## 1. Introduction

HF radar is increasingly being employed for determining surface currents in the near-shore ocean environment. Originally identified by Crombie (1955), and developed by Barrick (1972, 1980; Barrick and Snider 1977), the retrieval of surface current information from the Bragg scatter of 3–30-MHz radar signals have been performed using both phased antenna arrays and compact, collocated antenna arrays. Doppler shift from the known Bragg frequency determines current velocity, while either beam forming (Teague 1986), least squares (Lipa and Barrick 1983), or the MUSIC algorithm developed by Schmidt (1982, 1986) is used to determine a bearing estimate for that current velocity (D. E. Barrick and B. J. Lipa, Radar angle determination with MUSIC direction finding, U.S. Patent No. 5990834). The predominant system in use today is the commercially available SeaSonde by CODAR Ocean Systems, which employs the compact antenna array and the Multiple Signal Characterization (MUSIC) algorithm.

Validation of HF radar-derived measurements of surface currents against other measurement approaches have a rich history and include campaigns employing ships, moored buoys with point current meters, and surface drifters. Experiments by Teague (1986) using spar buoys and shipboard HF radar show RMS current velocity differences of ∼5 cm s^{−1}. Paduan and Rosenfeld (1996) compare CODAR measurements with moored current meters, drifters, and acoustic Doppler current profiler (ADCP) data, providing RMS current velocity differences between 6 and 16 cm s^{−1} when comparing the various instruments. Graber et al. (1997) compare phased array HF radar measurements with moored current meters and separate the total RMS difference of 10–20 cm s^{−1} into its major constituents. Ohlmann et al. (2006) recently employed drifters to further validate the HF radar velocities and found RMS difference to be ∼7 cm s^{−1} or less. While this list of studies is not exhaustive [see Graber et al. (1997) for a comprehensive review], consensus exists that multiple factors influence the observed differences including 1) depth differences between the HF radar–measured surface currents (often 1 m or less) to the fixed current meter or ADCP bin (typically 5–10 m is the norm); 2) HF radar is a temporally and spatially integrated value versus the in situ point time series, which is subject to finescale gradients; and 3) HF radar has inherent errors in angle and speed determination. As a result, measurements from HF radar are often attributed uncertainties of 5–10 cm s^{−1} with no definitive insight from whence these values arise because of the complexities of analyzing field data. Despite the present difficulty in separating velocity error from velocity uncertainty, there is increasingly growing acceptance and usage of the technology within the oceanographic community, due in part to its capability to map large areas of the ocean surface rapidly and efficiently.

In the most basic terms, the MUSIC algorithm takes power spectrum input from multiple antennas, at a specific Doppler shift (current velocity) away from the zero current Bragg frequency (defined by the radio frequency), and inverts that input to a bearing (or bearings) from where the signal originated. This is done by comparing the received antenna voltages to expected antenna voltages, which comprise an antenna manifold, over all bearings. The bearings of the antenna manifold that best match the input are returned as results (Schmidt 1982, 1986).

Simulations for assessing the capabilities of the MUSIC algorithm in ocean current applications are limited to a technical report by Barrick and Lipa (1996), which applies MUSIC to a limited wind wave/current scenario, and subsequent work by Laws et al. (2000), which included MUSIC simulations in an error analysis of the Multifrequency Coastal Radar (MCR) phased array HF radar. Motivated to understand the limitations of MUSIC processing in compact direction-finding-type HF radars to different coastal current scenarios, a simulation of the response of an idealized compact array system is developed herein, with the work by Barrick and Lipa (1996) providing the basic framework. The goals of this paper are to assess the inherent limitations of the MUSIC algorithm, provide insight in how to interpret data from existing observations, and provide guidance for areas of further research that may improve the processing of radar data, and plan for future system deployments. At the time of this writing, 100 compact-antenna-style systems are either in operation, or are funded for installation on U.S. shorelines. Since the analysis and findings in this document are intended to examine the skill of the MUSIC algorithm within the compact geometry of a three antenna system, the results are not intended to exactly mimic or simulate all system performance issues of existing available HF radar systems.

## 2. Simulation definition

Various currents scenarios are input to the MUSIC algorithm under different wind wave conditions. Currents are modeled in a Cartesian half-plane, spanning from the land-based HF radar location toward the west, out to the maximum range cell 31 (a West Coast radar with a straight north–south coastline). Range cell resolution, while set to approximately 1.5 km (31 range cells = 46.5 km), is not relevant to this analysis since range cell number is used only to artificially set the signal-to-noise ratio (SNR). Scattered HF signal strength versus range is well documented, as presented by Barrick (1971) and recently validated by Gurgel et al. (1999). Because of the need to examine the sensitivity of the MUSIC algorithm to a wide range of SNRs, the SNR is set in a linear fashion over 31 range cells, starting at 40 dB in range cell 1, and reducing by 2 dB per range cell thereafter. While in practice, this linear 2-dB/1.5-km range cell degradation in SNR is not exactly what is observed in the field at 25 MHz, it does provide a simple, yet rational method for simulating a wide dynamic range. More complex simulations that include a model of the full radar equation for the transmitted and scattered signals would also provide a range-dependent attenuation.

*u*,

*υ*) vectors on a Cartesian grid of (

*x*,

*y*) points, covering all radial range cells, and translated to a polar coordinate representation of (

*u*,

*υ*) current vectors, computed on a polar grid (

*R*,

*θ*), across all range cells and over water bearing angles (180°–360°). These (

*u*,

*υ*) current vectors on the polar grid are subsequently projected onto the radial directions defined by the location of the HF radar site to give scalar current quantities at each (

*R*,

*θ*) point at discrete 5° bearing increments. At each range cell

*R*, these are the input current velocities

*V*(

_{i}*θ*). The scalar current is defined positive toward the HF radar and negative away from the HF radar. These scalar currents are used at each range cell to determine the Doppler shift on the Bragg scatter received from each bearing

*θ*using the equation where

*k*

_{0}is the radar wavenumber.

*θ*

_{w}is the wind direction bearing; and 0.01 provides a nonzero value in the null of the cardiod.

Gaussian noise is added to the three time series signals to set the received signal-to-noise ratio. The time series are multiplied by a Hamming window, converted to spectra by FFT, and normalized by bandwidth to produce power spectral densities. The power spectral densities are cross-multiplied to produce what are commonly referred to as cross spectra (first-order scatter only). Three cross spectra are then averaged; hence, the first 512-point-averaged spectra cover 12 min and 48 s of received signal. These are the exact cross spectra that are input to the MUSIC algorithm in this paper, commonly referred to in actual HF radar operation as 10-min cross spectra. The resulting radial current velocities at each range and bearing are then compared to the simulated input and a skill metric is computed to analyze the accuracy of the MUSIC algorithm under the different current, wind, and SNR scenarios. The flowchart provided in Fig. 1 illustrates this processing flow.

The MATLAB source code used for these simulations was originally published by Barrick and Lipa (1996). The public domain “CSSim” MATLAB software was modified to introduce noise at the time domain for these simulations. While the MUSIC algorithm inversion from cross spectra to radial current velocities remains unchanged, the methods of error mitigation and time/space averaging were changed for this study. While no proprietary software was used, the HF radar parameters used in this simulation correspond to an operational SeaSonde system maintained on the Coronado Islands by Scripps Institution of Oceanography and are given in Table 1. The ideal antenna beam patterns are defined to have 5° bearing resolution, which constrains the MUSIC algorithm to have the same resolution.

## 3. Simulation input

This section describes the idealized scenarios of wind wave directions and surface current patterns that are simulated. For each wind wave scenario, a wind direction is chosen that governs the orientation of the generated cardiod signal power function. This function determines the variance for the power of the Bragg scatter signal received at the HF radar.

### a. Wind wave scenario A

The first wind wave scenario represents onshore winds blowing normal to shore from a bearing of *θ _{w}* = 270, with the onshore wind waves producing positive Bragg scatter and the reflected/receding waves at bearing 090 producing negative Bragg scatter.

### b. Wind wave scenario B

The second wind wave scenario represents wind blowing from *θ _{w}* = 185 resulting in alongshore wind waves from bearing 185 producing positive Bragg scatter, and the reflected/receding waves at bearing 355 producing negative Bragg scatter.

### c. Current scenarios

Four different surface current scenarios are chosen to represent a broad range of conditions that realistically represents the arena in which the radar might operate.

#### 1) Surface current scenario 1—Onshore uniform current

The first current model is a uniform current of 20 cm s^{−1} with an onshore bearing angle of 270°, shown in a Cartesian plane in Fig. 2a, and its radial projection in Fig. 3a.

#### 2) Surface current scenario 2—Alongshore uniform current

The second current scenario is a uniform, alongshore 20 cm s^{−1} current at bearing 180°, shown in a Cartesian plane in Fig. 2b, and its radial projection in Fig. 3b.

#### 3) Surface current scenario 3—Alongshore shear current

This current scenario represents linear alongshore shear (40 cm s^{−1} max to 5 cm s^{−1} min, across 46.5 km) at bearing 180°. This scenario is shown in a Cartesian plane in Fig. 2c, and its radial projection in Fig. 3c.

#### 4) Surface current scenario 4—Eddy current

This scenario represents an eddy with a maximum radial current velocity of ∼20 cm s^{−1}, directly offshore (bearing 270°), centered at range cell 17 (25.5 km), 30-km diameter, superimposed on weak alongshore uniform current of 10 cm s^{−1}. This scenario is shown in a Cartesian plane in Fig. 2d, and its radial projection in Fig. 3d.

## 4. MUSIC algorithm error analysis

To illustrate the error analysis procedure used in all simulations, the input for wind wave scenario A/surface current scenario 1 (Fig. 2a) is detailed as an example. The radial scalars shown in Fig. 3a are input to the signal synthesis equations given in section 2. An example of the resulting monopole antenna spectrum across all 512 Doppler cells at range cell 6 is shown in Fig. 4a. This example shows all the Bragg energy present to the right of the Bragg frequency lines, indicating that all the currents are positive, toward the radar antenna.

The raw MUSIC output for the uniform current input is shown in Fig. 5, and is found to have several large erroneous results. The scale of the current velocity vectors has been reduced (for Fig. 5 only) from the previous vector plots to show the large errors that are generated. The magnitude of these errors suggests that some level of simple postprocessing can be used to identify and remove them.

### a. Classes of error

The errors are categorized into three groups.

#### 1) Outliers

Outliers are produced when Doppler cell inputs that are far from the Bragg frequencies pass the signal detection threshold criteria of 6-dB SNR will be processed by MUSIC. These inputs are typically noise, yet are interpreted as signals resulting from an unusually large radial current velocity. The possible range of radial current velocities for the radar simulated herein (see Table 1) is (−2.9804, +2.9567) m s^{−1}. Considering all 512 Doppler cells, and therefore allowing for radial current velocities between and including the minimum and maximum over the range, MUSIC produces results as in Fig. 5. While the very large vectors are errors, the underlying small vectors over water are (in general) correct.

The first example of an MUSIC error is the large overland outlier of −279.12 cm s^{−1} at a bearing of 135° in range cell 24 in Fig. 5. To see how this outlier is produced, the input signal and Doppler cell that was input to MUSIC is examined in Fig. 4b (the Doppler cell determines the velocity of −279.12 cm s^{−1}—since the Doppler cell is on the negative side of the negative Bragg peak a large negative velocity is produced). The MUSIC direction of arrival (DOA) functions are then examined in Fig. 6a to see how this noise gets interpreted as an overland velocity. Since there is only one large eigenvalue in the signal covariance matrix, MUSIC considers only a single-bearing solution. Figure 6a shows the DOA metric under both single-bearing (upper plot) and dual-bearing (lower plot) hypotheses. Under the single-bearing hypothesis, there is a maximum in the DOA function at (*R*, *θ*) = (24, 135) with a radial current velocity of −279.12 cm s^{−1}, which MUSIC determines to be the most probable DOA. However, the maximum in the single-bearing DOA function is a relatively small number, perhaps indicating a poor bearing estimation (this topic is left for further research). The output at (*R*, *θ*) = (24, 135) is over land, and in practice these velocities are removed through a land mask.

Overwater outliers are also present in Fig. 5, for example, 180° away from the previous outlier at (*R*, *θ*) = (23, 315) with a velocity of −234.17 cm s^{−1}. These gross outliers are eliminated in practice by two methods: 1) restricting the Doppler search window to frequencies that are within the expected value found at the observation site, and 2) applying a spectral analysis algorithm that identifies the first-order Bragg scatter region of the cross spectra and eliminating the rest from the MUSIC analysis. It should be noted, however, that under environmental conditions where the expected values of the surface current velocities are quite high; for example in the Columbia River plume, Gulf Stream, or Florida Current, the limitation of the Doppler search is not practical. In those cases, actual Bragg scatter energy may be present in those high Doppler cells versus the Gaussian noise used in this simulation.

#### 2) Dual-bearing determination errors

The MUSIC algorithm, due to the three (*N*) independent antennas inherent in the compact antenna design, has the degrees of freedom to determine two (*N* − 1) bearing solutions from which a given radial current velocity is detected. In Fig. 5 and the previous section, an example of a single-bearing solution has been identified as an outlier. Dual-bearing solution outliers are also generated by MUSIC, often with one overland bearing and one oversea bearing. It is important to note here that the inputs that generate dual-bearing errors can be in Doppler cells that are well within the expected radial current velocities, which in this current scenario is approximately ±20 cm s^{−1}. This is illustrated by the input Doppler cell on the positive side of the negative Bragg peak that corresponds to a velocity of +18.92 cm s^{−1} shown in Fig. 4c. Figure 6b shows the DOA functions for this input Doppler cell. Since the two largest eigenvalues of the signal covariance matrix are close in magnitude, the MUSIC algorithm generates a dual-bearing solution of (*R*, *θ*) = (26, 105) and (*R*, *θ*) = (26, 295). The first of the MUSIC results for this Doppler cell input can be seen as a small vector over land in Fig. 5 at (*R*, *θ)* = (26, 105). The companion dual-bearing solution (*R*, *θ*) = (26, 295) is visible over water, slightly overlapping with another result from range cell 25 at the same bearing (*R*, *θ*) = (25, 295). In Fig. 6b, the asterisks (*) in the plot show the bearings at which the input radial current velocity is within 1.18 cm s^{−1} (which is half the system velocity resolution of 2.36 cm s^{−1}) of the output radial current velocity. The input between (*R*, *θ*) = (26, 245) and (*R*, *θ*) = (26, 295) is between 18.92 ± 1.18 cm s^{−1}. Thus, the other dual-bearing solution over water is relatively close to an input point.

In this case, the overwater solution is relatively close to the known input, yet there are other examples where the dual-bearing solutions produce two poor results. While overland errors are easy to flag, erroneous dual-bearing results that occur over water and are within the expected speed range cannot be easily identified and eliminated. The three cases relating to dual-bearing solutions within the expected current velocity range are as follows: 1) both results are over land and are erased, 2) one result is over land and is erased, and one is over water and is not erased, and 3) both results are over water and are not erased. In the situation where the results are overwater bearings and within the expected radial current velocity range, the accuracy of the results vary and require statistical analysis. In a subsequent section, statistics are collected as a function of range cell (SNR) and bearing of 1) the RMS error normalized by the input and its first and second moments over a large data sample, 2) the frequency of occurrence of overland solutions and dual-bearing MUSIC solutions, and 3) an overall skill metric.

#### 3) Gaps

The converse to producing results at bearings with no input signal is the lack of a MUSIC solution over water. An example can be seen as gap in solutions in Fig. 5 around (*R*, *θ*) = (17, 290). In practice, this phenomenon can occur because of distortions in measured (nonideal) gain patterns of the three radar antennas (see Toh 2005). In the idealized antenna case herein, this phenomenon occurs when there are more than two bearing angles over sea that have nearly the same input radial current velocity. For a given range cell and a given Doppler cell (and thus a given radial current velocity), the MUSIC algorithm can produce a maximum of two bearing solutions. Any more bearings in that range cell with that same radial current velocity will be left out, producing a gap where there is no solution. This is an inherent limitation of using MUSIC with the compact antenna design, with the statistics of the gaps depending on the environmental input. The previous section provides a perfect example, where there are 11 different bearings, denoted by the asterisks in Fig. 6b, with a radial current velocity between 18.92 and ±1.18 cm s^{−1}. One common approach used to mitigate this influence, albeit crude, is to azimuthally average data along a given range cell within a 10° wedge. The effects of this spatial averaging approach will be examined in the following simulations to allow us to compare the simulation results with the earlier work described in Barrick and Lipa (1996).

### b. Metrics of performance

Various metrics and statistics are generated to evaluate the MUSIC algorithm. First, these metrics and statistics are computed on the raw output of the algorithm to give the worst-case results. In the subsequent sections of this paper, the same metrics will be used to assess the performance of MUSIC after easily identified errors (outliers and overland vectors) are removed, and MUSIC results are postprocessed (azimuthally smoothing to mitigate gaps and time averaging). The metrics were chosen so that the analysis would 1) reflect the inherent errors that result from the MUSIC algorithm itself, 2) identify trends that depend on the wind and current inputs, and 3) illustrate benefits of postprocessing MUSIC results. While the received signal power is randomized by the wind wave pattern, and Gaussian noise is added at the system level to simulate electrical noise, there is no noise added to the input current velocity. This approach was chosen to restrict the number of variables influencing MUSIC algorithm performance. The effects of decreasing SNR by range cell are examined; however, no other noise or distortion sources (interfering signals, ships, second-order Bragg scatter, distorted beam patterns) are considered in keeping with the objectives of this paper. If these additional sources of distortion are included, performance would be, in general, worse than presented herein.

#### 1) Computation of sample SNR

*k*is the number of Doppler cells with signal detections in range cell

*R, s*

_{i}are the signal power samples, and

*n*

_{i}are the noise power samples.

#### 2) Normalized RMS error and standard deviation

*R*,

*θ*) point. For each set of 10-min cross spectra, at each (

*R*,

*θ*) point, there can be no solution (a gap), a single solution, or multiple solutions of which the mean is reported. To examine the spatial (

*R*,

*θ*) error characteristics, a normalized RMS error metric is defined in the following manner and computed over all ranges and bearings: where

*R*is the range cell number,

*θ*is the bearing angle,

*V*

_{i}is the input radial velocity from current model at each (

*R*,

*θ*), the subscript

*i*indicating input, and

*V*

_{m}is the mean output radial velocity from MUSIC at each (

*R*,

*θ*), the subscript

*m*indicating MUSIC.

While there is always a known input radial current velocity, though is sometimes small in magnitude, there is not always an output radial current velocity (a gap). In these cases, the MUSIC output radial current velocity *V _{m}* is set to zero in the normalized RMS error computation, producing a result of unity. The normalization was performed to show error as a percentage of the input since a 2 cm s

^{−1}error will not be as significant for large inputs versus small input currents of similar magnitude.

To provide insight on the spatial dependencies of the RMS error and standard deviation of the MUSIC solutions, simulations were conducted for a time period representing 4 h of simulated 10-min cross spectra. While ocean currents are rarely steady state for 4 h at a time, this exercise provides a method to assess the spatial performance of MUSIC. For the example of onshore wind and onshore uniform current, with no radial velocity restrictions (examining all Doppler cells), and without erasing results over land, the normalized RMS error metric over the entire spatial grid is shown in Fig. 7. The color bar on the normalized RMS error plots is set to a maximum value of 0.5, while the normalized RMS error ranges from 0 to 1. This aggregates all normalized RMS error values greater than 0.5 into the color red, allowing a clearer presentation of the normalized RMS error.

Figure 7 indicates that the normalized RMS error is on the low end of the scale through range cell 14 (above 20-dB SNR). Second, beyond range cell 14 the normalized RMS error is lower where the input currents are higher, and higher where the input currents are lower because of normalization by a smaller input currents. This feature is most prominent near the edges of the sea bearings, between 185°–190° and 350°–355°. Third, on the boundary between sea and land bearings, between 180°–185° and 0°–5°, the normalized RMS error is 0.5 or greater at almost all ranges. This is due to MUSIC often producing results (at higher SNR) at bearings just beyond the edges of the input.

The standard deviation (around the RMS error used in Fig. 7) ranges from 0–20 cm s^{−1}. Points with a standard deviation of zero are often due to there only being one result at those points on the spatial grid, but there are also a few times when a given point has exactly the same solution from different 10-min cross spectra. In general, the standard deviation is on the low end of the scale through range cell 14 oversea bearings. There are mixed results beyond range cell 14. For a more favorable example, the normalized RMS error at (*R*, *θ*) = (30, 260) is 0.05 and the standard deviation is 0. For a less-favorable example, the normalized RMS error at (*R*, *θ*) = (30, 315) is well above 1.0 (the actual unnormalized RMS error is 66 cm s^{−1}) and the standard deviation of the results is 14.5 cm s^{−1}.

#### 3) Frequency of dual-bearing results, outliers, over land

For the 4 h of data analyzed above, the number of MUSIC dual-bearing solutions, known velocity errors (outliers), and known spatial errors (results over land) are counted. A MUSIC result is considered to be a radial velocity outlier if it is outside the range [min(*V _{i}*) − 2.36, max(

*V*) + 2.36], where

_{i}*V*is the input radial current velocity and 2.36 cm s

_{i}^{−1}is the system radial velocity resolution. The data are tabulated in Table 2. Because of the symmetric nature of this wind wave/current scenario, there are many instances where the same radial current velocity is coming from two or more bearings, hence the number of dual-bearing solutions is relatively high (46%). With this, a relatively high occurrence of velocity outliers (7.73% of the total number of results) as well as a relatively high occurrence of overland vectors (12.26% of the total number of results) can be seen.

#### 4) Model skill

_{θ}denotes average over all bearings.

The skill metric for a single 10-min cross spectra and the sample SNR are plotted together in Fig. 8. Again, including all the velocity outliers, vectors over land, and not performing any spatial or temporal averaging, the skill is maximum at 0.77 in range cell 8. As is seen in previous sections, there is an anomaly in range cell 15 due to the statistical noise being high. The skill metric levels off at approximately 0.5 in the farthest range cells due to half the results being correct (no velocity vectors over land) and half of the results being incorrect (all erroneous vectors over sea).

The results and statistics in this first case study are certainly not optimal. There are different types of postprocessing: 1) limiting the radial current velocity (Doppler search), 2) erasing vectors over land, 3) spatial averaging of results, and 4) time averaging of results all play a role in improving the utility of solutions from the MUSIC algorithm.

## 5. MUSIC performance: Wind wave/current scenarios

The performance of MUSIC to resolve varying scenarios of ocean surface current structure is examined, using simple scenarios of wind waves and ocean currents, and different combinations of the two. Using the radar signal generation model and MUSIC algorithm (see flowchart in Fig. 1), the results are analyzed after 1) limiting the Doppler search by using a maximum radial current velocity of 50 cm s^{−1} (roughly twice the input) and 2) removing vectors over land. This approach is justified since any HF radar operator can supply the necessary expected velocities and bearings for any given system. The simulations are based upon a single set of covariance matrices from the 10-min cross spectra.

The MUSIC results of the four current structure scenarios for both wind wave scenarios A and B are shown in Figs. 9a–d and 10a–d, respectively. A comparison of the skill metrics for these different environmental inputs is shown in Figs. 11 and 12.

### a. Summary of simulations

The most apparent difference is that the simple alongshore uniform current (scenario 2) produces the best skill metric while the eddy (scenario 4) produces the worst under both wind wave scenarios. This bias in performance is due to the relative distribution of the magnitude of the input currents over all ranges and bearings between these two current scenarios. The alongshore uniform current provides the strongest input in the most (*R*, *θ*) cells; conversely, the eddy provides the least, suggesting a prejudice of the chosen skill metric to scenarios that have high input current over the (*R*, *θ*) space. This prejudice can be compounded by the system radial velocity resolution of 2.36 cm s^{−1} (Table 1), whereby input current less than the resolution will bias the metric downward. Not only do low currents produce low skill metrics, but low signal strength due to tangential wind wave direction in some of the (*R*, *θ*) grid also give this result. Hence the combination of stronger currents and wind wave energy benefit the MUSIC algorithm skill metric.

The simulations also indicate a general rapid decrease in skill an SNR of 11–12 dB. This is slightly lower than results presented in internal documentation provided by Codar Ocean Sensors (Barrick 2005) that states 14 dB of SNR is necessary for good MUSIC algorithm performance. This can be attributed to different methods in computing SNR between the two studies.

## 6. Spatial error statistics

In this section, spatial error patterns produced by MUSIC are examined using the normalized RMS error and the standard deviation of the MUSIC results (not normalized) over the entire spatial field, for 4 h of simulated current input and MUSIC algorithm output. Outliers that exceed the 50 cm s^{−1} velocity limit as well as any vectors resulting over land are not included in the statistical data. A skill metric is not computed for each range cell over all bearings, but instead spatial error patterns are presented over the entire (*R*, *θ*) space. The normalized RMS error is shown for wind wave scenarios A and B in Figs. 13a–d and 14a–d. As before, the normalized RMS error is color coded between 0 and 0.5 to bring attention to the smaller errors. Since the Doppler search is limited and overland vectors are filtered out, the standard deviation now typically ranges between 0 and 5 cm s^{−1}. Overall statistics for single- versus dual-bearing results, radial velocity outliers, and overland vectors (integrated over the spatial grid) are provided in Table 3.

### a. Summary of spatial error statistics

#### 1) Surface current scenario 1—Onshore uniform current

Figure 13a (scenario 1) indicates three sectors of low normalized RMS error (dark blue) between the bearings of 200°–225°, 265°–275°, and 320°–345°. Also present is a standard deviation out to range cell 20 ranging between 0 and 2 cm s^{−1}. When the wind waves switch to alongshore, and the currents remain the same, there is an increase in performance over the spatial field in Fig. 14a. The normalized RMS error is more uniform and relatively lower over a larger area. One cause of this increase in performance is seen by looking at the columns in Table 3 corresponding to current scenario 1. Note that the percentage of single-bearing solutions goes up (and the percentage of dual-bearing solutions goes down) by 34% with the alongshore wind, suggesting that dual-bearing solutions are more prone to error than single-bearing solutions.

The significant change in single- versus dual-bearing results can be attributed to two things: 1) the effect of the alongshore wind increasing the SNR in low current velocity Doppler cells, and 2) the distribution of these low velocities between the two Bragg regions. Consider a symmetrical pair of small input currents (Fig. 3a) in any given range cell, one at bearing 185° and the other at bearing 355°. Under the onshore wind wave scenario (A), both currents will be received in the same Doppler cell, at the same SNR, in the positive Bragg region. In addition, both currents will be received in the same Doppler cell, at the same (lower) SNR, in the negative Bragg regions. Both of these spectral contributions give rise to dual-bearing solutions. However, under the alongshore wind wave scenario B, the input from bearing 185° will be received at a higher SNR but only in the positive Bragg region, and the input from bearing 355° will be received at the same SNR but only in the negative Bragg region. Not only is the SNR higher for these smaller currents under the alongshore wind wave scenario, but they are separated among the Bragg regions and now give rise to single-bearing solutions.

There is a difference in the percentage of velocity outliers in Table 3 between the two wind wave scenarios, alongshore wind producing a higher percentage (1.64% versus 0.14%). The cause of this is unknown. However, the onshore wind produces more overland vectors (6.15% versus 4.79%), again attributed to the greater percentage of dual-bearing results.

#### 2) Surface current scenario 2—Alongshore uniform current

Both alongshore current scenarios (2 and 3) have mostly single-bearing inputs balanced between both sides of the Bragg peaks under both wind wave scenarios. The results are tabulated in Table 3.

#### 3) Surface current scenario 3—Alongshore shear current

Though similar to the results from the scenario 2, this scenario has higher magnitude input at long range (low SNR), producing more double bearing results. The results in Table 3 illustrate this increase in variability.

#### 4) Surface current scenario 4—Eddy current

The results from the eddy simulation are the poorest, as illustrated in Figs. 13d and 14d, with the area in and around the eddy showing the largest errors. These regions are defined between bearings 210°–240°, 265°–275°, and 300°–330° (see Fig. 3d), where the radial current transitions from positive to negative or vice versa and has a low velocity. MUSIC can only produce two bearings with the same radial current velocity for a given Doppler cell, and one (or perhaps two) gets left out (a gap). As expected, there is a region of high standard deviation within the eddy, between 3 and 5 cm s^{−1}, due to the high variability of the current input across the three current transition regions. Again, there is a higher percentage of dual-bearing results (23% for both wind wave scenarios) producing a greater amount of higher normalized RMS errors. The percentage of velocity outliers and overland bearings are very close for both wind wave scenarios. The wind appears to have little or no impact on the results for these complex input currents.

## 7. Spatial (azimuthal) averaging

Spatial averaging is commonly used in operational HF radars of this type. To show the effects of spatial averaging of MUSIC output, a 10° wide averaging wedge is applied over the field of results, to examine the impact to the skill metric. The 10° wedge is rotated in azimuth over the sea bearings at 5° increments, creating a boxcar average of three MUSIC results each step, the center bearing result and the two neighboring results ±5° away. The spatial average of the three results then replaces the center bearing result. This average is only conducted azimuthally. This spatial average is applied to the 10-min cross spectra results from section 5. The results for wind wave scenarios A and B are shown in Figs. 15 and 16.

### a. Summary of spatial averaging results

These figures indicate an increase in skill using spatial averaging when compared to Figs. 11 and 12. Under both wind wave scenarios, results for all four current scenarios show a rise in the skill metric. Table 4 quantifies this increase by tabulating the overall increase in skill for all input scenarios. The overall increase in skill is computed by taking the skill increase in each range cell, and averaging over all 31 range cells, thus integrating the entire spatial field into one metric. Using 10° spatial averaging, the overall skill increases between 55% and 100% looking across all the scenarios. A relatively small bearing sector has been chosen for averaging. In practice, the angular width of this sector can be specified by the user with performance dictated by the true gradients of the surface currents.

## 8. Time averaging

Time averaging results over multiple sets of 10-min cross spectra is another technique almost always implemented in practice. While the previous skill metrics produced for all the simulations involved no time averaging of the radial data generated by the MUSIC algorithm, we now examine and compare results from time averages of 1, 2, 3, and 4 h. Cross spectra averaging is performed in a slightly different manner than is performed on the first three sets of cross spectra, which is also different than used by Barrick and Lipa (1996). Every third (4 min, 16 s) set of cross spectra are saved, and used as the first cross spectra for the subsequent average. Hence the averaged cross spectra that are input to MUSIC have a temporal overlap. Averaging is done this way to better mimic spectral averaging that takes place in the commercially available SeaSonde system. The averaged cross spectra are then analyzed by the MUSIC algorithm.

Attention is focused on the surface current scenario of the eddy since it would potentially benefit the most from temporal averaging. Wind wave scenario A is used, and also the spatial averaging (in azimuth) described in the previous section is applied. The simulations were repeated the following number of iterations, with all input variables held constant except for the synthesized received radar signal in which noise is added with random amplitudes and phases: 1) six 10-min cross spectra, 1 h of data; 2) twelve 10-min cross spectra, 2 h of data; 3) eighteen 10-min cross spectra, 3 h of data; and 4) twenty-four 10-min cross spectra, 4 h of data. The results are shown in Fig. 17.

### a. Summary of time-averaging results

Results for the temporal averaging show a rapid improvement at areas of low skill after 1 h of averaging, with additional improvement after 2 h. The 1-h improvement represents the averaging of radial solutions from MUSIC processing of 14 independent cross spectra. Inherent to interpreting the improvement is a stationary environment during the observation period.

A nonintuitive result is the apparent increase in system range. This result is statistical in nature, in that the signal strength and SNR from cross spectra to cross spectra is variable. While MUSIC may find no results for one averaged cross spectra, it will find results in the next one, and conditionally average those results. These far range results occur less frequently, but benefit from the spatial averaging, filling in the far range cell results. In general, time averaging is still limited by the presence of small input radial current velocities. Since much of the benefit of time averaging is seen after 1 h, additional overall statistics were computed for the 1-h average. Referring again to Table 4, there is a 29%–33% increase in overall skill using 1 h of time averaging of the eddy results. Similar improvements in skill are found, ranging between 14% and 33%, when applying temporal averaging to the other current scenarios (the figures are omitted for brevity).

Finally, the output of the MUSIC algorithm under wind wave scenario A, current scenario 4 (eddy) after 10° spatial averaging and 1-h time averaging is presented. This gives the best results possible in our simulation and is shown in Fig. 18. Only every third range cell is plotted for visual clarity. Comparing this output to the input in Fig. 3d, there is a similar reproduction of the eddy current. The skill metrics across the eddy feature between range cells 10–20 range from 0.53 to 0.72.

## 9. Conclusions

In these simulations, the random perturbation of the input current was not included to allow an isolated analysis of the MUSIC algorithm performance. In Table 4, the overall RMS error is as high as 10.5 cm s^{−1} when processing a single set 10-min cross spectra with no spatial or temporal averaging, and as low as 2.5 cm s^{−1} including the averaging. The lower value of overall RMS error is consistent with previously published results by Barrick and Lipa (1996). To achieve these results, raw errors in the MUSIC algorithm output are mitigated by 1) restricting the Doppler cell search to expected current speeds, and 2) removing overland results. Without implementation of these simple error flags, the skill of the MUSIC algorithm is found to be relatively flat (Fig. 8) starting out around 0.7 in the near range cells, and reaching a steady state of approximately 0.5 at far range cells (since the land bearings are included). While these results point out the anomalous behavior of unrestricted cross spectra to radial inversion, it does not provide insight to the operational performance of MUSIC in deployed systems. After employing the simple error mitigation techniques, the skill metric sheds more light on the MUSIC results under current inputs of varying complexity in the known area of interest (Figs. 15 and 16). The scenarios providing the best skill metric were uniform input currents with radial current velocities well above the system velocity resolution that did not give rise to dual-bearing solutions. The skill metrics were found to decrease at SNRs of 11–12 dB and lower. The skill metric asymptotes to zero at the far range cells, as SNR decreases.

In general, the skill metric is worse for small input currents. This is because of the radial current velocity resolution of the MUSIC algorithm, spatial averaging, and because of the chosen definition of the skill metric used in this paper. The radial current velocity resolution of 2.36 cm s^{−1} affects skill where the input is less than that resolution. When the input currents are small at a given (*R*, *θ*), the spatial averaging will smooth the larger results over those cells. Since the denominator of the skill metric is the magnitude of the input current, the metric has limited utility at low velocities.

Single- and double-bearing outlier results are easily removed by setting a threshold on the expected value of the radial current velocity results. This expected value must be selected carefully depending on the current structure one is trying to measure. Setting the threshold too high will result in an increase in outlier data, while setting it too low will limit the MUSIC algorithm’s ability to determine large radial currents. Without this restriction, 7.7% of the results are velocity outliers (Table 2). Using a radial velocity limit of roughly twice the expected value of the input currents, the percentage of velocity outliers are found to range from 0.03% to 1.64% (Table 3). The signal detection (power) threshold also plays a critical role in generating outliers. In the simulation, any Doppler cell having an overall signal-plus-noise power that is 6 dB over the noise floor is considered to be valid input for radial inversion. However, this is sometimes not the case in a noisy environment and careful measurement of the (variable) noise floor of the system is required. An adaptive approach to setting this threshold should be considered in further research.

Dual-bearing solutions are found to lead to suspect bearing retrievals, even if their speeds are within an expected range. This is clearly demonstrated by the production of overland radial solutions where there are no simulated input signals. Radar operators often attribute over land results to reflections from cars and planes; however, the simulations presented generate overland results in the absence of these targets. In Fig. 5 (without the current velocity restriction) 12.26% of the MUSIC results are over land. Applying the restriction, the percentage of overland results is reduced to between 4.12% and 6.47%. In practice, signals are always received from land bearings, but these tend to be stationary target scatter with zero Doppler. However, other interfering signal sources may remain in the averaged cross spectra. If MUSIC can produce dual-bearing results from land areas providing no input, it will also produce dual-bearing solutions over the sea where there is no input as well. At present, there are no consistent methods to remove these erroneous dual-bearing results and they are averaged into the final radial current velocities. Identification and flagging of these potential errors are suggested as areas of additional focus for the HF radar community.

In general, dual-bearing inputs and results tend to generate higher variability and less accuracy in the output. This is exemplified in section 6 with the onshore uniform current scenario. This is the most prominent result showing how an environmental variable, the wind wave bearing, has a major impact on the resulting spatial error distribution. The effect of changing the wind waves from onshore to alongshore redistributes the input from a majority of dual-bearing inputs to a majority of single-bearing inputs per Doppler cell. The reduction in RMS error and standard deviation over a larger area of the spatial grid shows the benefit of this effect. The simulations show that in scenarios that generate large numbers of dual-bearing solutions, the error statistics are higher and more variable.

Gaps in MUSIC results arise from more than two bearings in a given range cell with the same radial current velocity. In practice, these can be crudely mitigated by a spatial average in azimuth. Considering the eddy current, this may not always be a good thing to do. On one side of the eddy, the radial currents are positive. Sweeping in azimuth, the radial currents quickly go to zero, and then change direction (sign) entirely. A 10° spatial averaging wedge has been successfully used to increase overall skill herein between 0.19 and 0.31 (an average 74% increase), however spatial averaging of complex structures may negatively affect skill in practice, depending on the size of the averaging wedge. The application of spatial averaging should be conducted with some caution and should be application dependent.

Time averaging increases the skill of the MUSIC algorithm, assuming the input currents are stationary during the averaging time. Increases in skill are apparent after 1 h of averaging between 0.09 and 0.18 (an average 25% increase), and seem to reach a relatively steady state thereafter. The benefit of 1-h temporal averaging of results has been shown, as is commonly done in operational HF radars. Using all of the aforementioned error mitigation techniques (limited radial current velocity, erasing results over land, spatial averaging, and temporal averaging) skill metrics of 0.94 for alongshore uniform currents with high SNR, and skill metrics between 0.53 and 0.72 across the center of an eddy current with high SNR are achieved.

The following areas of research were identified while conducting the reported research and are shared with the motivation to improve HF radar operations and the interpretation of their data.

- Definition of the distance properties of the antenna manifold and their effect on the direction of arrival (DOA) function, along with a statistical examination on the magnitude of the maxima of the DOA functions that produce MUSIC bearing results, focusing on measured antenna patterns.
- Additional statistical analysis of the frequency of occurrence of single- versus dual-angle solutions from the MUSIC algorithm, and its impact on skill. Particular focus should be directed toward measured antenna patterns.
- Adaptive error mitigation (rather than hard limits) that depend on measured noise, the expected value of the radial current, and the expected spatial complexity of the measured currents.
- Development of current and RF models that include subgrid-scale motions and second-order scattering effects.
- Analysis of the condition, eigenvalues, and eigenvectors of the signal covariance matrix, and their influence on the skill of MUSIC for retrieving surface currents.

This manuscript was significantly improved through discussions with Lisa Lelli, Mark Otero, Sung Yong Kim, and Tom Cook of the Coastal Observing Research and Development Center at Scripps Institution of Oceanography. We thank both JOATO reviewers, who provided excellent positive criticism lending greatly to the accuracy and clarity of the manuscript. A special thanks is extended to Sung Yong Kim for the MATLAB code to compute radial projections from a given vector field. This work was supported by Grant NA17RJ1231, “Southern California Coastal Ocean Observing System—Shelf to Shoreline Observatory Development,” managed by the National Oceanic and Atmospheric Administration.

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HF radar parameters.

Raw statistics of dual-bearing solutions, outliers, and overland results.

Postprocessed statistics of dual-bearing solutions, outliers, and overland results.

Statistical effects of spatial and time averaging.