A new parameter scheme for radar contours of hail cells

Atmospheric Research, 25 ( 1 9 9 0 ) 4 8 5 - 4 9 8

485

Elsevier Science Publishers B.V., A m s t e r d a m

A new parameter scheme for radar contours of hail cells W. Schmid and A. Waldvogel Atmospheric Physics. ETH, CH-8093 Ziirich (Switzerland) (Received September 2, 1989; accepted after revision November 28, 1989)

ABSTRACT Schmid, W. and Waldvogel, A., 1990. A new parameter scheme for radar contours of hail cells. Atmos. Res., 25: 485-498. High-resolution 10 cm radar data from 209 severe convective cells are investigated. The areal distribution of radar reflectivity is fitted by a three-parameter power law. It is found that the exponent of the power law is on the order of two. This means, that a simplified a r e a - Z relationship exists, which contains only two radar parameters: a maximum reflectivity parameter and a reflectivity gradient parameter. The agreement between hailfall parameters derived directly from radar measurements and computed using the two radar parameters is excellent. We conclude that the two parameters represent important aspects of the structure of hail cells and suggest that consideration be given to use of this parameterization scheme in all types of hail cell models.

RESUME On a proc6d6 h l'6tude des donn6es d'un radar 10 cm/~ haute r6solution sur 209 cellules convectives puissantes. On repr6sente la distribution g6ographique de la r6flectivit6 radar h l'aide d'une loi en puissance ~ trois param6tres. On trouve que l'exposant de la loi en puissance est de l'ordre de deux. Cela veut dire qu'il existe une relation simplifi6e surface-Z qui contient seulement deux param6tres radar: un param6tre de r6flectivit6 maximale, et un param6tre de gradient de r6flectivit6. L'accord entre les param6tres des chutes de gr~le d6riv6s directement des mesures radar et ceux calcul6s ~i l'aide des deux param6tres est excellent. On conclut que les deux parambtres repr6sentent des aspects importants de la structure des cellules h gr~le, et on sugg6re d'utiliser ce sch6ma de param6trisation dans tousles types de mod61es de eellule h gr61e.

INTRODUCTION

Radar-derived precipitation parameters give a good idea about the extent and intensity of convective cells. Such parameters have been defined and discussed by many authors. Waldvogel and Federer ( 1976 ) introduced the socalled AZTI-quantity (the Area of reflectivity Z Time Integrated), which turned out to be very useful for discriminating between rain and hail cells. 0169-8095/90/$03.50

© 1 9 9 0 - - Elsevier Science Publishers B.V.

486

W. SCHMID AND A. WALDVOGEL

Similar parameters have been studied by Foote et al. (1979), Calheir.os and Zawadzki ( 1983 ) and Doneaud et al. (1984). Another way to define precipitation parameters is to transform the measured values of radar reflectivity into values of rain or hail intensity, such as the rainfall rate (Marshall and Palmer, 1948 ), the flux of hail mass, or the flux of hail kinetic energy (Waldvogel et al., 1978a; Ulbrich, 1978). In this paper our main interest will be restricted to hailfall parameters, but the analyses given below are applicable to rainfall parameters as well. Hailfall parameters have been used in two major research topics: the comparison of radar and ground network hail data (Waldvogel et al., 1978b; Waldvogel and Schmid, 1983 ) and the evaluation of hail suppression experiments (Foote et al., 1979; Federer et al., 1986 ). Precipitation parameters normally are computed from many individual measurements of radar reflectivity. This procedure is straightforward and gives accurate numbers for areal and temporal integrals. The disadvantage of this technique, however, is that one neglects the complex shapes of the radar reflectivity contours, which contain essential information about the structure of hail cells. The goal of the present study is to fit the areal distribution of radar reflectivity by an area-Z relationship, which Can be treated very easily in further computations of radar-derived precipitation parameters. The areaZ relationship "replaces" the many individual reflectivity measurements. This is an efficient way to model the structure of hail cells. The crux will be to find an area-Z relationship which contains as few parameters as possible, but still gives a reliable picture of the areal distribution of radar reflectivity. It will be shown in this paper that a very simple relationship exists, which contains only two radar parameters: a maximum reflectivity parameter and a reflectivity gradient parameter. The reliability of this scheme is studied with a large data set of 209 severe convective cells. The kinetic energy of hail is selected as a "test" quantity. A comparison between the classical method for computing this quantity and the new "two-parameter" technique yields excellent agreement. THEORY The power model

We suggest a power law for the approximation of the relationship between the area A (Z) of a given radar reflectivity contour and the reflectivity Z, which forms the boundary of the radar contour:

(Zm-Z)

(1)

The units ofA (Z) and of Z are given in km / and dBZ, respectively. The max-

RADAR CONTOURS OF HAIL CELLS

487

imum reflectivity factor in the core of a radar echo is represented by Zm, and b and g are additional parameters of the relationship. Several authors have found a quadratic relationship ( b = 2 ) between A(Z) and Zm-Z (e.g. Altman, 1970; Hudlow and Scherer, 1975; Konrad, 1978). Altman(1972) also proposed a linear relationship (b = 1 ), whereas Wiggert et al. ( 1976 ) fitted a bivariate normal frequency distribution to rain intensities derived from radar reflectivities. Rogers et al. ( 1983 ) investigated the relationship between area and radar reflectivity at two different altitudes. At 2 km msl the quadratic power law turned out to be reasonable, whereas at 7 km msl an exponential approximation appeared more realistic. We conclude from these studies, and from our own experience (Schmid et al., 1984), that the power law may be a useful approximation of the distribution of radar reflectivity within convective cells, and that the exponent b often is on the order of two. It is clear, however, that the parameters of eq. 1 may vary considerably under different environmental conditions. The reliability of eq. 1 has not yet been proved for a large sample of hail cells, but will be addressed in a subsequent section of this paper. Eq. 1 can also be written in the following form:

f(Z)=~ ( z , n - z )

b-'

(2)

w h e r e f ( Z ) represents the "ring" area (given in km2/dB ) of the reflectivities in the range of Z - - - Z + dZ. The two quantities A (Z) a n d f ( Z ) are related by: Zm

A(Z)= f f(Z') dZ'

(3)

Z

We prefer to use eq. 2 rather than eq. 1 in the following discussion, since the errors of observation o f f ( Z ) can be considered as statistically independent, which is apparently not the case for the corresponding errors ofA (Z). As an example, Fig. 1 shows the relationship between the radar measurements of Zm- Z and the measured area values f (Z). The data have been taken from the observations of a strong hail cell, which appeared on July 15, 1982 in central Switzerland. The measured maximum reflectivity Zm was set to the highest registered reflectivity level plus 1/2 dB, since the resolution of the radar is 1 dB. One can recognize a linear relationship between the log-values o f f ( Z ) and (Zm--Z). Applying a linear regression procedure to the log-values of the data yields estimates of the parameters b and g. The resulting regression lines also are shown in Fig. 1. They fit the data reasonably well. The correlation coefficient between the log-values o f f (Z) and Z m - Z is 0.987 in this example.

488

W. SCHMID AND A. WALDVOGEL

15.7.1982 16.21H

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Fig. 1. The measured area values f(Z) as a function of Zm-Z, in linear scale (left) and in logarithmic scale (right).

Application to hailfall parameters The purpose of this section is to derive theoretically hailfall parameters from the power law as introduced in the last section. For this purpose we define a "local" hailfall parameter (Q) in the following general form: a-lO (~z/l°)

Z>Zo

(~(Z) =

(4) 0

Z < Zo

The numerical values of the constants a, a and Zo depend on the hailfall parameter considered. Integrated hailfall parameters (Qzo and Qzo ) can be written as follows: Zm

(2Zo =

f Q_(Z)f(Z) dZ

(5)

ZO le

Qzo = J- Q_zodt

(6)

ts

The quantity (2Zo represents the integral over the area of a convective cell at a specific time, whereas Qzo is the time integral of (2Zo, and therefore may be denoted as a "global" quantity (Waldvogel et al., 1978b). The two time indications (& and te) may be arbitrary, but often are chosen to b o u n d the lifetime of a severe convective cell. Most of the hailfall parameters found in the literature can be written in the general form of eqs. 4-6. The AZTI-quantity, for example, corresponds to Qzo when the parameters of eq. 4 are set to a = 1 and ot = 0. Another example

489

RADAR CONTOURS OF HAIL CELLS

is the flux of hail kinetic energy (/~), which normally is given in units of J m -2 s-1. Waldvogel et al. (1978b) give the following numerical values of the parameters, a, a and Zo: a = 5 " 1 0 -6, a = 0 . 8 4 , Z o _ 5 6 or 61 dBZ. The area and area-time integrals of/~ may be denoted by Ezo and Ezo, respectively. The quantity E56 (global kinetic hail energy) served as the primary response variable of the Grossversuch IV hail suppression experiment (Federer et al., 1978/79). Inserting eqs. 2 and 4 into eq. 5 yields an expression for Qzo as a function of the model parameters Zm, b and g:

0 O-g2(ca b*)b e¢~.zm~.F(b, wo) wo=a*(Zm-Zo);

a*=a

(7)

ln(10) -a-0.23026 10

where F(b, wo) is the incomplete G a m m a function. Assuming b = 2 one obtains: 2a 0 z o - a(g* - - ~

e(c~*Zm)" (1 --

(1 +Wo)e -w°)

(8)

Eq. 7 defines an alternative way to calculate hailfall parameters from radar data. Instead of inserting the measured values o f f ( Z ) into eq. 5, one can insert estimated values of the parameters Zm, b and g into eq. 7. Assuming b = 2, one can also use the simplified eq. 8. A subsequent section will present a comparison of the two given methods to compute hailfall parameters. TEST O F THE T H E O R Y

Radar and data An examination of the theoretical framework given in the last two sections should include two important aspects. First, one must validate the power model of eq. 2; second, one must examine the agreement of the two given methods to estimate hailfall parameters. These two issues will be addressed in the next two sections. This section gives an overview of the radar data, which will be analyzed in the following. The data were obtained using an Sband radar, which has been operated in central Switzerland since 1975. Each PPI-scan (obtained at a standard elevation angle of 5.5 ° ) was stored on magnetic tape with a resolution of 1 ° in azimuth, 1/3 km in range, 1 dB in intensity and 1 min in time. The m a x i m u m recording distance is 60 km. Values of equivalent radar reflectivity (Ze) were averaged in logarithmic scale from 25 independent pulse measurements. The calibration procedure revealed an overall uncertainty of _+0.5 dB. In this paper the simplification:

490

Ze = Z

W. SCHMID AND A. WALDVOGEL

(9)

is used, where Z represents the radar reflectivity factor for Rayleigh scatterers. Further details concerning the specifications of the radar and the calibration procedure can be found in the article by Waldvogel and Schmid ( 1982 ). A total number of 510 hail cells were identified during the period of Grossversuch IV (Federer et al., 1978/79). The criteria for identification of hail cells are given by Waldvogel and Schmid ( 1982 ). The most important points are summarized as follows. A hail cell is defined as a succession of reflectivity contours of 45 dBZ, inside of which a series of RHI measurements indicate that the hail cell seeding criterion (Waldvogel et al., 1979) is met. Merger and split cases are treated as one cell only. The following analyses are restricted to a "homogeneous" data set of 209 cells which fulfil the following additional criteria. Each cell must have been scanned by the radar throughout its entire lifetime. Furthermore, only "simply definable" cells (Waldvogel and Schmid, 1982 ) are considered. A simply definable cell is a cell which contains one single branch of a developing and decaying 45 dBZ contour. "Merger-split" cells are eliminated. It was found that merger-split cells have a larger size and last longer than simply definable cells. Hail production, however, is essentially the same for both cell groups (Schmid, 1988 ). The 209 selected cells were observed during 1978-82. Some of these cells were observed outside of the experimental area of the Grossversuch IV experiment, differentiating this data set from the one used in the confirmatory analysis of Grossversuch IV. For each cell, area values f ( Z ) have been computed and stored in a celloriented data bank. The resolution of these data is 1 dB in intensity and 1 rain in time. The lowest registered reflectivity level is 46 dBZ. For the analysis given in the next section, we have selected those time periods during which the cells reached a maximum reflectivity of at least 60 dBZ. This proved to be the case during a total of 1329 min; thus, the data sample used in the next section consists of 1329 sets o f f ( Z ) - v a l u e s in the range between 46 dBZ and the maximum radar reflectivity.

Test of the power model Several methods exist for examining the validity of the power model. One possibility is to apply a linear regression procedure (as suggested in the example of Fig. 1 ) to each sample o f f (Z)-values individually, and to examine the goodness of fit with statistical methods. Another method is to average first the areas f ( Z ) over categories of ( Z m - Z ) , and then apply the same regression procedure. The advantage of this method is that the residuals of the regression equation are reduced nearly to zero, if the model of eq. 2 fits the

491

RADAR CONTOURS OF HAIL CELLS

data structure, i.e. the mean areas lie on a straight line in logarithmic scale. We prefer this second method to the first because of its comparative simplicity• Fig. 2 presents linear and logarithmic plots of the average area values. Most of the defined data points lie on a straight line in the log-log diagram. The exception is the point at the highest reflectivity level ( 1/2 dB below Zm ). We conclude that the power model is an excellent approximation of the mean area values in the range of Zm - 1 to Zm -- 15 dB. The outlying point, however, merits some additional comments, since similar p h e n o m e n a - at least to the knowledge of the authors - have not been reported up to now in the literature. The effect is attributed to instrumental rather than to natural reasons. This can be seen when the average reflectivity gradients within convective echoes (5 d B / k m , see e.g. Torlaschi and Humphries, 1983 ) are related to the size of the radar measuring elements (1/3 km to 1 ° in our case). The reflectivity field within a radar measuring element may vary by more than 1 dB, implying that the m a x i m u m reflectivity value within the core of a radar echo may be underestimated, since the radar only yields an average reflectivity value from each radar measuring element. As a consequence, the a r e a s f ( Z m - 1/2 ) may be overestimated, since the areas of the reflectivities larger than Zm are also included in the measurements o f f ( Z m - 1/2 ). The situation is more favourable in the regions farther away from the core of a radar echo, since there the resulting "element-averaged" reflectivity values are less biased than the corresponding reflectivity values within the core of a radar echo. Zawadzki ( 1982 ) theoretically computed the bias of the averaged reflectivity values due to the instrumental effects of "beam smoothing" and "post detection integration". The calculations have shown that this bias is less than 1 dB within a "constant-gradient" reflectivity field, as long as the reflectivity gradients are on the order of less than 10 dB within a radar measuring element. This condition is fulfilled in our case in most observational situations; thus, we can assume that the resulting areas also have small 3.5 3.0 2.5 "~

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492

W. SCHMID AND A. WALDVOGEL

biases. The extent of the bias, however, may depend on the distance between the cell and the radar site, since the size of the radar measuring elements depends on the distance from the radar. These considerations justify omitting the outlying data point when computing the regression equation from the data given in Fig. 2. Table I presents the resulting values of b and g, and of the corresponding correlation coefficients. Estimates of these parameters also are given for different categories of distance between cell and radar. One can see that the correlation coefficient is almost one, as long as the radar range is less than 40 km. The power model fits the data closely at these ranges, whereas farther away the validity of the power model is less evident. One reason for this may be that the previously mentioned effects of "beam smoothing" and "post detection integration" become more important when the distance between cell and radar exceeds 40 km. Another important result shown in Table I is that the exponent b varies within a narrow range around an average value of 2.1, implying that the approximation b = 2 presumably can be accepted for estimates of hailfall parameters. This hypothesis will be tested in the next section. The parameter g, on the other hand, varies over a larger range between 2 and 5.5. One reason for this is that parameter g is very sensitive to small variations in the value of the exponent b. The assumption b = 2 means that the square root ofA (Z) depends linearly on Zm-Z; therefore, the parameter g (g2 hereafter) can be interpreted as a "gradient" parameter, since the units of g2 are dB/km. Schmid et al. (1984) suggested a simple method for estimating g2; TABLE 1 The coefficients b and g of the power model; the number of the radar echoes considered is n, the correlation coefficient between the mean values of f and Z m - Z is r*' Distance cellradar (kin)

n

b

g

r

g2 (dB/km)

0-10 10-20 20-30 30-40 40-50 50-60

111 376 380 243 126 93

2.272 2.302 2.148 1.960 1.718 1.762

5.43 4.78 3.25 3.29 2.67 2.17

0.999 0.999 1.000 0.997 0.991 0.989

3.728 3.317 2.703 3.670 4.010 2.976

0-60

1329

2.094

3.53

1.000

3.221

*tThe data points of the level Z m - Z = 0.5 dB were omitted for calculation of the three parameters, for reasons given in the text. The gradient parameter g2, as estimated from eq. 10, is also given in the table.

RADARCONTOURSOFHAILCELLS

493

5 g2 - A ( Z I - 5 d B ) l / 2 - A ( Z 1 ) 1/2

(10)

The reflectivity Z~ bounds a radar echo, whose area is 1 km 2, i.e.: A ( Z , ) = I km 2

(11)

Eq. 10 indicates that the slope of the relationship ofA (Z) ~/2 versus Zm - Z is estimated from the values of A (Z) ~/2 at the reflectivity level ZI and Z I - 5 dB, respectively. This definition is similar to the one given by Konrad ( 1978 ), who used the levels Zm and Z m - 6 dB for the definition of the slope. We prefer to use Z~ rather than Zm in the definition of g2, since the latter parameter apparently suffers from smoothing effects as described above. Estimates of g2 are given in Table I. One can see that this parameter varies from 2.5 to 4 dB/km, with a mean value of 3.2 dB/km. This variation is considerably smaller than the variation in the parameter g. One can obtain an estimate of the mean reflectivity gradients by multiplying the gradient parameter by a factor ~z~/2, since the expression g2" ~ ~/2 is an exact estimate of the reflectivity gradients, when b is assumed to be two, and when the contours of constant reflectivity are circles whose centers coincid7 with the core of a radar echo. Average values of the reflectivity gradients computed from the information listed in Table I are on the order of 5-7 dB/km, which is in good agreement with the results obtained by Torlaschi and Humphries ( 1983 ). Summarizing this section, we state that the power model is an excellent approximation of the frequency distribution of the radar reflectivities within hail cells. In addition, the assumption b = 2 is found to be a useful simplification, at least for the conditions under which these radar measurements have been obtained.

Comparison of different estimates of hailfall parameters The purpose of this section is to compare the two methods for estimating hailfall parameters from the radar data. The flux of hail kinetic energy (E56) and the global hail kinetic energy (E56) of hail cells are used as test.quantities. The estimated values of these quantities will be referred to as E56(J) and E56 (f) ("classical" method ), and E46 (Zl,g2) and E56 ( Z 1,g2 ) ( " t w o - p a r a m eter" method), respectively. In the two-parameter method, the parameters g2 and Z~ are estimated using eqs. 9 and 10, and the maximum reflectivity (Zm) is estimated as follows: Z m =Z~ +g2 " A ( Z , ),/2

( 12 )

The estimated values of Zm and g2 then are inserted into eq. 8. Note that the parameter g2 replaces the parameter g in eq. 8. As an example, Fig. 3 presents the time evolution of the defined parameters

494

W. S C H M I D A N D A. W A L D V O G E L

July A

15,

1982

"

70 Zm

Cell

",

1452

:"

,~

r-"

60

~

0

m

i

i

i

i

i

i

i

i

i

50

i00

150

50

i00

150

4 2 0

2500 Sol'id':

"~2000

'E5'6 ('ZI" g'2)'j~A . . . . .

1500 ~oi000

500 '~

0

0

50 Elapsed

time

I00 (min)

150

Fig. 3. The time evolution of several characteristic radar parameters, for a severe hail cell which has been observed on July 15, 1982, in central Switzerland.

for a severe hail cell which occurred in central Switzerland in summer 1982. The cell produced a hailswath 50 km long and 8 km wide. One can see that the parameters Zm and Z, are highly correlated, suggesting that both parameters contain essentially the same information. We will refer to Z, in the following discussion. One can also see that Z~ and g2 pulsate in a similar manner throughout the lifetime of the hail cell. The peaks of g2, however, do not correspond to the peaks of Z~. The two parameters are poorly correlated (r= 0.10). Each parameter therefore explains an important feature of the distribution of the reflectivity values within a hail cell. This can also be seen when the time evolution of the flux of hail kinetic energy (/~s6) is investigated (see the lowest diagram of Fig. 3 ). The third peak of E56 (some 90 min after the development of the cell) is the strongest one, although the corresponding peak of Zt is less pronounced than the two preceding peaks. This is explained by a marked drop in the gradient parameter 80 min after the cell development. The area v a l u e s f ( Z ) therefore become larger at this time and contribute in a significant manner to the flux of energy values. This feature confirms the importance OfZl andg2for estimates of hailfall parameters, but also calls for a microphysical and dynamical interpretation. We refer to Schmid ( 1988 )

RADARCONTOURSOF HAILCELLS

495

for this topic, since the present contribution is restricted to statistical aspects of radar measured hail cells. The two methods for computing E56 yield excellent agreement in the case of the cell discussed above (see the lowest diagram of Fig. 3 ). The correlation coefficient between the two time series is 0.99 for this example. More quantitative information about the agreement between the two different estimates of hail energy can be obtained from Figs. 4 and 5, which depict scatter diagrams o f E 5 6 ( f ) v s . / ~ 5 6 ( Z l , g 2 ) , a n d o f E s 6 ( f ) v s . E 5 6 ( Z I , g 2 ) . The data were obtained from the 209 severe convective cells, as described above; however, only the non-zero values are plotted in these diagrams. The agreement is best for the large energy values. The random errors (given in percent of the energy values) to be taken into account when using the (Z2,g2)-estimates are on the order of 10-20% of the "best possible" (f)-estimates, as long as the energy flux and the energies are larger than 30 mJ/min and 100 MJ, respectively. The random errors increase for energies smaller than the given thresholds. 10000 < ~ 1000 ~n m

~o

2

iOO F

4

t7

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~ N

.5

.M .i

1 E56(f)

I00 {IC 6 d / m i n l

13000

!CO (106 J / m i n )

"

E56(f)

10000

Fig. 4. The scatter diagrams of two different estimates of the flux of hail kinetic energy. Only non-zero values are plotted. 3185 radar echoes have been evaluated.

i00000

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496

W. SCHMID AND A. WALDVOGEL

This should be expected, since low energies are computed only from few and smallf(Z)-values. Note the striking similarity of Fig. 5 with Fig. 6 given in Waldvogel and Schmid ( 1982 ). As a whole, the correlation coefficients between the two given estimates of /~56 and E56 are on the order of 0.99 and do not change when the log-values of kinetic energy are considered in place of the untransformed energy values. Nearly the complete natural variability of hail kinetic energy (as measured by radar) can be explained from the variations of only two parameters: a maximum reflectivity parameter (e.g. ZI ) and a gradient parameter. We suggest that this finding is true for the entire ensemble of radar-derived hailfall parameters. CONCLUSIONS

The area-Z relationship of severe convective cells has been investigated. It was found that the relationship follows a quadratic power law. We conclude that hailfall parameters of severe convective cells can be computed rather simply and with high accuracy utilizing only two parameters: a maximum reflectivity parameter and a reflectivity gradient parameter. This result simplifies many investigations, for which the behaviour of hailfall parameters is of interest. We cite two examples below. ( 1 ) Hailpad-radar comparisons. Waldvogel et al. (1978b) and Waldvogel and Schmid (1983) found excellent agreement between hailpad- and radarderived hailfall parameters of hail cells. This agreement was observed under conditions which also are valid for application of the technique discussed in this paper: the radar should be a carefully calibrated 10 cm radar, the distance between hail cell and radar should be less than 40 km, and the radar should scan the hail cells below the level of zero degrees temperature. Thus, one can combine these earlier findings with the results given in the present study. Doing this, the temporal behaviour of hailfall parameters can be assessed rather easily from the temporal behaviour of the two defined radar parameters. This has many applications in hail climatology and in numerical and statistical models of hail cells. For instance, if a model can be used to predict the time development of ZI and g2, one can predict the time development of hailfall parameters as well. A detailed knowledge of the temporal behaviour of the two radar parameters, however, is absolutely necessary for the development and the verification of any kind of prediction model. The next step, therefore, will be to analyze time series of the two radar parameters. Such a study is currently being conducted. (2) Evaluation of weather modification experiments. Exploratory analyses of the radar data of seeding experiments have become very important in recent years (e.g. Foote et al., 1979; Federer et al., 1986). It is common to analyze a large number ofhailfall parameters, with the hope of finding significant

RADAR CONTOURS OF HAIL CELLS

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seeding sigflatures for at least a few of the many parameters. The findings of this paper suggest that the number of such analyses can be reduced drastically when severe convective cells are treated by the seeding operations. In our case, for example, it is sufficient to analyse the parameters of the area-Z relationship within severe convective cells. Possible seeding effects on the radar data should be recognizable by differences in the behaviour of these parameters, as long as the a r e a - Z relationship itself is not modified through the seeding operations. But even if this is the case, one can still apply the following procedure. First, investigate the reliability of the area-Z relationship independently by means of the seeded and non-seeded radar data. Second, select, if necessary, an alternative area-Z relationship, which is equally reliable for the complete data set. Last, repeat the statistical tests on seeding effects with the new set of parameters for the alternative area-Z relationship. Finally, we would like to state that the results given in this paper are valid for severe convective cells in central Switzerland, which are scanned at low levels by a carefully calibrated 10 cm radar. One should be cautious when transferring our findings to other radar devices, to other types of radar echoes and to other climatic regimes. We predict, however, that the quadratic relationship between the area and area-bounding reflectivity (i.e. b = 2 in eq. 1 ) will prove to be valid in many climatic regimes around the world. The validity of the quadratic relationship has received support so far from studies by Konrad ( 1978 ) and Rogers et al. ( 1983 ), who analyzed radar data from the East coast and the Montreal area, respectively. We therefore strongly recommend consideration of this relationship in future investigations of severe convective cells.

REFERENCES Altman, F.J., 1970. Storm reflectivity models. Preprints, 14th Conf. Radar Meteorology, Tucson, Ariz. Am. Meteorol. Soc., pp. 291-296. Airman, F.J., 1972. Storm cell models from digital radar data. Preprints, 15th Conf. Radar Meteorology, Champaign-Urbana. Am. Meteorol. Soc., pp. 41-44. Calheiros, R.V. and Zawadzki, I., 1983. Reflectivity rain rate relationship for radar hydrology in Brazil. Preprints, 21st Conf. Radar Meteorology, Edmonton. Alta. Am. Meteorol. Soc., pp. 687-692. Doneaud, A.A., Ionescu-Niscov, S. and Priegnitz, D.L., 1984. The area-time integral as an indicator for convective rain volumes. J. Climate Appl. Meteorol., 23:555-561. Federer, B., Waldvogel, A., Schmid, W., Hampel, F., Rosini, E., Vento, D., Admirat, P. and Mezeix, J.F., 1978/79. Grossversuch IV: Design of a randomized hail suppression experiment using the Soviet method. Pure Appl. Geophys., 117: 548-571. Federer, B., Waldvogel, A., Schmid, W., Schiesser, H.H., Hampel, F., Schweingruber, M., Stahel, W., Bader, J., Mezeix, J.F., Doras, B., d'Aubigny, G., DerMegreditchian, G. and Vento, D., 1986. Main results of Grossversuch IV. J. Climate Appl. Meteorol., 25: 917-957. Foote, G.B.. Rinehart, R.E. and Crow, E.L., 1979. Results of a randomized hail suppression

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experiment in Northeast Colorado, Part IV. Analysis of radar data for seeding effect and correlation with hailfali. J. Appl. Meteorol., 18:1569-1582. Hudlow, M.W. and Scherer, W.D., 1975. Precipitation Analysis for BOMEX Period III. NOAA Tech. Rept. EDS 13, 40 pp. Konrad, T.G., 1978. Statistical models of summer rainshowers derived from fine-scale radar observations. J. Appl. Meteorol., 17:171-188. Marshall, J.S. and Palmer, W.M., 1948. The distribution of raindrops with size. J. Meteorol., 5: 165-166. Rogers, R.R., Keen, K.J. and Schwartz, A.P., 1983. Statistics on the sizes and internal structures of rainshowers. Pure Appl. Geophys., 121: 133-166. Schmid, W., 1988. Hagelvorhersage mit Radar: Wolkenphysikalische Untersuchungen und ein statistisches Vorhersagemodell. Ph.D. Thesis Nr. 8684, ETH, Ziirich, 113 pp. Schmid, W., Waldvogel, A. and Beaud, P., 1984. Radar reflectivity gradients in hail echoes. Preprints, 22nd Conf. Radar Meteorology, Ziirich. Am. Meteorol. Soc., pp. 123-128. Torlaschi, E. and Humphries, R.G., 1983. Statistics of reflectivity gradients. Preprints, 21st Conf. Radar Meteorology, Edmonton, Alta. Am. Meteorol. Soc., pp. 173-175. Ulbrich, C.W., 1978. Relationships of equivalent reflectivity factor to the vertical fluxes of mass and kinetic energy of hail. J. Appl. Meteorol., 17:1803-1808. Waldvogel, A. and Federer, B., 1976. Large raindrops and the boundary between rain and hail. Preprints, 17th Conf. Radar Meteorology, Seattle, Wash. Am. Meteorol. Soc., pp. 167-172. Waldvogel, A. and Schmid, W., 1982. The kinetic energy of hailfalls, Part III. Sampling errors inferred from radar data. J. Appl. Meteorol., 21: 1228-1238. Waldvogel, A. and Schmid, W., 1983. Single wavelength radar measurements of hailfall kinetic energy. Preprints, 21 st Conf. Radar Meteorology, Edmonton, Alta. Am. Meteorol. Soc., pp. 425-428. Waldvogel, A., Schmid, W. and Federer, B., 1978a. The kinetic energy of hailfalls, Part I. Hailstone spectra. J. Appl. Meteorol., 17:515-520. Waldvogel, A., Federer, B., Schmid, W. and Mezeix, J.F., 1978b. The kinetic energy of hailfalis, Part II. Radar and hailpads. J. Appl. Meteorol., 17:1680-1693. Waldvogel, A., Federer, B. and Grimm, P., 1979. Criteria for the detection of hail cells. J. Appl. Meteorol., 18:1521 -1525. Wiggert, V., Ostlund, S.S., Lockett, G.J. and Stewart, J.V., 1976. Computer Software for the Assessment of Growth Histories of Weather Radar Echoes. NOAA Tech. Memo. ERL WMPO35, Boulder, Colo., 86 pp. Zawadzki, I., 1982. The quantitative interpretation of weather radar measurements. Atmos. Ocean, 20: 158-180.