Signal Sampling Impacts on HF Radar Wave Measurement

Lucy R. Wyatt Department of Applied Mathematics, University of Sheffield, and Seaview Sensing Ltd., Sheffield, United Kingdom

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J. Jim Green Department of Applied Mathematics, University of Sheffield, and Seaview Sensing Ltd., Sheffield, United Kingdom

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Andrew Middleditch Seaview Sensing Ltd., Sheffield, United Kingdom

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Abstract

Averaging is required for the measurement of ocean surface wave spectra and parameters with any measurement system in order to reduce the variance in the estimates. Sampling theory for buoy measurements is well known. The same theory can be applied to the impact of sampling on the estimation of high-frequency (HF) radar power spectra from which wave measurements are derived. Some work on the impacts on the HF radar wave measurements themselves is reviewed and applied to datasets obtained with three different radar systems, operating at different radio frequencies in different geographical locations. Comparisons with collocated buoy measurements are presented showing qualitative agreement with the sampling impact predictions but indicating that there are more sources of differences than can be explained by sampling. Increased averaging is applied to two of these datasets to demonstrate the improvement in data quality and quantity that can be obtained.

Corresponding author address: Lucy R. Wyatt, Dept. of Applied Mathematics, University of Sheffield, Hounsfield Rd., Sheffield S3 7RH, United Kingdom. Email: l.wyatt@sheffield.ac.uk

Abstract

Averaging is required for the measurement of ocean surface wave spectra and parameters with any measurement system in order to reduce the variance in the estimates. Sampling theory for buoy measurements is well known. The same theory can be applied to the impact of sampling on the estimation of high-frequency (HF) radar power spectra from which wave measurements are derived. Some work on the impacts on the HF radar wave measurements themselves is reviewed and applied to datasets obtained with three different radar systems, operating at different radio frequencies in different geographical locations. Comparisons with collocated buoy measurements are presented showing qualitative agreement with the sampling impact predictions but indicating that there are more sources of differences than can be explained by sampling. Increased averaging is applied to two of these datasets to demonstrate the improvement in data quality and quantity that can be obtained.

Corresponding author address: Lucy R. Wyatt, Dept. of Applied Mathematics, University of Sheffield, Hounsfield Rd., Sheffield S3 7RH, United Kingdom. Email: l.wyatt@sheffield.ac.uk

1. Introduction

The use of high-frequency (HF) radar for measuring waves and currents in coastal regions has been under development for over 30 yr. High-frequency radar current measurement is now a well-accepted technology and there are many systems of different types in operation around the world. These systems are located on the coast, and some of them can also provide simultaneous measurements of the ocean wave directional spectrum from close to the coast to more than 100 km offshore. Measurements of both waves and currents have been made from every 10 min to every hour (or more) and with spatial resolutions of 1 to 15 km as needed. Work at the University of Sheffield has focused on wave measurement techniques, although we have also developed current and wind measurements methods. These methods have been validated in numerous short- and long-term deployments at many different locations (e.g., in the United Kingdom, Norway, Spain, United States) with three different radar systems: an ocean surface current radar [OSCR; Wyatt and Ledgard (1996); this system is no longer available], the Wellen Radar [WERA; Gurgel et al. (1999); developed at the University of Hamburg, Germany, and available from Helzel GmbH], and Pisces [Wyatt et al. (2006), developed from a University of Birmingham, Birmingham, United Kingdom, prototype and available from Neptune Radar Ltd.]. The OSCR system had a fixed sampling strategy designed for surface current measurement. Additional averaging was carried out before wave data were extracted, taking into account the analysis, which is reviewed below. The averaging used for the original WERA wave measurements was designed to achieve a similar reduction in variance. In all experiments to date, the Pisces system has used hardware-controlled receive antenna beams and the averaging used was designed to provide spatial coverage over 1–3 h. The CODAR SeaSonde system (Barrick et al. 1994) is perhaps the most widely known HF radar system but is not discussed further here since, although it does provide wave outputs, the measurement principle is different.

Some currently operational WERA radar systems have been configured primarily for surface current measurement with no consideration given to the averaging requirements for waves. Data from such systems have recently been analyzed to provide wave measurements without realizing that reduced averaging was being used. Averaging requirements for currents are less than those for waves and the noisiness of the wave measurements obtained have prompted a revisit of some of the work carried out several years ago on estimating and reducing sampling variability impacts on wave measurements. Since not all of this work has been published before, it is timely to review it now. The sampling variability in HF radar wave measurements can be inferred from the temporal sampling variability in the radar Doppler spectra used to obtain those measurements. The implications of various sampling strategies that have been used for radar wave measurements are discussed here and are compared with the corresponding sampling variability of wave buoy measurements. The relative impacts of the sampling variability and other factors that influence the comparison between radar and buoy measurements will be discussed. The impact of increased averaging is demonstrated using WERA data from the European Radar Ocean Sensing (EuroROSE) experiment near the island of Fedje in Norway (Wyatt et al. 2003) and from the system operated by the University of Miami located on the northeastern Florida Keys (Haus et al. 2007).

2. Sampling statistics

The sampling theory for sea-state parameters derived from the frequency spectrum, S( f ), is discussed in Krogstad et al. (1999). For wave buoy data, S( f )is usually estimated from a smoothed periodogram giving Ŝ( f ), which is a χ2-distributed variable with ν degrees of freedom (for a Gaussian sea surface) and thus has the following properties:
i1520-0426-26-4-793-eq1
Significant wave height is given by Hs = 4m0½, where and the estimated significant wave height for a particular dataset is therefore Hs = 401/2. The variance of this estimate is given by 4m00/m0, where mrs = Cov(mr,ms). The covariance can be estimated from the spectrum using (Krogstad 1982)
i1520-0426-26-4-793-eq2
where T is the recording interval and N the number of points in the time series from which S( f ) and its moments have been estimated. The estimated covariance is given by
i1520-0426-26-4-793-eq3
and this can be used to estimate the variance in the wave-height estimate.

A similar wave-height estimate can be obtained from HF radar–measured frequency spectra. However, these spectra are obtained from a complicated numerical inversion procedure (Wyatt 2000; Green and Wyatt 2006). The statistics of these estimates were investigated by Sova (1995) using Monte Carlo simulations and he defined the degrees of freedom for each spectral estimate to be twice its mean squared divided by its variance, that is, equivalent to χ2-distributed variables. However, the estimates were found to depend on radio frequency, sample rate, and Doppler averaging, and estimating the degrees of freedom for a new set of such parameters, for example, for a new radar system, requires additional simulations. It is also not yet clear that the range of conditions used in the simulations is sufficient to get reliable estimates of the statistics.

Sova (1995) also looked at the sampling statistics associated with empirical methods to extract significant wave height from the HF radar Doppler spectrum. Barrick (1980a) also considered this problem. The Doppler spectrum is obtained with a smoothed periodogram and, on the assumption of approximate normality of HF radar backscatter [discussed by Barrick and Snider (1977) and also demonstrated by Vizinho (1998)], each spectral estimate is a χ2-distributed variable with ν degrees of freedom, that is, statistically similar to a buoy-measured wave spectrum. The number of degrees of freedom depends on the number of individual spectra that are averaged and on the degree of overlap of their respective time series [see Harris (1978), Nuttall (1981), and Sova (1995) for details]. The analysis here will assume that the time series of length M is subdivided into N-point sequences for which a periodogram is obtained using fast Fourier transform (FFT) methods and they are overlapped by 75% with a Blackman–Harris four-point window to minimize the correlation between successive time sequences. Note that M and N may be different for the different experimental datasets included in this analysis.

The empirical methods for wave-height estimation (Barrick 1977; Maresca and Georges 1980; Wyatt 1988, 2002; Graber and Heron 1997; Heron and Heron 1998) are of the form
i1520-0426-26-4-793-eq4
where F is an empirically determined function of σ2(η), the normalized (by the integral of the first-order peak region) second-order part of the averaged Doppler spectrum, summed over a range ηr of normalized Doppler frequencies, η. Because the normalized spectrum is involved, the Hs estimates involve ratios of sums of χ2-distributed variables and these ratios are therefore Fν2, ν1 distributed with ν2 degrees of freedom for the sum over the second-order spectrum and ν1 over the first-order peak.
The formulas used at Sheffield (Wyatt 1988, 2002) take the following form:
i1520-0426-26-4-793-eq5
where k0 is the radio wavenumber, and α and β are engineering parameters and are different for the cases when waves are traveling perpendicular to the radar beam and otherwise. The integration ranges below, L, and above, U, the main Bragg peak are separated in this formula. Barrick (1977, 1980a) carried out the analysis for the case where the two ranges are added together before applying a power. Using a Taylor series expansion, the variance of the above Hs estimate can be obtained using the approximate relationship,
i1520-0426-26-4-793-eq6
which gives the following:
i1520-0426-26-4-793-eq7
i1520-0426-26-4-793-eq8

The above wave-height variance has to be estimated of course by replacing the expectations in the above formulas by the calculated values for each spectrum. Similar expressions can be derived for other functional forms for the significant wave-height calculation.

Maresca and Georges (1980) used a different functional form for their empirical relationship between Hs and the Doppler spectrum:
i1520-0426-26-4-793-eq9
This form is similar to that originally proposed by Barrick (1977) except that he applied a weighting function to the second-order part of the Doppler spectrum and in this case found that a = 0.5 was a suitable exponent. Maresca and Georges estimated the parameters b and a using simulated Doppler spectra. The Barrick paper, as well as that of Maresca and Georges, expressed the relationship in terms of RMS amplitude, h, but this is easily adapted to Hs using Hs = 4h.
Writing , the variance in the above estimate can be estimated in the same way as was done for the Sheffield expression, giving
i1520-0426-26-4-793-eq10
or equivalently
i1520-0426-26-4-793-eq11
Barrick (1980a) used an asymptotic form for the moments of an F distribution to get the following expression for the variance (note that the analysis in Barrick is in normalized variables):
i1520-0426-26-4-793-eq12
This expression was also used by Maresca and Georges. In this expression, K is the number of Doppler spectra in the average and M and N the “equivalent numbers” (see below) of Doppler bins used for the second- and first-order parts of the spectrum, respectively.
To complete these calculations, the degrees of freedom in the first- and second-order parts of the spectrum need to be estimated. Sova estimated these from the Doppler spectrum D, using ν = νR(mean2/variance), where νR is the degrees of freedom for each Doppler spectral estimate (equal to 2 if no averaging has been done). When windowed, overlapped, and averaged Doppler spectra are used, νR is determined using (Harris 1978)
i1520-0426-26-4-793-eq13
where N is the number of overlapped series and ρ(x%) is the x% overlap correlation. For the minimum four-sample Blackman–Harris window (Harris 1978; Nuttall 1981) used for all the data presented here, ρ(75%) = 0.460, ρ(50%) = 0.038, and ρ(25%) is effectively zero.
The mean and variance are determined over the Doppler spectral range being considered. The expression Sova derived (after a tortuous algebraic analysis) is
i1520-0426-26-4-793-eq14
where ai, i = −λ,λ are the 2λ + 1 coefficients used in the window applied to the data during Doppler processing and fL,U are the Doppler spectral indices, so that D( f ) is shorthand for D( fΔ), where Δ is the frequency sampling in the Doppler spectrum This expression accounts for the loss of statistical independence of neighboring Doppler bins due to the windowing.
Barrick (1980a) derived an expression for what he termed the equivalent number of spectral samples, LE, for a given range of Doppler bins used in the estimation of Hs. Making use of the asymptotic behavior of the χ2 distribution, that is, that for large numbers of samples it tends toward a Gaussian, LE is estimated as
i1520-0426-26-4-793-eq15
where
i1520-0426-26-4-793-eq16
where Pi is the mean power at the ith spectral bin and PM is the maximum in the region. As with the Sova estimate, this gives a degrees of freedom estimate associated with the larger power Doppler power bins. But, in this case, there is no accounting for the statistical dependence between neighboring bins due to the windowing so its use is likely to overestimate the degrees of freedom and, consequently, underestimate the wave-height variance.

3. Application to measured data

The implications of these results for various sampling strategies that have been adopted with met-ocean radars will now be discussed. Data from three different radars will be considered: OSCR (Wyatt et al. 1999, 2005), Pisces (Wyatt 1991; Wyatt et al. 2006), and WERA (Wyatt et al. 1999; Haus et al. 2007). These radars operate at a range of frequencies in different geographical locations. Table 1 shows the cases that have been considered and the parameters that are relevant to the statistical analysis. In addition to these radars, part of the Fedje dataset has also been further averaged by factors of 2 and 3 (referred to as Fedje 2 and 3) and the Miami data by a factor of 3 (referred to as Miami 3) by averaging the appropriate number of consecutive incoherent datasets. The corresponding degrees of freedom for these longer averages can be obtained by multiplying the original figure by the additional averaging factor. Fedje is on the west coast of Norway and was the location of the WERA radars used during one of the EuroROSE experiments (Wyatt et al. 1999). For each deployment, the values of single-radar Hs have been determined using the Wyatt (1988, 2002) and Maresca and Georges (1980) methods and for each of these the variances have been estimated using the Sova (1995) and Barrick (1980a) methods. For all cases, except for the Miami dataset, cotemporal, collocated buoy data have been used to assess the validity of the Wyatt and Maresca–Georges relationships between Hs and the integrated spectral estimate used.

The discussion will focus on the coefficient of variation defined as cov = Std(Hs)/Hs expressed as a percentage, where Std(Hs) is the standard deviation obtained from the square root of the variance estimated using the methods in section 2. The results are presented in Table 2. Note that these show only the nonperpendicular Wyatt estimates of cov. The perpendicular values are slightly higher than the Maresca and Georges ones included in the table. The radar figures in Table 2 should be compared with a 4%–6% cov range for buoy wave-height data (Tucker 1991; Krogstad et al. 1999). The Barrick method figures in Table 2 are always lower than the Sova method figures, consistent with the fact referred to above that the Barrick method does not account for the statistical dependence between neighboring Doppler frequency bins. Use of the Maresca and Georges (MG) method gives larger cov estimates. This is because the ratio used is less sensitive to wave height than that of Wyatt, particularly at low k0Hs. This can be seen in Fig. 1. This is also the reason for the larger cov estimates for the Wyatt perpendicular formula. In addition, Fig. 1 shows that the data are poorly described by the Maresca and Georges relationship at low values of k0Hs. Some of the same datasets were considered in Wyatt (2002) where the directional characteristics of the waves were taken into account to show the need for the two relationships discussed here. The Fedje dataset was found to be anomalous in Wyatt (2002) in that the perpendicular cases fell above and the nonperpendicular cases below the two expressions that do describe reasonably well all the other datasets considered. That is also seen here and suggests that the Wyatt formulas may need to be modified for high k0Hs cases. The higher k0Hs cases are all from the Fedje dataset and are predominantly perpendicular cases. The Maresca and Georges relationship agrees better with these data.

Because of the limitations of the Maresca and Georges method at low k0Hs and of the Barrick analysis discussed above, we will use the Wyatt–Sova method to draw some conclusions about the sampling requirements for wave measurement. The Wyatt–Sova nonperpendicular results are plotted in Fig. 2. Here, it can be seen that to reduce the sampling variability impacts on wave-height estimates to the level achieved by wave buoys, it is necessary to have more than about 35 Doppler degrees of freedom. Only the NURWEC2 deployment (Wyatt 1991) used this level of averaging. The Sova estimates of variance for wave height estimated from radar-measured directional spectra for the Holderness dataset give a cov of 3%–5% (Krogstad et al. 1999). However, as mentioned earlier, we do not have as much confidence in this estimate, which was based on limited Monte Carlo modeling. Certainly, the scatter between radar and buoy wave-height estimates at Holderness, in the United Kingdom, discussed further below, was significantly greater than can be attributed to such a low cov (Krogstad et al. 1999).

4. Buoy comparisons

One statistic that can be used to assess the difference between two different time series of wave-height measurements, h1, h2, is the scatter index defined as
i1520-0426-26-4-793-eq17
where h1 is the mean of h1. If we can assume that h1 and h2 are independent random variables, then
i1520-0426-26-4-793-eq18
Hence,
i1520-0426-26-4-793-eq19
Making the assumption that h1 = h2, we can estimate the contribution to the measured scatter index from the sampling variability using SI = (cov)12 + (cov)22. This comparison is shown in Fig. 3 below. We do not have buoy comparisons for the Miami case. Note that the scatter index shown here is that obtained by comparing the wave height derived using integral inversion with the buoy whereas the theoretical values were obtained using the empirical wave-height method. In the following discussion it has been assumed that the theoretical values for integral inversion will be similar to or better than those for the empirical method.

There are two obvious things to notice in Fig. 3. First, all the measured scatter indices are higher than can be attributed to sampling variability. Second, the measured indices seem to fall into two classes: the Holderness and Fedje cases following one curve and the Shoaling Experiment (SHOWEX), the European Wave Energy Thematic Network (WAVENET), and NURWEC2 cases forming another higher curve, both curves meeting at about the Fedje 3 level. The SHOWEX result is at first sight surprising since the radar was operating at a similar frequency to those at Holderness and Fedje, and is the same radar system as that used at Holderness. The WAVENET and NURWEC2 cases are at lower radio frequencies, for which ionospheric impacts are greater and which are noisier in low sea states and therefore the scatter index might be expected to be higher. The higher radio frequency cases cannot measure high sea states (Wyatt 1995) and these are therefore not included in the scatter index measurements. However, the measured difference for the SHOWEX dataset is heavily influenced by a period of a couple of days with very strong sidelobe contamination (Wyatt et al. 2005). If these data are removed, the resulting scatter index is very similar to the other higher-frequency cases, as can be seen in Fig. 4.

The impact of increased averaging of the Fedje data on wave height and mean direction can be seen in Figs. 5 –8. Note that although the empirical wave-height algorithm does not describe this dataset adequately, the inverted wave height compares well with the buoy even without additional averaging. In both cases the correlation coefficient between the radar and buoy increases and the RMS difference decreases with the increased averaging; see Table 3. The differences are small but are clearly seen in the time series plots. The averaging also increases the amount of data that can be used for wave measurement (Table 3).

The Miami 3 case gave improved performance during the morning with significantly more, and less noisy, wave measurements (see Fig. 9), but in late morning and in the afternoon when the ionosphere was beginning to influence the radar backscatter in an intermittent manner, the averaging tended to smear out the ionospheric contamination in time, resulting in significantly less data (Fig. 10). This radar was operating at 16 MHz; a lower-frequency system would have a similar problem at a different time of day. Note that the full range of the Miami WERA radar is not shown on these maps; a subset of the data was used for this analysis.

5. Discussion

The need to average HF radar data for wave measurement for periods rather longer than are typically used for current measurements has been demonstrated here. Averaging for periods longer than are currently used for wave measurement is also recommended. Although the analysis has been carried out for empirical wave-height estimates, the patterns of behavior of estimates obtained from the full inversion are qualitatively similar. The differences between radar and buoy wave-height estimates are larger than can be attributed to sampling variability. Part of this will be because the buoy is measuring at a single point over time and the radar is deriving wave measurements from backscatter over an area and over time, so the sampling is not strictly comparable. Other reasons for the differences will be noise in the radar signals and radio-frequency-dependent inversion limitations in very high or very low seas.

There are limitations to temporal averaging. Stationarity in oceanographic and/or interference conditions is required. The Miami 3 case was an example where lack of stationarity caused problems. Increasing the averaging does require care when there is significant current variability. Current Doppler shifts need to be removed before averaging to avoid contamination of the second-order spectrum by multiple first-order peaks. This was done for this work by using a simple Doppler shift of the original averaged spectra; other methods could be used (e.g., Middleditch and Wyatt 2006).

An alternative to averaging Doppler spectra before inversion would be to average the wave measurements obtained over two or more standard measurement periods. The inversion procedure is nonlinear and it is not immediately obvious which approach will give the best results. Since we have not carried out the variance analysis on the inversion itself, merely inferring it from our empirical wave-height algorithm, we are not (yet) in a position to assess this approach theoretically. One advantage of Doppler spectral averaging is that the noise in the spectrum is also averaged and this will have an impact on signal to noise, an important indicator used to remove poor quality data before inversion. As has already been noted, for the Fedje dataset Doppler spectral averaging leads to a small (∼9% in this case) increase in available wave data when averaged over two consecutive measurement periods. Averaging two consecutive wave measurements has a smaller impact on the statistics than was the case for Doppler spectral averaging (in Table 3); a similar improvement requires three consecutive wave measurements. However, in both cases the amount of acceptable quality wave data is reduced by 12% when averaging two and by 19% when averaging three consecutive measurements, indicating that Doppler spectral averaging is preferable where possible in order to maximize good quality data return.

Reductions in variance are also possible in principle using Doppler and/or spatial averaging. Barrick (1980b) analyzed the impact of both of these methods and referred to the need to normalize (using a logarithmic method) each Doppler spectrum before spatial averaging to remove path-loss effects. To analyze the impact of Doppler averaging (smoothing), the correlation between neighboring bins due to the windowing in the spectral analysis must be taken into account. The impact of spatial averaging needs to account for correlations associated with range correlation due to windowing in the spectral analysis that is required for frequency-modulated continuous-wave (FMCW) systems and to any interpolation that might be applied in gridding the radar data. Azimuthal correlation may also be an issue when gridding since grid spacing is not usually matched to beamwidth. These issues are all being explored but it is too early to report definitive conclusions about their impacts. Preliminary work on spatial sampling has not yielded any clear benefit, probably because of the correlation in azimuth and range referred to above. The theoretical work needs to be developed to guide an appropriate averaging strategy.

Acknowledgments

The Fedje data were obtained during the EU-funded EuroROSE project and we thank Klaus-Werner Gurgel and all of the EuroROSE team for their contributions. The Miami radar data were provided by the University of Miami’s WERA group (L. K. Shay, B. K. Haus, T. M. Cook, and J. Martinez-Pedraja) in support of the Southeast Atlantic Coastal Observing System. Other datasets used to derive the statistical information are acknowledged in the associated references and we are grateful to all who contributed. We are grateful to an anonymous referee for suggesting we also include some remarks on averaging the wave measurements themselves.

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Fig. 1.
Fig. 1.

(a) Wyatt ratio R plotted against k0Hs for all of the datasets: solid line, nonperpendicular case; dashed line, perpendicular case. (b) Maresca and Georges ratio: solid line, their relationship.

Citation: Journal of Atmospheric and Oceanic Technology 26, 4; 10.1175/2008JTECHO614.1

Fig. 2.
Fig. 2.

Coefficient of variation plotted against degrees of freedom in the Doppler estimates. Individual datasets are identified by their first letter and where appropriate the number of additional averages.

Citation: Journal of Atmospheric and Oceanic Technology 26, 4; 10.1175/2008JTECHO614.1

Fig. 3.
Fig. 3.

Scatter index plotted against degrees of freedom in the Doppler estimates: theoretical, +; measured, ◊. Individual datasets are identified by their first letter (large for theoretical and smaller for measured) and where appropriate the number of additional averages.

Citation: Journal of Atmospheric and Oceanic Technology 26, 4; 10.1175/2008JTECHO614.1

Fig. 4.
Fig. 4.

As in Fig. 3, but for SHOWEX cases modified to exclude known antenna sidelobe cases.

Citation: Journal of Atmospheric and Oceanic Technology 26, 4; 10.1175/2008JTECHO614.1

Fig. 5.
Fig. 5.

Time series comparisons of significant wave height measured with the buoy (gray) and radar (black): (top) standard averaging and (bottom) the Fedje 2 case. These data are from March 2000.

Citation: Journal of Atmospheric and Oceanic Technology 26, 4; 10.1175/2008JTECHO614.1

Fig. 6.
Fig. 6.

As in Fig. 5, but for mean direction.

Citation: Journal of Atmospheric and Oceanic Technology 26, 4; 10.1175/2008JTECHO614.1

Fig. 7.
Fig. 7.

Scatterplots of significant wave height: (left) standard averaging and (right) the Fedje 2 case.

Citation: Journal of Atmospheric and Oceanic Technology 26, 4; 10.1175/2008JTECHO614.1

Fig. 8.
Fig. 8.

As in Fig. 7, but for mean direction.

Citation: Journal of Atmospheric and Oceanic Technology 26, 4; 10.1175/2008JTECHO614.1

Fig. 9.
Fig. 9.

Maps of significant wave height and direction measured with the Miami WERA system at 0805 UTC 25 Sep 2004: (top) standard averaging and (bottom) the Miami 3 case.

Citation: Journal of Atmospheric and Oceanic Technology 26, 4; 10.1175/2008JTECHO614.1

Fig. 10.
Fig. 10.

As in Fig. 9, but for 1125 UTC.

Citation: Journal of Atmospheric and Oceanic Technology 26, 4; 10.1175/2008JTECHO614.1

Table 1.

Datasets used in the analysis.

Table 1.
Table 2.

Coefficient of variation comparisons.

Table 2.
Table 3.

Statistics for the Fedje comparisons.

Table 3.
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  • Fig. 1.

    (a) Wyatt ratio R plotted against k0Hs for all of the datasets: solid line, nonperpendicular case; dashed line, perpendicular case. (b) Maresca and Georges ratio: solid line, their relationship.

  • Fig. 2.

    Coefficient of variation plotted against degrees of freedom in the Doppler estimates. Individual datasets are identified by their first letter and where appropriate the number of additional averages.

  • Fig. 3.

    Scatter index plotted against degrees of freedom in the Doppler estimates: theoretical, +; measured, ◊. Individual datasets are identified by their first letter (large for theoretical and smaller for measured) and where appropriate the number of additional averages.

  • Fig. 4.

    As in Fig. 3, but for SHOWEX cases modified to exclude known antenna sidelobe cases.

  • Fig. 5.

    Time series comparisons of significant wave height measured with the buoy (gray) and radar (black): (top) standard averaging and (bottom) the Fedje 2 case. These data are from March 2000.

  • Fig. 6.

    As in Fig. 5, but for mean direction.

  • Fig. 7.

    Scatterplots of significant wave height: (left) standard averaging and (right) the Fedje 2 case.

  • Fig. 8.

    As in Fig. 7, but for mean direction.

  • Fig. 9.

    Maps of significant wave height and direction measured with the Miami WERA system at 0805 UTC 25 Sep 2004: (top) standard averaging and (bottom) the Miami 3 case.

  • Fig. 10.

    As in Fig. 9, but for 1125 UTC.

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