The Development of an Inversion Technique to Extract Vertical Current Profiles from X-Band Radar Observations
1. Introduction
Upper-ocean currents and current shear at the air–sea boundary play important roles in the vertical mixing of entrained atmospheric gasses; advection of pollution plumes, including spilled oil, and wave dynamics; and influencing other navigational and scientific concerns (Phillips 1966; Halpern 1977; Davis et al. 1981). Despite their importance, accurate in situ measurements of near-surface currents remain difficult using traditional oceanographic tools due to platform motion complications and wave contamination of the current signal. Traditional point measurement current meters also lack spatial coverage (Paduan and Rosenfeld 1996). Remote sensing techniques have recently received considerable attention with the use of high-frequency (HF) radar, which provides maps of ocean currents with spatial resolutions of O(1–6) km out to distances of tens of kilometers and is used operationally around the United States (e.g., Terrill et al. 2006; Harlan et al. 2009, 2011). Shifting to higher-frequency electromagnetic (EM) signals, such as incoherent marine X-band radar, offers the advantage of providing sea surface backscatter maps from a relatively large range (out to approximately 3–8 km, depending on user-defined parameters and environmental conditions). The large spatial scales of remote sensing techniques, however, makes comparison to in situ current measurements in the open ocean difficult because of potentially weak currents and the presence of horizontal current shear from eddies or current fronts (Kohut et al. 2006; Ohlmann et al. 2007). The strong periodic currents within the tidally forced mouth of the Columbia River (MCR) make the region an ideal test bed for the development and validation of X-band current estimation techniques. Furthermore, the dynamics of tidally forced river inlets are a concern for navigation, civil, and scientific communities, which will directly benefit from the ability to retrieve accurate current and current shear information remotely.
The use of radar as a tool to study wave–current interaction began with Crombie’s (1955) observations of Doppler shifts using high-frequency radar. Since then, many studies have taken advantage of the sensitivity of different types of radar to ocean wave celerity to estimate both surface current velocity and bathymetry (e.g., Barrick 1972; Shuchman 1979; Alpers et al. 1981b; Young et al. 1985; Bell 1999). Marine X-band provides an advantage over other radar types by offering near-real-time imaging of a broad range of ocean wavelengths, ranging between approximately 20 and 200 m. The simultaneous measurement of time–space properties of the backscatter allow for exploration in the frequency–wavenumber domain. Where HF radar techniques involve direct observations of radio wave Doppler shifts, X-band backscatter images are processed to supply a broadband wavenumber–Doppler shift relationship from which currents can be estimated.
Incoherent radars, such as the one used in this study, differ from coherent radars in that incoherent systems supply only backscatter magnitude and not the phase of the returned EM signal. Thus, a single scan from an incoherent radar supplies information about the locations but not velocities of individual scatterers. The modulation of incoherent marine X-band backscatter is a result of Bragg scatter with centimeter-scale ocean roughness that is modified by the underlying surface gravity waves via three mechanisms: 1) hydrodynamic modulation, which, due to orbital velocities and wave–wave interactions, leads to the collection of capillary waves near the crests of longer waves; 2) shadowing by wave crests in low-grazing-angle geometries (typical of marine radar deployment); and 3) geometric scatter that occurs from the forward face of the ocean waves (Alpers et al. 1981a; Nieto Borge et al. 2004). A series of sequential backscatter images therefore captures the spatial and temporal evolution of the surface wavefield, which can then be transformed into the wavenumber–frequency domain using a Fourier transform (Young et al. 1985). Surface current information is present in the observed Doppler shift–wavenumber relationship and can be extracted using various techniques (e.g., Stewart and Joy 1974; Ha 1979; Young et al. 1985).
In previous studies, currents are extracted from wavenumber-dependent Doppler shift measurements under strict assumptions of the underlying current–depth profile shape (e.g., that currents are either depth uniform or vary linearly with depth) (Young et al. 1985; Senet et al. 2008; Teague et al. 2001; Trizna and Xu 2006; Hessner and Bell 2009). One aim of this work is to carry out a rigorous comparison of X-band-derived currents under both uniform and linear current profile assumptions to in situ current profile measurements. These comparisons are made over multiple tidal cycles to evaluate the success of each method in the presence of varying degrees of current shear.
There has been limited success in estimating depth-varying currents from wavefield measurements. Qualitative observations of current shear have been made using broadband Doppler shift–wavenumber measurements (Ha 1979; Fernandez et al. 1996; Dugan et al. 2008) by noting fluctuations in the Doppler shift–wavenumber relationships, suggesting the presence of vertical current shear. Ha (1979) attempted to use the wavenumber dependence of Doppler shift measurements to solve for the arbitrary current–depth profile (i.e., without an assumption of its shape) using a four-frequency HF system. This small number of discrete wavenumbers, however, did not adequately constrain the inversion, yielding noisy results. In this study the concept of extracting currents from the observed Doppler shifted dispersion relationship is extended. The ability of the X-band radar to image a broad range of ocean wavelengths is used in combination with a constrained least squares technique of the wavenumber-dense data to invert for current profiles.
2. Data collection
As a part of the Office of Naval Research (ONR)-funded Riverine and Estuarine Transport Experiment 2 (RIVET2) field campaign, X-band backscatter was collected at the MCR near Astoria, Oregon, from 24 May 2013 to 4 June 2013. The land-based radar system was located inside the mouth at the south jetty (Fig. 1).
A map of the Columbia River mouth, where gray is land and white is water. Light gray contours show bathymetry collected by USGS surveys. The dashed circle denotes the approximate maximum range of the X-band radar, and the triangle indicates the location of a bottom-mounted ADCP. The variance ellipse of the currents during RIVET2 is shown in black.
Citation: Journal of Atmospheric and Oceanic Technology 33, 9; 10.1175/JTECH-D-15-0145.1
Supporting current and water depth measurements were collected using a bottom-mounted acoustic Doppler current profiler [ADCP; 1000-kHz Nortek acoustic wave and current (AWAC) sensor] within the radar field of view (Fig. 1), collecting vertical profiles of three-dimensional currents at 15-min intervals in 0.5-m bins. The overlaid current variance ellipse, representing the depth-average variances of channel and cross-channel currents, shows the currents were strongly polarized due to the channel geometry. This led to the adoption of the convention that current directions were limited to positive (up channel) and negative (down channel). The vertical structure of the streamwise current (Fig. 2) shows strong tidally forced currents, ranging between −3.5 and 1.3 m s−1. Stronger currents occurred during ebb tides when the tidal and river forcing were aligned. The vertical current shear structure was also tidally dependent (Fig. 3). During slack conditions, tidal forcing became small and the flow was dominated by the river forcing, resulting in slightly negative depth-uniform currents. Flood and ebb conditions, however, were dominated by strong tidal forcing and complicated saltwater–freshwater flow, resulting in strong current shear.
ADCP measurements of the tidally forced currents over a 3-day period. Warm colors indicate upriver flow and cool colors indicate downriver flow.
Citation: Journal of Atmospheric and Oceanic Technology 33, 9; 10.1175/JTECH-D-15-0145.1
Ebb, flood, and slack current profiles measured by the ADCP on 31 May 2013.
Citation: Journal of Atmospheric and Oceanic Technology 33, 9; 10.1175/JTECH-D-15-0145.1
An example of a single radar scan (0800 UTC 3 June 2013) shows strong reflection from land features that illuminate the coastline, as well as the north and south jetties bounding the inlet (Fig. 4). Each radar scan was georeferenced using known stationary points in the field of view. Backscatter from incoming waves can be seen within the inlet. Overlaid on this scan is a 750 m × 750 m box centered over the position of the ADCP, which represents the region over which data were processed to study wave–current interactions. Bathymetry contours under the inspection square (Fig. 5) collected by the U.S. Geological Survey (USGS) show waves within the inspection square were propagating in water depths ranging between −10 and −15 m. To capture the large spatial scale of the MCR, X-band operating parameters (Table 1) were tuned to maximize the usable radar range by collecting a large number of samples (1024 samples per beam) with a coarse range resolution (approximately 7.5 m).
A section of a single X-band scan of the MCR, where light and dark shading represent high and low signal return, respectively. The radar is located in the middle of the white disk in the lower right, which denotes the 500-m blanking range. The study region (white square) is centered over the bottom-mounted ADCP (triangle).
Citation: Journal of Atmospheric and Oceanic Technology 33, 9; 10.1175/JTECH-D-15-0145.1
Bathymetry [North American Vertical Datum of 1988 (NAVD88)] from USGS surveys shows that water depth under the inspection square varies between −10 and −15 m. The ADCP (asterisk) was mounted at a depth of −11 m.
Citation: Journal of Atmospheric and Oceanic Technology 33, 9; 10.1175/JTECH-D-15-0145.1
Operational parameters for the land-based X-band radar during the RIVET2 campaign that collected data from 24 May 2013 to 4 Jun 2013.
Measurements of the relevant environmental conditions during RIVET2 include the wind conditions measured at the radar site and incident wave conditions collected approximately 8 km offshore by a California Data Information Program buoy (Figs. 6a and 6b).
Environmental conditions data collected at MCR near Astoria, including (a) offshore wave conditions and (b) wind conditions at the radar site. (c) The X-band SNR (solid), resulting from the wave and wind conditions. The SNR threshold (dotted) used to separate low-quality noise. (d) Relative wave–current direction displayed as the cosine of the absolute difference between wave and current directions, ΔθW−C. This study focuses on the time period of sustained SNR and small ΔθW−C from 30 May to 1 Jun.
Citation: Journal of Atmospheric and Oceanic Technology 33, 9; 10.1175/JTECH-D-15-0145.1
The signal strength of radar backscatter is influenced by a combination of wave height and wind conditions. A signal-to-noise ratio (SNR) was calculated by defining noise as the mean energy return in a high-wavenumber, low-frequency region of the backscatter spectrum, far removed from expected ocean wave information (Young et al. 1985). The evolution of the SNR for data recorded within the inspection box region demonstrates the radar signal’s complicated relationship to wind and incident wave conditions approaching the MCR (Fig. 6c). Sections of sustained high SNR correspond to times of large magnitude and near-incident (northwest) wind and waves.
The study of wave–current interaction is dependent on the relative angle between the current and overlying wave propagation directions. By estimating the local wave direction as the direction of the maximum SNR, the relative wave–current direction 
The 3-day time period of sustained SNR and collinear waves and currents [
3. Methods
a. Doppler shift extraction
Examples of the location of returned backscatter energy in wavenumber–frequency space during (a) ebb and (b) flood current conditions. Darker colors indicate higher energy return than lighter colors. The depth-corrected dispersion relationship (solid blue) and first harmonic relationship (dashed blue) indicate expected energy locations in the zero-current condition. The solid red line denotes selected 
Citation: Journal of Atmospheric and Oceanic Technology 33, 9; 10.1175/JTECH-D-15-0145.1
The remaining ridge of energy in the wavenumber–frequency domain was isolated, and wavenumber-dependent Doppler shifts were calculated by subtracting the dispersive frequency [(1)] from the observed frequency.
b. Depth-uniform and linear current estimation
An example of a linear fit to Doppler shift observations, where the −1.9 m s−1 slope represents an estimate of a depth-uniform current. These data were selected during an ebb tide at 0520 UTC 31 May 2013.
Citation: Journal of Atmospheric and Oceanic Technology 33, 9; 10.1175/JTECH-D-15-0145.1
The strong, depth-dependent current shear measured by the ADCP (Fig. 3) suggests that the accuracy of radar-derived current estimates would be enhanced by removing assumptions of the current–depth structure. To account for the effects of arbitrary current profiles on the wavefield, the integral in (3) can be used in either the “forward” or the “inverse” problem. The forward problem involves transforming measured current profiles (e.g., from an ADCP) into Doppler shift–wavenumber profiles to be compared to X-band measurements. The inverse problem is the extraction of current profiles from X-band measurements of the wavefield.
c. The forward problem
d. The inverse problem
The inverse problem involves extracting current profiles u(z) from the Doppler shift–wavenumber observations using (3). The inversions of Laplacian-type expressions, such as this, are plagued by the amplification of inherent measurement noise and truncation error (Ha 1979). Two methods were used in this study to stabilize the inversion: 1) the Gauss–Legendre method was applied to reduce the integral to quadrature (Weeks 1966; Cohen 2007) and 2) a constrained least squares approach was used to invert the resulting matrix expression (Twomey 1977; Wunsch 1996).
4. Results
a. Depth-uniform and linear current estimation
Currents were estimated under the depth-uniform current assumption using the linear regression in (4) over the selected 3-day time period. These results were compared to depth-averaged ADCP currents (Fig. 9). Time periods of missing X-band data correspond to low SNR (less than 5 dB), for example, due to low wind speed (below 3 m s−1) (Fig. 6c). The time series shows the agreement was good during slack tides but reduced as the currents reached flood and ebb conditions (Fig. 9a). The scatterplot shows the best agreement occurred when the depth-averaged current was between −0.25 and −0.5 m s−1, corresponding to times when the current was dominated by the river outflow (Fig. 9b).
(a) ADCP depth-averaged currents (dashed) and X-band bulk current estimates (dotted) vs time for the 3-day period. (b) Scatterplot of X-band bulk currents vs ADCP depth-averaged currents. Solid line indicates a 1-to-1 relationship and dashed line denotes linear fit.
Citation: Journal of Atmospheric and Oceanic Technology 33, 9; 10.1175/JTECH-D-15-0145.1
Current profiles were estimated under the linear current–depth assumption using (5) and the results were compared with ADCP measurements (Fig. 10). The skill of the current estimates was calculated using (9) (Fig. 10c). The skill ranges between 0.7 and 0.9 during ebb tides, whereas during flood conditions the skill ranges between −1 and 0.5. This behavior is consistent with the truncated wavenumber range during flood tides due to the harmonic contamination (e.g., Fig. 7b). Because solutions from this method exist only at depths defined by the effective depth 
(a) X-band-derived current profiles assuming a linear current–depth relationship, or 
Citation: Journal of Atmospheric and Oceanic Technology 33, 9; 10.1175/JTECH-D-15-0145.1
b. The forward problem
To compute the forward problem, the weighted sum in (6) was used to transform ADCP current measurements to Doppler shift–wavenumber profiles. The results were then directly comparable to X-band Doppler shift observations (Fig. 11). Missing X-band data in the higher wavenumbers during flood tides were due to the presence of the harmonics, which overwhelm the fundamental signal (e.g., Fig. 7). Results from the forward problem qualitatively agree with X-band profiles over the 3-day time period, exhibiting similar structure in the wavenumber dependence of the observed Doppler shifts.
(a) ADCP depth–current profile measurements. (b) The Doppler shift–wavenumber profiles resulting from transforming the ADCP current profiles via the forward problem. (c) Observed X-band Doppler shift–wavenumber profiles.
Citation: Journal of Atmospheric and Oceanic Technology 33, 9; 10.1175/JTECH-D-15-0145.1
To quantitatively assess the results of the forward problem (Fig. 11b), the average slope of each 
Time series of the slopes of the 
Citation: Journal of Atmospheric and Oceanic Technology 33, 9; 10.1175/JTECH-D-15-0145.1
c. The inverse problem
1) Model tuning
The Doppler shift–wavenumber profiles from the forward problem were used as input into (8) to tune the inversion method. The Legendre polynomial order and the tunable parameter λ were varied, until the best match with the measured ADCP current profiles was found (Fig. 13), which occurred with n = 7 and λ = 0.04. Residual discrepancy between these inversion results and measured current profiles were caused by 1) the truncation of the current profiles near the surface to avoid surface contamination and 2) the finite range of wavenumbers used in the forward problem calculation, defined by the resolution and box size imaged by the X-band radar.
The outcome of inverting the forward problem result (solid) is compared to measured current profiles (dashed) to tune the inversion method. In this example, Legendre order n = 7 and λ = 0.01.
Citation: Journal of Atmospheric and Oceanic Technology 33, 9; 10.1175/JTECH-D-15-0145.1
2) Inversion of X-band backscatter
The tuned inversion model [(8)] was used to estimate the current profiles from the X-band 
(a) The result of the inversion of current profiles from X-band backscatter. Black lines bound the region measured by the ADCP. (b) ADCP current measurements in the along-channel direction. (c) Time series of the skill of the X-band inversion method.
Citation: Journal of Atmospheric and Oceanic Technology 33, 9; 10.1175/JTECH-D-15-0145.1
The depth dependence of the skill of the X-band inversion method.
Citation: Journal of Atmospheric and Oceanic Technology 33, 9; 10.1175/JTECH-D-15-0145.1
5. Discussion
Results of the depth-uniform analysis show that error current estimates under the depth-uniform assumption exceeded 20% within one hour of slack currents (Fig. 9). This confirmed that the presence of current shear plays an important role in estimating currents from wavefield measurements. The simplest way to include the effects of current shear was to assume a linear depth–current relationship, which resulted in a model to directly map effective velocities of individual wavenumbers to currents of unique effective depths (Fig. 10). Although current estimates appeared to capture some evidence of current shear, solutions were limited to a small depth range of approximately 2.5 m. Furthermore, because of the direct mapping between individual wavenumbers to individual water depths, missing data and truncated Doppler shift data led to discontinuous and truncated current–depth profiles. Because of the small depth range and inconsistency of the current estimates, a more rigorous solution was sought after that involved the entire depth–current profile.
The forward problem investigated the sensitivity of the wavefield to the arbitrary current profile shapes measured by the ADCP. The Doppler shift–wavenumber profiles resulting from the forward problem compared well with those observed by the X band (Figs. 11 and 12), implying that the model developed by Kirby and Chen (1989) appropriately describes the observed wave–current relationship in the MCR. Furthermore, the result of the forward problem shows that the observed current shear significantly altered the Doppler shift–wavenumber relationship, suggesting that the retrieval of current shear information from Doppler shift measurements is possible. A method was therefore developed to calculate the inversion of the model used in the forward problem. Free parameters were tuned by forcing the inversion of the forward problem results to match ADCP current profiles (Fig. 13). Residual error between the two was partially due to error in the forward problem, which arose from the truncation of ADCP profiles near the riverbed (due to instrument mounting and blanking range) as well as below the surface (to avoid surface contamination).
The inversion method resulted in current estimates that successfully captured the evolution of the current profile throughout multiple tidal cycles in the MCR, with model skill exceeding 0.8 during ebb and slack current conditions (Fig. 14). The drop in model skill during flood tides (Fig. 14c), when harmonic contamination resulted in a narrower usable wavenumber range, shows the success of the inversion was dependent on broadband Doppler shift observations. The depth dependence of skill (Fig. 15) shows agreement between inverted and measured currents were best in the top 9 m of the ADCP range. The skill of current estimates below −10 m declined rapidly, as waves within the observed wavenumber range were less sensitive to deeper currents as indicated by (3).
One source of error in the results of the inversion was the bathymetric variation under the inspection square, which varied over a range of 
Water depth uncertainty contributed to error in the calculation of Doppler shifts and in the process of inverting the Doppler shifts to estimate currents. To examine the effects of error in water level on each of these parts of the inversion method separately, the water level was varied by 
Sensitivity of skill to error in water-level estimate. Solid line describes water-level sensitivity of the inversion process assuming perfect Doppler shift estimates. Dotted line includes water depth sensitivity of the process of estimating Doppler shifts.
Citation: Journal of Atmospheric and Oceanic Technology 33, 9; 10.1175/JTECH-D-15-0145.1
Both the time and depth dependence of skill suggest that the results of this study could be improved by expanding the wavenumber range used in the inversion. Raising the antenna height, for example, would minimize wave shadowing and therefore avoid the harmonic contamination during flood tides. Increasing the range resolution of the radar backscatter measurements would extend the Nyquist wavenumber cutoff by sampling smaller waves. Increasing the size of the inspection square would result in a smaller low wavenumber cutoff, potentially increasing the inversion skill below −10-m depth. However, a larger inspection area would be sensitive to a larger range of water depths, resulting in more uncertainty in Doppler shift estimation.
The accuracy of Doppler shift estimates depends on the time window of the FFT to exceed the wavefield’s decorrelation time scale. At a minimum one should ensure that time windows capture multiple wave periods of the longest measurable wave. However, increasing the FFT time window to sample multiple wave groups would ensure the maximum energy in each frequency band is observed, increasing the SNR. A potential improvement to the inversion example shown in this study, therefore, is to explore the effect of much longer time windows to sample across several time scales across which wave groups pass.
The success of future applications of the current inversion method will rely on both the presence and the ability to observe a broadbanded wavefield. Wavefields comprised of only short waves are largely unaffected by deeper currents, limiting the depth range of inversion results. Conversely, the inversion of wavefield measurements that contain only low wavenumber information will limit the ability to resolve near-surface currents.
6. Conclusions
In this work, X-band backscatter was collected in the mouth of the Columbia River, Oregon, to investigate the validity of current and current shear estimations from Doppler shift measurements. Although the well-established method that assumes uniform depth–current profiles provided reasonable current estimates during times of low shear, discrepancies exceed 20% within one hour of slack tide. The estimation of current profiles under a linear depth–current profile assumption was shown to be limited to a depth range of approximately 2.5 m and to be sensitive to observation noise. Through the use of the integral transform (Kirby and Chen 1989) that takes current shear into account, ADCP current profiles were transformed into Doppler shift–wavenumber profiles for comparison with X-band observations. The agreement between ADCP and X-band observations was improved by the inclusion of the observed current shear information. This result indicated the X-band observations of the wavefield contained information about the underlying current shear, which should be taken into account when estimating currents from Doppler shift observations.
A new inversion model to estimate current–depth profiles from X-band observations of the wavefield’s Doppler shift–wavenumber relationship was introduced. Applying the model to estimate current profiles in the MCR showed good agreement to concurrent ADCP measurements, with a model skill exceeding 0.8 during ebb and slack current conditions. The X-band-derived results were shown to accurately depict depth and time variable phenomena such as the tidally forced countercurrent at depth during tidal transition. The ability to remotely sense such phenomena suggest this new model can be used to address many types of nearshore scientific, navigational, and civil concerns. To support studies of open ocean currents on the submesoscale, future work can expand this inversion method to be applicable for use in deep-water environments. Expansion to the open ocean introduces geometric complications, with relative directions of waves of various wavenumbers and currents of various depths becoming important. Furthermore, smaller current magnitudes in the open ocean make it more difficult to extract Doppler shift–wavenumber profiles from wavefield observations.
Acknowledgments
The authors thank ONR for funding support (Award N00014-13-1-0198), and collaborators Dr. Rocky Geyer and Dr. Peter Traykovski at WHOI for the ADCP measurements. We also thank USACE for CDIP buoy data support and USGS for the bathymetric survey data.
REFERENCES
Alpers, W., Ross D. , and Rufenach C. , 1981a: On the detectability of ocean surface waves by real and synthetic aperture radar. J. Geophys. Res., 86, 6481–6498, doi:10.1029/JC086iC07p06481.
Alpers, W., Schröter J. , Schlude F. , Müller H.-J. , and Koltermann K. P. , 1981b: Ocean surface current measurements by an L band two-frequency microwave scatterometer. Radio Sci., 16, 93–100, doi:10.1029/RS016i001p00093.
Barrick, D. E., 1972: First-order theory and analysis of MF/HF/VHF scatter from the sea. IEEE Trans. Antennas Propag., 20, 2–10, doi:10.1109/TAP.1972.1140123.
Bell, P. S., 1999: Shallow water bathymetry derived from an analysis of X-band marine radar images of waves. Coast. Eng., 37, 513–527, doi:10.1016/S0378-3839(99)00041-1.
Bogden, P. S., Malanotte-Rizzoli P. , and Signell R. , 1996: Open-ocean boundary conditions from interior data: Local and remote forcing of Massachusetts Bay. J. Geophys. Res., 101, 6487–6500, doi:10.1029/95JC03705.
Cohen, A. M., 2007: Numerical Methods for Laplace Transform Inversion. Numerical Methods and Algorithms, Vol. 5, Springer, 252 pp.
Crombie, D. D., 1955: Doppler spectrum of sea echo at 13.56 Mc./s. Nature, 175, 681–682, doi:10.1038/175681a0.
Dankert, H., and Rosenthal W. , 2004: Ocean surface determination from X-band radar-image sequences. J. Geophys. Res., 109, C04016, doi:10.1029/2003JC002130.
Davis, R. E., deSzoeke R. , Halpern D. , and Niiler P. , 1981: Variability in the upper ocean during MILE. Part I: The heat and momentum balances. Deep-Sea Res., 28A, 1427–1451, doi:10.1016/0198-0149(81)90091-1.
Dugan, J., Piotrowski C. , Zuckerman S. , Chinn C. , Yi M. , and Vierra K. , 2008: Near-surface current profile measurements using time series optical imagery. Proceedings of the IEEE/OES/CMTC Ninth Working Conference on Current Measurement Technology, J. Rizoli White and S. P. Anderson, Eds., IEEE, 174–180, doi:10.1109/CCM.2008.4480864.
Fernandez, D. M., Vesecky J. F. , and Teague C. C. , 1996: Measurements of upper ocean surface current shear with high-frequency radar. J. Geophys. Res., 101, 28 615–28 625, doi:10.1029/96JC03108.
Golub, B. G. H., and Welsch J. H. , 1969: Calculation of Gauss quadrature rules. Math. Comput., 23, 221–230, doi:10.1090/S0025-5718-69-99647-1.
Ha, E. C., 1979: Remote sensing of ocean surface current shear by HF backscatter radar. Ph.D. thesis, Stanford University, 134 pp.
Halpern, D., 1977: Description of wind and of upper ocean current and temperature variations on the continental shelf off northwest Africa during March and April 1974. J. Phys. Oceanogr., 7, 422–430, doi:10.1175/1520-0485(1977)007<0422:DOWAOU>2.0.CO;2.
Harlan, J., Terrill E. , and Burnett B. , 2009: National high frequency radar network: update. OCEANS 2009: MTS/IEEE Biloxi—Marine Technology for Our Future; Global and Local Challenges, IEEE, 463–469.
Harlan, J., and Coauthors, 2011: National IOOS high frequency radar search and rescue project. Proc. OCEANS’11: MTS/IEEE Kona, Waikoloa, HI, IEEE, 9 pp. [Available online at http://ieeexplore.ieee.org/document/6107090.]
Hessner, K., and Bell P. S. , 2009: High resolution current and bathymetry determined by nautical X-Band radar in shallow waters. Proc. OCEANS’09 IEEE Bremen: Balancing Technology with Future Needs, Bremen, Germany, IEEE, 5 pp., doi:10.1109/OCEANSE.2009.5278333.
Hetland, R. D., 2006: Event-driven model skill assessment. Ocean Modell., 11, 214–223, doi:10.1016/j.ocemod.2004.12.001.
Kirby, J. T., and Chen T.-M. , 1989: Surface waves on vertically sheared flows: Approximate dispersion relations. J. Geophys. Res., 94, 1013–1027, doi:10.1029/JC094iC01p01013.
Kohut, J. T., Roarty H. J. , and Glenn S. M. , 2006: Characterizing observed environmental variability with HF Doppler radar surface current mappers and acoustic Doppler current profilers: Environmental variability in the coastal ocean. IEEE J. Oceanic Eng., 31, 876–884, doi:10.1109/JOE.2006.886095.
Nieto Borge, J. C., Rodríguez G. R. , Hessner K. , and González P. I. , 2004: Inversion of marine radar images for surface wave analysis. J. Atmos. Oceanic Technol., 21, 1291–1300, doi:10.1175/1520-0426(2004)021<1291:IOMRIF>2.0.CO;2.
Ohlmann, C., White P. , Washburn L. , Terrill E. , Emery B. , and Otero M. , 2007: Interpretation of coastal HF radar-derived surface currents with high-resolution drifter data. J. Atmos. Oceanic Technol., 24, 666–680, doi:10.1175/JTECH1998.1.
Paduan, J. D., and Rosenfeld L. K. , 1996: Remotely sensed surface currents in Monterey Bay from shore-based HF radar (Coastal Ocean Dynamics Application Radar). J. Geophys. Res., 101, 20 669–20 686, doi:10.1029/96JC01663.
Phillips, O. M., 1966: The Dynamics of the Upper Ocean. Cambridge University Press, 344 pp.
Senet, C. M., Seemann J. , and Ziemer F. , 2001: The near-surface current velocity determined from image sequences of the sea surface. IEEE Trans. Geosci. Remote Sens., 39, 492–505, doi:10.1109/36.911108.
Senet, C. M., Seemann J. , Flampouris S. , and Ziemer F. , 2008: Determination of bathymetric and current maps by the method DiSC based on the analysis of nautical X-band radar image sequences of the sea surface. IEEE Trans. Geosci. Remote Sens., 46, 2267–2279, doi:10.1109/TGRS.2008.916474.
Shuchman, R., 1979: The feasibility of measurement of ocean current detection using SAR data. Proceedings of the Thirteenth International Symposium on Remote Sensing of the Environment, IEEE, 93–103.
Stewart, R. H., and Joy J. W. , 1974: HF radio measurements of surface currents. Deep-Sea Res. Oceanogr. Abstr., 21, 1039–1049, doi:10.1016/0011-7471(74)90066-7.
Teague, C. C., Vesecky J. F. , and Hallock Z. R. , 2001: A comparison of multifrequency HF radar and ADCP measurements of near-surface currents during COPE-3. IEEE J. Oceanic Eng., 26, 399–405, doi:10.1109/48.946513.
Terrill, E., and Coauthors, 2006: Data management and real-time distribution for HF-radar national network. OCEANS 2006, Boston, MA, IEEE, 060331-220, doi:10.1109/OCEANS.2006.306883.
Trizna, D., and Xu L. , 2006: Target classification and remote sensing of ocean current shear using a dual-use multifrequency HF radar. IEEE J. Oceanic Eng., 31, 904–918, doi:10.1109/JOE.2006.886198.
Twomey, S., 1977: Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements. Dover Publications, 237 pp.
Weeks, W. T., 1966: Numerical inversion of Laplace transforms using Laguerre functions. J. Assoc. Comput. Mach., 13, 419–426, doi:10.1145/321341.321351.
Wunsch, C., 1996: The Ocean Circulation Inverse Problem. Cambridge University Press, 442 pp.
Young, I. R., Rosenthal W. , and Ziemer F. , 1985: A three-dimensional analysis of marine radar images for the determination of ocean wave directionality and surface currents. J. Geophys. Res., 90, 1049–1059, doi:10.1029/JC090iC01p01049.
