Short-Wave Directional Distribution for First-Order Bragg Echoes of the HF Ocean Radars

Transcription

1 105 Short-Wave Directional Distribution for First-Order Bragg Echoes of the HF Ocean Radars YUKIHARU HISAKI Department of Physics and Earth Sciences, Faculty of Science, University of the Ryukyus, Okinawa, Japan (Manuscript received 8 December 001, in final form April 003) ABSTRACT Directional distributions of Hz ocean waves were investigated using 4.5-MHz HF ocean radars. Recent observations of surface waves reported that the wave directional distribution was bimodal rather than unimodal in fetch-limited conditions. Therefore, the feasibility of using HF ocean radars to identify the bimodal distribution was investigated based on a Monte Carlo simulation. The accuracy of wave parameter estimation and the selection of the model function to describe the wave directional distribution were examined. The number of beam directions is critical to the estimation of wave parameters in the case of the bimodal distribution. It was found that the wave parameter estimation was good if the secondary directional distribution peak amplitude was small. The accuracy of estimation was poor if the secondary distribution peak amplitude was close to the primary distribution peak amplitude. It is necessary to check the error of estimated parameters by simulating first-order Bragg ratios from the estimated wave parameters and degrees of freedom of integrated Doppler spectra. Akaike s information criterion (AIC) was useful in selecting the model function, although there were some failures. As many radars as possible should be used to identify the bimodal distribution. Wave parameters were estimated from observed Doppler spectra, when an atmospheric front passed over the observation area. The estimated wave distributions were almost unimodal. However, since the wave fields were spatially inhomogeneous, it cannot be concluded that real wave distributions were unimodal. 1. Introduction High-frequency (HF) radars are useful tools for measuring surface currents and surface waves. An HF radar system that measures surface currents and waves was developed in Japan in 1988, and it has been applied to physical oceanography (Takeoka et al. 1995; Hisaki et al. 001). The wave directional spectrum can be estimated from the Doppler spectrum by solving a nonlinear integral equation, which relates the Doppler spectrum to the wave spectrum (Hisaki 1996). The backscattered signal is dominated by first-order Bragg scattering. In addition to the strong first-order Bragg echo, a weaker echo at displaced Doppler frequencies is seen in the Doppler spectrum. This is called second-order scattering and is used to estimate a wave spectrum. However, measurement of the wave directional spectrum requires good signal to noise in the Doppler spectrum and this region is quite limited. For our HF radar observation, the signal to noise was only sufficient to identify the first-order scattering. The wave directional distribution at the Bragg wavelength (half of Corresponding author address: Dr. Yukiharu Hisaki, Physical Oceanography Laboratory, Dept. of Physics and Earth Sciences, Faculty of Science, University of the Ryukyus, 1 Aza-Senbaru, Nishiharacho, Nakagami-gun, Okinawa , Japan. hisaki@sci.u-ryukyu.ac.jp the radio wavelength) can be estimated only from the first-order scattering. To estimate the short-wave (wave of Bragg wavelength) directional distribution, a model function of the short-wave directional distribution is assumed, and the unknown parameters in the model function are extracted from the ratios of observed first-order scattering (e.g., Long and Trizna 1973; Heron and Rose 1986; Georges et al. 1993; Hisaki 00). For example, Hisaki (00) estimated spread parameters and mean wave directions of short waves (waves of Bragg wavelength) by assuming the cos s ( w /) form (e.g., Mitsuyasu et al. 1975; Hasselmann et al. 1980), where w is the wave direction with respect to the mean wave direction and s is the spread parameter. However, a more appropriate model of the directional distribution may exist. The modeling of the short-wave directional distribution is important for the estimation of the wave directional spectrum for both nonlinear inversion (Hisaki 1996) and linear inversion (Wyatt 1990). Until recently, wave energy directional distributions have been treated as a unimodal function (e.g., Mitsuyasu et al. 1975; Hasselmann et al. 1980; Donelan et al. 1985), However, recent studies showed that the unimodality of the directional distribution is not always observed. It is becoming clear that short-scale waves are not aligned in the wind direction but rather they straddle the wind vector, displaying directional bimo- 004 American Meteorological Society

2 106 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 1 dality with two peaks under fetch-limited conditions (e.g., Young et al. 1995; Ewans 1998). Banner and Young (1994) showed that the presence of bimodal directional spreading was a robust feature at wave frequencies above the spectral peak. The mechanism responsible for this phenomenon is the energy transfer through nonlinear wave wave interactions. (e.g., Banner and Young 1994). Although the bimodality may have little impact for engineering applications, it is important for wave research, because the existence of the bimodality supports the validity of the four-wave nonlinear interaction theory of Hasselmann (196). Furthermore, the bimodality may affect the wind direction estimation from HF ocean radars. Wyatt et al. (1997) developed the maximum likelihood method not only to extract wave parameters but also to determine the best model to describe short-wave directional distribution. This method is used to extract the parameters of only two-parameter directional distribution models, and the unimodal angular distribution is assumed. However, the bimodal angular distribution may be a better description of a short wave in the case of the fetch-limited condition or sudden wind change. The number of parameters used to describe a bimodal angular distribution is greater than 3, while the number of parameters needed to describe a unimodal angular distribution is. We must select the best model function of the wave directional distribution from models with different numbers of parameters. We consider first the least squares method in order to identify important aspects of model fitting. We estimate the parameters of the model function from the observed values by seeking to minimize the residual as M I [ f ( 1,..., N) r i], (1) i1 where f is the model function, i (i 1,...,N) are parameters to be estimated, and r i (i 1,..., M, M N) are observed values. If the model with many parameters is an extension of the model with fewer parameters, the minimum value of I decreases with the increase of N, and it is 0 for M N. This fact seems to indicate that the model with many parameters provides a better description of the observed values. However, error is contained in the observed r i, and the model with many parameters may be a result of noisy data. Akaike s information criterion (AIC; Akaike 1974) is generally used to determine the optimal number of model parameters. For example, the order of the autoregressive model is determined by the AIC (Chan et al. 1996), although it is not always true that AIC gives a good estimate of the order. The use of the AIC in determining model parameter numbers will be investigated in this work. One of the main objectives of this study was to investigate the possibility of bimodal short-wave distributions from limited data. The radar successively measures in one direction then changes its direction for the next measurement. It is necessary to use at least four beam directions to estimate four-parameter wave distribution models. Therefore, if we used only two radars, the homogeneity of the wave field over the wide area must be assumed. The other objective of the study was to apply the AIC to identify the model function of the wave distribution from first-order Bragg echoes. In particular, we focus on the determination of whether the wave directional distribution is unimodal or bimodal. The formulations are described in section. The simulation result is presented in section 3. The application to observed data is presented in section 4. The conclusions are presented in section 5.. Formulation a. First-order scattering The first-order Bragg echoes in the Doppler spectrum for deep water are written as 6 4 1( D) k0 S(mk 0)(D m B), () m1 where m denotes the sign of the Doppler shift, k 0 is the incident radio wavenumber vector, k 0 k 0, D is a radian Doppler frequency, B (gk 0 ) 1/ is a radian Bragg frequency, g is the acceleration of gravity, and S(k) is the ocean wave spectrum composed of free waves for wavenumber vector k (k cos, ksin) (e.g., Barrick 197). The effect of the finite footprint on firstorder Bragg echoes (Hisaki and Tokuda 001) is not considered. The integral of the Doppler spectrum under the first-order peaks is denoted by P for negative Doppler frequencies and P for positive Doppler frequencies. The ratio of the first-order peaks r i for the beam n b (n b 1,..., N T, where N T is the total number of beam directions) is written as P S(k 0) D[(n b) ] r(n b), (3) P S(k 0) D[(n b)] where D() is the directional distribution at wavenumber k 0, and (n b ) is the radar beam direction for the beam n b. b. AIC The AIC is defined as AIC AIC(N) log(l g ) N, (4) where L g is the maximum likelihood and N is the number of parameters to be extracted (Akaike 1974). The model in which the AIC is smaller is the better model. The first term of the right-hand side of Eq. (4) becomes smaller with increasing N. The values P,i /A,i and P,i /A,i obey chi square dis-

3 107 tributions whose degrees of freedom are, respectively, 1,i and,i, where P,i and P,i are, respectively, the positive and negative integrated measured first-orderspectra; A,i and A,i are, respectively, the positive and negative integrated true first-order spectra; and i (i 1,...,M) is the Doppler spectrum data number. Therefore, the value of (P,i/A,i)/1,i,i ri Zi (5) (P,i/A,i)/,i 1,i r t[n b(i)] obeys an F distribution with degrees of freedom ( 1,i,,i ), where r i P,i /P,i is the measured Bragg ratio [Eq. (3)] for the ith Doppler spectrum data whose radar beam number n b n b (i). The ratio A,i D[(n b) ] r t[n b(i)] rt,i (6) A,i D[(n b)] [Eq. (3)] is the true Bragg ratio, which is expressed in terms of the unknown wave parameters to be extracted. The wave parameters estimated by the maximumlikelihood method minimize the function as M log(l ) [ log(c ) 1 1,i g i logz i i 1 (1,i,i) log(,i 1,iZ i) where 1,i ] t,i,i log r, (7) 1,i,i log(c ) log log i 1,i,i [ ] [ ] [ ] 1 log (1,i,i) 1,i,i log log (8) and is the gamma function (Wyatt et al. 1997). We express the directional distribution as L k k1 s D() a cos k, (9) where L 1 denotes the unimodal distribution, L denotes the bimodal distribution, s is the spread parameter, a 1 1, a ( a a1 ) denotes the (relative) amplitude of the secondary directional distribution peak, and 1 and are the dominant directions of the primary and secondary peaks, respectively. The number of parameters for the bimodal distribution is N 4. The author also tried to evaluate five parameters but encountered difficulties. Although the solution of Eq. (11) was estimated, the estimated wave parameters were significantly different from the true wave parameters in almost all cases. It is difficult to estimate parameters from noisy data for a five-parameter model. It should be noted that the wave direction may not be bimodal even for L in Eq. (9). The true Bragg ratio [Eq. (6)] is written as L s t b ak sin k1 L s k k1 r [n (i)] (n b) k [ ] (n b) k a cos. (10) [ ] Our radar system is available as 896 coherent I and Q signals, and is processed using a 56-point fast Fourier transform (FFT) after applying the Hamming window and overlapping by 50%, and the degree of freedom at one spectral point is 11 (Hisaki 1999). If K spectra are averaged, the degree of freedom is 11K. The degrees of freedom 1,i and,i for Z i [Eq. (5)] are estimated from k,i 1.3M k,h 11K (k 1, ). The value M k,h (k 1, ) is the number of spectral points within positive (k 1) or negative (k ) Bragg echo regions, where the spectral values are greater than half of the maximum Bragg peak level (Barrick 1980). We can estimate wave parameters s, k, and a (for L ) from values of the measured Bragg ratio r i based on Eqs. (5), (7), and (10), by solving these equations as [log(l g)] [log(l g)] [log(l g)] 0. (11) s k a c. Error estimation We can evaluate the error of the wave parameters associated with the errors of the Bragg ratio r i. We write Eqs. (11) as F j(r 1,t,...,r t,m, 1,..., N) 0, ( j 1,...,N), (1) where j ( j 1,...,N) is the wave parameter to be estimated. Let r i (i 1,..., M) be an error of the measured Bragg ratio, and j (j 1,...,N) bean error in the wave parameter j associated with errors of r i, and r i r t,i r i. The equation F j(rt,1 r 1,...,rt,M r M,,..., ) 0, 1 1 N N 1 ( j 1,...,N) (13) is also satisfied. We obtain M N Fj Fj ri i 0 (14) r i1 t,i i1 from Eqs. (1) and (13). The wave parameter error vector ( 1,..., N ) T is expressed in terms of the error vector r (r 1,...,r M ) T as i

4 108 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 1 1 F Frr. (15) The ( j, i) component of the N N matrix F is F ( j, i) F j / i and the (j, i) component of the N M matrix F r is F r (j, i) F j /r t,i. The variance of the wave parameters j (j 1,...,N) is estimated from Eqs. (5) and (15) as M 1,i k A t,i i i1,i F (k, i) r Z, (16) where denotes ensemble averaging, F A (k, i) is the 1 (k, i) component of the matrix F r, and F,i(1,i,i ) i 1,i,i,i Z. (17) ( ) ( 4) Equations (11) [or Eq. (1)] were calculated from the analytic expression of the derivatives of the function log(l g ) [Eq. (7)], while the second derivatives of the function (derivatives of F j ) were calculated by numerical differentiation. 3. Simulation a. Processing of Doppler spectra The homogeneity of the wave fields over the wide area is assumed in order to estimate the bimodal wave directions as described in section 1. Therefore, it is reasonable to use several Doppler spectra in the same beam direction. We have options for processing the Doppler spectra: the Doppler spectra are either radially averaged or not. The degrees of freedom ( 1,i,,i )(i 1,..., M) are increased by radially averaging the Doppler spectra, although the amount of data (M) decreases. The error levels ( 1/ k ) were investigated from Eq. (16) for radially averaged Doppler spectra and raw (nonaveraged) Doppler spectra. Figure 1 shows examples of the results. In this calculation, two radars are used, and the total number of beam directions is N T 1. The beam step is 7.5, and the most-rightward beam directions for the two radars are 0 and 5.5. The beam directions of one radar are (n ) 7.5(n 1) () b b 0 n b 1,...,, (18) while those of the other radar are N T N T b d b 0 (n ) 7.5 n 1 () N T n b 1,...,N T (19) (counterclockwise is positive), where 0 0 and d 5.5. The mean beam direction is b [(1) (N T )]/. The difference between the two most-rightward beam directions is 5.5, which obeys the experimental design described in section 4. The value M k,h (k 1, ) for the Bragg echo regions is 3, and the degree of freedom is k,i 43 (k 1,, i 1,...,M) for one Doppler spectrum. The number of original Doppler spectra per beam direction is N D 8. If the Doppler spectra are averaged radially, M N T and k,i 43N D 344, while M N T N D 96 if the Doppler spectra are not averaged. It is assumed that the value of S associated with error of r i (r i )is negligible. If this assumption is not considered, the value of s 1/ is too large. Figure 1 shows the errors in the primary peak direction [ 1 in Eq. (9)] and in the secondary peak direction ( ) as a function of the true primary peak direction and the secondary peak direction with respect to the primary peak direction ( 1 ). Other wave parameters are (s, a ) (7.5, 0.5) or (s, a ) (7.5, 0.8). The error predicted by Eq. (16) is large for 5 1 b 135 or 45 1 b 45, and for , where both the primary and secondary peak directions are almost parallel to a beam direction. The error of the secondary peak direction is larger than that of the primary peak direction, except at those peak directions where errors are large. There are no significant differences of errors for not radially averaged Doppler spectra and for radially averaged Doppler spectra. The difference in the predicted errors for various wave parameters is small (Figs. 1a,b,e,f). The Doppler spectra are not radially averaged in this study. b. Examples of wave estimation Wave parameters were estimated from simulated firstorder Bragg ratios. The true wave parameters (N, 1,, s, a ), radar parameters (N T, N D, M, (n b ), n b 1,..., N T ), and degrees of freedom ( 1,i,,i )(i 1,...,M) are used as input. The true first-order Bragg ratios r t [n b (i)] are determined from Eq. (10). The F-distributed values Z i with degrees of freedom ( 1,i,,i )(i 1,..., M) are numerically generated by the computer, and the observed first-order Bragg ratios r i are simulated from Eq. (5). Then, the author estimated wave parameters by solving Eq. (11) numerically. The seed for generating uniform random numbers using the computer is different in each sample. Figure shows examples of wave parameter estimation for the bimodal wave directional distribution. The number of Doppler spectra per beam direction for the calculation is N D 8, and 1,i,i 43 (i 1,...,M). Wave parameters were calculated for N T 6 (M 48) and N T 1 (M 96). The beam directions of the two radars are expressed by Eqs. (18) and (19) ( 0 0 and d 5.5). The true wave parameters are 1 30, 90, s 7.5, and a 0.5. The wave parameters for the unimodal distribution (N, L 1) are also calculated. The simulated first-order Bragg ratios r i [Eq. (5)] are also shown as dots. For N T 6, the estimated wave parameters are 1

5 109 FIG. 1. Errors of wave parameters as a function of true primary peak direction ( 1 ) and secondary peak direction with respect to the primary peak direction ( 1 ) for the bimodal wave distributions predicted by Eq. (16). The upper axis is the direction with respect to the mean beam direction ( 1 b ). (a) Errors of primary peak wave directions ( 1, ). The radar and wave parameters are N T 1, k,i 43 (k 1,, i 1,...,M), N D 8, M 96, s 7.5, and a 0.5. (b) Same as in (a) but the radar and wave parameters are N T 1, k,i 43N D 344(k 1,, i 1,...,M), N D 8, M 1, s 7.5, and a 0.5. (c) Errors of secondary peak wave directions (, ). The radar and wave parameters are the same as those in (a). (d) Same as in (c) but the radar and wave parameters are the same as those in (b). (e) Same as in (a) but for a 0.8. (f) Same as in (b) but for a 0.8.

6 110 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 1 FIG.. Examples of wave parameter estimation for the bimodal wave directional distribution. The upper axis is the direction with respect to the mean beam direction ( 1 b ). (a) True Bragg ratio (solid line), estimated Bragg ratio from the two-parameter model (dotted line), estimated Bragg ratio from the four-parameter model (dashed line), and simulated Bragg ratio (dots). The radar and wave parameters are N D 8, 1,i,i 43(i 1,...,M), N T 6(M 48), 1 30, 90, s 7.5, and a 0.5. (b) True wave distribution (solid line), estimated wave distribution for the two-parameter model from the simulated Bragg ratio in (a) (dotted line), estimated wave distribution for the four-parameter model (dashed line). (c) Same as in (b) but N T 1 (M 96). (d) Same as in (b) but for (c). 4., 69, s 1.8, and a The agreement of wave parameters is very poor, although the retrieved first-order ratio seems to be close to the true first-order ratio (Fig. a). The retrieved wave directional distribution for N T 6 and N 4(L) is close to the retrieved unimodal distribution N (L1) (Fig. b). On the other hand, the estimated wave parameters are , 99.6, s 10.5, and a 0.4 for N T 1. The agreement of the wave parameters is good. The normalized difference defined as e t R D () D () d D () d, (0) d [ ][ ] t where D t () and D e () are, respectively, the true and estimated directional distributions, is 0.3 for N 4(L ) in Fig. d. The values of R d are 0.66 for N (L 1) and 0.65 for N 4(L ) in Fig. b. The value of R d is 0.68 for N in Fig. d. The value of R d is small, if the agreement between D t () and D e () is good. If the number of beam directions is not suffi- 1

7 111 FIG. 3. Root-mean-square difference between true and estimated wave parameters as a function of the true primary peak direction ( 1 ) and secondary peak direction with respect to the primary peak direction ( 1 ) for bimodal wave distributions. The upper axis is the direction with respect to the mean beam direction ( 1 b ). (a) Primary peak wave direction ( 1, ). The radar and wave parameters are N T 1, k,i 43(k 1,, i 1,...,M), N D 8, M 96, s 7.5, and a 0.5. (b) Same as in (a) but for N T 8(M64). (c) Secondary peak wave direction (, ). The radar and wave parameters are the same as those in (a). (d) Same as in (c) but for N T 8. The wave parameters are s 7.5 and 0.5. a cient, the iterative method for solving Eq. (11) numerically converges to other solutions. The number of beam directions is critical to resolving the bimodal wave directional distribution. c. Comparison of wave parameters To investigate the probability of converging to other solutions, wave parameters were estimated for bimodal wave distributions. Figure 3 shows examples of the results. Wave parameters were estimated for 1 35, 65, 95, 15, and 155, and 1 90, 10, 114, 16, 138, and 150. Other wave parameters are s 7.5 and a 0.5. The author calculated 0 times for each pair of ( 1, ), and calculated the rms difference between the true and estimated wave parameters from 0 samples: The total number of samples is The beam directions are expressed by Eqs. (18) and (19) ( 0 0 and d 5.5). In general, the errors of the estimated wave parameters are reduced with increasing 1. Furthermore, the accuracies of the estimated 1 are better than those of the estimated, although the accuracies of the estimated 1 are related to those of the estimated. The agreement of the wave parameters for N T 8 is poor. For example, the rms difference of the secondary peak wave directions at 1 65 and is greater than 75 (Fig. 3d). The rms differences of the wave parameters shown in Fig. 3 are large at 1 65 and 1 16, while the error levels predicted by Eq. (16) seem to be small in Figs. 1a and 1c. This indicates that the iteration often converges to other solutions of Eq. (11). The fact that Eq. (16) cannot predict wave parameter errors suggests that an error check is necessary. The error check

8 11 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 1 FIG. 4. (a) Same as in Fig. 3a but for a 0.. (b) Same as in Fig. 3a but for a 0.8. (c) Same as in Fig. 3c but for a 0.. (d) Same as in Fig. 3c but for a 0.8. The wave parameter is s 7.5. proposed here is as follows. 1) The wave parameters are estimated from observed Bragg ratios. ) The true Bragg ratios r t,i (i 1,...,M) are calculated from estimated wave parameters [Eq. (3)]. 3) The F-distributed random variables Z i are generated based on the degrees of freedom, which are evaluated from M k,h (k 1, ). 4) The Bragg ratios r i are simulated from Eq. (5). 5) Wave parameters are estimated from Bragg ratios simulated in step 4) by solving Eq. (11). 6) The process from step 3 is repeated for other sets of F-distributed random numbers. By repeating steps 1 6, we can estimate the rms difference between the estimated and simulated wave parameters. This rms difference is the simulated error by the error check. The allowable error depends on the purpose of the wave analysis. Figure 4 shows wave parameter estimation errors for a 0. and a 0.8. Other wave and radar parameters are the same as those in Figs. 3a and 3c. Errors are relatively large for 0 1 b 60. We can also see the reduction of error with increasing 1. These features can also be seen in Fig. 3. Although the agree- ment of the wave parameters is good for a 0., the agreement is poor for a 0.8. In the case of a 0.8, the solution of Eq. (11) is not determined uniquely, and the iteration often converges to other solutions. Figure 5 shows (using examples from Fig. 4) that the iteration converges to the true solution and other solutions. The wave parameter estimation for N [L 1 in Eq. (9)] is also presented. The wave parameters are 1 35, 1 16, ( 161), s 7.5, and a 0.8. The simulated first-order Bragg ratios r i in Fig. 5a are different from those in Fig. 5c. The estimated wave parameters are 1 8.4, 115.4, s 1.9, and a 0.5 in Fig. 5b. The agreement between the true and retrieved wave parameters is poor, while it is good in the case of Fig. 5d. The estimated wave parameters for N [L 1 in Eq. (9)] have almost the same values in both cases. There may be cases in which the estimated 1 corresponds to the true and the estimated corresponds to the true 1, when the primary peak amplitude is comparable with the secondary peak amplitude. However, the example in Fig.

9 113 FIG. 5. (a) Same as in Fig. c but for 1 35, 161, and a 0.8. (b) Same as in Fig. 3d but for (a). (c) Same as in (a) but with a different pair of simulated Bragg ratios. (d) Same as in (b) but for (c). The wave parameter is s b does not apply to such a case. The values of R d [Eq. (0)] are 0.61 for N (L 1) and 0.80 for N 4 (L ) in Fig. 5b. The values of R d are 0.61 for N (L 1) and 0.03 for N 4(L ) in Fig. 5d. It is difficult to estimate the bimodal wave directional distribution as the secondary peak amplitude is larger. d. Comparison of AIC The AIC [Eq. (4)] is used to answer the question Which model is better? for different numbers of parameters. Wave parameters are estimated by assuming both the two-parameter wave model [N orl 1 in Eq. (9)] and the four-parameter wave model (N 4 or L ), while the true number of wave parameters used to simulate first-order Bragg ratios r i [Eq. (5); i 1,..., M] isn orn 4. The value of AIC(4) was compared with the value of AIC() [Eq. (4)] to verify that the AIC is the index for determining the number of model parameters. Figure 6 is the comparison of the AIC in the case where the true wave directional distribution is bimodal (N 4orL). This figure shows the percentage of failures in order to identify the true wave distribution model from 0 samples as a function of the primary peak direction ( 1 ) and the secondary peak direction with respect to the primary peak direction ( 1 ). The failure here means that AIC(4) AIC() for estimated wave parameters. The wave and radar parameters are the same as those for Figs. 3b,d (N T 8, a

10 114 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 1 FIG. 6. Percentage (%) of failures to identify model functions (N 4) from AIC as a function of true primary peak direction ( 1 ) and secondary peak direction with respect to the primary peak direction ( 1 ) for bimodal wave distributions. (a) The radar and wave parameters are N T 8, k,i 43(k 1,, i 1...,M), N D 8, M 64, s 7.5, and a 0.5. (b) Same as in (a) but for N T 1 (M 96). (c) Same as in (b) but for a 0.. (d) Same as in (b) but for 0.8. a 0.5), 3a,c (N T 1, a 0.5), 4a,c (N T 1, a 0.), and 4b,d (N T 1, a 0.8). The total numbers of failures for Figs. 6a d are, respectively, 164, 89, 9, and 76, while the total number of samples for each panel is 600 (section 3c). The AIC sometimes fails to identify the true wave distribution model (Fig. 6a), and the accuracy of the wave parameter estimation is poor (Figs. 3b,d) for N T 8. However, wave directions where the accuracies of the wave parameter estimation are poor are different from those where the AIC failed to identify the number of model wave parameters. If the secondary peak amplitude is small, the AIC often succeeds in identifying the true wave model (Fig. 6c). Although the accuracy of the wave parameter estimation for a 0.5 is better than that for a 0.8, the number of failures for a 0.5 is larger. Figure 7 shows the comparison of the AIC in the case where the true wave directional distribution is unimodal [N orl 1 in Eq. (9)]. The range of true mean direction ( 1 ) for the calculation is from 55 to 165 at a 0 step, and s 1.. The number of pairs of simulated Bragg ratios r i [Eq. (5); i 1,...,M] is0 for each 1. The total number of pairs of r i is , and the total number of failures is 19. The failure to identify the true wave distribution model means that AIC() AIC(4) for estimated wave parameters. There are two different types of failures. Figures 7a and 7b show examples in which the AIC cannot identify the number of wave parameters and the estimated wave distribution is far different from the true wave distribution. The value of R d is 0.68 for N 4(L ) in Fig. 7b. On the other hand, Figs. 7c and 7d show examples in which the AIC cannot identify the number of wave parameters but the estimated wave distribution is close to the true wave distribution. The value of R d is

11 115 FIG. 7. (a) Same as in Fig. 5a but for N (true wave model), 1 15, and s 1.. (b) Same as in Fig. 5a but for (a). (c) Same as in (a) but for (d) Same as in (b) but for (c). (e) Percentage (%) of failures to identify model functions (N ) from AIC as a function of the true mean wave direction ( 1 ).

12 116 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME for N 4(L ) in Fig. 7d. There are no failures in the identification between 1 5 and 1 85 in Fig. 7e. The wave direction where the Bragg ratio is a minimum exists in the observed beam direction. In other words, the Bragg ratio-minimum wave direction is between (1) and (N T ). On the whole, the AIC can identify the number of wave model parameters. However, there are possibilities for failure in the identification as shown in Figs. 6 and 7. e. Sensitivity to directional distributions The sensitivity of the wave parameter estimation errors to directional distributions is investigated. The simulations are performed with different values of s and a. Furthermore, the directional distribution expressed as L D() ak sech [( k)] (1) k1 (Donelan et al. 1985) is also used for the simulation. The parameter corresponds to the spread parameter s. If the half angular width of Eq. (1) is equal to that of Eq. (9), the relation 1 1/ 1/(s) 1 log(1 ){Arccos[ ]} () is satisfied. If the true directional wave distribution is the sech ( w ) model [such as Eq. (1)], the estimated wave distribution is also the sech ( w ) model. Figure 8 shows errors of primary peak directions for various wave parameters. The wave directional distribution is expressed by Eq. (9). In the examples, errors are large for 0 1 b 60 and are reduced with increasing 1. By comparing Fig. 8a with Fig. 8c and comparing Fig. 8b with Fig. 8d, we can see that the peak direction estimation is better for small secondary peak amplitudes. Figure 8e shows the errors in the primary peak directions as a function of s and a at ( 1, 1 ) (65, 114). Other wave parameters are s 5, 6, 7, 8, 9, 10, 11, and 1 and a 0., 0.4, 0.6, 0.8, and 1. The wave parameter estimation error is larger with the increase of a. There is a reduction in error at large s for the case of a large secondary amplitude. Figure 9 shows the errors in the primary peak directions for the sech ( w ) model [Eq. (1)]. The parameter 1.46 corresponds to s 7.5 for the distribution of Eq. (9), and 1.68 corresponds to s 10 for the distribution of Eq. (9) from Eq. (). In many cases, accuracies of the primary peak direction estimations for the sech ( w ) model are poorer than those for the cos s ( w /) model. For example, if we compare Fig. 9a with Fig. 3a, which is a counterpart to Fig. 9a, we can find that accuracies in Fig. 9a are poorer than those in Fig. 3a. Although the reason why the method of estimating wave parameters is better for the cos s ( w /) model is not clear, the method (the quasi-newton method) often falls into a local minimum of log(l g ) [Eq. (7)] with a sech ( w ) model. Even if the method is more accurate with a cos s ( w /) model, this does not mean that the measured data behave in this way. The large error peak directions in Fig. 9a are different from those in Fig. 3a. Accuracies of wave estimation for a 0. are better than those for a 0.8, as shown in Figs. 9b e. We do not observe the reduction of the error with increasing 1. Figure 9f shows errors of primary peak directions as a function of and a at 1 65 and Other wave parameters are 1., 1.9, 1.38, 1.47, 1.57, 1.66, 1.75, and 1.84 and a 0., 0.4, 0.6, 0.8, and 1. The parameter 1. corresponds to s 5 for the distribution of Eq. (9), and 1.84 corresponds to s 1 for the distribution of Eq. (9) from Eq. (). Figure 9f shows that the peak wave direction estimation is better for a smaller secondary peak amplitude. On the other hand, the dependency of the wave direction estimation errors on is not clear. 4. Observation a. Experimental design The HF ocean radars of the Okinawa Radio Observatory (Okinawa Subtropical Environment Remote Sensing Center), Communications Research Laboratory, were deployed along the east coast of Okinawa Island (Fig. 10). The radars were located at site A (618.63N, E) and site B (67.19N, E) as shown in Fig. 10. Measurement of ocean currents and waves by HF ocean radars was conducted from 15 April to 15 May The radio frequency was 4.5 MHz; the corresponding Bragg frequency was Hz. The beam direction is electronically controlled by hardware in real time, and 1 beam directions for each radar are used in the observation. The temporal resolution of the radar system is h. The step of the beam direction is 7.5, and the range resolution is 1.5 km. The details of the observation are described in Hisaki et al. (001). The footprints for the radars are spaced on radial grids, while the Doppler spectra are not spatially interpolated on square grids. Therefore, the Doppler spectra are statistically independent of each other. The positions of footprints to calculate wave parameters from Doppler spectra are indicated in Fig. 10. The radar parameters are N T 1, N D 8, 0 81 (with respect to the eastward direction), and d 5.5 in Eq. (19). The surface winds are measured in location I (Itokazu, 609N, 1746E) at 10-min intervals by the Japan Meteorological Agency (JMA). The resolution of the wind speed was 1 m s 1 and that of the wind direction was.5 (16 directions). b. Example of observation Figure 11 shows an example of the observation from 0000 Japan standard time (JST) 5 April to 00 JST

13 117 FIG. 8. (a) Same as in Fig. 4a but for a 0. and s 5. (b) Same as in Fig. 4a but for a 0. and s 10. (c) Same as in Fig. 4a but for a 0.8 and s 5. (d) Same as in Fig. 4a but for a 0.8 and s 10. (e) Root-mean-square difference between true and estimated 1 as a function of s and a at ( 1, 1 ) (65, 114).

14 118 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 1 FIG. 9. (a) Same as in Fig. 3a but for Eq. (1) with a 0.5 and (b) Same as in Fig. 8a but for Eq. (1) with a 0. and 1.. (c) Same as in Fig. 8b but for Eq. (1) with a 0. and (d) Same as in Fig. 8c but for Eq. (1) with a 0.8 and 1.. (e) Same as in Fig. 8d but for Eq. (1) with a 0.8 and (f) Same as in Fig. 8e but for Eq. (1) as a function of and. a

15 119 FIG. 10. Map of the observation area A and B, locations of the HF ocean radars; I, and meteorological station. Dots are centers of footprints of Doppler spectra for the calculation of wave parameters. 6 April The cos s ( w /) model [Eq. (9)] is used to estimate wave distributions, because the accuracy of the wave direction estimation for this model is better than that for the sech ( w ) model [Eq. (1)]. An atmospheric front passed over the observation area on 5 April 1998 (Hisaki 00), and the wind turned southeastward (Fig. 11c). The fetch-limited winds were observed from 1000 JST 5 April to 0800 JST 6 April. In general, directional distributions for N [or L 1 in Eq. (9); Fig. 11a] seem to be similar to those for N 4 (Fig. 11b), although AIC(4) AIC() during the observation period. However, directional distributions for N 4 are narrower than those for N in many cases. The bimodal wave distribution was clearly observed only at 0000 JST 6 April in this example, which is associated with a sudden change of the shortwave direction. Even when fetch-limited winds were observed, the wave distributions were unimodal. The estimated wave parameters for the four-parameter model at 0000 JST 6 are 1 137, s 33, 38, and a 0.4. The value of s is larger than the examples of the simulations, and we can expect that the AIC is successful in identifying the number of wave model parameters in this case. The author conducted the error check (section 3c) in all cases. The number of samples for one wave estimation is 0. The rms difference between wave parameters obtained from observations and wave parameters estimated from Monte Carlo simulations was calculated from 0 samples. In all of the cases of the error check, AIC(4) AIC(), and the fourparameter model provided the better description of the wave directional distributions. Figure 11d is the rms difference of the primary wave directions ( 1 ). The rms difference is large only at 1400 JST 5 April, when the estimated wave parameters for the four-parameter model (N 4, L ) are 1 44, s 1.7, 68, and a 0.73, and the wave distribution changes significantly. The estimated wave parameters are not reliable at that time. Figure 11e shows a time series of the standard deviation (Z e ) of estimated Z i [i 1,...,M; Eq.(5)] and the predicted standard deviation calculated from M 1 i M i1 1/ Z Z, (3) p where Z i is estimated from Eq. (17). If Z e is close to Z p, the wave field is statistically homogeneous. In most cases, estimated standard deviations, Z e, were larger than predicted standard deviations, Z p, which suggests that the wave field was statistically in homogeneous. We expect that wave parameter errors can be estimated by multiplying simulated errors as shown in Fig. 11e by Z e /Z p, if the wave parameter errors are approximated to be the linear combination of the standard deviation of the estimated Z i [Eq. (16)]. In particular, Z e was large at 000 JST 6 April, when the wave distribution changes significantly. The value of Z e was also large at 1400 JST 5 April, when the error estimated from the error check was large. This shows that the wave distribution over the observed area did not change simultaneously. 5. Concluding remarks The possibility of short-wave directional distribution estimation from HF ocean radars was investigated based on Monte Carlo simulations, when the distribution was bimodal. The possibility was investigated from two points of view. The first is to investigate the accuracy of the wave estimation, and the other is to identify the optimal model function of the wave distribution. With regard to the former, the number of beam directions is critical to the estimation of the wave parameters, because the parameters that maximize the likelihood function are not uniquely determined. The minimum number of beam directions equals the number of model parameters; however, it is insufficient. We should use as many beam directions as possible, which requires more than three radars or the assumption that the wave field is statistically homogeneous over the wide area. The advantage of the HF ocean radar is that it can measure ocean waves with fine resolution. Although the spatial resolution is decreased by the assumption of homogeneity, there is still an advantage in using the HF ocean radar system; the land-based radar system is easier to maintain than the sea-based measurement system. It was found that wave estimation was good for small secondary peak amplitudes. On the other hand, if the secondary peak amplitude was comparable with the pri-

16 10 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 1 FIG. 11. (a) Time series of wave directional distributions for the two-parameter model (normalized by maximum values), (b) same as in (a) but for the fourparameter model, (c) winds at I in Fig. 10, (d) root-meansquare difference between estimated primary wave directions and primary wave directions from simulated Bragg ratios (), and (e) Z e (solid line) and Z p (dashed line). mary peak amplitude, wave estimation was poor. If the s value becomes larger for the bimodal distribution and the large secondary amplitude, the wave estimation error is reduced in the cos s ( w /) model [Eq. (9)]. In many cases, wave estimation errors are large in 0 1 b 60 for the cos s ( w /) model. It is necessary to check the error of the estimated parameters. The AIC has been used to select one of two model directional distributions. In general, the AIC can identify the model function; however, the AIC sometimes fails to identify the true model function. If the true model function is the four-parameter model, the probability of successfully identifying the model is high when the secondary peak amplitude is small.

17 11 Wave parameters were estimated for observed Doppler spectra, when an atmospheric front passed over the observation area and fetch-limited winds were dominant. It was found that the four-parameter model provided a better description of the directional distribution than the two-parameter model, from the comparison of the AIC. However, the bimodal distribution was observed only when the short-wave direction shifted. Even in fetch-limited conditions, the bimodal distribution was not observed. However, since the wave fields were spatially in homogeneous, we cannot conclude that the wave distributions in fetch-limited conditions were unimodal. To investigate the bimodal properties of waves, it is desirable to use as many radars as possible located appropriately, in order to avoid spatial inhomogeneity limitations and to provide the best estimates of parameters and their number. However, as a practical matter it is difficult to use many radars, because the costs of the system become high and it is difficult to obtain sufficient radar sites. Even with the use of many radars, we still cannot guarantee that estimates of the bimodal wave distributions are true because of all the reasons presented in this simulation work. We must at least conduct an error check to confirm the validity of the estimated wave parameters. Acknowledgments. The author acknowledges the anonymous reviewers for their insightful comments that have contributed to the improvement of the manuscript. The author acknowledges the staff of the Okinawa Radio Observatory (Okinawa Subtropical Environment Remote-Sensing Center), Communications Research Laboratory, for providing the Doppler spectrum data. The author acknowledges the Japan Meteorological Agency for providing the meteorological data. 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