Editorial Type:
Article
Article Type:
Research Article

A Noninterpolated Estimate of Horizontal Spatial Covariance from Nonorthogonally and Irregularly Sampled Scalar Velocities

Jang Gon Yoo Environmental Fluid Mechanics Laboratory, Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Yuseong-gu, Daejeon, South Korea

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Sung Yong Kim Environmental Fluid Mechanics Laboratory, Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Yuseong-gu, Daejeon, South Korea

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Bruce D. Cornuelle Climate, Atmospheric Science and Physical Oceanography, Scripps Institution of Oceanography, La Jolla, California

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P. Michael Kosro College of Earth, Ocean, and Atmospheric Sciences, Oregon State University, Corvallis, Oregon

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Alexander L. Kurapov College of Earth, Ocean, and Atmospheric Sciences, Oregon State University, Corvallis, Oregon

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Print Publication:
01 Nov 2017
DOI:
https://doi.org/10.1175/JTECH-D-17-0100.1
Page(s):
2407–2430
Full access

Abstract

This paper presents a least squares method to estimate the horizontal (isotropic or anisotropic) spatial covariance of two-dimensional orthogonal vector components, without introducing an intervening mapping step and biases, from the spatial covariance of the nonorthogonally and irregularly sampled raw scalar velocities. The field is assumed to be locally homogeneous in space and sampled in an ensemble so the unknown spatial covariance is a function of spatial lag only. The transformation between the irregular grid on which nonorthogonal scalar projections of the vector are sampled and the regular orthogonal grid on which they will be mapped is created using the geometry of the problem. The spatial covariance of the orthogonal velocity components of the field is parameterized by either the energy (power) spectrum in the wavenumber domain or the lagged covariance in the spatial domain. The energy spectrum is constrained to be nonnegative definite as part of the solution of the inverse problem. This approach is applied to three example sets of data, using nonorthogonally and irregularly sampled radial velocity data obtained from 1) a simple spectral model, 2) a regional numerical model, and 3) an array of high-frequency radars. In tests where the true covariance is known, the proposed direct approaches fitting to parameterizations of the nonorthogonally and irregularly sampled raw data in the wavenumber domain and spatial domain outperform methods that map the data to a regular grid before estimating the covariance.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Sung Yong Kim, syongkim@kaist.ac.kr

Abstract

This paper presents a least squares method to estimate the horizontal (isotropic or anisotropic) spatial covariance of two-dimensional orthogonal vector components, without introducing an intervening mapping step and biases, from the spatial covariance of the nonorthogonally and irregularly sampled raw scalar velocities. The field is assumed to be locally homogeneous in space and sampled in an ensemble so the unknown spatial covariance is a function of spatial lag only. The transformation between the irregular grid on which nonorthogonal scalar projections of the vector are sampled and the regular orthogonal grid on which they will be mapped is created using the geometry of the problem. The spatial covariance of the orthogonal velocity components of the field is parameterized by either the energy (power) spectrum in the wavenumber domain or the lagged covariance in the spatial domain. The energy spectrum is constrained to be nonnegative definite as part of the solution of the inverse problem. This approach is applied to three example sets of data, using nonorthogonally and irregularly sampled radial velocity data obtained from 1) a simple spectral model, 2) a regional numerical model, and 3) an array of high-frequency radars. In tests where the true covariance is known, the proposed direct approaches fitting to parameterizations of the nonorthogonally and irregularly sampled raw data in the wavenumber domain and spatial domain outperform methods that map the data to a regular grid before estimating the covariance.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Sung Yong Kim, syongkim@kaist.ac.kr
Keywords: In situ oceanic observations; Radars/Radar observations; Fourier analysis; Inverse methods; Statistical techniques

1. Introduction

Covariances of Earth processes are often used for mapping of irregular or gappy data (e.g., Moffet 1968; Bretherton et al. 1976; Wilkin et al. 2002; Kim et al. 2007; Babu and Stoica 2010). Estimates of scalar covariances can be made from irregular data by bin averaging (e.g., Roemmich and Gilson 2009), but for vector fields that are sampled nonorthogonally and irregularly, the bin averaging of observations requires more care. Here, nonorthogonally means orthogonal projections of the vector are not available, and irregularly means the sample spacing is irregular and independent vector projections are not available at the same point. For instance, the raw scalar observations from high-frequency radar (HFR) are (nonorthogonal) radial projections of a two-dimensional vector current field with respect to bearing angles at irregularly sampled radial grid points (red and blue arrows in Fig. 1). In the past, radial velocities at a diversity of angles in a region were first mapped to a regular grid of vector currents and then the covariances of these orthogonal vector data (or currents) were formed in a standard way (e.g., Kim et al. 2007; Roesler et al. 2013). Mapping before averaging to make the covariance will incorporate the bias and error of mapping into the covariance statistics. For this reason, it is better to average before mapping as opposed to mapping before averaging. We explore a method for computing covariances of nonorthogonally and irregularly sampled velocity data in a region without a mapping step, which will be called “direct” estimation. The results will be compared to the covariance of the true velocity field and will be shown to avoid some sources of bias in the mapped estimates.

Fig. 1.
Fig. 1.

An example of radial velocity maps (red and blue arrows) nonorthogonally and irregularly sampled from an idealized model vector current field (black arrows). The sampling radial grid points at individual radar sites (black circles; R1 and R2) are marked with colored dots, which are not collocated, and the sampling area of the vector currents is indicated with a black box.

Citation: Journal of Atmospheric and Oceanic Technology 34, 11; 10.1175/JTECH-D-17-0100.1

In oceanographic data analysis, it is challenging to derive orthogonal vector components and their covariances from nonorthogonally and irregularly sampled velocity data (e.g., Chelton and Schlax 2003; Shay et al. 2007; Kaplan and Lekein 2007; Kim et al. 2008; Yaremchuk and Sentchev (2009, 2011). Kim et al. (2008) investigated the mapping of nonorthogonally and irregularly sampled radial velocity maps obtained from HFRs into a vector current field in terms of the structure of the correlation functions. In particular, the correlation function (or covariance function) has been assumed to be Gaussian or exponential with parameters estimated from the mapped vector current fields. In a similar context, Shay et al. (2007), Kaplan and Lekein (2007), and Yaremchuk and Sentchev (2009, 2011) have reported techniques to map HFR-derived radial velocity maps into the vector current fields using a least squares fit and to generate gap-free vector current fields using mode analysis and variational analysis using dynamic constraints, such as divergence and curl of the given current fields. Conversely, the spatial covariance in the spatial domain and/or the energy spectrum in the wavenumber domain are estimated from irregularly spaced observations with a least squares fit (e.g., Zetler et al. 1965; Wunsch 1996) and expectation maximization (Dempster et al. 1977; Do and Batzoglou 2008; Rubin and Szatrowski 1982). The expectation maximization method iteratively estimates missing data in otherwise regular sampling by maximum likelihood estimation using the current estimate of the covariance, recomputing the covariance until the estimates for the missing data have converged. Additionally, the multitaper method (Percival and Walden 1993; Erdöl and Günes 2005) is applicable to regularly spaced observations in the energy spectrum estimate, and it minimizes the leakage of spectral contents by multiplying the orthogonal tapers to the data. Although a least squares fit has been considered expensive because of the size of the matrices involved, present-day computing resources permit least squares fitting in practical applications.

This paper adapts a standard form of spectral estimation (e.g., Mills et al. 1958, 1960, 1961) to directly compute the spatial covariance of a two-dimensional vector field from nonorthogonally and irregularly sampled radial velocities by converting the sample covariance of the radial velocity measurements. This is compared to mapping from radial velocities to the two-dimensional vector field and then computing covariances from the mapped field. To allow for spatially varying variance, the correlation is used, which is the covariance normalized by the standard deviations at the two points, and the estimate of covariance in this work is framed as the estimate of the spatial correlation function.

The methodological descriptions of the covariance estimates for one-dimensional scalar data and two-dimensional vector data are presented in sections 2 and 3, respectively. The proposed method is applied to the data obtained from a simple spectral model, a numerical model, and observations of HFR-derived surface radial velocity maps. The discussion and conclusions follow in sections 4 and 5, respectively. Note that this paper focuses on the direct estimates of spatial covariance without introducing an intervening step of vector current mapping, and we do not discuss the vector current mapping in the rest of paper. The performance of vector current mapping has been reported elsewhere (e.g., H. S. Soh et al. 2017, manuscript submitted to J. Atmos. Oceanic Technol. Kim et al. 2008; Shay et al. 2007; Chapman et al. 1997).

2. One-dimensional covariance of scalar data

a. Formulation

As an introduction, we present an example of estimating the spatial covariance (or correlation) of a scalar field in a one-dimensional spatial domain. Assuming ergodicity, an ensemble of assumed independent realizations of the data field over time is generated from the true correlation at unevenly spaced but unchanging locations ( or ; p or q = 1, 2, … , L, where and are the two points for the correlation sample, and L denotes the number of sampling points in space). Each realization, labeled by a time, is generated by a sum of randomized spectral components [Eq. (1)], so the scalar data at times (, r = 1, 2, , R, where R denotes the number of realizations) are given by the Fourier series expansion:
e1
e2
where indicates the basis functions () and denotes a set of the amplitudes given by zero-mean Gaussian random variables [] sampled at each realization . The symbol † denotes the matrix transpose, and is the square root of the true energy spectrum of the data field [] at the wavenumber of (where Lx is the domain size), which is converted from the true covariance (or correlation) of the field [see Eq. (3)] and ϵ denotes the variability that is uncorrelated between samples, even at very short length scales, usually interpreted as white noise in the data (e.g., Gaussian random noise). The shape of the true spectrum (or correlation) is chosen to represent realistic cases.
The conversion between a homogeneous covariance function [] in the spatial domain (x) and an energy spectrum [] in the wavenumber domain (k) (e.g., Brigham 1988; Cohen 1992; Oppenheim et al. 1998) is done by a Fourier transform:
e3
For instance, an exponential covariance function in the one-dimensional spatial domain corresponds to a one-dimensional wavenumber energy spectrum with a slope of :
e4
where λ denotes the decorrelation length scale, which is defined as an e-folding scale ].

b. Covariance estimate

If the data are ergodic and stationary in time and the associated processes are locally homogeneous in space, the one-dimensional covariance can be estimated either in the wavenumber domain using a least squares fit [see Colosi et al. (2013) for an example] in the spatial domain using bin averaging, which is a form of least squares; or as a hybrid, using both spectral and local spatial parameterizations, whichever is simplest. Here we will use both spectral and binning methods.

Assuming homogeneity, the sample data covariance matrix in space [] is computed in the spatially lagged domain (, where ) by ensemble averaging the observed point-to-point covariances over realizations, and is represented by a one-dimensional covariance function []:
e5
e6
e7
where p and q are the sampling indices (p, ), and is the point-to-point covariance of the short-scale processes, sometimes called the noise covariance. This could be set to zero so that all correlations must be described by the spectral representation, but this is inconvenient for white noise, which requires an infinite spectrum but is only a single parameter in lag space. In this case we will assume that , so it is zero except at zero lag. This could be generalized to include correlation parameters in lag space for a few very short lags to model significant decorrelation at short lags and to model the longer scales in spectral space. In the wavenumber domain, the one-dimensional sample correlation function [] is modeled with trigonometric basis functions and coefficients (), estimated using a least squares fit:
e8
where is the wavenumber (, j = 1, 2, , J) and J is the number of basis functions. A term denotes the uncertainty in the averaged correlation data, and is the delta function. If the observations are regularly spaced, have no statistical uncertainty, and are chosen properly, the least squares fit becomes a discrete Fourier transform. As the correlation function is symmetric for a locally homogeneous real field, the sine basis functions are excluded and only cosine basis functions are considered. The optimal solutions can be found using a standard weighted and regularized least squares fit. The derivation of this least squares solution to the constrained optimization problem [Eq. (8)] is not included in this paper, as it is described in many places.
The estimated coefficients () in the cosine series together with the localized correlation then can be used to calculate the estimated correlation in the spatial lag domain. We show it at regularly spaced points ():
e9
where denotes the estimated white noise in the data (correlated only at zero lag). The uncertainty in these estimates can be calculated using the least squares formalism.

To estimate the correlation purely in the spatial domain, the spatial correlations can be represented by a sum of local functions such as splines. The simplest model assumes a constant correlation within each region (bin) of a set defined in lag space, so the parameters are the correlation values in each bin. In these examples, the bins are regularly spaced in lag space (), as defined above. The parameters (correlations in each bin) are estimated by binned averages of the spatial lags [] falling into each bin. The bin-averaged one-dimensional correlation function [] has higher statistical uncertainty for the bins with fewer samples given equal variances, which tend to be at longer lags. This uncertainty obscures the true covariance.

c. Data generation

Using a simple spectral model as in Eq. (4), a true spatial correlation function having a decorrelation length scale of λ and exponentially decaying shape ( km; Fig. 2a) in a continuous one-dimensional spatial domain corresponds to a wavenumber energy spectrum []:
e10
where denotes the mth wavenumber.
Fig. 2.
Fig. 2.

(a) A true correlation function () with an exponentially decaying shape of a decorrelation length scale of 2 km [ km in Eqs. (4) and (10)]. (b) Five examples (, , , , and ) of the irregularly sampled scalar data at 250 locations in the spatial domain ( and ), which are generated under the given true correlation function in (a), are presented as a function of the sampling index (p or q; ). (c) One-dimensional correlations of the sampled data are presented as a function of the sampling index (p and q). (d) The sampling locations ( and ) and the sampling index (p or q). (e) Spacing () between sampling locations as a function of the sampling index (p or q).

Citation: Journal of Atmospheric and Oceanic Technology 34, 11; 10.1175/JTECH-D-17-0100.1

Using the wavenumber-domain energy spectrum [Eqs. (2) and (10)], the data are generated (or sampled) at 250 unevenly spaced locations [Eq. (1), L = 250] with 500 realizations (R = 500) with zero short-scale variability [ in Eq. (1)]. This sampling is sufficient to characterize the data field. A discussion of the effects of the number of spatial samples and realizations is presented in section 2e. For instance, only five realizations are shown in Fig. 2b as a function of the sampling index. The sample data covariance (or correlation) matrix is calculated (Fig. 2c), and its off-diagonal terms and corresponding spatial lags (; Fig. 2e) are used to estimate the one-dimensional correlation. The diagonal (zero lag) terms are not used in order to avoid having to estimate the short-scale component ϵ. The spatial sampling locations and sampling indices are shown in Fig. 2d.

d. Estimated one-dimensional correlation functions

The coefficients () are estimated by a weighted, regularized least squares fit with a constraint that the coefficients are nonnegative because they are the wavenumber-domain energy spectrum of the data field. The correlation function [ in Eq. (9)] in the regularly spaced domain () (Fig. 3a) is computed from the estimated coefficients (). Alternatively, the correlation functions [ in Fig. 3b] can be estimated purely in lag space by bin-averaging off-diagonal terms of the sample data correlation matrix (Fig. 2c). In this case the energy spectrum is not required to be nonnegative definite. The error bars in Fig. 3b show the standard error of the averages at individual spatial bins. In both estimates, the off-diagonal terms (e.g., the lower or upper diagonal terms) of the sample data correlation matrix are used only once because of symmetry of the covariance matrix.

Fig. 3.
Fig. 3.

(a) An estimated correlation function using a parameterization in the wavenumber domain (). (b) An estimated correlation function using a parameterization in the spatial domain (bin averaging) (). Error bars correspond to standard errors for individual spatial bins. (c) A comparison of the true correlation function (; km) and the estimated correlation functions (, , and ), where denotes the correlations of the optimally interpolated data from irregularly sampled data on the regular grid (see section 2f for more details). The estimated decorrelation length scales using parameterizations in the wavenumber and spatial domains are 2.03 and 2.09 km, respectively ( km; km). The optimal interpolation is conducted with an exponential correlation function with a decorrelation length scale of 1.5 km. The estimated correlation function () has a Gaussian shape, different from that of the true correlation function, and a decorrelation length scale of 2.78 km ( km) associated with the accumulated bias.

Citation: Journal of Atmospheric and Oceanic Technology 34, 11; 10.1175/JTECH-D-17-0100.1

The true correlation function and the estimated correlation functions by a least squares fit and bin averaging (, , and , respectively, in Fig. 3c) have comparable decorrelation length scales within 3% difference ( km and km), which demonstrates the feasibility of estimating the one-dimensional correlation functions with either method. The bias in the correlation functions, which are estimated from the gridded data using a biased estimator (e.g., optimal interpolation), will be highlighted in section 2e. The covariance estimates from nonorthogonal and irregular observations will be evaluated for a two-dimensional vector field in section 3 that is more realistic than the one-dimensional case.

e. Influence of the number of realizations, the density of unevenly spaced samples, and the noise

The number of (temporal) realizations (R) and the density of unevenly spaced samples—that is, the number of spatial samples (L) within the domain—can affect the performance of the correlation estimates. For example, increasing the number of realizations reduces statistical noise, while coarsely spaced samples can degrade the covariance estimate because of the lack of information to characterize the data field at short lags. To examine these effects, the performance of the proposed approach is evaluated with a comparison of averaging of mapped fields and mapping of an averaged field (section 2f).

Although estimates with an infinite number of realizations have zero statistical noise, we chose 25 cases varying the number of realizations (R = 667, 695, , 50 000) and three cases varying of the number of spatial samples (L = 125, 250, and 500) to evaluate the performance in reconstruction of the one-dimensional covariance (Fig. 4). All estimates are based on the spatial lags between the nearest sampling locations and the corresponding correlations, obtained from the first off-diagonal terms of the covariance matrix; that is, the off-diagonal terms of the data covariance matrix at the spatial lags with only adjacent sampling locations (white-circled components in Fig. 5) to save calculation time by ignoring longer spatial lags, although these would have improved the estimate. If bin averaging is used to estimate the spatial correlations, coarser sampling requires larger bin sizes, or there may be bins without samples. The ensemble mean of differences between expected (true) correlations and sample correlations at individual bins and their standard deviations show that coarser sampling and fewer realizations lead to more unstable estimates and higher uncertainty (Fig. 4) (see section 4 for a discussion on the statistical noise and reconstruction error). Thus, we chose the number of realizations as 50 000 () and the number of samples as 250 (), which is enough to characterize the data field and to reconstruct the given one-dimensional covariance perfectly (section 2d).

Fig. 4.
Fig. 4.

(a),(c),(e) The sample correlations (colored crosses) and true correlations (black line) of the sampled data as a function of the number of randomly chosen realizations (R; R = 667, , 8334, 16 667, and 50 000) under a given number of samples (L). (b),(d),(f) The sample correlations are bin averaged at individual sampling bins () and presented as a function of the number realizations (x axis has a log scale unit). The number of samples is (a),(b) 500; (c),(d) 250; and (e),(f) 125. Note that all estimates are based on the correlations and spatial lags between the nearest sampling locations, which are obtained from the first off-diagonal terms of the square matrix.

Citation: Journal of Atmospheric and Oceanic Technology 34, 11; 10.1175/JTECH-D-17-0100.1

Fig. 5.
Fig. 5.

A lower triangle of a square covariance matrix [; Eqs. (13)–(15)], including diagonal terms, is marked with colored column vectors [Eq. (39)] that are concatenated into a single-column vector [; Eq. (16)] via vectorization. Cross-covariance terms [Eq. (37)] at the adjacent spatial lags of the square covariance matrix are denoted with white-circled components. The diagonal terms [Eq. (36)] of the covariance matrix are indicated with black-circled components, and its off-diagonal terms [Eq. (38)] are denoted with both white-circled and noncontoured components. The indices of the diagonal terms, the off-diagonal terms at the adjacent spatial lags, the off-diagonal terms, and the lower-triangle terms of the square covariance matrix can be found in Eqs. (36)–(39), respectively.

Citation: Journal of Atmospheric and Oceanic Technology 34, 11; 10.1175/JTECH-D-17-0100.1

The sensitivity of the one-dimensional correlation estimate to the short-scale variability (“noise”) in the data is examined with the estimated decorrelation length scale of the correlation function, computed from the data sampled at 250 unevenly spaced locations with 50 000 realizations ( and ), as a function noise level; that is, the inverse of the ratio of the variance to noise [κ, in Eq. (8); Fig. 6]. When the data have nonzero white noise [ in Eq. (1)], there is a discontinuous drop of correlation between zero and the first nonzero spatial lag. Thus, when we include the correlations at the zero lag (; ) in the estimates of the one-dimensional correlation function using parameterizations in the wavenumber and spatial domains, we must include the ϵ parameter in the fit, or spurious correlation estimates can appear at the zero lag (Fig. 6a). As mentioned above, we estimate the correlation functions by excluding the correlations at zero lag (Fig. 6b) and compute the decorrelation length scales of the correlation functions from the nonzero lags only (Fig. 6c), which makes the estimate insensitive to the amount of short-scale variability (noise) in the true correlation function. Both estimates of the correlation functions ( and ) have a consistent exponential shape. Their decorrelation length scales ( and ) are nearly consistent with the true decorrelation length scale, and they are nearly constant regardless of the amount of noise (Fig. 6d), confirming the insensitivity to the noise in the data. However, the estimated correlation function () computed from the mapped data has a Gaussian shape, and its decorrelation length scale () is approximately 25%–30% longer than the true decorrelation length scale (Fig. 6d).

Fig. 6.
Fig. 6.

(a)–(c) An example of the one-dimensional true correlations (, km) and the estimated correlation functions using parameterizations in the wavenumber domain () and spatial domain ( and ) from the data having nonzero noise [, where in Eq. (8)]. (a) Estimated correlations when including perfect correlations at the zero lag (; ). (b) Estimated correlations when excluding perfect correlations at the zero lag (; ). (c) Estimated correlations when excluding perfect correlations at the zero lag (; ) and scaling up the estimated correlations with the maximum value of correlations at the zero lag. (d) Estimated decorrelation length scales (, , and ) as a function of the noise level (κ), which is an inverse of the ratio of the signal amplitude to noise. In (a) and (b), α and β denote the noise level of the data by including and excluding the perfect correlation at the zero lag, respectively.

Citation: Journal of Atmospheric and Oceanic Technology 34, 11; 10.1175/JTECH-D-17-0100.1

f. A comparison of averaging of mapped fields and mapping of an averaged field

In estimating the correlation of the irregularly sampled data on an evenly spaced grid, the order of computations in mapping the data and in estimating the correlation matters. In other words, the bias in the mapping of data on a regular grid is accumulated in the correlation estimate when the data are mapped before averaging. Thus, we compare the following two estimated correlations: 1) mapped correlations of the irregularly sampled data into a regularly spaced grid using a least squares fit () and 2) correlations of the mapped data on the regular grid from irregularly sampled data (). Both estimates are based on the same number of realizations and the same number of samples (section 3e). To compare two estimates in a reasonable way, an exponential correlation function (ϱ) having a decorrelation length scale of 1.5 km is used to map the irregularly sampled data for and irregularly estimated correlations for on the regular grid ():
e11
where is the subset of the irregularly sampled data or irregularly estimated correlations participating in the correlation estimate () at the ith regular grid point ( and ), and denote the distances between the sampling locations and the ith grid point and between the sampling locations, respectively, and and indicate the model and error variances of the field, respectively.

The bias embedded in the mapped data field is clearly visible when it is compared with the true correlation () and other estimated correlations ( and ) (Fig. 3c). The estimated correlation function () has a Gaussian shape, different from that of the true correlation function, and its decorrelation length scale is equal to 2.78 km ( km), which can be associated with the accumulated bias.

Then, we extend the proposed approach into the estimation of two-dimensional (horizontal) correlation functions in the following section.

3. Two-dimensional covariance of vector data

a. Formulation

1) A geometric relationship from radial grids to vector grids

To estimate a two-dimensional horizontal covariance of the orthogonal vector fields directly from nonorthogonally and irregularly sampled scalar radial velocities, we use the multiple HFR-derived radial velocity maps as an example (Fig. 1). The radial velocity (r) at a radial grid point with a bearing angle (θ) is a sum of projections of two orthogonal components (u and υ) onto that angle plus an observational error (ϵ) at a given time (t):
e12
where is the directional unit vector, θ is the bearing angle, and is the vector current at a location.
The estimate of covariance between radial velocities at two locations (p and q) is a geometric transformation of the covariances between orthogonal velocity components at the two locations:
e13
e14
e15
where and for the covariance estimate using diagonal and lower-triangle terms in the square radial covariance matrix (colored components in Fig. 5), and and for the variance estimate using only its diagonal terms (black-circled components in Fig. 5).
The expansion of each covariance term in Eq. (15) into a single-column vector (Fig. 5) becomes
e16
where , , , and are the matrices of the product of directional unit vectors to constitute the pair of bearing angles. The symbol refers to a column vector made by concatenating the columns of the lower triangle of a covariance matrix [see an example of and in Fig. 5].

2) Inverse methods

The least squares estimate of covariance (or variance) is formulated as an inverse problem:
e17
where includes both diagonal terms and lower-triangle terms of the square covariance matrix of radial velocities [left-hand side terms in Eq. (16)], and denotes the residuals. The unknown () is the vector of parameters determining the covariance matrix: either the energy spectrum of the vector current field in the wavenumber domain or the covariance of the vector field specified on a regular grid of spatial lags. For the spectral representation, the mapping matrix () consists of the trigonometric functions with respect to the bearing angles of the pairs of radial velocities in the wavenumber domain [, , , and in Eq. (16); see section 3b(1)]. For the spatial domain grid, the elements of the matrix are linear combinations of grid values weighted by spatial distance [section 3b(2)]. The sizes of vectors and matrices in Eq. (17) for the estimates in the wavenumber and spatial domains are summarized in Table 1.
Table 1.

Dimensions of vectors and matrices in the direct estimate of variance and covariance. In the wavenumber domain, M and N denote the number of basis functions in the x and y directions, respectively, and L indicates the number of radial velocities [Eqs. (20), (21), and (30)]. In the spatial domain, denotes the number of radial velocities within each spatial bin [Eqs. (31) and (32)].

Table 1.
The estimated coefficients () are given by
e18
e19
where and denote the prior covariances of the uncertainty in the estimated model parameters and in the observations, respectively, and they are nonnegative definite (e.g., Wunsch 2006). The estimated coefficients should be nonnegative based on their physical meaning as the energy spectrum of the vector current field in the wavenumber domain and its covariance at regular spatial lags in the spatial domain.

Although and have different dimensions, if one of them is assumed to be a scaled identity matrix, then the scaling factor is pulled out so that regularization depends on a matrix with a single parameter, which is an inverse of the signal-to-noise ratio diagonal matrices. For simplicity, we suggest some fraction of the mean eigenvalue of an inverting matrix ( or ) in Eqs. (18) and (19) (e.g., Kim et al. 2007).

b. Covariance estimates

1) Parameterization in the wavenumber domain

The horizontal vector components (u and υ) are modeled in the wavenumber domain at a given time (t):
e20
e21
where the wavenumber ( and ) is defined in the x and y directions, respectively:
e22
e23
with corresponding wavenumber indices of m and n ( and ), and denote the length of the domain in the x and y directions, respectively, and M and N denote the number of basis functions to describe the velocity field in two-dimensional space.
If the data field is locally homogeneous, for instance, then the spatial covariance between a u component at and a u component at is only a function of spatial lags (, and it is modeled in the wavenumber domain:
e24
where
e25
e26
In a similar way, the other covariance terms are modeled as
e27
e28
e29
Equations (24), and (27)–(29) are formulated using Eq. (17):
e30
where L denotes the number of radial velocities [see Table 1 for the sizes of vectors and matrices in Eq. (30)].

Then, the coefficients (, , , and ) are estimated using Eqs. (18) and (19) under a constraint that the coefficients should be nonnegative values because they are a squared quantity.

Then, the individual covariance terms are computed with the estimated coefficients (, , , and ) and Eqs. (24), (27), (28), and (29). The correlation functions estimated in the wavenumber domain contain a physically repeated shape because the covariance terms are modeled with Fourier basis functions. Thus, the size of the domain should be chosen to avoid the periodicity of the estimated spatial correlations.

2) Parameterization in the spatial domain

The individual covariance terms between velocity components (u and υ) are alternatively modeled by interpolation from a table of values at discrete and regular spatial lags (, ). For the interpolation the spatial correlation function between regular grid points is assumed to be piecewise constant, linear (bilinear in two dimensions), or polynomial (which enforces the continuity of spatial derivatives to some order) functions. For example, assuming a constant correlation within a given range of spatial lags is called bin averaging.

The covariance terms of the orthogonal vector components (, , , and ) at a regular grid point are estimated from the covariances of observed radial velocities within a two-dimensional bin corresponding indices of (m, n) ( and ), centered on a rectangular grid point (, ),
e31
e32
making equations in four unknowns (, , , and ) at each spatial bin.
Similarly, the unknown covariance terms in Eq. (16) are formulated using Eq. (17) as follows:
e33
[see Table 1 for the sizes of vectors and matrices in Eq. (33)]. As this estimate is based on the spatial domain, the complications of periodicity are absent.

c. Cautionary remarks on inverse methods

1) Cross-covariance terms

The off-diagonal terms of the square covariance matrix of the vector data are not typically identical (),
e34
However, the condition of identical cross-covariance terms () is applicable to estimate the variance using the diagonal terms and some of the off-diagonal terms of the square covariance matrix,
e35
Then, in the wavenumber domain, Eqs. (27) and (28) are merged into a single equation. Conversely, in the spatial domain, one of the rows of and in the column vector and the corresponding columns for or in Eqs. (30) and (33) needs to be ignored. Note that the off-diagonal terms of the square covariance matrix of the scalar data (b) are symmetric ().

2) Indices and prior smoothing

As the inverting matrix [ or in Eqs. (18) and (19)] is a symmetric matrix, it is convenient to use indices indicating the diagonal and off-diagonal terms in constituting a data vector () in Eq. (17). The indices of diagonal terms (; black-circled components in Fig. 5), off-diagonal terms at the adjacent spatial lags (; white-colored components in Fig. 5), off-diagonal terms (; white-circled and noncontoured components in Fig. 5), and a lower-triangle terms (; colored components in Fig. 5) in a square matrix () are
e36
e37
e38
e39
where and .
The prior model covariance () is smoothed into ,
e40
where and are the scaling factors in the wavenumber domain. The prior padding value (ς) is designed to avoid singularity in case of the inverse of the prior itself:
e41

d. Data generation

Horizontal spatial covariance functions are estimated with nonorthogonally and irregularly sampled scalar velocities obtained from three sets of resource, such as a simple spectral model, a regional numerical model, and an array of HFRs (see appendix A for a practical approach to estimate the covariance of huge datasets).

1) A simple spectral model

A simple spectral model is formed, based on the energy spectrum in the wavenumber domain or spatial covariance in the spatial domain, to generate the vector current fields. Particularly, the normalized structure of the spatial covariance function—that is, spatial correlation coefficients—is characterized with decorrelation length scales. For instance, the spatial correlation coefficients (ρ) at a local grid point (x, y) are expressed with an exponential function (e.g., Kim et al. 2007),
e42
or a Gaussian function,
e43
and the semimajor axis () of the spatial correlations is rotated into the relative angle of the semimajor axis in the original spatial lag axes ( and ) because the spatial correlation function [] is oriented in the different directions from the x and y axes.

2) A regional numerical model

Numerical simulations using the Regional Ocean Modeling System (ROMS) off the coast of Oregon provide maps of surface currents (at the surface layer) with hourly temporal and 2-km horizontal resolution for a period of about one year [August 2008–August 2009; see Kim et al. (2014) for more details], which implemented freshwater discharge of the Columbia River for a realistic simulation. A subset of the model domain is chosen to enclose the effective spatial coverage of an array of HFRs off the coast of Oregon (red boxes in Figs. 8a and 8b). The ROMS-derived (surface) currents well represent the variability of regional circulation, including variance in the low frequency ( cycles per day), the near-inertial frequency, and the barotropic and baroclinic tidal frequency bands (e.g., Osborne et al. 2011, 2014; Kim et al. 2014). Thus, the ROMS-simulated surface currents are taken as the true vector currents in this experiment.

3) Observations

The hourly averaged surface radial velocities and their optimally interpolated vector currents are provided by the array of HFRs off the coast of Oregon for a period of 2 years (2007–08) (e.g., Kosro 2005; Kim and Kosro 2013). Different from the radial velocity data constructed from the spectral model and numerical simulations below [sections 3e(1) and 3e(2), respectively], the observed radial velocities have missing data in both time and space as well as sampling and measurement errors. Moreover, the optimal interpolation (OI)-mapped vector currents include a mapping bias associated with an assumed covariance function and decorrelation length scales even though they were computed with an exponential correlation function having a relatively short decorrelation length scale to minimize spatial smoothing.

e. Estimated two-dimensional correlation functions

1) A simple spectral model

Based on a simple spectral model in the wavenumber domain, we generate vector current maps with 5000 realizations under a Gaussian correlation function on a regular grid and project them onto polar grid points with bearing angles of radars to yield radial velocity maps (see Table 2 for a detailed configuration of the spectral model). The number of realizations is chosen to capture the designed wavenumber-domain energy spectrum of the vector current field. A single snapshot of the vector current map and radial velocity maps sampled from two radars (R1 and R2) are shown in Fig. 1. Each realization is independent in time. However, we could modify the spectral model in the wavenumber domain to include variance in the frequency domain as well (H. S. Soh et al. 2017, manuscript submitted to J. Atmos. Oceanic Technol).

Table 2.

Definition of variables used in the model current simulation and covariance estimates.

Table 2.

The true correlation function () and the estimated correlation function () are compared in Fig. 7. The true correlation function has a Gaussian shape, and its decorrelation length scales are 4 and 3 () and 4 and 5 km () in the x and y directions, respectively (Figs. 7a and 7b).

Fig. 7.
Fig. 7.

(a),(b) Assumed two-dimensional true (Gaussian spatial) correlation functions with zero cross-correlation terms between vector components (): (a) ( km; km) and (b) ( km; km). (c)–(f) Estimated correlations () using a parameterization in the wavenumber domain [section 3b(1)]. (g)–(j) Estimated correlations () using a parameterization in the spatial domain [section 3b(2)]. The following terms are shown: (c),(g) ; (d),(h) ; (e),(i) ; and (f),(j) .

Citation: Journal of Atmospheric and Oceanic Technology 34, 11; 10.1175/JTECH-D-17-0100.1

The autocorrelation terms ( and ) of the estimated spatial correlation functions using wavenumber-domain parameterization have very similar shapes compared with those of true correlation functions, having approximately 2% difference in the decorrelation length scales when they are estimated as the e-folding scale [i.e., ] (Figs. 7c and 7d). Although the cross-correlation terms ( and ) of the true correlation functions do not exist, the estimated cross-correlation terms ( and ) are nearly zeros when the single cross-covariance term () is assumed (not shown), and they have weak spatial structures when the two cross-covariance terms are assumed () [see section 3c(1)] (Figs. 7e and 7f). The estimated correlation functions using spatial domain parameterization (Figs. 7g–j) have consistent spatial structure compared with those obtained from wavenumber-domain parameterization even though the data have coarse spatial resolutions as a result of the sampling resolution in the spatial domain. On the contrary, the decorrelation length scales of the estimated correlation functions ( and ) using bin averaging may not be easily quantified because the estimated correlations can have spatial structures instead of simply decaying structure in space. In other words, the estimated correlations may have multiple crossings [] and may have values higher than for all spatial lags (Figs. 7g and 7h). These spatial domain estimates of the cross-correlation terms also show a similar tendency in terms of the equality and inequality of cross-covariance terms (Figs. 7e, 7f, 7i, and 7j).

2) A regional numerical model

The spatial correlation functions of ROMS-simulated surface currents near two grid points (A and B; see Fig. 8c.) are estimated from 1) the bin averaging of spatial correlations of vector currents () (Figs. 9a–d and 10a–d, respectively), and covariances of radial velocities using parameterization in the 2) wavenumber domain () (Figs. 9e–h and 10e–h, respectively) and 3) spatial domain (bin averaging) () (Figs. 9i–l and 10i–l, respectively).

Fig. 8.
Fig. 8.

A study domain of the ROMS-simulated surface currents and HFR-derived surface current observations off the coast of Oregon. (a),(b) A ROMS domain and its spatial subset in a coastal region that encloses the effective spatial coverage of the operational HFRs (a yellow curve). The bottom bathymetry is contoured with 50, 100, 250, 500, 1000, 1500, 2000, 2500, and 3000 m. As a reference, major coastal regions are denoted by abbreviated two letter names from south to north: Cape Blanco (CB), Winchester Bay (WB), Newport (NP), and Loomis Lake (LL). (c) Sampling radial grid points of the operational HFRs (gray dots) and the radial grid points (black dots) within 24 km of point A (red dot; 46°N, 124.5°E) and point B (red dot; 44°N, 124°E). Two red rectangular boxes in (a) and (b) indicate the same region. All the radial grid points within the red box in (b) are shown in (c) (gray dots).

Citation: Journal of Atmospheric and Oceanic Technology 34, 11; 10.1175/JTECH-D-17-0100.1

Fig. 9.
Fig. 9.

Spatial correlations of the ROMS-simulated surface currents at point A in Fig. 8c are estimated from (a)–(d) the bin averaging of the vector currents (), (e)–(h) the covariances of radial velocities using a parameterization in the wavenumber domain () [section 3b(1)], and (i)–(l) the covariances of radial velocities using a parameterization in the spatial domain (bin averaging) () [section 3b(2)]. The following terms are shown: (a),(e),(i) ; (b),(f),(j) ; (c),(g),(k) ; and (d),(h),(l) . The shared color bar on the bottom ranges from 0 to 1 for and , and from −0.5 to 0.5 for and .

Citation: Journal of Atmospheric and Oceanic Technology 34, 11; 10.1175/JTECH-D-17-0100.1

Fig. 10.
Fig. 10.

Spatial correlations of the ROMS-simulated surface currents at point B in Fig. 8c are estimated from (a)–(d) the bin averaging of the vector currents (), (e)–(h) the covariances of radial velocities using a parameterization in the wavenumber domain () [section 3b(1)], and (i)–(l) the covariances of radial velocities using a parameterization in the spatial domain (bin averaging) () [section 3b(2)]. The following terms are shown: (a),(e),(i) ; (b),(f),(j) ; (c),(g),(k) ; and (d),(h),(l) . The shared color bar on the bottom ranges from 0 to 1 for and , and from −0.5 to 0.5 for and .

Citation: Journal of Atmospheric and Oceanic Technology 34, 11; 10.1175/JTECH-D-17-0100.1

In the comparison of spatial correlation terms ( and ), the overall shape of the correlation functions and their directional dominance are consistent, although the estimated correlation functions using bin averaging ( and ) have a bit reduced decorrelation length scales (Figs. 9i and 9j, respectively) and those obtained from wavenumber-domain parameterization ( and ) have a more significant portion of weak correlation (less than 0.4; blue, Figs. 9e and 9f, respectively) compared with the correlation functions estimated from the bin averaging ( and ) (Figs. 9a and 9b, respectively).

The cross-correlation terms are estimated with two ways of assumptions on the cross-covariance terms [see section 3c(1)]. The cross-correlation terms have mostly positive values (less than 0.3, exceptionally 0.5 in larger spatial lags) and do not exhibit the consistent spatial structures between cross terms of the estimated correlation functions (the third and fourth columns in Figs. 9 and 10).

3) Observations

The spatial correlation functions using observations are estimated in a similar way, conducted in section 3e(2): 1) the bin averaging of the OI-mapped vector currents (Figs. 11a–d and 12a–d), and the covariance of radial velocities using parameterizations in the 2) wavenumber domain (Figs. 11e–h and 12e–h) and 3) spatial domain (Figs. 11i –l and 12i–l).

Fig. 11.
Fig. 11.

Spatial correlations of the HFR-derived surface currents at point A in Fig. 8c are estimated from (a)–(d) the bin averaging of the vector currents (), (e)–(h) the covariances of radial velocities using a parameterization in the wavenumber domain () [section 3b(1)], and (i)–(l) the covariances of radial velocities using a parameterization in the spatial domain (bin averaging) () [section 3b(2)]. The following terms are shown: (a),(e),(i) ; (b),(f),(j) ; (c),(g),(k) ; and (d),(h),(l) . The shared color bar on the bottom ranges from 0 to 1 for and , and from −0.5 to 0.5 for and .

Citation: Journal of Atmospheric and Oceanic Technology 34, 11; 10.1175/JTECH-D-17-0100.1

Fig. 12.
Fig. 12.

Spatial correlations of the HFR-derived surface currents at point B in Fig. 8c are estimated from (a)–(d) the bin averaging of the vector currents (), (e)–(h) the covariances of radial velocities using a parameterization in the wavenumber domain () [section 3b(1)], and (i)–(l) the covariances of radial velocities using a parameterization in the spatial domain (bin averaging) () [section 3b(2)]. The following terms are shown (a),(e),(i) ; (b),(f),(j) ; (c),(g),(k) ; and (d),(h),(l) . The shared color bar on the bottom ranges from 0 to 1 for and , and from −0.5 to 0.5 for and .

Citation: Journal of Atmospheric and Oceanic Technology 34, 11; 10.1175/JTECH-D-17-0100.1

The bin averaging () of spatial correlations of the OI-mapped vector currents exhibits discrete spatial structures depending on the resolution of the vector current maps (the first row in Figs. 11 and 12). Moreover, the spatial correlations () estimated using spatial domain parameterization also depend on the available radial velocities and spatial distribution of spatial lags (the second row in Figs. 11 and 12). Thus, both estimates have limitations in the precise estimation of spatial correlation functions. The spatial correlations () estimated using wavenumber-domain parameterization are considered to be better estimates compared with the other two ( and ), based on the spatial resolution and the spatial structure of the correlations estimated from numerical model outputs (Figs. 9e and 9f, 10e and 10f, 11e and 11f, and 12e and 12f). The spatial consistency of the estimated correlation functions (, , and ) is relatively low, which can be caused by the inconsistent observations and spatial bias instead of the noise level and the number of missing data (see appendix B for examples of inconsistent observations and spatial bias embedded in the observations).

4. Discussion

a. Estimates of cross-covariance terms

There are two independent sources of error in the estimates of the covariance terms: statistical error caused by the finite number of realizations (R) used to make the sample covariance and reconstruction error caused by insufficient sampling (L) in the spatial domain, including angles of the radial velocities. The statistical error decreases with the increasing number of realizations, which can be made as small as needed in the simulation. The reconstruction error represents how well the covariance or spectrum is reconstructed from the observations, and it is determined by sampling in the spatial domain, by the number of unique measurement points, the location of the measurements, and the orientation of the projections. Sampling on the appropriate regular grid results in a perfect reconstruction, just as in the discrete Fourier transform. The sampling on the irregular grid is characterized by a sampling matrix [ in Eq. (44)]. If it is full rank and there is no noise in the observations, then the reconstruction will be perfect.

For instance, the two-dimensional analytic energy spectra of vector components [, , , and in Eq. (45)] in the wavenumber domain are inversely Fourier transformed into the two-dimensional covariances [, , , and in Eq. (44)] in the spatial lagged domain [Eq. (3)] without error (e.g., Oppenheim et al. 1998):
e44
e45
where
e46
When either or both of the energy spectra of the cross terms in the covariance of vector components ( and ) are zero,
e47
the corresponding cross-covariance terms are also zero (Fig. 13), and
e48
Fig. 13.
Fig. 13.

Two-dimensional true (Gaussian) correlations (), estimated correlations () computed from an inverse Fourier transform of the given energy spectra of the vector components, and their difference (). (a)–(d) True covariance terms of vector components, where (a) ( km and km), (b) ( km and km), (c) , and (d) . (e)–(h) Estimated covariance terms of vector components, where (e) , (f) , (g) , and (h) . (i)–(l) Differences between true and estimated covariance terms of vector components, where (i) , (j) , (k) , and (l) .

Citation: Journal of Atmospheric and Oceanic Technology 34, 11; 10.1175/JTECH-D-17-0100.1

Thus, the nonzero cross-covariance terms in Figs. 7e, 7f, 7i, and 7j are attributed to the reconstruction error from insufficient sampling in the spatial domain as long as there is no statistical error.

b. Spatial homogeneity and temporal stationarity of the data

In this paper, the data are assumed to be locally homogeneous in space and stationary in time to estimate the spatial correlations using the proposed methods from the nonorthogonally and irregularly sampled data. The spatial homogeneity has been evaluated with the data sampled from the different size of the domain by changing the radius from the center of the domain of interest. The spatial structure of the estimated correlation functions is nearly identical when the radius is greater than 25 km. The temporal stationarity has been investigated with the data sampled from the different size of the time window from the entire data by changing the record length of the data. The spatial structure of the estimated correlation functions is constantly maintained when the more than a half year of data are used. This can be also discussed with the influence of the number of realizations: The greater the number of realizations, the greater the number of robust estimates of the spatial correlations that are possible having less statistical noise. For instance, the estimated correlations under more realizations are closer to the true correlations (Fig. 4), and the error bar of the correlations (or covariances) can be found in Priestley (1981). In addition, a comparison between estimated correlation functions can be made in terms of the number of realizations and their degrees of smoothness of the estimated correlations (Figs. 3a, 3b, and 6a).

c. Reduction of bias in the covariance estimate

The proposed approach to directly estimate covariance functions avoids an intermediate mapping step to estimate the vector currents from radial velocities, so it can minimize the cumulative spatial and temporal biases, such as the influence of the assumed mapping covariance functions (e.g., unweighted least squares fit or optimal interpolation). Although the proposed approach is influenced by the noise and error levels in the raw data, the shape of the covariances of the vector currents, directly estimated from nonorthogonally and irregularly sampled velocity data, is well matched with the shape of the true covariance.

This method is applicable to both scalar and vector data fields, such as in situ Lagrangian data [e.g., Argo floats and drifters] and unevenly sampled Eulerian data [e.g., altimeter- and HFR-derived observations]. Additionally, it allows us to quantify the spatial structures of the sampled data effectively from limited in situ observations (e.g., Colosi et al. 2013).

d. Performance of individual methods

The direct estimates of the spatial covariance of a given vector data of interest using parameterizations in the wavenumber and spatial domains outperform the indirect approach to estimate the covariance of the mapped data on the regular grid from the raw data because the direct estimate can minimize the potential biases in the intervening step and to avoid the propagation of the bias in the derived products. The shapes of the covariance functions estimated from the direct and indirect methods may not be quantitatively compared. In addition, the decorrelation length scales and shapes of the covariance functions estimated from the direct approaches using parameterizations in the wavenumber and spatial domains depend on the choice of basis functions and the density of the raw data. Thus, we provide a qualitative comparison of covariance functions estimated indirect and direct approaches.

5. Conclusions

We report a method for directly estimated isotropic and anisotropic spatial covariance functions of the orthogonal vector components from nonorthogonally and irregularly sampled scalar velocity observations. Using assumptions of local homogeneity and temporal stationarity of the current fields, the data covariance functions are represented as a function of only spatial lags within the domain of interest, which is made as small as possible by setting the covariance to zero outside the region.

The estimates of the one- and two-dimensional covariance functions are examined by a comparison of the true covariance functions and the estimated covariance functions using parameterizations in either the wavenumber or spatial domains. A simple spectral model, based on a covariance function with decorrelation length scales in the spatial domain, generates an ensemble of nonorthogonally and irregularly sampled velocity fields to make velocity observations. In addition, more realistic evaluations of spatial covariance estimates were conducted with the data, obtained from a regional numerical model and an array of high-frequency radars. The spatial covariance terms of velocity data in a regular grid and an unstructured grid are related by a mapping matrix representing the geometric transformation of two grids. The estimated spatial covariance functions correspond to nonnegative energy (power) spectra in the wavenumber domain, which can be used as a constraint of the given inverse problems. In the evaluation of individual approaches, the shape of the covariance functions and the decorrelation length scales estimated from the indirect approach are different from those of the true correlation. We compare two direct approaches using parameterizations in the wavenumber and spatial domains, the number of basis functions, and the spatial density of the raw data matter on the successful estimates of the noninterpolated covariance estimates using nonorthogonally and irregularly sampled scalar velocities. Thus, we present the qualitative comparison of estimated correlation functions.

Often observations sampled at nonorthogonally and irregularly spaced grid points are mapped to a regular grid before computing the covariance. These gridded values contain biases, which propagate through products derived from the gridded data, such as time integration and covariance estimates from the gridded velocity data. The proposed method provides a direct way to estimate the covariance with a minimum level of bias and is applicable to scalar and vector quantities.

The direct estimates of the spatial covariances allow us to quantify the decorrelation length scales. The spatial scales in the oceanic processes are relevant to the temporal scales (e.g., Woods 1980). For instance, the spatial scales of baroclinic tides on the shelf are O(10) km and they become longer for low-frequency currents, including geostrophic currents (e.g., Kim et al. 2010; Ponte and Cornuelle 2013). The spatial covariances and their structures, including the decorrelation length scales reported in this paper, depend on the dominant energy, which are low-frequency currents (greater than a 5-day period) off the coast of Oregon. Thus, the estimates of the spatial covariances and corresponding scales of currents in of interest frequency bands can be addressed in a similar way. Moreover, the direct covariance estimate can be applicable to the gap filling in the observations and data assimilation in the numerical modeling.

Acknowledgments

Jang Gon Yoo and Sung Yong Kim were supported by a research project titled “Research for Applications of Geostationary Ocean Color Imager” through Korea Institute of Marine Science and Technology Promotion (KIMST), Ministry of Oceans and Fisheries, and a grant through the Disaster and Safety Management Institute, Ministry of Public Safety and Security (KCG-01-2017-05), South Korea. This work is a part of the graduate studies of the first author. Bruce Cornuelle was supported by the Office of Naval Research Grant N00014-15-1-2285.

APPENDIX A

Time Incremental Estimates of the Variance and Covariance in Space

In the estimate of the spatial variance and covariance of huge data (e.g., spatial time series), a cumulative and incremental way in time minimizes the computational expense by allocating a part of all the data in memory within available computing resources. For instance, HFR-derived vector current maps at a resolution of 6 km off the U.S. West Coast are sampled at approximately 9000 grid points and archived as individual hourly data files. Thus, it would be convenient to read individual data files one at a time and to estimate the temporal mean and spatial variance and covariance in a cumulative way as below.

a. Estimates of temporal mean and variance

The temporal mean and variance of the data (r), sampled at a given location over N realizations, are defined as
ea1
ea2
where N and denote the number of total records and the number of observations, respectively. The covariance matrix of the data at M sampling locations has the elements of . The number of diagonal and off-diagonal terms in the covariance matrix is M and , respectively.

b. Estimates of spatial covariance with zero time lag

The sample covariance between and at the pth and qth locations is computed with
ea3
where denotes the number of concurrent observations of and (), and
ea4
ea5
and and denote the number of observations at the pth and qth locations, respectively. Note that the variables of N, , and should be maintained and updated cumulatively as the data are read incrementally in time.

APPENDIX B

Uncertainty and Signal-to-Noise Ratio of the Radial Velocity Data

The uncertainty and the signal-to-noise ratio of the radial velocities are quantified with the standard deviation (λ) of the sum of paired radial velocities ( and ) and their correlations (ρ), respectively [Figs. B1a and B1b, respectively] (e.g., Lipa et al. 2006; Kim et al. 2008), obtained from nearby radial grid points whose distance is very close. They are presented as a function of the difference between bearing angles of the paired radial velocities:
eb1
eb2
where the radial velocities are given by
eb3
eb4
and and denote the bearing angles, and and indicate the observational error at the location of and , respectively.
Fig. B1.
Fig. B1.

(a) Standard deviations (λ) of the sum of the paired ROMS-simulated surface radial velocities off the Oregon coast and (b) their cross correlation (ρ). The two radial grid points are chosen to have a distance of less than 200 m. Expected λ and ρ are plotted with colored curves under the conditions of = 0.25 (blue), 0.50, 0.75, 1.00, 1.25, 1.50, 1.75, and 2.00 (red) and , where and . (c),(d) Estimated cross correlations (β) and variance ratios () of the vector current components; they share the color bar on the bottom, and the individual ranges are marked separately. (e)–(g) A joint PDF ( scale) of and β, and their one-dimensional PDFs.

Citation: Journal of Atmospheric and Oceanic Technology 34, 11; 10.1175/JTECH-D-17-0100.1

For these estimates, Kim et al. (2008) and Kim (2015) assumed that the current fields are isotropic (, where denotes a variance ratio of vector components) and there is no correlation between vector components ().

Assuming that the current field is anisotropic () and its orthogonal vector components (u and υ) have nonzero correlation (),
eb5
eb6
The vector current components of u and υ are formulated as
eb7
where , and the variance of the noise time series (ϵ) is given by
eb8
The true vector current fields (u and υ) are generated by an idealized wavenumber-domain model to satisfy two conditions in Eqs. (B7) and (B8), and the paired radial velocity time series ( and ) at two nearby radial grid points are sampled by projecting the true vector current fields with respect to the arbitrary bearing angles (θ) of two radars. The standard deviation (λ) of the sum of the paired radial velocities obtained from an idealized wavenumber-domain model and their correlations (ρ) are considered as the reference values.

The ROMS-simulated surface radial velocity maps, mathematically converted from the true vector current fields, do not contain the sampling error and the mapping error. Thus, we evaluate λ and ρ with a condition having zero sampling error. The vector components are more correlated nearshore than offshore (Fig. B1c). The surface circulation tends to be anisotropic in the coastal region (within 50 km from the shoreline) as a result of bathymetry and coastline, and it becomes more isotropic offshore (Fig. B1d). The dominant variation ratio () and the cross correlation (β) of the vector components appear between 0.5 and 2 and between −0.3 and 0.7, respectively, based on their joint probability density function (Fig. B1e). The expected λ and ρ of the paired radial velocities are computed with seven cases of ( 0.50, 0.75, 1.00, 1.25, 1.50, 1.75, and 2.00) with a constant cross correlation ( 0.25) (Figs. B1a and B1b).

Additionally, the variance ratios and cross correlations of the vector components are investigated with the HFR-derived vector currents off Oregon for a period of 2 years (2007–08) (Figs. B2c and B2d). The expected λ and ρ of the paired radial velocity data are shown with four cases of ( 0.50, 0.75, 1.00, and 1.25) with a constant β () (Figs. B2a and B2b).

Fig. B2.
Fig. B2.

(a) Standard deviations (λ) of the sum of the paired HFR-derived surface radial velocities off the Oregon coast and (b) their cross correlation (ρ). The two radial grid points are chosen to have a distance of less than 200 m. Expected λ and ρ are plotted with colored curves under conditions of = 0.25 (blue), 0.50, 0.75, 1.00, 1.25, and 1.50 (red); and , where and . The dotted lines in Figs. B2a and B2b indicate the expected values with zero cross correlations of the vector current components. (c) Estimated cross correlations (β) and (d) variance ratios () of the vector current components; they share the color bar on the bottom, and the individual ranges are marked separately. (e)–(g) A joint PDF ( scale) of and β, and their one-dimensional PDFs.

Citation: Journal of Atmospheric and Oceanic Technology 34, 11; 10.1175/JTECH-D-17-0100.1

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  • Kim, S. Y., and P. M. Kosro, 2013: Observations of near-inertial surface currents off Oregon: Decorrelation time and length scales. J. Geophys. Res. Oceans, 118, 3723–3736, doi:10.1002/jgrc.20235.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Kim, S. Y., E. J. Terrill, and B. D. Cornuelle, 2007: Objectively mapping HF radar-derived surface current data using measured and idealized data covariance matrices. J. Geophys. Res., 112, C06021, doi:10.1029/2006JC003756.

    • Search Google Scholar
    • Export Citation

  • Kim, S. Y., E. J. Terrill, and B. D. Cornuelle, 2008: Mapping surface currents from HF radar radial velocity measurements using optimal interpolation. J. Geophys. Res., 113, C10023, doi:10.1029/2007JC004244.

    • Crossref
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    • Export Citation

  • Kim, S. Y., B. D. Cornuelle, and E. J. Terrill, 2010: Decomposing observations of high-frequency radar-derived surface currents by their forcing mechanisms: Decomposition techniques and spatial structures of decomposed surface currents. J. Geophys. Res., 115, C12007, doi:10.1029/2010JC006222.

    • Crossref
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    • Export Citation

  • Kim, S. Y., P. M. Kosro, and A. L. Kurapov, 2014: Evaluation of directly wind-coherent near-inertial surface currents off Oregon using a statistical parameterization and analytical and numerical models. J. Geophys. Res. Oceans, 119, 6631–6654, doi:10.1002/2014JC010115.

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    • Export Citation

  • Kosro, P. M., 2005: On the spatial structure of coastal circulation off Newport, Oregon, during spring and summer 2001 in a region of varying shelf width. J. Geophys. Res., 110, C10S06, doi:10.1029/2004JC002769.

    • Search Google Scholar
    • Export Citation

  • Lipa, B., B. Nyden, D. S. Ullman, and E. Terrill, 2006: SeaSonde radial velocities: Derivation and internal consistency. IEEE J. Oceanic Eng., 31, 850–861, doi:10.1109/JOE.2006.886104.

    • Crossref
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    • Export Citation

  • Mills, B., O. Slee, and E. Hill, 1958: A catalogue of radio sources between declinations +10° and −20°. Aust. J. Phys., 11, 360–387, doi:10.1071/PH580360.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Mills, B., O. Slee, and E. Hill, 1960: A catalogue of radio sources between declinations −20° and −50°. Aust. J. Phys., 13, 676–699, doi:10.1071/PH600676.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Mills, B., O. Slee, and E. Hill, 1961: A catalogue of radio sources between declinations −50° and −80°. Aust. J. Phys., 14, 497–507, doi:10.1071/PH610497.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Moffet, A., 1968: Minimum-redundancy linear arrays. IEEE Trans. Antennas Propag., 16, 172–175, doi:10.1109/TAP.1968.1139138.

  • Oppenheim, A. V., R. W. Schafter, and J. R. Buck, 1998: Discrete-Time Signal Processing. 2nd ed. Prentice Hall Signal Processing Series, Prentice-Hall, Inc., 870 pp.

  • Osborne, J. J., A. L. Kurapov, G. D. Egbert, and P. M. Kosro, 2011: Spatial and temporal variability of the M2 internal tide generation and propagation on the Oregon shelf. J. Phys. Oceanogr., 41, 2037–2062, doi:10.1175/JPO-D-11-02.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Osborne, J. J., A. L. Kurapov, G. D. Egbert, and P. M. Kosro, 2014: Intensified diurnal tides along the Oregon Coast. J. Phys. Oceanogr., 44, 1689–1703, doi:10.1175/JPO-D-13-0247.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Percival, D. B., and A. T. Walden, 1993: Spectral Analysis for Physical Applications: Multipaper and Conventional Univariate Techniques. Cambridge University Press, 583 pp., doi:10.1017/CBO9780511622762.

    • Crossref
    • Export Citation

  • Ponte, A. L., and B. D. Cornuelle, 2013: Coastal numerical modelling of tides: Sensitivity to domain size and remotely generated internal tide. Ocean Modell., 62, 17–26, doi:10.1016/j.ocemod.2012.11.007.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Priestley, M. B., 1981: Spectral Analysis and Time Series. Academic Press, 890 pp.

  • Roemmich, D., and J. Gilson, 2009: The 2004–2008 mean and annual cycle of temperature, salinity, and steric height in the global ocean from the Argo Program. Prog. Oceanogr., 82, 81–100, doi:10.1016/j.pocean.2009.03.004.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Roesler, C., W. Emery, and S. Y. Kim, 2013: Evaluating the use of high-frequency radar coastal currents to correct satellite altimetry. J. Geophys. Res. Oceans, 118, 3240–3259, doi:10.1002/jgrc.20220.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Rubin, D. B., and T. H. Szatrowski, 1982: Finding maximum likelihood estimates of patterned covariance matrices by the EM algorithm. Biometrika, 69, 657–660, doi:10.1093/biomet/69.3.657.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Shay, L. K., J. Martinez-Pedraja, T. M. Cook, B. K. Haus, and R. H. Weisberg, 2007: High-frequency radar mapping of surface currents using WERA. J. Atmos. Oceanic Technol., 24, 484–503, doi:10.1175/JTECH1985.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Wilkin, J. L., M. M. Bowen, and W. J. Emery, 2002: Mapping mesoscale currents by optimal interpolation of satellite radiometer and altimeter data. Ocean Dyn., 52, 95–103, doi:10.1007/s10236-001-0011-2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Woods, J. D., 1980: Do waves limit turbulent diffusion in the ocean? Nature, 288, 219–224, doi:10.1038/288219a0.

  • Wunsch, C., 1996: The Ocean Circulation Inverse Problem. Cambridge University Press, 442 pp.

    • Crossref
    • Export Citation

  • Wunsch, C., 2006: Discrete Inverse and State Estimation Problems: With Geophysical Fluid Applications. Cambridge University Press, 384 pp.

    • Crossref
    • Export Citation

  • Yaremchuk, M., and A. Sentchev, 2009: Mapping radar-derived sea surface currents with a variational method. Cont. Shelf Res., 29, 1711–1722, doi:10.1016/j.csr.2009.05.016.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Yaremchuk, M., and A. Sentchev, 2011: A combined EOF/variational approach for mapping radar-derived sea surface currents. Cont. Shelf Res., 31, 758–768, doi:10.1016/j.csr.2011.01.009.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Zetler, B., M. Schuldt, R. Whipple, and S. Hicks, 1965: Harmonic analysis of tides from data randomly spaced in time. J. Geophys. Res., 70, 2805–2811, doi:10.1029/JZ070i012p02805.

    • Crossref
    • Search Google Scholar
    • Export Citation
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  • Kaplan, D. M., and F. Lekein, 2007: Spatial interpolation and filtering of surface current data based on open-boundary modal analysis. J. Geophys. Res., 112, C12007, doi:10.1029/2006JC003984.

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  • Kim, S. Y., 2015: Quality assessment techniques applied to surface radial velocity maps obtained from high-frequency radars. J. Atmos. Oceanic Technol., 32, 1915–1927, doi:10.1175/JTECH-D-14-00207.1.

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  • Kim, S. Y., and P. M. Kosro, 2013: Observations of near-inertial surface currents off Oregon: Decorrelation time and length scales. J. Geophys. Res. Oceans, 118, 3723–3736, doi:10.1002/jgrc.20235.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Kim, S. Y., E. J. Terrill, and B. D. Cornuelle, 2007: Objectively mapping HF radar-derived surface current data using measured and idealized data covariance matrices. J. Geophys. Res., 112, C06021, doi:10.1029/2006JC003756.

    • Search Google Scholar
    • Export Citation

  • Kim, S. Y., E. J. Terrill, and B. D. Cornuelle, 2008: Mapping surface currents from HF radar radial velocity measurements using optimal interpolation. J. Geophys. Res., 113, C10023, doi:10.1029/2007JC004244.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Kim, S. Y., B. D. Cornuelle, and E. J. Terrill, 2010: Decomposing observations of high-frequency radar-derived surface currents by their forcing mechanisms: Decomposition techniques and spatial structures of decomposed surface currents. J. Geophys. Res., 115, C12007, doi:10.1029/2010JC006222.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Kim, S. Y., P. M. Kosro, and A. L. Kurapov, 2014: Evaluation of directly wind-coherent near-inertial surface currents off Oregon using a statistical parameterization and analytical and numerical models. J. Geophys. Res. Oceans, 119, 6631–6654, doi:10.1002/2014JC010115.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Kosro, P. M., 2005: On the spatial structure of coastal circulation off Newport, Oregon, during spring and summer 2001 in a region of varying shelf width. J. Geophys. Res., 110, C10S06, doi:10.1029/2004JC002769.

    • Search Google Scholar
    • Export Citation

  • Lipa, B., B. Nyden, D. S. Ullman, and E. Terrill, 2006: SeaSonde radial velocities: Derivation and internal consistency. IEEE J. Oceanic Eng., 31, 850–861, doi:10.1109/JOE.2006.886104.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Mills, B., O. Slee, and E. Hill, 1958: A catalogue of radio sources between declinations +10° and −20°. Aust. J. Phys., 11, 360–387, doi:10.1071/PH580360.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Mills, B., O. Slee, and E. Hill, 1960: A catalogue of radio sources between declinations −20° and −50°. Aust. J. Phys., 13, 676–699, doi:10.1071/PH600676.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Mills, B., O. Slee, and E. Hill, 1961: A catalogue of radio sources between declinations −50° and −80°. Aust. J. Phys., 14, 497–507, doi:10.1071/PH610497.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Moffet, A., 1968: Minimum-redundancy linear arrays. IEEE Trans. Antennas Propag., 16, 172–175, doi:10.1109/TAP.1968.1139138.

  • Oppenheim, A. V., R. W. Schafter, and J. R. Buck, 1998: Discrete-Time Signal Processing. 2nd ed. Prentice Hall Signal Processing Series, Prentice-Hall, Inc., 870 pp.

  • Osborne, J. J., A. L. Kurapov, G. D. Egbert, and P. M. Kosro, 2011: Spatial and temporal variability of the M2 internal tide generation and propagation on the Oregon shelf. J. Phys. Oceanogr., 41, 2037–2062, doi:10.1175/JPO-D-11-02.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Osborne, J. J., A. L. Kurapov, G. D. Egbert, and P. M. Kosro, 2014: Intensified diurnal tides along the Oregon Coast. J. Phys. Oceanogr., 44, 1689–1703, doi:10.1175/JPO-D-13-0247.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Percival, D. B., and A. T. Walden, 1993: Spectral Analysis for Physical Applications: Multipaper and Conventional Univariate Techniques. Cambridge University Press, 583 pp., doi:10.1017/CBO9780511622762.

    • Crossref
    • Export Citation

  • Ponte, A. L., and B. D. Cornuelle, 2013: Coastal numerical modelling of tides: Sensitivity to domain size and remotely generated internal tide. Ocean Modell., 62, 17–26, doi:10.1016/j.ocemod.2012.11.007.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Priestley, M. B., 1981: Spectral Analysis and Time Series. Academic Press, 890 pp.

  • Roemmich, D., and J. Gilson, 2009: The 2004–2008 mean and annual cycle of temperature, salinity, and steric height in the global ocean from the Argo Program. Prog. Oceanogr., 82, 81–100, doi:10.1016/j.pocean.2009.03.004.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Roesler, C., W. Emery, and S. Y. Kim, 2013: Evaluating the use of high-frequency radar coastal currents to correct satellite altimetry. J. Geophys. Res. Oceans, 118, 3240–3259, doi:10.1002/jgrc.20220.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Rubin, D. B., and T. H. Szatrowski, 1982: Finding maximum likelihood estimates of patterned covariance matrices by the EM algorithm. Biometrika, 69, 657–660, doi:10.1093/biomet/69.3.657.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Shay, L. K., J. Martinez-Pedraja, T. M. Cook, B. K. Haus, and R. H. Weisberg, 2007: High-frequency radar mapping of surface currents using WERA. J. Atmos. Oceanic Technol., 24, 484–503, doi:10.1175/JTECH1985.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Wilkin, J. L., M. M. Bowen, and W. J. Emery, 2002: Mapping mesoscale currents by optimal interpolation of satellite radiometer and altimeter data. Ocean Dyn., 52, 95–103, doi:10.1007/s10236-001-0011-2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Woods, J. D., 1980: Do waves limit turbulent diffusion in the ocean? Nature, 288, 219–224, doi:10.1038/288219a0.

  • Wunsch, C., 1996: The Ocean Circulation Inverse Problem. Cambridge University Press, 442 pp.

    • Crossref
    • Export Citation

  • Wunsch, C., 2006: Discrete Inverse and State Estimation Problems: With Geophysical Fluid Applications. Cambridge University Press, 384 pp.

    • Crossref
    • Export Citation

  • Yaremchuk, M., and A. Sentchev, 2009: Mapping radar-derived sea surface currents with a variational method. Cont. Shelf Res., 29, 1711–1722, doi:10.1016/j.csr.2009.05.016.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Yaremchuk, M., and A. Sentchev, 2011: A combined EOF/variational approach for mapping radar-derived sea surface currents. Cont. Shelf Res., 31, 758–768, doi:10.1016/j.csr.2011.01.009.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Zetler, B., M. Schuldt, R. Whipple, and S. Hicks, 1965: Harmonic analysis of tides from data randomly spaced in time. J. Geophys. Res., 70, 2805–2811, doi:10.1029/JZ070i012p02805.

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

    An example of radial velocity maps (red and blue arrows) nonorthogonally and irregularly sampled from an idealized model vector current field (black arrows). The sampling radial grid points at individual radar sites (black circles; R1 and R2) are marked with colored dots, which are not collocated, and the sampling area of the vector currents is indicated with a black box.

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

    (a) A true correlation function () with an exponentially decaying shape of a decorrelation length scale of 2 km [ km in Eqs. (4) and (10)]. (b) Five examples (, , , , and ) of the irregularly sampled scalar data at 250 locations in the spatial domain ( and ), which are generated under the given true correlation function in (a), are presented as a function of the sampling index (p or q; ). (c) One-dimensional correlations of the sampled data are presented as a function of the sampling index (p and q). (d) The sampling locations ( and ) and the sampling index (p or q). (e) Spacing () between sampling locations as a function of the sampling index (p or q).

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

    (a) An estimated correlation function using a parameterization in the wavenumber domain (). (b) An estimated correlation function using a parameterization in the spatial domain (bin averaging) (). Error bars correspond to standard errors for individual spatial bins. (c) A comparison of the true correlation function (; km) and the estimated correlation functions (, , and ), where denotes the correlations of the optimally interpolated data from irregularly sampled data on the regular grid (see section 2f for more details). The estimated decorrelation length scales using parameterizations in the wavenumber and spatial domains are 2.03 and 2.09 km, respectively ( km; km). The optimal interpolation is conducted with an exponential correlation function with a decorrelation length scale of 1.5 km. The estimated correlation function () has a Gaussian shape, different from that of the true correlation function, and a decorrelation length scale of 2.78 km ( km) associated with the accumulated bias.

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

    (a),(c),(e) The sample correlations (colored crosses) and true correlations (black line) of the sampled data as a function of the number of randomly chosen realizations (R; R = 667, , 8334, 16 667, and 50 000) under a given number of samples (L). (b),(d),(f) The sample correlations are bin averaged at individual sampling bins () and presented as a function of the number realizations (x axis has a log scale unit). The number of samples is (a),(b) 500; (c),(d) 250; and (e),(f) 125. Note that all estimates are based on the correlations and spatial lags between the nearest sampling locations, which are obtained from the first off-diagonal terms of the square matrix.

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

    A lower triangle of a square covariance matrix [; Eqs. (13)–(15)], including diagonal terms, is marked with colored column vectors [Eq. (39)] that are concatenated into a single-column vector [; Eq. (16)] via vectorization. Cross-covariance terms [Eq. (37)] at the adjacent spatial lags of the square covariance matrix are denoted with white-circled components. The diagonal terms [Eq. (36)] of the covariance matrix are indicated with black-circled components, and its off-diagonal terms [Eq. (38)] are denoted with both white-circled and noncontoured components. The indices of the diagonal terms, the off-diagonal terms at the adjacent spatial lags, the off-diagonal terms, and the lower-triangle terms of the square covariance matrix can be found in Eqs. (36)–(39), respectively.

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

    (a)–(c) An example of the one-dimensional true correlations (, km) and the estimated correlation functions using parameterizations in the wavenumber domain () and spatial domain ( and ) from the data having nonzero noise [, where in Eq. (8)]. (a) Estimated correlations when including perfect correlations at the zero lag (; ). (b) Estimated correlations when excluding perfect correlations at the zero lag (; ). (c) Estimated correlations when excluding perfect correlations at the zero lag (; ) and scaling up the estimated correlations with the maximum value of correlations at the zero lag. (d) Estimated decorrelation length scales (, , and ) as a function of the noise level (κ), which is an inverse of the ratio of the signal amplitude to noise. In (a) and (b), α and β denote the noise level of the data by including and excluding the perfect correlation at the zero lag, respectively.

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

    (a),(b) Assumed two-dimensional true (Gaussian spatial) correlation functions with zero cross-correlation terms between vector components (): (a) ( km; km) and (b) ( km; km). (c)–(f) Estimated correlations () using a parameterization in the wavenumber domain [section 3b(1)]. (g)–(j) Estimated correlations () using a parameterization in the spatial domain [section 3b(2)]. The following terms are shown: (c),(g) ; (d),(h) ; (e),(i) ; and (f),(j) .

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

    A study domain of the ROMS-simulated surface currents and HFR-derived surface current observations off the coast of Oregon. (a),(b) A ROMS domain and its spatial subset in a coastal region that encloses the effective spatial coverage of the operational HFRs (a yellow curve). The bottom bathymetry is contoured with 50, 100, 250, 500, 1000, 1500, 2000, 2500, and 3000 m. As a reference, major coastal regions are denoted by abbreviated two letter names from south to north: Cape Blanco (CB), Winchester Bay (WB), Newport (NP), and Loomis Lake (LL). (c) Sampling radial grid points of the operational HFRs (gray dots) and the radial grid points (black dots) within 24 km of point A (red dot; 46°N, 124.5°E) and point B (red dot; 44°N, 124°E). Two red rectangular boxes in (a) and (b) indicate the same region. All the radial grid points within the red box in (b) are shown in (c) (gray dots).

  • <sc>Fig</sc>. 9.
    View in gallery
    Fig. 9.

    Spatial correlations of the ROMS-simulated surface currents at point A in Fig. 8c are estimated from (a)–(d) the bin averaging of the vector currents (), (e)–(h) the covariances of radial velocities using a parameterization in the wavenumber domain () [section 3b(1)], and (i)–(l) the covariances of radial velocities using a parameterization in the spatial domain (bin averaging) () [section 3b(2)]. The following terms are shown: (a),(e),(i) ; (b),(f),(j) ; (c),(g),(k) ; and (d),(h),(l) . The shared color bar on the bottom ranges from 0 to 1 for and , and from −0.5 to 0.5 for and .

  • <sc>Fig</sc>. 10.
    View in gallery
    Fig. 10.

    Spatial correlations of the ROMS-simulated surface currents at point B in Fig. 8c are estimated from (a)–(d) the bin averaging of the vector currents (), (e)–(h) the covariances of radial velocities using a parameterization in the wavenumber domain () [section 3b(1)], and (i)–(l) the covariances of radial velocities using a parameterization in the spatial domain (bin averaging) () [section 3b(2)]. The following terms are shown: (a),(e),(i) ; (b),(f),(j) ; (c),(g),(k) ; and (d),(h),(l) . The shared color bar on the bottom ranges from 0 to 1 for and , and from −0.5 to 0.5 for and .

  • <sc>Fig</sc>. 11.
    View in gallery
    Fig. 11.

    Spatial correlations of the HFR-derived surface currents at point A in Fig. 8c are estimated from (a)–(d) the bin averaging of the vector currents (), (e)–(h) the covariances of radial velocities using a parameterization in the wavenumber domain () [section 3b(1)], and (i)–(l) the covariances of radial velocities using a parameterization in the spatial domain (bin averaging) () [section 3b(2)]. The following terms are shown: (a),(e),(i) ; (b),(f),(j) ; (c),(g),(k) ; and (d),(h),(l) . The shared color bar on the bottom ranges from 0 to 1 for and , and from −0.5 to 0.5 for and .

  • <sc>Fig</sc>. 12.
    View in gallery
    Fig. 12.

    Spatial correlations of the HFR-derived surface currents at point B in Fig. 8c are estimated from (a)–(d) the bin averaging of the vector currents (), (e)–(h) the covariances of radial velocities using a parameterization in the wavenumber domain () [section 3b(1)], and (i)–(l) the covariances of radial velocities using a parameterization in the spatial domain (bin averaging) () [section 3b(2)]. The following terms are shown (a),(e),(i) ; (b),(f),(j) ; (c),(g),(k) ; and (d),(h),(l) . The shared color bar on the bottom ranges from 0 to 1 for and , and from −0.5 to 0.5 for and .

  • <sc>Fig</sc>. 13.
    View in gallery
    Fig. 13.

    Two-dimensional true (Gaussian) correlations (), estimated correlations () computed from an inverse Fourier transform of the given energy spectra of the vector components, and their difference (). (a)–(d) True covariance terms of vector components, where (a) ( km and km), (b) ( km and km), (c) , and (d) . (e)–(h) Estimated covariance terms of vector components, where (e) , (f) , (g) , and (h) . (i)–(l) Differences between true and estimated covariance terms of vector components, where (i) , (j) , (k) , and (l) .

  • <sc>Fig</sc>. B1.
    View in gallery
    Fig. B1.

    (a) Standard deviations (λ) of the sum of the paired ROMS-simulated surface radial velocities off the Oregon coast and (b) their cross correlation (ρ). The two radial grid points are chosen to have a distance of less than 200 m. Expected λ and ρ are plotted with colored curves under the conditions of = 0.25 (blue), 0.50, 0.75, 1.00, 1.25, 1.50, 1.75, and 2.00 (red) and , where and . (c),(d) Estimated cross correlations (β) and variance ratios () of the vector current components; they share the color bar on the bottom, and the individual ranges are marked separately. (e)–(g) A joint PDF ( scale) of and β, and their one-dimensional PDFs.

  • <sc>Fig</sc>. B2.
    View in gallery
    Fig. B2.

    (a) Standard deviations (λ) of the sum of the paired HFR-derived surface radial velocities off the Oregon coast and (b) their cross correlation (ρ). The two radial grid points are chosen to have a distance of less than 200 m. Expected λ and ρ are plotted with colored curves under conditions of = 0.25 (blue), 0.50, 0.75, 1.00, 1.25, and 1.50 (red); and , where and . The dotted lines in Figs. B2a and B2b indicate the expected values with zero cross correlations of the vector current components. (c) Estimated cross correlations (β) and (d) variance ratios () of the vector current components; they share the color bar on the bottom, and the individual ranges are marked separately. (e)–(g) A joint PDF ( scale) of and β, and their one-dimensional PDFs.

Fig. 1.

An example of radial velocity maps (red and blue arrows) nonorthogonally and irregularly sampled from an idealized model vector current field (black arrows). The sampling radial grid points at individual radar sites (black circles; R1 and R2) are marked with colored dots, which are not collocated, and the sampling area of the vector currents is indicated with a black box.
Fig. 2.

(a) A true correlation function () with an exponentially decaying shape of a decorrelation length scale of 2 km [ km in Eqs. (4) and (10)]. (b) Five examples (, , , , and ) of the irregularly sampled scalar data at 250 locations in the spatial domain ( and ), which are generated under the given true correlation function in (a), are presented as a function of the sampling index (p or q; ). (c) One-dimensional correlations of the sampled data are presented as a function of the sampling index (p and q). (d) The sampling locations ( and ) and the sampling index (p or q). (e) Spacing () between sampling locations as a function of the sampling index (p or q).
Fig. 3.

(a) An estimated correlation function using a parameterization in the wavenumber domain (). (b) An estimated correlation function using a parameterization in the spatial domain (bin averaging) (). Error bars correspond to standard errors for individual spatial bins. (c) A comparison of the true correlation function (; km) and the estimated correlation functions (, , and ), where denotes the correlations of the optimally interpolated data from irregularly sampled data on the regular grid (see section 2f for more details). The estimated decorrelation length scales using parameterizations in the wavenumber and spatial domains are 2.03 and 2.09 km, respectively ( km; km). The optimal interpolation is conducted with an exponential correlation function with a decorrelation length scale of 1.5 km. The estimated correlation function () has a Gaussian shape, different from that of the true correlation function, and a decorrelation length scale of 2.78 km ( km) associated with the accumulated bias.
Fig. 4.

(a),(c),(e) The sample correlations (colored crosses) and true correlations (black line) of the sampled data as a function of the number of randomly chosen realizations (R; R = 667, , 8334, 16 667, and 50 000) under a given number of samples (L). (b),(d),(f) The sample correlations are bin averaged at individual sampling bins () and presented as a function of the number realizations (x axis has a log scale unit). The number of samples is (a),(b) 500; (c),(d) 250; and (e),(f) 125. Note that all estimates are based on the correlations and spatial lags between the nearest sampling locations, which are obtained from the first off-diagonal terms of the square matrix.
Fig. 5.

A lower triangle of a square covariance matrix [; Eqs. (13)–(15)], including diagonal terms, is marked with colored column vectors [Eq. (39)] that are concatenated into a single-column vector [; Eq. (16)] via vectorization. Cross-covariance terms [Eq. (37)] at the adjacent spatial lags of the square covariance matrix are denoted with white-circled components. The diagonal terms [Eq. (36)] of the covariance matrix are indicated with black-circled components, and its off-diagonal terms [Eq. (38)] are denoted with both white-circled and noncontoured components. The indices of the diagonal terms, the off-diagonal terms at the adjacent spatial lags, the off-diagonal terms, and the lower-triangle terms of the square covariance matrix can be found in Eqs. (36)–(39), respectively.
Fig. 6.

(a)–(c) An example of the one-dimensional true correlations (, km) and the estimated correlation functions using parameterizations in the wavenumber domain () and spatial domain ( and ) from the data having nonzero noise [, where in Eq. (8)]. (a) Estimated correlations when including perfect correlations at the zero lag (; ). (b) Estimated correlations when excluding perfect correlations at the zero lag (; ). (c) Estimated correlations when excluding perfect correlations at the zero lag (; ) and scaling up the estimated correlations with the maximum value of correlations at the zero lag. (d) Estimated decorrelation length scales (, , and ) as a function of the noise level (κ), which is an inverse of the ratio of the signal amplitude to noise. In (a) and (b), α and β denote the noise level of the data by including and excluding the perfect correlation at the zero lag, respectively.
Fig. 7.

(a),(b) Assumed two-dimensional true (Gaussian spatial) correlation functions with zero cross-correlation terms between vector components (): (a) ( km; km) and (b) ( km; km). (c)–(f) Estimated correlations () using a parameterization in the wavenumber domain [section 3b(1)]. (g)–(j) Estimated correlations () using a parameterization in the spatial domain [section 3b(2)]. The following terms are shown: (c),(g) ; (d),(h) ; (e),(i) ; and (f),(j) .
Fig. 8.

A study domain of the ROMS-simulated surface currents and HFR-derived surface current observations off the coast of Oregon. (a),(b) A ROMS domain and its spatial subset in a coastal region that encloses the effective spatial coverage of the operational HFRs (a yellow curve). The bottom bathymetry is contoured with 50, 100, 250, 500, 1000, 1500, 2000, 2500, and 3000 m. As a reference, major coastal regions are denoted by abbreviated two letter names from south to north: Cape Blanco (CB), Winchester Bay (WB), Newport (NP), and Loomis Lake (LL). (c) Sampling radial grid points of the operational HFRs (gray dots) and the radial grid points (black dots) within 24 km of point A (red dot; 46°N, 124.5°E) and point B (red dot; 44°N, 124°E). Two red rectangular boxes in (a) and (b) indicate the same region. All the radial grid points within the red box in (b) are shown in (c) (gray dots).
Fig. 9.

Spatial correlations of the ROMS-simulated surface currents at point A in Fig. 8c are estimated from (a)–(d) the bin averaging of the vector currents (), (e)–(h) the covariances of radial velocities using a parameterization in the wavenumber domain () [section 3b(1)], and (i)–(l) the covariances of radial velocities using a parameterization in the spatial domain (bin averaging) () [section 3b(2)]. The following terms are shown: (a),(e),(i) ; (b),(f),(j) ; (c),(g),(k) ; and (d),(h),(l) . The shared color bar on the bottom ranges from 0 to 1 for and , and from −0.5 to 0.5 for and .
Fig. 10.

Spatial correlations of the ROMS-simulated surface currents at point B in Fig. 8c are estimated from (a)–(d) the bin averaging of the vector currents (), (e)–(h) the covariances of radial velocities using a parameterization in the wavenumber domain () [section 3b(1)], and (i)–(l) the covariances of radial velocities using a parameterization in the spatial domain (bin averaging) () [section 3b(2)]. The following terms are shown: (a),(e),(i) ; (b),(f),(j) ; (c),(g),(k) ; and (d),(h),(l) . The shared color bar on the bottom ranges from 0 to 1 for and , and from −0.5 to 0.5 for and .
Fig. 11.

Spatial correlations of the HFR-derived surface currents at point A in Fig. 8c are estimated from (a)–(d) the bin averaging of the vector currents (), (e)–(h) the covariances of radial velocities using a parameterization in the wavenumber domain () [section 3b(1)], and (i)–(l) the covariances of radial velocities using a parameterization in the spatial domain (bin averaging) () [section 3b(2)]. The following terms are shown: (a),(e),(i) ; (b),(f),(j) ; (c),(g),(k) ; and (d),(h),(l) . The shared color bar on the bottom ranges from 0 to 1 for and , and from −0.5 to 0.5 for and .
Fig. 12.

Spatial correlations of the HFR-derived surface currents at point B in Fig. 8c are estimated from (a)–(d) the bin averaging of the vector currents (), (e)–(h) the covariances of radial velocities using a parameterization in the wavenumber domain () [section 3b(1)], and (i)–(l) the covariances of radial velocities using a parameterization in the spatial domain (bin averaging) () [section 3b(2)]. The following terms are shown (a),(e),(i) ; (b),(f),(j) ; (c),(g),(k) ; and (d),(h),(l) . The shared color bar on the bottom ranges from 0 to 1 for and , and from −0.5 to 0.5 for and .
Fig. 13.

Two-dimensional true (Gaussian) correlations (), estimated correlations () computed from an inverse Fourier transform of the given energy spectra of the vector components, and their difference (). (a)–(d) True covariance terms of vector components, where (a) ( km and km), (b) ( km and km), (c) , and (d) . (e)–(h) Estimated covariance terms of vector components, where (e) , (f) , (g) , and (h) . (i)–(l) Differences between true and estimated covariance terms of vector components, where (i) , (j) , (k) , and (l) .
Fig. B1.

(a) Standard deviations (λ) of the sum of the paired ROMS-simulated surface radial velocities off the Oregon coast and (b) their cross correlation (ρ). The two radial grid points are chosen to have a distance of less than 200 m. Expected λ and ρ are plotted with colored curves under the conditions of = 0.25 (blue), 0.50, 0.75, 1.00, 1.25, 1.50, 1.75, and 2.00 (red) and , where and . (c),(d) Estimated cross correlations (β) and variance ratios () of the vector current components; they share the color bar on the bottom, and the individual ranges are marked separately. (e)–(g) A joint PDF ( scale) of and β, and their one-dimensional PDFs.
Fig. B2.

(a) Standard deviations (λ) of the sum of the paired HFR-derived surface radial velocities off the Oregon coast and (b) their cross correlation (ρ). The two radial grid points are chosen to have a distance of less than 200 m. Expected λ and ρ are plotted with colored curves under conditions of = 0.25 (blue), 0.50, 0.75, 1.00, 1.25, and 1.50 (red); and , where and . The dotted lines in Figs. B2a and B2b indicate the expected values with zero cross correlations of the vector current components. (c) Estimated cross correlations (β) and (d) variance ratios () of the vector current components; they share the color bar on the bottom, and the individual ranges are marked separately. (e)–(g) A joint PDF ( scale) of and β, and their one-dimensional PDFs.
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