Editorial Type:
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Article Type:
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Optimum Estimation of Coastal Currents Using Moving Vehicles

KuanYu Chen aInstitute of Oceanography, National Taiwan University, Taipei, Taiwan
bDepartment of Power Vehicle and Systems Engineering, National Defense University, Taoyuan, Taiwan

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Chen-Fen Huang aInstitute of Oceanography, National Taiwan University, Taipei, Taiwan

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https://orcid.org/0000-0001-8736-9545
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Zhe-Wen Zheng cDepartment of Earth Science, National Taiwan Normal University, Taipei, Taiwan

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Sheng-Fong Lin dDepartment of Marine Environmental Engineering, National Kaohsiung University of Science and Technology, Kaohsiung, Taiwan

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Jin-Yuan Liu eDepartment of Electrical and Computer Engineering, Tamkang University, New Taipei City, Taiwan

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Jenhwa Guo fDepartment of Engineering Science and Ocean Engineering, National Taiwan University, Taipei, Taiwan

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Online Publication:
12 Dec 2023
Print Publication:
01 Dec 2023
DOI:
https://doi.org/10.1175/JTECH-D-23-0039.1
Page(s):
1619–1629
Open access
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Abstract

Ocean acoustic tomography (OAT) deploys most moored stations on the periphery of the tomographic region to sense the solenoidal current field. Moving vehicle tomography (MVT), an advancement of OAT, not only samples the region from various angles for improving the resolution of mapped currents but also acquires information about the irrotational flow due to the sampling points inside the region. To reconstruct a complete two-dimensional current field, the spatial modes derived from the open-boundary modal analysis (OMA) are preferable to the conventional truncated Fourier series since the OMA technique describes the solenoidal and irrotational flows efficiently in which all modes satisfy the coastline and open boundary conditions. Comparisons of the reconstructions are presented using three different representations of currents. The first two representations explain only the solenoidal flow: the truncated Fourier series and the OMA Dirichlet modes. The third representation, accounting for the solenoidal and irrotational flows, uses all the OMA modes. For reconstructing the solenoidal flow, the OMA representation with the Dirichlet modes performs better than the Fourier series. A large difference appears near the bay mouth, where the OMA-Dirichlet reconstruction shows a better fit to the uniform currents. However, considerable uncertainty exists outside the bay mouth where the irrotational currents dominate. This can be improved by the third representation with the inclusion of the Neumann and boundary modes. The reconstruction results using field data were validated against the acoustic Doppler current profiler (ADCP) measurements. Additionally, incorporating constraints from ADCP measurements enhances the accuracy of the reconstruction.

Significance Statement

This study contributes toward improving our understanding of accurately measuring oceanic circulation patterns over large areas without relying solely upon stationary sensors or satellite imagery. The study combines multiple sources, such as shipboard ADCP and tomographic techniques, to obtain a complete picture of what is happening beneath surface waters across entire regions under investigation. It has important implications for fields such as climate science, marine biology, and fisheries management, where accurate knowledge of the movement and distribution of water masses is crucial for predicting future trends and making informed decisions.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Chen-Fen Huang, chenfen@ntu.edu.tw

Abstract

Ocean acoustic tomography (OAT) deploys most moored stations on the periphery of the tomographic region to sense the solenoidal current field. Moving vehicle tomography (MVT), an advancement of OAT, not only samples the region from various angles for improving the resolution of mapped currents but also acquires information about the irrotational flow due to the sampling points inside the region. To reconstruct a complete two-dimensional current field, the spatial modes derived from the open-boundary modal analysis (OMA) are preferable to the conventional truncated Fourier series since the OMA technique describes the solenoidal and irrotational flows efficiently in which all modes satisfy the coastline and open boundary conditions. Comparisons of the reconstructions are presented using three different representations of currents. The first two representations explain only the solenoidal flow: the truncated Fourier series and the OMA Dirichlet modes. The third representation, accounting for the solenoidal and irrotational flows, uses all the OMA modes. For reconstructing the solenoidal flow, the OMA representation with the Dirichlet modes performs better than the Fourier series. A large difference appears near the bay mouth, where the OMA-Dirichlet reconstruction shows a better fit to the uniform currents. However, considerable uncertainty exists outside the bay mouth where the irrotational currents dominate. This can be improved by the third representation with the inclusion of the Neumann and boundary modes. The reconstruction results using field data were validated against the acoustic Doppler current profiler (ADCP) measurements. Additionally, incorporating constraints from ADCP measurements enhances the accuracy of the reconstruction.

Significance Statement

This study contributes toward improving our understanding of accurately measuring oceanic circulation patterns over large areas without relying solely upon stationary sensors or satellite imagery. The study combines multiple sources, such as shipboard ADCP and tomographic techniques, to obtain a complete picture of what is happening beneath surface waters across entire regions under investigation. It has important implications for fields such as climate science, marine biology, and fisheries management, where accurate knowledge of the movement and distribution of water masses is crucial for predicting future trends and making informed decisions.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Chen-Fen Huang, chenfen@ntu.edu.tw
Keywords: Acoustic measurements/effects; In situ oceanic observations; Remote sensing

1. Introduction

Knowledge of coastal currents can benefit the management of coastal seas, which are the most biologically productive and vulnerable at the same time, as most of the fish is harvested from the coastal seas. In situ measurements of coastal currents are difficult to obtain due to their spatial and temporal complexities. For example, circulations in coastal seas are often formed by flow separation due to sudden changes in topography (Middleton 2001). Circulations also occur when a strong current contacts with the surrounding seawater of a slower velocity caused by topography. These features in the coastal current system cannot be easily observed by point measurements, such as acoustic Doppler current profiler (ADCP), due to their relatively small spatial scales.

Synoptic images of ocean states, such as ocean temperature and current fields, can be obtained by ocean acoustic tomography (OAT), proposed by Munk and Wunsch (1979). OAT has been applied to the coastal seas to obtain the horizontal structure of tidal currents in coastal seas (e.g., Park and Kaneko 2000; Yamoaka et al. 2002; Zhu et al. 2013). Reconstructing an accurate ocean image using OAT typically requires deploying many tomographic moorings on the periphery of the tomographic region to obtain a sufficient number of acoustic rays to sample the region.

Moving vehicle tomography (MVT) uses mobile platforms to increase the number of acoustic rays that sample the tomographic region from various angles and to improve the reconstructions of temperature (Cornuelle et al. 1989) and currents (Huang et al. 2019). Two types of currents are of interest: solenoidal flows and irrotational flows. Solenoidal flows can be obtained by both OAT and MVT, which both use travel-time measurements between two platforms. However, obtaining irrotational flows requires measurements within the tomographic region (Munk et al. 1995), and can only be obtained with MVT as the mobile platforms cruise within the region where current fields evolve gradually.

Reconstructing ocean current fields using tomographic techniques requires decomposing a continuous field into a chosen basis set. Many studies (e.g., Munk et al. 1995; Yamoaka et al. 2002; Gaillard 1992; Huang et al. 2013) represented the solenoidal flow component with a truncated Fourier series. However, this basis set might be inefficient in coastal seas, as it does not satisfy the coastline boundary conditions and cannot adequately describe localized flow features such as fronts. An ocean representation with prior environmental information could accurately describe ocean physics. Thus, this study explores the open-boundary modal analysis (referred to as OMA) (Lekien et al. 2004) to represent both the irrotational and solenoidal flows in coastal seas using three types of coastline-fitting normal modes, i.e., the Dirichlet, Neumann, and boundary modes. The OMA representation has been applied to the spatial interpolation and filtering of radar surface current measurements in Bodega Bay and Monterey Bay (Kaplan and Lekien 2007). For the application of OAT, the OMA representation has been applied to coastal seas (M. Chen et al. 2020).

An optimal estimation of ocean states has been achieved for temperature fields by combing tomographic data and direct point measurements, e.g., XBT/conductivity–temperature–depth (CTD), in an inverse procedure (Cornuelle et al. 1993; Dushaw et al. 1993a,b; Dushaw and Sagen 2016). For the current reconstruction, the tomographic data measure the integrated current velocity projected along the ray path, and thus, the integrated measurements are sensitive to the relatively low wavenumber characteristics. While the ADCP point measurements measure the variation at a single location, and thus, the data can localize a detailed spatial feature (Cornuelle and Worcester 1996). As the sampling properties between the tomographic and point measurements complement each other, employing both data types in an observation system would lead to an optimal estimate of ocean states.

In this study, we demonstrate the optimal estimate of ocean currents using the data collected during a coastal MVT experiment conducted in WangHaiXiang Bay, Keelung, Taiwan, in 2017 (K. Chen et al. 2020), in which both integral and point measurements of currents were taken. WangHaiXiang Bay, with an average water depth of about 25 m, is a semienclosed bay characterized by strong tidal currents flowing parallel to the bay mouth. These tidal currents create cyclonic and anticyclonic circulations inside the bay during the flood and ebb tidal phases, respectively. Throughout the experiment, several CTD casts were conducted at the site. These casts revealed uniform depth profiles for both temperature and salinity, which are likely attributed to the relatively shallow water depth and efficient tidal mixing within the water column. A separate analysis of currents measured by the shipboard and bottom-mounted ADCPs (C.-R. Ho 2023, personal communication) indicates that the depth variation of flow is relatively mild. Thus, the primary objective of this study is to map the depth-averaged horizontal current fields in the bay. To further evaluate the performance of the Fourier and OMA representations, numerical experiments of reciprocal acoustic transmission are performed using the simulated currents from the Coastal Ocean Model off Keelung (referred to as KCOM) for the period of this MVT experiment. The synthetic data [differential travel times (DTTs)] are obtained using the geometry of the acoustic rays from the MVT experiment.

The rest of this paper is organized as follows. Section 2 briefly describes the synthetic ocean currents and the synthetic DTT data of the 2017 MVT experiment. Section 3 introduces the current field representations using the truncated Fourier series and OMA modes. Section 4 formulates the MVT inverse problem. Section 5 presents the application of OMA to the tomographic reconstruction of ocean currents in WangHaiXiang Bay and discusses the model resolution. The results of synthetic tomographic experiments and the 2017 MVT experiment for the current reconstructions are examined in section 6, followed by concluding remarks in section 7.

2. Forward problem

The DTTs obtained from reciprocal acoustic pulses are proportional to the line integral of the current velocity projected along the ray path (Worcester 1977). For the jth transceiver pair, the DTT dj is related to the current velocity v and a reference sound speed c (Munk et al. 1995):
dj=−2c2∫Γjv⋅r′ds,
where r′ is the unit vector tangent to the ray path Γj for the reference sound speed c, and s is the arc length along the ray.
Applying the Helmholtz representation, a two-dimensional (2D) current field v(x, y) can be decomposed into solenoidal and irrotational components
v(x,y)=∇×(ψz^)+∇ϕ,
where ψ is the streamfunction and ϕ is the velocity potential, and z^ is the unit vector along the z axis. Both ψ and ϕ are scalar fields depending on (x, y), defined within a horizontal domain.
Substituting Eq. (2) into Eq. (1) and applying the gradient theorem, we obtain
dj=−2c2∫Γj[∇×ψ(x,y)z^]⋅r′ds−2c2[ϕ(rj,2)−ϕ(rj,1)],
where rj,1 and rj,2 are the beginning and end points of the ray path Γj. As shown by Norton (1988), the solenoidal flow can be uniquely reconstructed from the line integral of v(x, y) along the ray paths within a bounded area and further provides a direct measure of a relative vorticity field ∇×(∇×ψz^).

For the irrotational flow, only the values on ϕ at the locations of each transceiver pair does the line-integral measurement provide, see Eq. (3). To obtain the irrotational flow ∇ϕ and the resulting divergence field ∇2ϕ, the transceivers are required to be placed in the interior of the tomographic region (Munk et al. 1995). The MVT experiment is suitable for reconstructing the irrotational component, as the transceivers are towed by vehicles cruising inside the tomographic region.

3. Representation of the current field

To reconstruct the current field using the OAT method, two representations of the current field are discussed, including the truncated Fourier series and the spatial modes from the OMA.

a. Truncated Fourier series

Most OAT studies in coastal seas have used the truncated Fourier series (Yamoaka et al. 2002; Zhu et al. 2013) to represent the streamfunction for expressing the solenoidal flow v=∇×ψz^. The streamfunction could be represented by the following Fourier series truncated at the maximum wavenumber Kmax (Gaillard 1992; Huang et al. 2013),
ψ(x,y)=∑k=−KmaxKmax∑l=0l′(k)[mklcos⁡cos⁡(2πkxLinv+2πlyLinv)+mklsin⁡sin⁡(2πkxLinv+2πlyLinv)],
where l′(k)=(Kmax2−k2)1/2 with Kmax being high enough to depict the small-scale feature; mklcos⁡ and mklsin⁡ are the Fourier coefficients. The side length of the inversion domain Linv is twice larger than the tomographic area to avoid the periodic effect.
Substituting Eq. (4) into Eq. (1), the DTT for jth ray can be written as
dj=−2c2∫Γj(cos⁡θj∂ψ∂y+sin⁡θj∂ψ∂x)ds,
where θj is the angle between the jth ray path Γj and x axis. In this study, Γj is the straight line connecting the jth transceiver pair.

b. Spatial modes using OMA

OMA uses eigenmodes to project the flow onto a horizontal domain based on the modal decomposition in streamfunction and velocity potential (Lekien et al. 2004; Kaplan and Lekien 2007). Three types of eigenmodes are derived. First, streamfunction modes ψi satisfy Dirichlet boundary conditions (referred to as Dirichlet modes). Second, velocity potential modes ϕi satisfy Neumann boundary conditions (Neumann modes). Third, velocity potential modes ϕib satisfy the nonhomogeneous boundary conditions along the open boundary (boundary modes). Thus, a comprehensive 2D current field accounting for solenoidal and irrotational flows can be represented by a linear combination of those modes
v(x,y)=∑i=1Nψmiψ∇×ψi(x,y)z^+∑i=1Nϕmiϕ∇ϕi(x,y)+∑i=1Nϕbmib∇ϕib(x,y),
in which miψ, miϕ, and mib are the corresponding mode coefficients. To retain significant information, the current field is approximated with a finite number of each mode set Nψ, Nϕ, and Nϕb.
Substituting Eq. (6) into Eq. (1), the DTT for the jth ray can be expressed as
dj=−2c2∑i=1Nψmiψ∫Γj(​cos⁡θj∂ψi∂y+sin⁡θj∂ψi∂x​)ds​​−2c2∑i=1Nϕmiϕ[ϕi(rj,2)−ϕi(rj,1)]−2c2∑i=1Nϕbmib[ϕib(rj,2)−ϕib(rj,1)]⋅
As indicated by the second and third terms on the right-hand side of Eq. (7), the line-integral measurement only gives information about the values of ϕi and ϕib at the endpoints of each ray.

4. Inverse problem

For the reconstruction of the current field using line-integral measurements, Eq. (5) or Eq. (7) can be expressed by a linearized data rule as an inner product between the data kernel G and the model parameter vector m,
d=Gm+n,
where the data vector d is contaminated by the noise n due to observation and modeling errors. For the Fourier representation, the (j, i) element of the observation matrix Gtomo, Eq. (5), is given by the projection of current contributed by the ith Fourier basis function on the ray path Γj. Similarly for the OMA representation in Eq. (7), with the additional discrete values of ϕi and ϕib at the endpoints of the ray.
Incorporation of point measurements into line-integral data could further improve the current mapping. When using point measurements with the OMA representation, the (j, i) element of the observation matrix Gpoint is the ith OMA velocity mode sampled at the jth location. A combination of those two data types results in an augmented data vector (Dushaw and Sagen 2016)
d=[dtomodpoint],
and observation matrix
G=[GtomoGpoint].
Reconstruction of current fields is an ill-posed inverse problem. To stabilize the inverse problem solutions, we use the generalized Tikhonov regularization method and minimize the following objective function (Munk et al. 1995)
J(m)=(d−Gm)TCd−1(d−Gm)+βmTCm−1m,
where Cd and Cm are the data and model weight matrices, respectively. This study uses a diagonal Cd whose jth element is determined from the uncertainty in dj. A diagonal Cm is employed for both current field representations. For the Fourier representation, a red-noise spectrum with a rollover length (Huang et al. 2013) is used to enforce more weight on smaller wavenumbers. As for the OMA representation, Cm is a prior model covariance matrix obtained from the temporal variation of the modeled current fields. The regularization parameter β is determined by the L-curve criterion (Hansen and O’Leary 1993) to find a balance between the weighted data residual norm and the weighted model norm.
The generalized least squares solution is
m^=G†d,
where G†=(GTCd−1G+βCm−1)−1GTCd−1 is a generalized inverse of the observation matrix G. Then, the estimated current fields in eastward and northward directions (υx, υy) at the spatial grid are
v^(x,y)=(υx,υy)=(Em^,Nm^),
where the matrices E and N are for mapping the estimated model parameter vector m^ to, respectively, the eastward and northward components of the current field. Each column of E or N is the differentiation of the corresponding mode. For the solenoidal part represented by the Fourier basis functions or the Dirichlet modes, the columns of E and N are the differentiation of the functions (for Fourier basis functions) or modes (for Dirichlet modes) with respect to y and x, respectively. As for the irrotational part represented by the Neumann and boundary modes, the columns of E and N are the differentiation of the modes with respect to x and y, respectively.
Assuming that there is no correlation between m and d, the model uncertainty is described by the covariance matrix
P^=⟨(m^−m)(m^−m)T⟩
=1β(R−I)Cm(R−I)T+G†CdG†T,
where R=G†G is the model resolution matrix and R ≈ I indicates that the model can be virtually perfectly resolved. The first term in Eq. (15) indicates the uncertainty resulting from the limited resolution that characterizes the bias of the predicted model, and the second term is due to the propagation of data error.
The performance of different current field representations is evaluated by comparing the reconstructed currents with the true using the root-mean-square difference (RMSD) and fractional-residual-error variance (FREV) defined as follows:
RMSD=⟨|v^−vtrue|2⟩1/2,
FREV=⟨|v^−vtrue|2⟩⟨|vtrue|2⟩,
where 〈⋅〉 denotes a spatial average over the tomographic area surrounded by the outer edge of all ray paths; 〈|vtrue|2〉 is the variability of the true tidal current speeds in the tomographic area, referred to as the tidal current variance. For the simulation study, vtrue is the depth-averaged velocity calculated from the synthetic ocean. While for the experimental data, vtrue is replaced by the depth-averaged ADCP measurements, i.e., vtrue = vadcp.

5. Applications of OMA to WangHaiXiang Bay

To illustrate the OMA method presented in section 3b and its applicability in WangHaiXiang Bay, synthetic data are generated using the simulated currents from ocean models and the ray distribution (yellow lines in Fig. 1b) from the 2017 MVT experiment (K. Chen et al. 2020).

Fig. 1.
Fig. 1.

(a) Model domain of the KCOM. Background colors indicate the bathymetry with a contour interval of 10 m. The purple star near the Keelung Harbor (KH) indicates the location of Keelung Tidal Station. Blue solid and dashed lines indicate computational domains of the OMA and truncated Fourier series, respectively, for the WangHaiXiang Bay (WB). (b) Observation area for the 2017 MVT experiment. Yellow lines indicate the acoustic rays collected using two mobile and one moored transceivers.

Citation: Journal of Atmospheric and Oceanic Technology 40, 12; 10.1175/JTECH-D-23-0039.1

a. Synthetic ocean

A three-dimensional (3D) KCOM was developed with the related initial and lateral boundary conditions to simulate the tidal currents for synthetic tomographic experiments. With tidal forcing, the KCOM is based on the Princeton Ocean Model (POM) (Blumberg and Mellor 1987). The domain includes WangHaiXiang Bay (WB), Keelung Harbor (KH), and the adjacent waters, from 121°43.8′E to 121°52.2′E and from 25°7.2′N to 25°12.6′N, with a horizontal grid resolution of 50 m (Fig. 1a). For vertical grids, 26 sigma levels were used with the spacing decreasing near the surface and bottom to resolve the momentum flux in the upper and lower boundaries, respectively. The simulation started on 1 June 2017 and lasted for 30 days with an interval of 1 min.

The output of KCOM is validated by comparing the modeled sea surface height with the measurements at the Keelung Tidal Station (purple star in Fig. 1a) with a root-mean-square error less than 6 cm. The resulting depth-averaged current fields (Figs. 2a,b) indicate that, during the flood (ebb) tide, the strong tidal currents stream northwestward (southeastward) passing through the bay mouth with a cyclonic (anticyclonic) circulation formed inside the bay.

Fig. 2.
Fig. 2.

Depth-averaged current fields for (a) flood and (b) ebb tides modeled by the KCOM.

Citation: Journal of Atmospheric and Oceanic Technology 40, 12; 10.1175/JTECH-D-23-0039.1

b. Synthetic DTTs

Numerical reciprocal acoustic propagation experiments were performed using the Gaussian-beam acoustic model (Porter and Bucker 1987) to simulate the acoustic propagation between paired transceivers in a moving medium. The effects of ocean currents on acoustic propagation are accounted for by adding or subtracting the 3D KCOM projected currents along the ray from the actual sound speed (Taniguchi and Huang 2014). The acoustic parameters used in the simulations are listed in Table 1.

Table 1.

Parameters used in the acoustic simulations.
Table 1.

From the simulated arrival patterns in reciprocal directions of the transceiver pair, the synthetic DTT is obtained by calculating the cross-correlation function (CCF) of the reciprocal arrival patterns (Huang et al. 2019). The DTT data are further contaminated by the additive zero-mean Gaussian noise with the standard deviation determined from the measured DTTs collected in the 2017 MVT experiment.

c. OMA modes for the WangHaiXiang Bay

The computational domain for the OMA analysis is specified as 5 km × 5 km (blue solid line in Fig. 1a). The obtained modes are ordered from the lowest to the highest corresponding eigenvalues. A higher-order mode exhibits a relatively small-scale variability. The total number of OMA modes, which includes Dirichlet, Neumann, and boundary modes, in the current reconstruction is determined by examining how well the reconstruction captures variation in the KCOM-modeled current field over a 24-h period. The number of individual modal sets (Nψ, Nϕ, and Nϕb) is chosen based on an eigenvalue threshold. When reducing the eigenvalue threshold, more OMA modes are included. Assuming that the KCOM-modeled currents represent the true field, the fraction of the current variance captured by including additional modes increases and eventually levels off at 93.2%. This is achieved with a total of 72 modes, consisting of 24 Dirichlet modes (Nψ = 24), 37 Neumann modes (Nϕ = 37), and 11 boundary modes (Nϕb=11). Thus, a total of 72 OMA modes, which account for 93.2% of the tidal current variance at the site, are included in the following current reconstruction.

Figure 3 displays a set of the selected modes. The corresponding velocity modes are tangent to the streamlines of the interior Dirichlet modes (first column), and they are orthogonal to the level sets of interior Neumann (second column) and boundary modes (third column). Note that the boundary modes are derived based on a discrete Fourier basis defined on the open boundary; ϕ1b corresponds to the constant boundary condition, ϕ2b to ϕ6b correspond to the boundary conditions of the Fourier cosine functions, and ϕ7b to ϕ11b correspond to those of the Fourier sine functions. Compared to ϕ6b (Fig. 3f), a relatively small spatial variability is observed for ϕ7b (Fig. 3e) as the open boundary of this mode satisfies the lowest Fourier sine function.

Fig. 3.
Fig. 3.

Spatial patterns of the selected OMA modes. (a),(b) The Dirichlet modes ψ6 and ψ23, (c),(d) the Neumann modes ϕ12 and ϕ31, and (e),(f) the boundary modes ϕ7b and ϕ6b. The background color is nondimensional, with dark blue to off-white to dark red representing negative to zero to positive values. All panels use the color bar next to (f). The arrows indicate nondimensional velocity vectors. The green and magenta solid lines indicate the ray distribution in the observation region bounded by the red rectangle.

Citation: Journal of Atmospheric and Oceanic Technology 40, 12; 10.1175/JTECH-D-23-0039.1

d. Resolution of the OMA modes

Due to the regularization applied to stabilize the current reconstruction, a bias exists between reconstructed and true currents. Here, we examine how much more reliable certain OMA modes are compared to the others using the relative diagonal values of the resolution matrix R (Fig. 4). Among all the three types of modes, Dirichlet ψ6 (circles) has the best resolution with a value of 0.45. For the Neumann and boundary modes, ϕ12 (triangles) and ϕ7b (squares) have the best resolution with a value of 0.42 and 0.09, respectively.

Fig. 4.
Fig. 4.

Diagonal elements of model resolution matrix for the Dirichlet (D), Neumann (N), and boundary (B) modes indicated by circle, triangle, and square symbols, respectively.

Citation: Journal of Atmospheric and Oceanic Technology 40, 12; 10.1175/JTECH-D-23-0039.1

The reliability/resolvability of Dirichlet modes ψ is related to the projection of the corresponding velocity field onto the rays. A Dirichlet mode is better resolved for a given ray distribution when most rays have a large line integral of the currents projected along the path. From the diagonal values of R (circle in Fig. 4), better resolution is observed for ψ6, ψ8, and ψ3, in descending order. The velocity pattern of ψ6 (Fig. 3a) shows a circulation feature inside the bay, similar to those of ψ8 and ψ3 (not shown here). With those three transceivers, the rays are triangular shapes pivoting on the moored transceiver located at the lower right corner. Thus, a portion of the rays along these three triangular sides are parallel to the current circulation inside the bay. However, the modes ψ2, ψ19, and ψ23 (Fig. 3b for the illustration of ψ23) have a relatively poor resolution due to most of the rays being perpendicular to the velocity modes.

For the Neumann modes ϕ, the resolution is related to the difference in ϕ at the locations of a transceiver pair. A Neumann mode is better resolved when most ray paths have a large difference in ϕ values. From the mode pattern of the best resolvable mode ϕ12 (Fig. 3c), we see that, except for the rays colored in magenta, most of the rays have blue and red colors at the endpoints of the ray. For the pattern of the poorly resolved Neumann mode, e.g., ϕ31 (Fig. 3d), most of the rays have a relatively small change in the color lightness at the endpoints.

Similar to the discussion for the Neumann modes, the resolvability of a boundary mode ϕb depends on the difference in the sampled potential values at the endpoints of a ray. The diagonal values of R (square in Fig. 4) decrease from ϕ2b to ϕ6b and from ϕ7b to ϕ11b. Compared to the Dirichlet and Neumann modes, a relatively small resolution value of 0.09 (square in Fig. 4) is observed for the best-resolved boundary mode ϕ7b. For the corresponding mode pattern (Fig. 3e), the endpoints of each ray have less contrast in the color lightness compared to those of the Neumann modes. Most boundary modes (e.g., Fig. 3f for ϕ6b) have a relatively poor resolution due to the small variation of the potential field inside the bay, resulting from the distant open boundary.

6. Results and discussion

All the DTT data collected during the experiment are used to improve the accuracy of the current estimate, provided that the ocean field is frozen over the data collection period. The reconstruction of the ocean currents using either synthetic or experimental data uses the following three different representations. The first two representations, accounting for only the solenoidal flow, are the truncated Fourier series and the OMA-derived Dirichlet modes (referred to as the OMA-Dirichlet). The third representation, including both the solenoidal and irrotational flows, is all three types of OMA-derived modes (referred to as the OMA-all).

a. Inversion of synthetic data

For the simulation study, the hourly DTT datasets based on the ray distribution of the 2017 MVT experiment are synthesized using the KCOM current field on 27 June 2017. For the solenoidal flow, the FREVs of the OMA-Dirichlet reconstruction (dashed line in Fig. 5b) are always smaller than those for the Fourier reconstruction (dotted line) over a period of 24 h. The time average of the FREV with the OMA-Dirichlet representation is 28.7%, compared to 48.2% with the Fourier. The OMA-Dirichlet representation results in an approximately 20% reduction of the time-averaged FREV as those modes satisfy the boundary condition and thus provide a more accurate representation of the oceanographic process.

Fig. 5.
Fig. 5.

Time series of (a) the tide height (solid line) and the spatial average of squared current magnitude 〈|v|2〉 (dashed line) output from the ocean model in the study area and (b) the fractional-residual-error variances (FREVs) for the reconstructed currents using the representations of the truncated Fourier series (dotted line), the OMA-Dirichlet (dashed line), and the OMA-all modes (solid line).

Citation: Journal of Atmospheric and Oceanic Technology 40, 12; 10.1175/JTECH-D-23-0039.1

The OMA-all representation is used to reconstruct the current field consisting of solenoidal and irrotational flows. With the additional inclusion of the Neumann and boundary modes that satisfy the boundary conditions, the OMA-all representation is expected to have better performance than the Fourier representation. Over the entire period of 24 h, the FREVs for the reconstructed currents using the OMA-all representation (solid line in Fig. 5b) are smaller than those using the Fourier (dotted line). The time-averaged FREV using the OMA-all representation is greatly reduced to 17.8%, compared to 48.2% using the Fourier representation.

The spatial distributions of the reconstructed currents at 7 and 16 h are shown in the top and bottom panels of Fig. 6, respectively. The current field at 7 h exhibits a lower spatial current variance of 0.06 m2 s−2 (downward arrow in Fig. 5a). Especially, the currents outside the bay are weakened and have slightly more spatial variations (see red arrows in the top panels of Fig. 6), which might result in considerable uncertainty in the current reconstruction. At this period, the current field reconstructed with the Fourier representation (black arrows in Fig. 6a-1) accounts for only 46.8% of the tidal current variance (FREV = 53.2%) due to a lack of crossing rays in resolving the current variations outside the bay. Using the OMA-Dirichlet representation, the reconstructed currents (Fig. 6a-2) explain 54.6% of the tidal current variance (FREV = 45.4%). The OMA-Dirichlet reconstructed currents can explain 7.8% more of the tidal current variance. This improvement might be due to incorporating the coastal boundary conditions inside the bay for the OMA-Dirichlet modes. Finally, using the OMA-all representation, the reconstructed currents (Fig. 6a-3) explain the maximum 67.7% of the tidal current variance (FREV = 32.3%) due to the inclusion of the Neumann and boundary modes. Of all three representations, the smallest RMSD of about 13 cm s−1 is observed using the OMA-all representation. Similarly, large FREVs are observed at about 2, 7, 13, and 19 h (Fig. 5b). This might result from small tidal current variances occurring at a period of half of the semidiurnal cycle (dashed line in Fig. 5a).

Fig. 6.
Fig. 6.

Simulated tomographic reconstructions using the synthetic data for (a) 7 and (b) 16 h. For each hour, the columns show the reconstruction results (black arrows) using the representations of (left) the truncated Fourier series, (center) the OMA-Dirichlet modes, and (right) OMA-all modes. Red arrows indicate the true currents. The background color denotes the difference between the true and reconstruction.

Citation: Journal of Atmospheric and Oceanic Technology 40, 12; 10.1175/JTECH-D-23-0039.1

Compared to the current field at 7 h, the current field at 16 h corresponds to a high current variance of 1.31 m2 s−2 (rightward arrow in Fig. 5a). Outside the bay, the current field exhibits a strong uniform current flowing southeastward. Inside the bay, it circulates in a clockwise direction (see red arrows in the bottom panels of Fig. 6). Using the Fourier representation, the tomographic inversion (black arrows in Fig. 6b-1) reconstructs the circulation inside the bay, but cannot explain the uniform currents outside; the reconstruction accounts for only 44.2% of the tidal current variance (FREV = 55.8%). A poor fit to the true currents is observed outside the bay where the irrotational component dominates, resulting in a large RMSD with a value of 52 cm s−1. Using the OMA-Dirichlet representation (Fig. 6b-2), the circulation inside the bay can also be resolved. An improvement is observed near the bay mouth, with the FREV of 29.8% and RMSD of 38 cm s−1. Compared to the Fourier representation, the reconstructed field yields approximately 26% reduction of the FREV since most of the inverted currents inside the bay are well constrained by the coastline. Finally, using the OMA-all representation, the tomographic inversion (Fig. 6b-3) reconstructs both the circulation and the uniform currents well; the reconstruction accounts for up to 92.0% of the tidal current variance (FREV = 8.0%). The RMSD is significantly reduced to 20 cm s−1.

The reasons for the noticeable performance of the OMA-all representation are as follows. First, the spatial modes inherently satisfy the boundary conditions along the shoreline and open boundaries, thus imposing more prior information on the tomographic inversion. Second, the irrotational flow outside the bay can be appropriately accounted for by including the Neumann and boundary modes. Thus, the reconstruction using the complete OMA representation results in consistency with the true currents. However, the Fourier representation is inefficient due to the absence of the irrotational flow. Therefore, the OMA-all representation is preferable for mapping coastal currents in a shallow-water environment.

b. Inversion of experimental data

Data acquired during the 2017 MVT experiment are used to illustrate those three representations of the current field. As a reference, the current field near the bay mouth was reconstructed (red arrows in Fig. 7d) by smoothing and filtering the shipboard ADCP measurements (white arrows) using the OMA-all representation. Note that the reconstructed currents cannot explain 9.8% of the ADCP-measured current variance. Below are the tomographic reconstructions using three different representations of the current field.

Fig. 7.
Fig. 7.

Reconstructions using the data collected in the 2017 MVT experiment. (top) The inverted currents (red arrows) using the DTT data for the ocean currents represented by (a) the truncated Fourier series, (b) the OMA-Dirichlet modes, and (c) the OMA-all modes. (bottom) The OMA-all reconstructions using (d) the ADCP data and (e) both the DTT and ADCP data. The background color denotes the predictive uncertainty in standard deviation (STD). Only the current estimate with a STD of less than 10 cm s−1 is displayed by the red arrow on the map. The white arrows and white lines indicate the ADCP measurements and the ray paths for the DTTs, respectively.

Citation: Journal of Atmospheric and Oceanic Technology 40, 12; 10.1175/JTECH-D-23-0039.1

First, using the Fourier representation for the solenoidal flow, the reconstructed currents (red arrows in Fig. 7a) show a fair agreement with the ADCP measurements (white arrows) except for the currents near the northernmost and easternmost tomographic boundaries. Due to the absence of ray crossing, the reconstruction results in a 49.7% of the ADCP-measured current variance unexplained. The circulation is revealed with a predictive uncertainty of 7 cm s−1 near the bay center. In contrast, considerable uncertainty is observed near the bay mouth since the Fourier representation has a better wavenumber resolution (K. Chen et al. 2020).

Second, using the OMA-Dirichlet representation for the solenoidal flow, the circulation inside the bay is also resolved (Fig. 7b). However, the estimated currents show a poor fit to the ADCP measurements outside the bay mouth due to lacking rays from various angles. Similar to the 49.7% unexplained variance using the Fourier representation, a FREV of 49.9% is obtained using the OMA-Dirichlet. In addition, the predictive uncertainty (background color in Fig. 7b) shows a relatively small uncertainty compared to that using the Fourier representation (Fig. 7a). This is because the OMA-Dirichlet representation provides more prior information about the boundary conditions along the coastline.

Third, using the OMA-all representation for the solenoidal and irrotational flows, a significant improvement in the reconstructed currents (Fig. 7c) is observed outside the bay, where the irrotational flow dominates. The uniform currents near the northeast area are better reconstructed using the Neumann and boundary modes for the irrotational flow. This improvement cannot be achieved using the truncated Fourier series or Dirichlet modes for the solenoidal flow alone. The FREV of OMA-all reconstruction is reduced to 23.0%, and the predictive uncertainty is reduced to 4 cm s−1 near the circulation center.

Incorporating the ADCP measurements in the tomographic reconstruction process further improves the current field using the OMA-all representation. Compared to the previous reconstruction results, the predictive uncertainty is the minimum, especially outside the bay mouth where the ADCP measurements are available (Fig. 7e), since the point measurements can compensate for the lack of current information due to the few crossing rays in that region. The FREV of this optimally reconstructed current field is reduced to 11.2%. Combining the DTT and ADCP data with the OMA-all representation may yield an optimal estimate of the current field.

From this optimal estimate of the current field (Fig. 7e), the solenoidal and irrotational flows can be obtained (Fig. 8) using the modal current decomposition; the solenoidal flow is determined by the Dirichlet modes, and the irrotational flow is determined by the Neumann and boundary modes. The reconstruction suggests that the solenoidal flow could account for a larger portion (about 54.2%) of the tidal current variance than the irrotational flow (45.8%). The variances for different flows are consistent with those of the synthetic ocean at 16 h (Fig. 6b-3). Furthermore, the relative vorticity and horizontal divergence fields (background color in Figs. 8a,b) can be derived from the estimated solenoidal and irrotational flows. Note that, due to the properties of spatial filtering using the OMA modes, both the relative vorticity and horizontal divergence fields have a small spatial variability. The relative vorticity pattern (Fig. 8a) shows negative relative vorticity (blue color) inside the bay, indicating an anticyclonic circulation. The minimum vorticity inside the bay is about −1.5 × 10−3 s−1 due to the semiclosed topography and the strong tidal currents dominating the ebb tide. During this ebb-tide period, the horizontal divergence pattern (Fig. 8b) shows that there is an extensive area of positive divergence off the western headland with a maximum value of 6.6 × 10−4 s−1. In contrast, an area of negative divergence exists off the eastern headland. As a result, the irrotational currents flow southeastward outside the bay.

Fig. 8.
Fig. 8.

(a) Solenoidal and (b) irrotational flows estimated from the optimal estimate of the current field shown in Fig. 7e with the background color indicating the corresponding relative vorticity and divergence fields.

Citation: Journal of Atmospheric and Oceanic Technology 40, 12; 10.1175/JTECH-D-23-0039.1

7. Concluding remarks

This study has demonstrated the optimum reconstruction of ocean currents using moving vehicles with the current representations satisfied the coastal boundary conditions and the combination of both tomographic (DTT) and point (ADCP) measurements. Simulation studies have been conducted using synthetic DTTs to evaluate the reconstruction performance using those three current representations: the Fourier, OMA-Dirichlet, and OMA-all representations. The OMA-Dirichlet representation performs better than the Fourier representation for the solenoidal flow, especially in the area near the shoreline and the uniform currents near the bay mouth. For the solenoidal and irrotational flows, the OMA-all representation achieves a complete and accurate current field, resolving both the circulation inside the bay and the uniform currents near the mouth. The data collected from the 2017 MVT experiment demonstrated the reconstruction method. The overall reconstructed currents show consistency with the ADCP measurements, indicating that using the complete basis functions to represent the ocean currents can improve the mapping.

Regarding the properties of tomographic line-integral data and ADCP-point measurements, the former provides an averaged current along the ray path (better resolution in the wavenumber domain), and the latter focuses on the local currents (better resolution in the spatial domain). Different sampling properties of the tomographic and ADCP measurements are combined to compensate for each other. The combined reconstruction leads to an optimal comprehensive image of ocean currents in coastal seas.

Acknowledgments.

This work is supported by the Ministry of Science and Technology (MOST) of Taiwan through Contracts 108-2638-M-002-002-MY2, 112-2611-M-002-017, and 112-2218-E-019-011.

Data availability statement.

The processed acoustic data are stored at the Institute of Oceanography, National Taiwan University. To access these data, one may reach out to C.F.H. for potential collaboration. The OMA was performed using an algorithm sourced from the open-access repository: https://github.com/rowg/hfrprogs.

REFERENCES

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    • Export Citation

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    • Export Citation

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    • Search Google Scholar
    • Export Citation

  • Munk, W., P. F. Worcester, and C. Wunsch, 1995: Ocean Acoustic Tomography. Cambridge University Press, 450 pp., https://doi.org/10.1017/CBO9780511666926.

  • Norton, S. J., 1988: Tomographic reconstruction of 2-D vector fields: Application to flow imaging. Geophys. J. Int., 97, 161–168, https://doi.org/10.1111/j.1365-246X.1989.tb00491.x.

    • Search Google Scholar
    • Export Citation

  • Park, J.-H., and A. Kaneko, 2000: Assimilation of coastal acoustic tomography data into a barotropic ocean model. Geophys. Res. Lett., 27, 3373–3376, https://doi.org/10.1029/2000GL011600.

    • Search Google Scholar
    • Export Citation

  • Porter, M. B., and H. P. Bucker, 1987: Gaussian beam tracing for computing ocean acoustic fields. J. Acoust. Soc. Amer., 82, 1349–1359, https://doi.org/10.1121/1.395269.

    • Search Google Scholar
    • Export Citation

  • Taniguchi, N., and C.-F. Huang, 2014: Simulated tomographic reconstruction of ocean current profiles in a bottom-limited sound channel. J. Geophys. Res. Oceans, 119, 4999–5016, https://doi.org/10.1002/2014JC009885.

    • Search Google Scholar
    • Export Citation

  • Worcester, P. F., 1977: Reciprocal acoustic transmission in a midocean environment. J. Acoust. Soc. Amer., 62, 895–905, https://doi.org/10.1121/1.381619.

    • Search Google Scholar
    • Export Citation

  • Yamoaka, H., A. Kaneko, J.-H. Park, H. Zheng, N. Gohda, T. Takano, X.-H. Zhu, and Y. Takasugi, 2002: Coastal acoustic tomography system and its field application. IEEE J. Oceanic Eng., 27, 283–295, https://doi.org/10.1109/JOE.2002.1002483.

    • Search Google Scholar
    • Export Citation

  • Zhu, X.-H., A. Kaneko, Q. Wu, C. Zhang, N. Taniguchi, and N. Gohda, 2013: Mapping tidal current structures in Zhitouyang Bay, China, using coastal acoustic tomography. IEEE J. Oceanic Eng., 38, 285–296, https://doi.org/10.1109/JOE.2012.2223911.

    • Search Google Scholar
    • Export Citation
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Cover Journal of Atmospheric and Oceanic Technology
Journal of Atmospheric and Oceanic Technology

  • Blumberg, A. F., and G. L. Mellor, 1987: A description of a three-dimensional coastal ocean circulation model. Three-Dimensional Coastal Ocean Models, N. S. Heaps, Ed., Amer. Geophys. Union, 1–16, https://doi.org/10.1029/CO004p0001.

  • Chen, K., C.-F. Huang, S.-W. Huang, J.-Y. Liu, and J. Guo, 2020: Mapping coastal circulations using moving vehicle acoustic tomography. J. Acoust. Soc. Amer., 148, EL353–EL358, https://doi.org/10.1121/10.0002031.

    • Search Google Scholar
    • Export Citation

  • Chen, M., and Coauthors, 2020: Mapping current fields in a bay using a coast-fitting tomographic inversion. Sensors, 20, 558, https://doi.org/10.3390/s20020558.

    • Search Google Scholar
    • Export Citation

  • Cornuelle, B. D., and P. F. Worcester, 1996: Ocean acoustic tomography: Integral data and ocean models. Modern Approaches to Data Assimilation in Ocean Modeling, P. Malanotte-Rizzoli, Ed., Elsevier Oceanography Series, Vol. 61, Elsevier, 97–115, https://doi.org/10.1016/S0422-9894(96)80007-9.

  • Cornuelle, B. D., W. Munk, and P. Worcester, 1989: Ocean acoustic tomography from ships. J. Geophys. Res., 94, 6232–6250, https://doi.org/10.1029/JC094iC05p06232.

    • Search Google Scholar
    • Export Citation

  • Cornuelle, B. D., and Coauthors, 1993: Ocean acoustic tomography at 1000-km range using wavefronts measured with a large-aperture vertical array. J. Geophys. Res., 98, 16 365–16 377, https://doi.org/10.1029/93JC01246.

    • Search Google Scholar
    • Export Citation

  • Dushaw, B. D., and H. Sagen, 2016: A comparative study of moored/point and acoustic tomography/integral observations of sound speed in Fram Strait using objective mapping techniques. J. Atmos. Oceanic Technol., 33, 2079–2093, https://doi.org/10.1175/JTECH-D-15-0251.1.

    • Search Google Scholar
    • Export Citation

  • Dushaw, B. D., P. F. Worcester, B. D. Cornuelle, and B. M. Howe, 1993a: On equations for the speed of sound in seawater. J. Acoust. Soc. Amer., 93, 255–275, https://doi.org/10.1121/1.405660.

    • Search Google Scholar
    • Export Citation

  • Dushaw, B. D., P. F. Worcester, B. D. Cornuelle, and B. M. Howe, 1993b: Variability of heat content in the central North Pacific in summer 1987 determined from long-range acoustic transmissions. J. Phys. Oceanogr., 23, 2650–2666, https://doi.org/10.1175/1520-0485(1993)023<2650:VOHCIT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Gaillard, F., 1992: Evaluating the information content of tomographic data: Application to mesoscale observations. J. Geophys. Res., 97, 15 489–15 505, https://doi.org/10.1029/92JC01295.

    • Search Google Scholar
    • Export Citation

  • Hansen, P. C., and D. P. O’Leary, 1993: The use of the L-curve in the regularization of discrete ill-posed problems. SIAM J. Sci. Comput., 14, 1487–1503, https://doi.org/10.1137/0914086.

    • Search Google Scholar
    • Export Citation

  • Huang, C.-F., T. C. Yang, J.-Y. Liu, and J. Schindall, 2013: Acoustic mapping of ocean currents using networked distributed sensors. J. Acoust. Soc. Amer., 134, 2090–2105, https://doi.org/10.1121/1.4817835.

    • Search Google Scholar
    • Export Citation

  • Huang, C.-F., Y.-W. Li, and N. Taniguchi, 2019: Mapping of ocean currents in shallow water using moving ship acoustic tomography. J. Acoust. Soc. Amer., 145, 858–868, https://doi.org/10.1121/1.5090496.

    • Search Google Scholar
    • Export Citation

  • Kaplan, D. M., and F. Lekien, 2007: Spatial interpolation and filtering of surface current data based on open-boundary modal analysis. J. Geophys. Res., 112, C12007, https://doi.org/10.1029/2006JC003984.

    • Search Google Scholar
    • Export Citation

  • Lekien, F., C. Coulliette, R. Bank, and J. Marsden, 2004: Open-boundary modal analysis: Interpolation, extrapolation, and filtering. J. Geophys. Res., 109, C12004, https://doi.org/10.1029/2004JC002323.

    • Search Google Scholar
    • Export Citation

  • Middleton, J., 2001: Topographic eddies. Encyclopedia of Ocean Sciences, J. H. Steele, Academic Press, 57–64, https://doi.org/10.1016/B978-012374473-9.00145-4.

  • Munk, W., and C. Wunsch, 1979: Ocean acoustic tomography: A scheme for large scale monitoring. Deep-Sea Res. I, 26, 123–161, https://doi.org/10.1016/0198-0149(79)90073-6.

    • Search Google Scholar
    • Export Citation

  • Munk, W., P. F. Worcester, and C. Wunsch, 1995: Ocean Acoustic Tomography. Cambridge University Press, 450 pp., https://doi.org/10.1017/CBO9780511666926.

  • Norton, S. J., 1988: Tomographic reconstruction of 2-D vector fields: Application to flow imaging. Geophys. J. Int., 97, 161–168, https://doi.org/10.1111/j.1365-246X.1989.tb00491.x.

    • Search Google Scholar
    • Export Citation

  • Park, J.-H., and A. Kaneko, 2000: Assimilation of coastal acoustic tomography data into a barotropic ocean model. Geophys. Res. Lett., 27, 3373–3376, https://doi.org/10.1029/2000GL011600.

    • Search Google Scholar
    • Export Citation

  • Porter, M. B., and H. P. Bucker, 1987: Gaussian beam tracing for computing ocean acoustic fields. J. Acoust. Soc. Amer., 82, 1349–1359, https://doi.org/10.1121/1.395269.

    • Search Google Scholar
    • Export Citation

  • Taniguchi, N., and C.-F. Huang, 2014: Simulated tomographic reconstruction of ocean current profiles in a bottom-limited sound channel. J. Geophys. Res. Oceans, 119, 4999–5016, https://doi.org/10.1002/2014JC009885.

    • Search Google Scholar
    • Export Citation

  • Worcester, P. F., 1977: Reciprocal acoustic transmission in a midocean environment. J. Acoust. Soc. Amer., 62, 895–905, https://doi.org/10.1121/1.381619.

    • Search Google Scholar
    • Export Citation

  • Yamoaka, H., A. Kaneko, J.-H. Park, H. Zheng, N. Gohda, T. Takano, X.-H. Zhu, and Y. Takasugi, 2002: Coastal acoustic tomography system and its field application. IEEE J. Oceanic Eng., 27, 283–295, https://doi.org/10.1109/JOE.2002.1002483.

    • Search Google Scholar
    • Export Citation

  • Zhu, X.-H., A. Kaneko, Q. Wu, C. Zhang, N. Taniguchi, and N. Gohda, 2013: Mapping tidal current structures in Zhitouyang Bay, China, using coastal acoustic tomography. IEEE J. Oceanic Eng., 38, 285–296, https://doi.org/10.1109/JOE.2012.2223911.

    • Search Google Scholar
    • Export Citation
  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    (a) Model domain of the KCOM. Background colors indicate the bathymetry with a contour interval of 10 m. The purple star near the Keelung Harbor (KH) indicates the location of Keelung Tidal Station. Blue solid and dashed lines indicate computational domains of the OMA and truncated Fourier series, respectively, for the WangHaiXiang Bay (WB). (b) Observation area for the 2017 MVT experiment. Yellow lines indicate the acoustic rays collected using two mobile and one moored transceivers.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    Depth-averaged current fields for (a) flood and (b) ebb tides modeled by the KCOM.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    Spatial patterns of the selected OMA modes. (a),(b) The Dirichlet modes ψ6 and ψ23, (c),(d) the Neumann modes ϕ12 and ϕ31, and (e),(f) the boundary modes ϕ7b and ϕ6b. The background color is nondimensional, with dark blue to off-white to dark red representing negative to zero to positive values. All panels use the color bar next to (f). The arrows indicate nondimensional velocity vectors. The green and magenta solid lines indicate the ray distribution in the observation region bounded by the red rectangle.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    Diagonal elements of model resolution matrix for the Dirichlet (D), Neumann (N), and boundary (B) modes indicated by circle, triangle, and square symbols, respectively.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    Time series of (a) the tide height (solid line) and the spatial average of squared current magnitude 〈|v|2〉 (dashed line) output from the ocean model in the study area and (b) the fractional-residual-error variances (FREVs) for the reconstructed currents using the representations of the truncated Fourier series (dotted line), the OMA-Dirichlet (dashed line), and the OMA-all modes (solid line).

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    Simulated tomographic reconstructions using the synthetic data for (a) 7 and (b) 16 h. For each hour, the columns show the reconstruction results (black arrows) using the representations of (left) the truncated Fourier series, (center) the OMA-Dirichlet modes, and (right) OMA-all modes. Red arrows indicate the true currents. The background color denotes the difference between the true and reconstruction.

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    Reconstructions using the data collected in the 2017 MVT experiment. (top) The inverted currents (red arrows) using the DTT data for the ocean currents represented by (a) the truncated Fourier series, (b) the OMA-Dirichlet modes, and (c) the OMA-all modes. (bottom) The OMA-all reconstructions using (d) the ADCP data and (e) both the DTT and ADCP data. The background color denotes the predictive uncertainty in standard deviation (STD). Only the current estimate with a STD of less than 10 cm s−1 is displayed by the red arrow on the map. The white arrows and white lines indicate the ADCP measurements and the ray paths for the DTTs, respectively.

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    (a) Solenoidal and (b) irrotational flows estimated from the optimal estimate of the current field shown in Fig. 7e with the background color indicating the corresponding relative vorticity and divergence fields.

Fig. 1.

(a) Model domain of the KCOM. Background colors indicate the bathymetry with a contour interval of 10 m. The purple star near the Keelung Harbor (KH) indicates the location of Keelung Tidal Station. Blue solid and dashed lines indicate computational domains of the OMA and truncated Fourier series, respectively, for the WangHaiXiang Bay (WB). (b) Observation area for the 2017 MVT experiment. Yellow lines indicate the acoustic rays collected using two mobile and one moored transceivers.
Fig. 2.

Depth-averaged current fields for (a) flood and (b) ebb tides modeled by the KCOM.
Fig. 3.

Spatial patterns of the selected OMA modes. (a),(b) The Dirichlet modes ψ6 and ψ23, (c),(d) the Neumann modes ϕ12 and ϕ31, and (e),(f) the boundary modes ϕ7b and ϕ6b. The background color is nondimensional, with dark blue to off-white to dark red representing negative to zero to positive values. All panels use the color bar next to (f). The arrows indicate nondimensional velocity vectors. The green and magenta solid lines indicate the ray distribution in the observation region bounded by the red rectangle.
Fig. 4.

Diagonal elements of model resolution matrix for the Dirichlet (D), Neumann (N), and boundary (B) modes indicated by circle, triangle, and square symbols, respectively.
Fig. 5.

Time series of (a) the tide height (solid line) and the spatial average of squared current magnitude 〈|v|2〉 (dashed line) output from the ocean model in the study area and (b) the fractional-residual-error variances (FREVs) for the reconstructed currents using the representations of the truncated Fourier series (dotted line), the OMA-Dirichlet (dashed line), and the OMA-all modes (solid line).
Fig. 6.

Simulated tomographic reconstructions using the synthetic data for (a) 7 and (b) 16 h. For each hour, the columns show the reconstruction results (black arrows) using the representations of (left) the truncated Fourier series, (center) the OMA-Dirichlet modes, and (right) OMA-all modes. Red arrows indicate the true currents. The background color denotes the difference between the true and reconstruction.
Fig. 7.

Reconstructions using the data collected in the 2017 MVT experiment. (top) The inverted currents (red arrows) using the DTT data for the ocean currents represented by (a) the truncated Fourier series, (b) the OMA-Dirichlet modes, and (c) the OMA-all modes. (bottom) The OMA-all reconstructions using (d) the ADCP data and (e) both the DTT and ADCP data. The background color denotes the predictive uncertainty in standard deviation (STD). Only the current estimate with a STD of less than 10 cm s−1 is displayed by the red arrow on the map. The white arrows and white lines indicate the ADCP measurements and the ray paths for the DTTs, respectively.
Fig. 8.

(a) Solenoidal and (b) irrotational flows estimated from the optimal estimate of the current field shown in Fig. 7e with the background color indicating the corresponding relative vorticity and divergence fields.
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