
 Cierniak, R.: Xray Computed Tomography in Biomedical Engineering. Springer, London (2011)CrossRefGoogle Scholar
 Curtin, T.B., Bellingham, J.G., Catipovic, J., Webb, D.: Autonomous oceanographic sampling networks. Oceanography 6(3), 86–94 (1993)CrossRefGoogle Scholar
 Davis, R.E., Eriksen, C.C., Jones, C.P.: Autonomous buoyancydriven underwater gliders. In: Griffiths, G. (ed.) Technology and Applications of Autonomous Underwater Vehicles, pp. 37–58. CRC Press, Boca Raton (2002)CrossRefGoogle Scholar
 Fornberg, B., Flyer, N.: A Primer on Radial Basis Functions with Applications to the Geosciences. Society for Industrial and Applied Mathematics, Philadelphia (2015)CrossRefzbMATHGoogle Scholar
 Fratantoni, D.M., Haddock, S.H.D.: Introduction to the autonomous ocean sampling network (AOSNII) program. Deep Sea Res. Part II Top. Stud. Oceanogr. 56(3–5), 61 (2009)CrossRefGoogle Scholar
 Girosi, F., Poggio, T.: Networks and the best approximation property. Biol. Cybern. 63(3), 169–176 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
 Gordon, R., Bender, R., Herman, G.T.: Algebraic reconstruction techniques (ART) for threedimensional electron microscopy and Xray photography. J. Theor. Biol. 29(3), 471–481 (1970)CrossRefGoogle Scholar
 Gurgel, K.W., Antonischki, G., Essen, H.H., Schlick, T.: Wellen radar (WERA): a new groundwave HF radar for ocean remote sensing. Coast. Eng. 37(3–4), 219–234 (1999)CrossRefGoogle Scholar
 Hackbarth, A., Kreuzer, E., Schröder, T.: CFD in the loop: ensemble Kalman filtering with underwater mobile sensor networks. In: ASME 2014 33rd International Conference on Ocean, Offshore and Arctic Engineering, p. V002T08A063 (2014)Google Scholar
 Haidvogel, D., Arango, H., Budgell, W., Cornuelle, B., Curchitser, E., Di Lorenzo, E., Fennel, K., Geyer, W., Hermann, A., Lanerolle, L., Levin, J., McWilliams, J., Miller, A., Moore, A., Powell, T., Shchepetkin, A., Sherwood, C., Signell, R., Warner, J., Wilkin, J.: Ocean forecasting in terrainfollowing coordinates: formulation and skill assessment of the regional ocean modeling system. J. Comput. Phys. 227(7), 3595–3624 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
 Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)CrossRefzbMATHGoogle Scholar
 Israelsen, B.W., Smith, D.A.: Generalized Laguerre reduction of the Volterra kernel for practical identification of nonlinear dynamic systems. arXiv:1410.0741 [cs.LG] (2014)
 Jazwinski, A.H.: Stochastic Processes and Filtering Theory. Dover Publications, New York (1970)zbMATHGoogle Scholar
 Kaczmarz, S.: Angenäherte auflösung von systemen linearer gleichungen. Bulletin International de l’Academie Polonaise des Sciences et des Lettres 35, 355–357 (1937)zbMATHGoogle Scholar
 Kaczmarz, S.: Approximate solution of systems of linear equations. Int. J. Control. 57(6), 1269–1271 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
 Kak, A.C., Malcom, S.: Principles of Tombgraphic Imaging. Society for Industrial and Applied Mathematics, Philadelphia (2001)CrossRefGoogle Scholar
 Krener, A.J., Ide, K.: Measures of unobservability. In: Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference, pp. 6401–6406 (2009)Google Scholar
 Kumar, S.: Neural Networks: A Classroom Approach. Tata McGrawHill Education, New York (2004)Google Scholar
 Kuznetsov, L., Ide, K., Jones, C.K.R.T.: A method for assimilation of Lagrangian data. Monthly Weather Rev. 131(10), 2247–2260 (2003)CrossRefGoogle Scholar
 Leonard, N.E., Paley, D.A., Davis, R.E., Fratantoni, D.M., Lekien, F., Zhang, F.: Coordinated control of an underwater glider fleet in an adaptive ocean sampling field experiment in Monterey Bay. J. Field Robot. 27(6), 718–740 (2010)CrossRefGoogle Scholar
 Luettich, R.A., Westerink, J.J., Scheffner, N.W.: ADCIRC: an advanced threedimensional circulation model for shelves, coasts, and estuaries. Report 1. Theory and methodology of ADCIRC2DDI and ADCIRC3DL. Tech. rep., Coastal Engineering Research Center, Vicksburg, Mississippi (1992)Google Scholar
 Merckelbach, L., Briggs, R., Smeed, D., Griffiths, G.: Current measurements from autonomous underwater gliders. In: 2008 IEEE/OES 9th Working Conference on Current Measurement Technology, pp. 61–67 (2008)Google Scholar
 Mokhasi, P., Rempfer, D., Kandala, S.: Predictive flowfield estimation. Physica D Nonlinear Phenomena 238(3), 290–308 (2009)CrossRefzbMATHGoogle Scholar
 Molcard, A.: Assimilation of drifter observations for the reconstruction of the Eulerian circulation field. J. Geophys. Res. 108(C3) (2003)Google Scholar
 Natterer, F.: The Mathematics of Computerized Tomography. Society for Industrial and Applied Mathematics, Philadelphia (1986)zbMATHGoogle Scholar
 Paduan, J., Petruncio, E., Barrick, D., Lipa, B.: Surface currents within and offshore of Monterey Bay as mapped by a multiplesite HF radar (CODAR) network. In: Proceedings of the IEEE Fifth Working Conference on Current Measurement, pp. 137–142 (1995)Google Scholar
 Park, J., Sandberg, I.W.: Universal approximation using radialbasisfunction networks. Neural Comput. 3(2), 246–257 (1991)CrossRefGoogle Scholar
 Pawlowicz, R., Beardsley, B., Lentz, S.: Classical tidal harmonic analysis including error estimates in MATLAB using T\_TIDE. Comput. Geosci. 28(8), 929–937 (2002)CrossRefGoogle Scholar
 Petrich, J., Woolsey, C.A., Stilwell, D.J.: Planar flow model identification for improved navigation of small AUVs. Ocean Eng. 36(1), 119–131 (2009)CrossRefGoogle Scholar
 Robinson, A.R., Lermusiaux, P.F.: Data Assimilation in Models. Encyclopedia of Ocean Sciences, 2nd Edn., pp. 1–12 (2008)Google Scholar
 Shadden, S.C., Lekien, F., Marsden, J.E.: Definition and properties of Lagrangian coherent structures from finitetime Lyapunov exponents in twodimensional aperiodic flows. Physica D Nonlinear Phenomena 212(3–4), 271–304 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
 Shay, L.K., MartinezPedraja, J., Cook, T.M., Haus, B.K., Weisberg, R.H.: Highfrequency radar mapping of surface currents using WERA. J. Atmos. Ocean. Technol. 24(3), 484–503 (2007)CrossRefGoogle Scholar
 Shchepetkin, A.F., McWilliams, J.C.: The regional oceanic modeling system (ROMS): a splitexplicit, freesurface, topographyfollowingcoordinate oceanic model. Ocean Modell. 9(4), 347–404 (2005)CrossRefGoogle Scholar
 Smith, R.N., Pereira, A., Yi Chao, Li, P.P., Caron, D.A., Jones, B.H., Sukhatme, G.S.: Autonomous underwater vehicle trajectory design coupled with predictive ocean models: A case study. In: 2010 IEEE International Conference on Robotics and Automation, pp. 4770–4777 (2010)Google Scholar
 Song, Z., Mohseni, K.: Longterm inertial navigation aided by dynamics of flow field features. IEEE J. Ocean. Eng. 43(4), 940–954 (2017)CrossRefGoogle Scholar
 Szwaykowska, K., Fumin, Z.: Trend and bounds for error growth in controlled Lagrangian particle tracking. IEEE J. Ocean. Eng. 39(1), 10–25 (2014)CrossRefGoogle Scholar
 Szwaykowska, K., Zhang, F.: Controlled Lagrangian particle tracking: error growth under feedback control. IEEE Trans. Control Syst. Technol. 26(3), 874–889 (2018)CrossRefGoogle Scholar
 Wu, W., Chang, D., Zhang, F.: Glider CT: reconstructing flow fields from predicted motion of underwater gliders. In: Proceedings of the Eighth ACM International Conference on Underwater Networks and SystemsWUWNet ’13, p. 47 (2013)Google Scholar
 Zhang, F.: Cybermaritime cycle: autonomy of marine robots for ocean sensing. Found. Trends Robot. 5(1), 1–115 (2016)MathSciNetCrossRefGoogle Scholar
 Zhang, F., Marani, G., Smith, R.N., Choi, H.T.: Future trends in marine robotics [TC Spotlight]. IEEE Robot. Autom. Magn. 22(1), 14–122 (2015)CrossRefGoogle Scholar
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