Estimators for the Standard Deviation of Horizontal Wind Direction

Transcription

1 1403 Estimators for the Stard Deviation of Horizontal Wind Direction RUDOLF O. WEBER Paul Scherrer Institute, Villigen, Switzerl (Manuscript received 17 December 1996, in final form 31 March 1997) ABSTRACT The stard deviation of horizontal wind direction is a central quantity in the description of atmospheric turbulence of great practical use in dispersion models. As horizontal wind direction is a circular variable, its stard deviation cannot be directly estimated by on-line methods. For a mathematically strict determination of the angular stard deviation, it is necessary to store all observations perform off-line calculations. A more practical approach is to calculate on-line moments of linear variables to parameterize angular stard deviation in terms of these moments. A variety of such estimators is compared by means of a large dataset from an ultrasonic anemometer. The paper systematically investigates which types of linear variables lead to the best estimators which parameterizations are best within each group of linear variables. Estimators based on moments of the sine cosine of the wind direction turned out to be most robust. The parameterizations based on an isotropic Gaussian model of turbulence gave the estimators with smallest error within the different groups. 1. Introduction The dispersion of airborne material is mainly due to turbulent diffusion inherent in atmospheric motion. For realistic estimates of dispersion, it is therefore of primary importance to have an accurate description of atmospheric turbulence. The stard deviation of horizontal wind direction, also called angular stard deviation, can give accurate estimates of the lateral (cross-wind) spread y of a plume by the simple expression y xf (x), where x is the downwind distance from the source (Hanna et al. 198; Pasquill Smith 1983). A variety of different forms of f(x) can be found in the literature. Irwin (1983) compared several models for f(x) by an analysis of field tracer data. Although other methods exist for the parameterization of lateral spread y in terms of atmospheric stability (Gifford 1976), the use of observed turbulence characteristics such as gives better agreement with tracer experiments (Hanna 1990; Cirillo Poli 199; Sharan et al. 1995; Yadav Sharan 1996). On the microscale, Hicks et al. (1987) used to describe the aerodynamic resistance in a model of dry deposition. As the stard deviation of wind direction is a fundamental quantity in the description of atmospheric turbulence is of central importance in the applications discussed above, it is of great interest to obtain simple methods for deriving from meteorological observations. However, the horizontal wind direction is a cir- Corresponding author address: Dr. Rudolf O. Weber, Paul Scherrer Institute, CH-53 Villigen PSI, Switzerl. rudolf.weber@psi.ch cular variable (Mardia 197), its stard deviation cannot be calculated by simply summing up the angle its square during the measurement interval (Essenwanger 1986; Fisher 1987). A possibility for the estimation of is to store all data during the measurement period then to calculate off-line with a so-called multipass procedure (elson 1984; Yamartino 1984; Essenwanger 1986). This requires both considerable memory computing capacities of the measurement system. To avoid these multipass methods, one can try to parameterize the angular stard deviation as a function of moments of linear variables, which in turn can be estimated by single-pass methods. The methods following these ideas differ in the type of linear variable considered, in the form of the parameterization, also in whether some turbulence model or theoretical probability distribution function or experimental data are used to fit the best parameterization. Several methods are based on moments of the horizontal wind components (Ackermann 1983; Weber 1991) or on the means of the sine cosine of the wind direction (Mardia 197; Verrall Williams 198; Yamartino 1984; Ibarra 1995). Other methods are based on the ratio of vector mean wind speed to scalar mean wind speed, the socalled persistence of wind direction (Yamartino 1984 ; Mori 1986; Weber 1991; Weber 199; Ibarra 1995; Leung Liu 1996). In this paper some refined parameterizations are given, which are based on a simple isotropic model of turbulent wind fluctuations. Using a large dataset of fast wind measurements, a systematic analysis is undertaken to find the linear variables that give the most robust estimate of American Meteorological Society

2 1404 JOURAL OF APPLIED METEOROLOGY VOLUME 36 Section recapitulates some properties of angular variables, such as the horizontal wind direction. In section 3, the wind data obtained from an ultrasonic anemometer are described. In section 4, an isotropic Gaussian model of turbulence is discussed several parameterizations of are given in terms of moments of linear variables. Section 5 summarizes some other estimators of taken from the literature. In section 6, the accuracy robustness of the different estimators are investigated by means of the observed wind data. Section 7 discusses some limitations of the isotropic Gaussian model.. Angular stard deviation For linear, nonperiodic, rom variables, the mean variance are defined in a stard way (see, e.g., Stuart Ord 1987). Let the linear rom variable X be described by a probability density function (PDF) p X (x). The mean x is then defined by the variance X xp (x) dx, (1) x x is defined by x x X (x ) p (x) dx. () It can easily be seen that the variance () is the minimum second-order moment in the following sense: If a general second-order moment is defined as x X (A) (x A) p (x) dx, (3) it attains its minimum for A x. Thus, the definition in () gives a minimum variance in the sense described above. As the integration limits in (1) () are fixed, both mean variance can be calculated in a straightforward way as long as the PDF p X (x) is known. For experimental data, it would be tedious to extract the PDF p X (x) from the data. However, for a sequence X 1, X,..., X of measured data, there exist unbiased estimators for the mean variance (Stuart Ord 1987). The sample mean x the sample variance s x are calculated from the observed data by 1 1 i x i i1 1 i1 x X, s (X x). (4) The sample variance can be rewritten as 1 1 x i i 1 i1 1 i1 s X X. (5) This formula shows that the sample variance of a linear variable can be estimated by continuously summing up the observed values X 1 their squares. Such a method is termed single-pass estimator. The vertical angle of a 3D wind vector can be treated as a linear variable, as it is uniquely defined on the interval [/, /], the single-pass estimators (4) (5) can be used to calculate its sample mean variance. The horizontal wind direction, however, is a circular (or periodic, or angular) variable. Any physical quantity, such as a Cartesian component of the wind vector, is periodic in the angle with a period of. The PDF of the angle is also periodic: p () p ( k) for all, (6) where k is an integer (Fisher 1987). It is sufficient to know the rom variable in an arbitrary interval [ c, c ] of length (see also Weber 1991 for more details). The starting point c of the interval is called the cut point. The PDF p () is normalized by c c The mean is defined by p()d1. (7) c c () p()d (8) c depends on the cut point c. The angular variance is defined in a minimum variance sense, analogously to (3), by requiring c c c c ( ) [ ( )] p () d (9) to become a minimum, where c is a free parameter, which is varied. This is equivalent to the minimum variance method for observed data, as described in Essenwanger (1986). The requirement ( c ) minimal to- gether with (8) (9) form a set of equations for c, ( c ), ( c ). If the PDF (6) is symmetric about 0, the cut point leading to minimum variance is either 0,, or a 0 of the PDF (Weber 1991). The sample mean variance cannot be as easily obtained as in the case of linear variables. As for circular variables, the integration limit in (8) (9) is not fixed; it must be varied until the variance attains its minimum value. Thus, there are no single-pass estimators for the angular mean (8) or the angular variance (9) (Essenwanger 1986; Weber 1991). elson (1984) Yamartino (1984) describe methods to obtain an approximation of the minimum variance by multipass methods. Both authors use 1 trig tan (sin/cos) (10) as an approximation of the mean angle, where sin 1/ i1 sin i cos 1/ i1 cos i. A method for estimating mean variance of an angular variable,

3 1405 together with the cut point c, is described in the appendix. 3. Data Fast wind measurements from an ultrasonic anemometer were used. On a building of the Paul Scherrer Institute a small mast (about 14 m high) was mounted, bearing the anemometer on top. The instrument, a Gill Instruments Limited sonic anemometer, measured stored the three Cartesian wind components every s (corresponding to a sampling frequency of about 0.83 Hz). From July 199 to March 1993, several observation periods of up to 6 days length took place. Data from 1431 h were available, with a total of about 10 8 measured values of each wind component. The 1431 h of observation cover a variety of meteorological conditions. For example, the 10-min means of wind speed (8588 values) range from 0.13 to 11.1 m s 1, with a mean of 1.7 m s 1 a median of 1.5 m s 1. Thus, data are taken from a wide range of mean wind speeds, but the bulk comes from low-speed conditions for which large values of can be expected. As the sampling frequency is quite high (0.83 Hz) all measured data are stored, it becomes possible to calculate the true minimum variance (9) mean (8) of the horizontal wind direction. This true angular mean stard deviation are computed by the multipass method described in the appendix. 4. Isotropic Gaussian model of turbulence The angular mean variance in the minimum variance sense are fixed if the angular PDF p () is given. In the model used in this section, the angular PDF p () is not defined directly, but rather a microscopic description of turbulence is given by prescribing the probability density of the longitudinal (along the mean wind direction) lateral wind components. It is assumed that 1) the longitudinal the lateral components of the horizontal wind vector are both rom variables described by a Gaussian distribution function, ) the longitudinal the lateral components are statistically independent, 3) the variances of the two components are equal. This model is termed the isotropic Gaussian model. It was used in McWilliams et al. (1979) to calculate the distribution function of wind speed wind direction. Ibarra (1995) used the same model to determine single-pass estimators for the angular stard deviation. An anisotropic Gaussian model, with unequal variances of the longitudinal lateral components, was used in Weber (1991) to obtain single-pass estimators of. In the isotropic Gaussian model of turbulent wind fluctuations, the longitudinal component u the lateral component are described by the PDFs [ ] [ ] 1 (u u) p (u) exp U ; u 1 p () exp V. (11) u As the two horizontal components u are assumed to be statistically independent, the joint PDF is given by the product p UV (u, ) p U (u)p V (), (1) which after transformation to polar coordinates (u r cos r sin) becomes p (r, ) R [ ] 1 (r cos u) r sin r exp. (13) u The PDF of the angle is obtained after integration over r is given by 1 1/ p () exp [1 exp( ) erfc()], (14) where cos/ u / u (15) is the longitudinal signal-to-noise ratio, or the inverse of the longitudinal turbulence intensity. Here, erfc (z) denotes the complementary error function u erfc(z) 1 erf(z) exp(t ) dt. (16) The circular PDF (14) is a special case of the offset normal distribution given in Mardia (197, Eq ) of the anisotropic PDF given in Weber (1991). As the circular PDF (14) is symmetric about 0, the angular variance can be calculated as z p () d. (17) For 0, the angular distribution (14) becomes uniform on the circle /3 1/. For, the angular distribution goes to zero everywhere except for 0 0. In the isotropic model of turbulent wind fluctuations, the joint PDF [(1) or (13)] describing the horizontal wind vector is given. Therefore, it is possible to calculate a variety of moments of linear variables study their relation to the angular variance (17) or its square root, the angular stard deviation. For the Cartesian wind components e n in a fixed horizontal coordinate system, the means e n the variances e n are related to the means variances of the longitudinal u lateral component by u u

4 1406 JOURAL OF APPLIED METEOROLOGY VOLUME 36 cos, sin ; (18) e u V n u V cos sin, 0; (19) u e V n V e n u, (0) where the angle of the horizontal mean wind vector is given by V tan 1 ( n / e ). (1) These transformations are the isotropic special cases of the anisotropic transformations given in Jones (1964) Weber (1991). The trace of the covariance matrix h u e n () is called the total variance of wind (Ibarra 1995) or the squared stard vector deviation of the wind (Essenwanger 1986). This quantity remains constant under rotations of the horizontal coordinate system. Considering the angle alone, the linear variables cos sin can be used to calculate the trigonometric means cosp () d cos sinp () d (3) sin the trigonometric variances cos cos sin sin (cos ) p () d (sin ) p () d. (4) Another way of characterizing the variability of wind direction is through the persistence of wind direction (Essenwanger 1986). The persistence P of wind direction is defined as the ratio of the vector mean V the scalar mean V s wind speed: P VV s (5) 1/ 1/ V (e n) (u ), (6) Vs (en ) 1/ (u ) 1/, (7) where x is the expectation value (1) of a variable x. Persistence ranges from 0 for a perfect rom motion with no preferred direction to 1 for a wind with constant direction. For observed data, these moments of linear variables can all be estimated by single-pass methods from the wind measurements. Therefore, the angular stard deviation can be estimated by a single-pass method from observed data if suitable parameterizations of in terms of these linear variables are found. Assuming that the same relationships hold for observed data the isotropic Gaussian model, such parameterizations are derived from the model results. To obtain the parameterizations, the parameter [in (15)] of the isotropic model was varied from zero to was calculated by numerical integration of (17). Simultaneously, the moments of the linear variables, as described above, were calculated for each value of, which allows to be a function of the moments of linear variables determines suitable fit functions. For convenience, all constants in the following expressions for the angular stard deviation are chosen such that is in degrees. In Fig. 1a the angular stard deviation is shown as a function of the persistence P in (5) together with the parameterization W (1 P 761 ) , (8) which was found as the best least squares fit among a variety of test functions. Figure 1b shows the angular stard deviation as function of scaled scalar wind speed V s / h, where h u is the trace of the covariance matrix () together with the fit [ ] h 180 V exp ln s W, (9) 3 where / is the lower bound for V s / h (Ibarra 1995) ln denotes the natural logarithm. Figure 1c shows as a function of the scaled vector wind speed V / h together with the parameterization [ ] h 180 V 1 tanh ln W3. (30) In Fig. 1d, is shown as dependent on the trigonometric means cos sin together with the parameterization W6 ( R R ) 3 arccos(r 1), (31) where R cos sin. The four parameterizations (8), (9), (30), (31) were obtained by testing a variety of model functions of the results of the isotropic Gaussian model. Thus, no fit to experimental data was made. 5. Other single-pass estimators Since Mardia s (197) derivation of the stard deviation for the wrapped Gaussian distribution, many methods have been proposed for single-pass estimators of the angular stard deviation. The methods can be grouped according to the linear variables whose moments are used in the estimator of. For convenience

5 1407 FIG. 1. Angular stard deviation from (17) as a function of different moments of linear variables, as obtained from the isotropic Gaussian model (solid lines) together with the parameterizations (dashed lines) of section 4: (a) as a function of the persistence P in (5) of wind direction the parameterization (8), (b) as a function of the scaled scalar mean wind speed V s / h the fit function (9), (c) as a function of the scaled vector mean wind speed V v / h together with the parameterization in (30), (d) as a function of the trigonometric means cos sin (R ) together with the parameterization in (31). cos sin all constants are given in the following expressions, such that the angular stard deviation is in degrees. The first group is based on moments of the Cartesian wind components e n, where e gives the wind component in the easterly direction n the wind component in the northerly direction. From the measured data, the sample means e n the sample variances s e s n of the two components are calculated as in (4). For some estimators, the cross covariance s en is also needed. Ackermann (1983) estimates the angular stard deviation by 1/ (e sn n se ens en) A, (3) (e n ) which is based on the propagation law of errors only holds for small values of A. After transformation to longitudinal lateral coordinates u, the estimator in (3) becomes A s /u (Weber 1991). Weber (1991) has derived another estimator based on

6 1408 JOURAL OF APPLIED METEOROLOGY VOLUME 36 TABLE 1. Various estimators of angular stard deviation in terms of the persistence P of wind direction. Estimator M1 180/ arctan{[(1 P)/P] 1/ } M 180/( lnp) 1/ Y1 180/(1 P ) 1/ Y 180/[ (1 P ) 3/ ] arcsin[(1 P ) 1/ ] W (1 P) CSI 81(1 P) 1/ I3 180/3 exp{1.158[(1 P.35 ) 1/.35 1] 0.58 } LL 97(1 P) 0.46 W (1 P 761 ) Source Mori (1986) Mori (1986) Yamartino (1984) Yamartino (1984) Weber (1991) Hauser et al. (1994) Ibarra (1995) Leung Liu (1995) Present paper, (8) the means variances of the Cartesian wind components by a fit to the results of an anisotropic version (s u s ) of the Gaussian model. The mean vector of the sampling period is determined by averaging the two horizontal wind components e n, the two dimensionless parameters u/s u s u /s are calculated, where the longitudinal direction is taken as the direction of the mean vector. The angular stard deviation is estimated by A B C W1, 1 D E A, B, C , D, E (33) Using the total variance of wind from (), the scalar mean wind speed from (7) the vector mean wind speed from (6) can be rescaled (Ibarra 1995) by Vˆ v V v /s h Vˆ s V s /s h. Ibarra (1995) gives two estimators based on these scaled scalar vector mean wind speeds: I1 exp[1.4(v s/sh ) 3 ] (34) /.35 exp1.4{[(v /s ) ] I v h }, (35) which were derived as fits to the numerically obtained curves of the same isotropic Gaussian model as described in section 4 ( /). In the present paper, two different estimators, W from (9) W3 from (30), are given using the scaled wind speeds. A second class of estimators for the angular stard deviation uses the persistence of wind direction P from (5). Some of them are reviewed in Weber (199). Table 1 lists various estimators based on persistence P. A third class of estimators for the angular stard deviation makes use of the trigonometric moments of the angle. Let cos sin denote the sample means s cos s sin the sample variances of cos sin. Mardia (197) defined a circular (or angular) stard deviation as 180 1/ Ma [ln(r )], (36) which is motivated by the fact that a wrapped normal distribution (Mardia 197) with a mean resultant length R, where R (cos) (sin), (37) can be obtained by wrapping a linear normal distribution with stard deviation Ma to the circle (Fisher 1987). The method of Verrall Williams (198); (see also the comment of Fisher 1983 the reply by Verrall Williams 1983) is based on geometrical considerations requires the calculation of two vectors z (sin s, cos s ) (38) 1 sin cos z (sin s, cos s ), (39) sin cos from which the angular stard deviation is computed by 90 z 1 z VW arccos. (40) z z 1 Yamartino (1984) gives a variant of the method of Verrall Williams (198), in which the vectors z 1 z are redefined as

7 1409 FIG.. Minimum variance mean angle from (8), as determined by the method of the appendix, vs (a) the vector mean angle from (1) (b) the trigonometric mean angle from (10) for all min samples. where g Z (sin gs, cos gs ) (41) 1 sin cos Z (sin gs, cos gs ), (4) sin cos 3/ the estimator for is given by 90 Z 1 Z Y3 arccos. (43) g Z Z 1 In Yamartino (1984), the following interpolation to as obtained from simulated data, is presented: Y4 (1 b ) arcsin(), (44) where 1 R 1 (cos sin ) b (/ 3) is a constant. In the present paper, a fit to the results of the isotropic Gaussian model gives a new estimator W6 from (31). Many of the methods discussed above are compared in the literature quoted in this section in other papers, such as Turner (1986). However, the comparisons are mostly done with a rather limited set of data, more importantly, no attempt has been made to determine which group of estimators provides the most robust estimate of under a variety of meteorological conditions. This question will be addressed in the next section. approximations of angular stard deviation discussed in section 5 are based on a that is centered around the vector mean angle from (1) or the trigonometric mean angle from (10). To see how well these different mean angles agree, all three of them were calculated for the min samples. In Fig., the minimum variance mean angle is plotted against the vector mean angle (Fig. a) the trigonometric mean angle (Fig. b). In both cases the agreement is on average very good (the correlation is , respectively). How- 6. Comparison of single-pass estimators true angular stard deviation The minimum variance from (9) is centered around the minimum variance mean angle from (8), which is determined together with the cut point c. Many of the FIG. 3. Minimum variance calculated by the method of the appendix vs the multipass estimate of Yamartino (1984) for all min samples.

8 1410 JOURAL OF APPLIED METEOROLOGY VOLUME 36 FIG. 4. Minimum variance as a function of different moments of linear variables for all min samples: (a) as a function of the persistence P in (5) of wind direction, (b) as a function of the scaled scalar mean wind speed V s / h [(7) ()], (c) as a function of the scaled vector mean wind speed V / h [(6) ()], (d) as a function of the trigonometric means cos sin, with R from (37). cos sin ever, there are few quite large outliers, where the difference between the angles is rather large (90 more), mainly for the vector mean angle. As the trigonometric mean angle the minimum variance angle agree well with each other, one expects that the multipass estimator of Yamartino (1984) would also agree well with the minimum variance estimator described in the appendix. Whereas the minimum variance estimator determines both the mean angle angular stard deviation by a multipass method, Yamartino (1984) uses the trigonometric mean angle for a multipass calculation of the stard deviation. In Fig. 3, the minimum variance is plotted versus the multipass estimate of Yamartino (198) for all min samples. The agreement between the two multipass methods is excellent, with small differences for large angular stard deviations. For 60, Yamartino s method underestimates the minimum variance somewhat. For 10-min averages the maximal difference is 8, for 1-h averages it becomes slightly larger than 11. For all practical applications, the two multipass methods can be considered as equal.

9 1411 FIG. 5. Same as Fig. 4 but for all h samples. In Fig. 4 the minimum variance, as estimated by the method of the appendix, is plotted versus several moments of linear variables for all min samples. A comparison with Fig. 1, where the corresponding curves of the isotropic Gaussian model are presented, shows that this simple model precisely describes the average behavior of the turbulent wind fluctuations observed in our meteorological measurements. In Fig. 5, the same scatterplots are shown for all h samples. The data points for this longer sampling interval fall on the same smooth curves as for the 10-min samples. As can be expected, there is a shift toward larger values of for the longer sampling interval. If is plotted versus the persistence P (Fig. 4a), large scatter is present, whereas the other three parameterizations show much less scatter of the data. We would therefore expect that estimators of based on persistence also show considerable differences from the true. Whether estimators based on V /s h, V s /s h,orr are in good agreement with the minimum variance depends on how well the necessary moments of linear variables can be estimated from the data. To see in detail how accurate the various estimators of angular stard deviation are, the root-mean-square error (rmse) between the estimators the minimum variance was calculated. These rmses are shown in Table, together with the maximum absolute error of

10 141 JOURAL OF APPLIED METEOROLOGY VOLUME 36 TABLE. Root-mean-square error maximum absolute error for various estimators of as determined for all min samples. The maximum absolute error is given for five classes of. The formulas of the persistence estimators are given in Table 1. Estimator equation rmse (deg) Cartesian components estimators A (3) W1 (33) I1 (34) W (9) I (35) W3 (30) Persistence estimators M1 M Y1 Y W4 CSI I3 LL W Maximum absolute error (deg) Trigonometric moments estimators Ma (36) VW (40) Y3 (43) Y4 (44) W6 (31) several classes of. All estimators have an rmse of less than 10, except for Ackermann s method A from (3), which has an extremely high rmse of 31. The singularity for small wind speeds inherent in that model causes this large rmse also the large absolute error for large. The other estimators based on Cartesian components or persistence have similar rmses. The very best rmse is observed for the two estimators Y4 from (44) W6 from (31), which make use of trigonometric moments. These two estimators are the only ones for which the absolute error remains within 10 for all ranges of. The other three estimators using trigonometric moments have much higher absolute errors for large. This can be attributed to the fact that these parameterizations do not fit the curve of the isotropic Gaussian model well (Fig. 1d). For example, Mardia s estimator Ma from (36) has a singularity for small R. For all estimators based on persistence, the absolute errors are considerably larger, in agreement with the impression obtained from Fig. 4a, where much more scatter is present than in Fig. 4d. In summary, the best estimators in terms of small rmses also the most robust ones in terms of small maximum absolute error are the two estimators Y4 (Yamartino 1984) W6 from (31), both based on R (cos sin ), where cos ( correspondingly sin) is the sample mean of cos used as an estimator of the expectation value. The new estimator W6 from (31) is slightly better in the rmse the maximum absolute error in most classes. Among the estimators using moments of Cartesian components, the estimator of Weber (1991), W1 from (33), is the best, closely followed by the two methods of I from (35) W3 from (30), which are based on the scaled vector mean speed. Several of the estimators based on persistence (see Table 1), W4 (Weber 1991), I3 (Ibarra 1995), LL (Leung Liu 1995), W5 from (8), perform about equally well, either in terms of rmse or in terms of maximum error. 7. Validity of the isotropic model The simple model of turbulent wind fluctuations of section 4 is based on three assumptions of 1) isotropy of the fluctuations ( u ), ) independence of u, 3) wind fluctuations following a normal (Gaussian) distribution. The parameterizations of in Fig. 3 show some scatter (in the case of the persistence, even quite large scatter) around the curves of the isotropic Gaussian model (Fig. 1). This scatter, or part of it, may be due to the fact that the model assumptions are not met in the observed data. The deviations from isotropy are described by u /, where u are the stard deviation of the longitudinal lateral wind components. For the 10-min samples, ranges from 0.5 to 4.6. To see the influence of anisotropy, the versus persistence P curves for an anisotropic model (Weber 1991) are shown in Fig. 6a for 0. 5 as extreme cases of the observed values. A comparison with Fig. 4a shows that the deviations from the isotropic curve are about the same as the ones observed in the data. Thus, a part of the large scatter of Fig. 4a may be due to the anisotropy of the wind fluctuations. As a measure for the independence of u, the cross correlation r between them is calculated (for normally distributed variables statistically independent uncorrelated are equivalent). The observed r ranges from 0.93 to As a turbulence model with r 0 depends only on r ; in Fig. 6b, the versus persistence P curve for r 0.95 is shown. The deviation from the model of section 4 is much smaller than that for the anisotropic model. To see how well the u components are normally distributed, the skewness kurtosis (Stuart Ord 1987) are calculated. For a Gaussian rom variable, both skewness kurtosis are zero; nonzero values indicate deviations from a normal distribution. For large sample size n, their stard errors become 6/ 6/ (Essenwanger 1986). The observed data show deviations from a normally distributed rom variable, therefore nonzero values of skewness kurtosis must be expected. Several filters were applied to the 10-min data samples to select only samples that fulfill the model assumptions well. The first filter accepts only data samples for which holds, the second filter accepts samples

11 1413 FIG. 6. Angular stard deviation vs persistence P for (a) anisotropic models with 1 (b) a model with correlated (r 0) u wind components. with a small cross correlation r 0.1, the third filter accepts only samples whose skewness of both u is less than 0.33 whose kurtosis of both u is less than The versus persistence P scatterplots of the filtered data are shown in Fig. 7. The number of accepted 10-min samples is about the same in all three cases: 641 in Fig. 7a, 704 in Fig. 7b, 736 in Fig. 7c. A comparison with the unfiltered data of Fig. a shows that all three filters reduce the scatter around the curve of the isotropic model. Thus, violation of the model assumption can lead to large scatter in the parameterization errors in the estimation of. The isotropy filter (Fig. 7a) reduces scatter most effectively. Hence, it seems that isotropy is the most relevant model assumption, at least for the persistence estimators. 8. Summary As the stard deviation of horizontal wind direction is a fundamental quantity in the description of atmospheric turbulence is used in many dispersion models to quantify lateral spread, it is of great practical use to have reliable estimators of. Wind direction is a circular variable, only defined up to multiples of. Therefore, it is not possible to obtain its stard deviation in the same simple manner as for a linear variable. Instead of storing all measured data performing off-line calculations of, parameterizations of in terms of linear variables can be used. As moments of these latter linear variables can be easily calculated on-line, single-pass estimators for the angular stard deviation result from these parameterizations. Based on a simple isotropic Gaussian model of turbulent wind fluctuations, various parameterizations of were developed. Together with many parameterizations from the literature, these were compared by application to a large dataset of wind measurements. As data from an ultrasonic anemometer were stored at a 1-Hz rate, calculation of the true angular stard deviation is possible this quantity can be compared with the various estimators. In a systematic way, an analysis was performed to see which group of linear variables leads to the best most robust estimators of. The sine cosine of horizontal wind direction (trigonometric moments) turned out to give the best estimators of. Estimators based on persistence, although feasible for many datasets, show larger errors are less robust under different meteorological conditions. For all types of parameterizations, the new fits based on the isotropic Gaussian turbulence model give the best root-mean-square error are among the most robust estimators. The sensitivity of the estimators to deviations from the model assumptions was tested. For the estimators based on persistence, the assumption of isotropic wind fluctuations is most important. If this assumption is violated, large differences between the estimators the true occur. Acknowledgments. The author thanks Dr. M. Furger G. Stefanicki for their help with the wind measurements. APPEDIX Fast Determination of Minimum Variance The minimum variance from (9) is defined by a variational principle together with the mean angle

12 1414 JOURAL OF APPLIED METEOROLOGY VOLUME 36 FIG. 7. Angular stard deviation vs persistence P for 10-min samples filtered according to the following conditions: (a) , (b) r 0.1, (c) skewness of u less than 0.33 kurtosis of u less than the cut point C. If all data from a sampling interval are stored, it becomes possible to find a sample minimum variance s by varying C. To make this search for the optimal C faster, the angles are discretized to 1. All observed wind directions of a sampling period are mapped to the interval (0.5, 360.5) by adding multiples of 360 are then rounded to the next integer, giving values of 1,,..., 360. The occurrence n j of each integer j, or bin, is counted. The cut point is first set to 0.5 then increased in steps of 1 until it reaches For each value of C, the sample mean the sample variance s are determined. This can be done by a fast iterative procedure. In the first step, C (1) (1) j j j (1) (1) j x n, (A1) S (x ) n j, (A) 1 j1 (1) (1) (1) s S ( ), (A3) 1 where j. (1) x j In the kth step, the angles, or values of the bins, are mapped to the interval [0.5 (k1), (k 1)]. The mean variance are updated from the former step by

13 (k) (k1) n k1, (A4) 360 (k) (k1) (k1) S S (360 x k1 )n k1, 1 (A5) (k) (k) (k) s S ( ). (A6) 1 (k) Among all 360 values of s the smallest one gives the minimum variance the corresponding mean angle. REFERECES Ackermann, G. R., 1983: Means stard deviations of horizontal wind components. J. Climate Appl. Meteor.,, Cirillo, M. C., A. A. Poli, 199: An intercomparison of semiempirical diffusion models under low wind speed, stable conditions. Atmos. Environ., 6A, Essenwanger, O. M., 1986: World Survey of Climatology. Vol. 1B, General Climatology 1 B: Elements of Statistical Analysis, Elsevier, 44 pp. Fisher,. I., 1983: Comment on A method for estimating the stard deviation of wind directions. J. Climate Appl. Meteor.,, 1971., 1987: Problems with the current definitions of the stard deviation of wind direction. J. Climate Appl. Meteor., 6, Gifford, F. A., 1976: Turbulent diffusion-typing schemes: A review. ucl. Saf., 17, Hanna, S. R., 1990: Lateral dispersion in light-wind stable conditions. uovo Cimento, 13, , G. A., Briggs, R. P. Hosker Jr., 198: Hbook on Atmospheric Diffusion. Department of Energy Tech. Rep. DOE/TIC-113, 10 pp. [Available from ational Technical Information Service, 585 Port Royal Road, Springfield, VA 161.] Hauser, R. K., C. D. Whiteman, K. J. Allwine, 1994: Short-tower measurements in the Colorado plateaus region. Preprints, Eighth Conf. on Applications of Air Pollution Meteorology, ashville, T, Amer. Meteor. Soc., Hicks, B. B., D. D. Baldocchi, T. P. Meyers, R. P. Hosker Jr., D. R. Matt, 1987: A preliminary multiple resistance routine for deriving dry deposition velocities from measured quantities. Water Air Soil Pollut., 36, Ibarra, J. I., 1995: A new approach for the determination of horizontal wind direction fluctuations. J. Appl. Meteor., 34, Irwin, J. S., 1983: Estimating plume dispersion A comparison of several sigma schemes. J. Climate Appl. Meteor.,, Jones, J. I. P., 1964: Continuous computation of the stard deviations of longitudinal lateral wind velocity components. Brit. J. Appl. Phys., 15, Leung, D. Y. C., C. H. Liu, 1996: Improved estimators for the stard deviations of horizontal wind fluctuations. Atmos. Environ., 30, Mardia, K. V., 197: Statistics of Directional Data. Academic Press, 357 pp. McWilliams, B., M. M. ewmann, D. Sprevak, 1979: The probability distribution of wind velocity direction. Wind Engin., 3, Mori, Y., 1986: Evaluation of several single-pass estimators of the mean the stard deviation of wind direction. J. Climate Appl. Meteor., 5, elson, E. W., 1984: A simple accurate method for calculation of the stard deviation of the horizontal wind direction. J. Air Pollut. Control Assoc., 34, Pasquill, F., F. B. Smith, 1983: Atmospheric Diffusion. 3d ed. Ellis Horwood, 437 pp. Sharan, M., A. K. Yadav, M. P. Singh, 1995: Comparison of sigma schemes for estimation of air pollutant dispersion in low winds. Atmos. Environ., 9, Stuart, A., J. K. Ord, 1987: Kendall s Advanced Theory of Statistics. Vol. 1, Distribution Theory, Charles Griffin Company, 604 pp. Turner, D. B., 1986: Comparison of three methods for calculating the stard deviation of the wind direction. J. Climate Appl. Meteor., 5, Verrall, K. A., R. L. Williams, 198: A method for estimating the stard deviation of wind directions. J. Appl. Meteor. 1, ,, 1983: Reply. J. Climate Appl. Meteor.,, 197. Weber, R., 1991: Estimator for the stard deviation of wind direction based on moments of the Cartesian components. J. Appl. Meteor., 30, , 199: A comparison of different estimators for the stard deviation of wind direction based on persistence. Atmos. Environ., 6A, Yadav, A. K., M. Sharan, 1996: Statistical evaluation of sigma schemes for estimating dispersion in low wind conditions. Atmos. Environ., 30, Yamartino, R. J., 1984: A comparison of several single-pass estimators of the stard deviation of wind direction. J. Climate Appl. Meteor., 3,