A Comparison of Algorithms for Extracting Significant Wave Height ...

OCTOBER 1998

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HERON AND HERON

A Comparison of Algorithms for Extracting Significant Wave Height from HF Radar Ocean Backscatter Spectra S. F. HERON

AND

M. L. HERON

Physics Department, James Cook University, Townsville, Australia (Manuscript received 27 December 1996, in final form 9 September 1997) ABSTRACT A comparison is made between three different but related algorithms for the extraction of rms wave heights from high frequency ocean backscatter radar spectra. All three methods are based on the ratio of second- to first-order energies as developed by Barrick, and each was scaled so that the mean values of the radar analysis results and the corresponding wave buoy data were zero. The rms difference between the radar wave heights and those from the buoy was taken as a measure of fit, and the recommended algorithm had an rms difference value of 7 cm. Barrick’s algorithm (after scaling), which uses a weighted second-order energy integral, performed marginally better than the others. The condition requiring wind directions other than close to orthogonal to the radar beam is retained in the recommended algorithm but is not evaluated because of sparsity of data. The algorithm for extraction of rms wave heights is validated against the buoy data over rms wave height ranges from about 0.2 to 0.7 m.

1. Introduction Groundwave high frequency (HF) ocean radars produce power density spectra that have two characteristically large first-order peaks and a continuum of second-order energy superposed on a small number of spectral lines. The first-order spectral peaks are produced by a Bragg spatial resonance effect, whereby energy is returned most strongly from sea surface waves that have wavelengths half that of the incident radar wave and are propagating in the direction of the radar beam. The firstorder peaks are typically 20–30 dB above the secondorder spectral densities, and the Doppler shift of the first-order peaks has been widely used to map sea surface current (see e.g., Prandle and Ryder 1989; Shay et al. 1995). Early theoretical work by Barrick (1972a,b) showed that the main second-order continuum of energy in HF radar ocean backscatter is produced by the combination of hydrodynamic nonlinearity and a double scatter from ocean waves with vector wavenumbers k1 and k 2 constrained by 2k 0 5 2k1 2 k 2 ,

(1)

where k 0 is the wavenumber of the incident radar wave. Barrick (1977) showed that, subject to some approximations, there is a direct relationship between the rms wave height and the ratio of the total second-order en-

Corresponding author address: Prof. Mal L. Heron, Physics Department, James Cook University, Townsville 4811, Queensland, Australia. E-mail: [email protected]

q 1998 American Meteorological Society

ergy to the total first-order energy in the spectra of the form

E

`

2 h2 5

s (2) (v d )/W(v d /v B ) dv d

2`

E

,

`

k02

(2)

s (1) (v d ) dv d

2`

where s(2) and s(1) are the second- and first-order scattering cross sections, respectively, and are functions of the Doppler shift, v d . The term W(v d /v B ) is a weighting function of the Doppler shift scaled by the Bragg frequency, v B , characteristic of the radar transmission wavelength. The weighting is assumed to be invariant with the directional wave spectrum and is included to remove the effects of nonlinear wave–wave coupling. Barrick pointed out that the error associated with this assumption is dominated by the angle between the wind direction and that of the radar beam, and he estimated that this error was of the order of 15%. In applying Barrick’s algorithm (2) we use the notation h5

(2)1/2 0.5 R , k0 w

(3)

where R signifies the ratio of second- to first-order energies and the subscript w denotes the use of the weighting function. Maresca and Georges (1980) carried out a series of numerical simulations of HF radar spectra and concluded that the variability with angle to the wind direction dominates over the correction for nonlinear cou-

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pling offered by the weighting function W(v d /v B ), provided we do not use spectra for which the radar beam direction is within 158–308 of orthogonality to the wind direction. For a wide range of simulated spectral conditions, they recommended the form h5

a b R , k0

(4)

where R is the unweighted ratio of second-order energy surrounding the highest first-order peak to the energy in that first-order peak, and a and b were empirically fitted and found to be 0.8 and 0.6, respectively. For the simulated HF radar spectra, Maresca and Georges were able to compare the rms wave height values from (4) with those from the directional wave spectrum models. The error associated with adopting the power-law form of (4) was 14%. Heron et al. (1985) used a hybrid approach in which the simplicity of Maresca and Georges (1980) was retained by using the unweighted R, but the parameter b was set to 0.5. One reason for limiting the number of degrees of freedom of the fitted algorithm was because their dataset was rather limited in its range of wind directions and the wave heights were low. In fact, the wave height conditions of these data did not satisfy the criterion k 0 h $ 0.15 required in Barrick’s theoretical development of (2). The empirical fit based on R 0.5 was still useful for the range of validation and the model used was h5

a 0.5 R , k0

(5)

where a 5 2.1. In applying each of these algorithms, we remove frequency shifts of the whole spectrum due to surface currents. The use of ratios of energy in different frequency bands within each spectrum removes all considerations of radar system gain, except for issues relating to signalto-noise ratio. There have been other significant approaches to the problem of inverting the HF ocean backscatter spectra to determine parameters of the directional ocean wave spectrum (Wyatt 1986; Howell and Walsh 1993). These methods have great promise for full directional analysis but do not currently have the speed or robustness of the one-parameter (rms wave height) analysis based on Barrick’s approach. In this paper we are evaluating the three algorithms represented by (3), (4), and (5) using a sevenday section of data archived during the DUCK94 deployment of the University of Miami’s HF ocean radar off the North Carolina on the Atlantic coast.

FIG. 1. A typical Doppler shift spectrum for the 25.4-MHz radar showing the first-order energy (F), second-order inner (SI) and outer (SO) energy bands, and the noise floor level (dashed line).

order region, and the second-order outer region, respectively, for the dominant side of the spectrum. The frequency of the first-order peak for the 25.4-MHz radar is at 60.514 Hz and the width of the first-order energy peak is defined to be 60.072 Hz as a trade-off between capturing all of the first-order energy and eliminating swell; this boundary excludes all wind–wave secondorder energy and swells with periods up to 14 s. The outer limit for the second-order region is set at 0.5 Hz from the first-order peak, which includes all wind wave energy with wave periods down to 2 s. Heron and Heron (1997, manuscript submitted to J. Remote Sens.) showed that the underlying noise in the HF radar spectra arises from Gaussian processes and the signal plus noise asymptotically approaches a Rayleigh distribution at low levels. This does not imply that there are no noise spikes or interference bands in the spectra because it is a conclusion based on statistics of the low noise bands. We adopt their asymptotic method to establish the level of the noise floor shown in Fig. 1. The signal energy in each band required for the rms wave height algorithms is taken as the integral of spectral density values minus the noise spectral density level. For the Barrick algorithm (3) the weighting function W(v d /v B ) is applied to the noise level in the form

Rw 5

E

S

[s (v) 2 N ]/W(v) dv

E

,

(6)

[s (v) 2 N ] dv

F

2. Application of the algorithms A typical HF ocean backscatter radar spectrum is shown in Fig. 1, where the regions SI, F, and SO are shown to mark the second-order inner region, the first-

where F and S denote the first- and second-order integrals and N is the spectral density of the noise. For the Maresca and Georges (1980) and the Heron et al. (1985) algorithms, (4) and (5), respectively, W(v) 5 1. Maresca and Georges (1980) discussed the option of

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FIG. 2. The site layout for the DUCK94 experiment shows the master and slave radar stations (large filled circles) and the site of the inner shelf (IS) wave buoy (shaded triangle). The dots mark the center points of radar observation cells.

evaluating the integrals in (6) over only the most energetic side of the spectrum rather than over both sides. They found that the algorithm performed slightly better if only the more energetic side is used. We confirm that this is also true for the present dataset, where the rms wave height values extracted from the two-sided integration are consistently more noisy than for the onesided evaluation and there is no mean shift with respect to the wave buoy data. This would be an odd result in a system with only signal and Gaussian noise, and it indicates that the lower energy side of the spectrum may be more susceptible to interference noise. All of the following analyses are done using only the dominant side of the HF radar spectrum. 3. Database from the DUCK94 experiment The layout of the DUCK94 experimental site is shown in Fig. 2, where the main points are the master and slave HF radar sites of the ocean surface current radar (OSCR) system and the inner shelf National Oceanic and Atmospheric Administration directional wave buoy. The radar master station is at 36.18298N, 75.75108W and the boresight bearing of the linear phased array antenna was 30.178T; the corresponding numbers for the slave station are 36.39238N, 75.82858W and 120.008T (Haus et al. 1995). The buoy was located at 36.26758N, 75.4986118W, 24.6 km from the master station at a bearing of 678E (32.7 km at 1158E from the slave station). The radar spatial resolution is 2 km in the range direction, determined by the pulse length, and 68 in azimuth, determined by the length of the receiving antenna, which at the range of the inner shelf wave buoy gives a spatial resolution of around 2 km 3 2.6 km. The radar measurement points, shown as dots in Fig. 2, are on a 1km grid inshore (with a few interstitial points at instru-

FIG. 3. Close-up of Fig. 2 showing the locality of the inner shelf wave buoy (shaded triangle) and the radar measurement points within a 2.5-km radius of the buoy (*). Spectra from these points were incoherently averaged at each sampling time.

ment locations) and a 2-km grid beyond about 25 km. The radar measurement points are chosen by phaseshifting the signals in the receive antenna elements to move the radar beam. While the spatial resolution is of the order of 2 km 3 2.6 km, the finer-scale sampling by the radar allows for noise reduction techniques to be applied in the analysis of Doppler shift spectra from the radar. A proper comparison between radar and wave buoy wave heights needs to take into consideration the time and space dimensions of the sampling in each case. The radar stations record sequentially for 5 min every 20 min and sample the 2 km 3 2.6 km patch around the marked points in Fig. 2. The wave buoy records for about 17 min every hour. To estimate an equivalent spatial dimension for the buoy we can use the celerity, c, of the dominant wave multiplied by the 17-min duration of the sample. For a 4- and 8-s dominant wave period the spatial scale is 6.4 and 12.8 km, respectively. To get a better match between the wave buoy wave height sampling and the radar sampling we chose to spread the radar spatial resolution by averaging data at each sampling time over the radar sites within the circle of radius 2.5 km centered at the inner shelf wave buoy shown on Fig. 3. Incoherent averaging of these 15 spectra also improves the signal-to-noise ratio in the backscatter spectra and produces better estimates of spectral energy in the first- and second-order bands of the spectra used to determine rms wave height. In this evaluation of the algorithms for rms wave height we are using data from the master station over the period 2–8 October 1996. The wave buoy samples

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TABLE 1. Scaling factors determined for the radar rms wave height data to equate their means to that of the wave buoy over the test period. The resulting root-mean-square differences between the rms wave heights from each scaled radar algorithm and that from the wave buoy are shown in the third column. Algorithm

Scaling factor (z)

rms difference (cm)

Barrick (1977) Maresca and Georges (1980) Heron et al. (1985)

0.542 0.647 0.307

4.4 6.5 4.8

from the 33d to the 50th min of each hour and the radar samples from the 15th–20th, 35th–40th, and 55th–60th min of each hour. If spatial resolution is critical, then we can use only the 35–40-min radar sample for comparison, and the effective spatial scales are as described in the preceding paragraph. If spatial scale is not critical, then we can use the 35–40-min sample averaged with the samples 20 min before and after. This latter scheme gives the radar an effective timescale of about twice that

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of the wave buoy but improves the stability of the radar data. The radar technique uses a monochromatic electromagnetic wave to probe the directional wavenumber spectrum of the sea surface. The Doppler-shifted spectrum of radar echoes includes self-interference fading as an intrinsic effect. Fast fading is reduced by the incoherent averaging over the 40-min time-sampling windows. Long-period fading over periods greater than 5 min is manifest as a random scaling of each individual radar spectrum. The procedure of taking ratios of energy bands within each 5-min spectrum provides immunity to long-period fading. Residual effects of random fading are not specifically addressed in this analysis and contribute to the errors in the comparison with buoy data. 4. Results For all three algorithms we found it necessary to include a scaling factor to match the mean radar wave

FIG. 4. Time series of radar-derived rms wave heights (marked with dots) with the solid line representing the corresponding wave buoy data using (a) the Barrick (1977) algorithm, (b) the Georges and Maresca (1980) algorithm, and (c) the Heron et al. (1985) algorithm.

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FIG. 5. Scatterplots for radar-derived rms wave heights vs wave buoy data. The solid line is at 458 using (a) the Barrick (1977) algorithm, (b) the Georges and Maresca (1980) algorithm, and (c) the Heron et al. (1985) algorithm.

heights to the mean buoy wave heights over the period of the test. The rms wave heights given by (3), (4), and (5) were all consistently and significantly higher than those from the wave buoy and a factor z was determined for each method, as shown in Table 1. The time series of radar and wave buoy data are shown in Fig. 4. The gaps in the data from the radar in Fig. 4 are due to field logistics and have no bearing on the scientific analysis. Since all three algorithms are based on a ratio of energies in the radar spectra, they have many common features and all are susceptible to interference noise. That said, it is also clear that all three algorithms follow the major features of the wave buoy data. The ability

of each algorithm to cope with the variations in conditions is measured by the rms difference between the radar and wave buoy values shown in Table 1. The Barrick algorithm is marginally better than the others and had a resultant rms deviation across the record of 4.4 cm for the rms wave heights. Barrick (1977) identified conditions in which the dominant wave direction (often taken to be the wind direction) is perpendicular to the radar beam as being error prone for this algorithm. In this experiment we were able to use buoy wind direction data to identify these conditions, but in applying the algorithm to only radar data, it is possible to use the ratio, R1 of energy

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FIG. 6. The recommended algorithm based on Barrick (1977) was used to produce the plotted rms wave heights (dots) and the wave buoy data (solid line). The error bars show the std dev of the radar data over a 10-point interval. Gaps occur where the wind was within 158 of orthogonal to the radar beam direction or when data were missing for logistical reasons.

in the two first-order peaks in each radar spectrum to exclude data from wave height analysis when the geometry is inappropriate. We use S 5 1 in the directional spreading model for ocean surface waves, which models the ratio of the first-order energies as R1 5 tan S

12 2 , u

(7)

where u is the angle between the wind and the radar beam. Maresca and Georges (1980) suggested that spectra with u between 158 and 308 produced unreliable results. For these data, we defined the exclusion angle to be 158 so as to maximize the included spectra. Hence the criterion for near orthogonality is where 0.77 # R1 # 1.30. During the period of data shown in Figs. 4a–c there are only a few relatively short periods in which the wind was within the orthogonality condition, and furthermore, these periods do not correspond to significantly worse fluctuations in radarproduced wave heights. When we removed these periods from the data, the Barrick (1977) algorithm had an rms deviation of 4.3 cm across the record compared to the 4.4 cm in Table 1. Although this is not a significant improvement, we do not have enough data to make the claim that the orthogonal condition will not pose a problem, so we have applied that exclusion to data taken when the wind was nearly orthogonal to the radar beam. Normally, in an operational deployment of an HF radar system for coastal ocean conditions, there would be two stations situated to give radar beams intersecting at large acute angles to allow the extraction of surface current vectors. Under these conditions there would be at least one acceptable radar beam direction for any wind direction. Data from the slave station during the DUCK94 deployment were not able to be analyzed for rms wave heights because the dynamic range

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of the signal was too great for the OSCR radar and the noise floor was not resolved. Scatterplots of radar rms wave heights against wave buoy rms wave heights are shown in Figs. 5a–c. Here it is more clear why the Barrick algorithm performs better than the other two. The Heron et al. algorithm underestimates the wave heights at the higher values, and the Maresca and Georges algorithm has a bias toward overestimating at the higher values. The Barrick algorithm has a (reduced) tendency to underestimate the wave heights at high values. One important test for the algorithm is its stability as the wave height varies. As it is the higher wave height regimes that are normally more important in geoscience and engineering, we define a second scaling parameter, j, to be evaluated for rms buoy wave heights of 0.4 m and higher. The value of j for the Barrick (1977) algorithm was 0.551, which, when applied to the full data record, gave an rms deviation of 6.7 cm. This scale factor is not greatly different from that shown in Table 1, and we propose to use this ‘‘high-wave’’ scale factor for the final algorithm for all wave heights in order to weight the accuracy to the larger wave heights. (See Fig. 6.) 5. Conclusions The comparison between the three algorithms and wave buoy data over a range of rms wave heights from 0.1 to 0.7 m indicates that a scaled form of the algorithm proposed by Barrick (1977) is best able to follow the range of conditions. The recommended algorithm for this range of rms wave heights is h5j

(2)1/2 0.5 R , k0 w

(8)

where Rw is the weighted ratio of second- to first-order energy, Eq. (2); k 0 is the radar wavenumber; and j 5 0.551 is a fitted parameter. The radar rms wave height algorithm was not used when the wind direction is within 158 of orthogonality to the radar beam direction. This condition is retained more in deference to previous works because the small number of data points subject to this condition was not significantly different from other data points. Using this algorithm over the period, we found that the rms difference between the radar rms wave heights and the wave buoy values was 7 cm. We estimate that setting the noise floor spectral density in the radar spectra contributes about 1 cm to the error, and the approximations in the algorithm (by ignoring influences of other parameters) contribute about 4 cm. The resolution of the wave buoy is 1 cm, and its response in these wave height conditions is likely to account for most of the remaining rms difference between the two datasets. The use of HF radar systems for determining rms wave heights is developing rapidly with the motivation being in economical monitoring of conditions around ports and other maritime facilities. This study shows that the technique is robust, even for rms wave heights

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as low as 0.2 m, and theoretical analysis indicates that the method is basically better for higher wave heights. More validation at higher wave conditions is required to establish the full benefit of the technique for routine monitoring. The developed algorithm for the calculation of significant wave heights is commercially available for use with the Coastal Ocean Surface Radar II system. Codar SeaSonde HF system also has a wave analysis package. Acknowledgments. We gratefully acknowledge the access to the DUCK94 database offered by the Rosenstiel School of Marine and Atmospheric Science (RSMAS) and the University of Miami, and we are particularly grateful to Dr. Hans Graber for his comments and encouragement. The manuscript was prepared while we were visiting RSMAS. REFERENCES Barrick, D. E., 1972a: First-order theory and analysis of MF/HF/VHF scatter from the sea. IEEE Trans. Antennas Propag., AP-20, 2–10.

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, 1972b: Remote sensing of the sea state by radar. Remote Sensing of the Troposphere, V. E. Derr, Ed., NOAA/Environmental Research Laboratories, 12-1–12-46. , 1977: Extraction of wave parameters from measured HF radar sea-echo Doppler spectra. Radio Sci., 12, 415–424. Haus, B. K., H. C. Graber, L. K. Shay, and J. Martinez, 1995: Ocean surface current observations with HF Doppler radar during the DUCK94 experiment. RSMAS Tech. Rep. 95-010, University of Miami, Miami, FL, 104 pp. Heron, M. L., P. E. Dexter, and B. T. McGann, 1985: Parameters of the air–sea interface by high-frequency ground–wave Doppler radar. Aust. J. Mar. Freshwater Res., 36, 655–670. Howell, R., and J. Walsh, 1993: Measurement of ocean wave spectra using narrow-beam HF radar. IEEE J. Oceanic Eng., 18, 295–305. Maresca, J. W., Jr., and T. M. Georges, 1980: Measuring rms wave height and the scalar ocean wave spectrum with HF skywave radar. J. Geophys. Res., 85, 2759–2771. Prandle, D., and D. K. Ryder, 1989: Comparison of observed (HF radar) and modeled nearshore velocities. Contin. Shelf Res., 9, 941–963. Shay, L. K., H. C. Graber, D. B. Ross, and R. D. Chapman, 1995: Mesoscale ocean surface current structure detected by high-frequency radar. J. Atmos. Oceanic Technol., 12, 881–900. Wyatt, L., 1986: The measurement of the ocean wave directional spectrum from HF radar Doppler spectra. Radio Sci., 21, 473–485.