Modeling the Variability of Drop Size Distributions in Space and Time

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Modeling the Variability of Drop Size Distributions in Space and Time GYUWON LEE* J. S. Marshall Radar Observatory, Department of Atmospheric and Oceanic Sciences, McGill University, Montreal, Quebec, Canada

ALAN W. SEED Australian Government Bureau of Meteorology Research Centre, Melbourne, Victoria, Australia

ISZTAR ZAWADZKI J. S. Marshall Radar Observatory, Department of Atmospheric and Oceanic Sciences, McGill University, Montreal, Quebec, Canada (Manuscript received 12 April 2006, in final form 17 October 2006) ABSTRACT The information on the time variability of drop size distributions (DSDs) as seen by a disdrometer is used to illustrate the structure of uncertainty in radar estimates of precipitation. Based on this, a method to generate the space–time variability of the distributions of the size of raindrops is developed. The model generates one moment of DSDs that is conditioned on another moment of DSDs; in particular, radar reflectivity Z is used to obtain rainfall rate R. Based on the fact that two moments of the DSDs are sufficient to capture most of the DSD variability, the model can be used to calculate DSDs and other moments of interest of the DSD. A deterministic component of the precipitation field is obtained from a fixed R–Z relationship. Two different components of DSD variability are added to the deterministic precipitation field. The first represents the systematic departures from the fixed R–Z relationship that are expected from different regimes of precipitation. This is generated using a simple broken-line model. The second represents the fluctuations around the R–Z relationship for a particular regime and uses a space–time multiplicative cascade model. The temporal structure of the stochastic fluctuations is investigated using disdrometer data. Assuming Taylor hypothesis, the spatial structure of the fluctuations is obtained and a stochastic model of the spatial distribution of the DSD variability is constructed. The consistency of the model is validated using concurrent radar and disdrometer data.

1. Introduction The transformation of remote sensing measurements of clouds and precipitation into quantities of interest requires the knowledge of drop size distribution (DSD). As a simplification, we often assume some deterministic relationship between a measurable quantity, such as radar reflectivity factor Z, and a quantity of interest, such as rain rate R. We assume that the parameters of this relationship can be stratified into a

* Current affiliation: Research Applications Laboratory, National Center for Atmospheric Research, Boulder, Colorado.

Corresponding author address: Dr. GyuWon Lee, National Center for Atmospheric Research, Research Applications Laboratory, P.O. Box 3000, Boulder, CO 80307-3000. E-mail: [email protected] DOI: 10.1175/JAM2505.1 © 2007 American Meteorological Society

JAM2505

finite number of categories related to different precipitation systems or clouds. Alternatively, we strive to increase the number of measured parameters to reduce the uncertainty in the transformation from the measurable parameter to the quantity of interest. Experience shows that the relationship between any two moments of the drop size distribution, for example, Z and R, can be approximated using a power law. However, the fluctuations around the line of least squares regression in log–log coordinates are significant, and therefore it is necessary to understand the nature of these fluctuations and their influence on practical issues, such as precipitation estimates by radar. For example, it would be interesting to understand the result of Wilson and Brandes (1979) who showed that radarto-gauge ratios had great spatial variability even if averages of radar and of gauges were taken over entire storms or days. Let us pose our questions by starting with an ex-

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FIG. 1. (a) The scattergram corresponds to 1-min data taken on 1 Jun 2004. The amplitude of the fluctuations in rain rate around the best-fit line is denoted by ␦R. (b) The same data are shown for successive hours after SIFT is applied. (c) The correlation function and the structure function of the raw data.

ample. Figure 1a shows an R–Z scattergram for 1 June 2004 obtained from disdrometer data operated by the J. S. Marshall Radar Observatory, Montreal, Quebec, Canada. Each point corresponds to the raw 1-min data. All data used in this work are described in section 2. The color in Fig. 1a represents the time line and it demonstrates that the scatter has temporal structure. In Fig. 1b the same data are plotted after some time filtering using the sequential intensity filtering technique (SIFT) described in Lee and Zawadzki (2005) (briefly, all DSDs within 1-h period are ordered by increasing Z and the average of 10 consecutive DSDs leads to one point in the scattergram). It is apparent that one could separate a number of R–Z relationships, one for each hour within the day with the exponent and coefficient smoothly changing with time. Figure 1c shows the time correlation function of the fluctuations, as well as the second-order structure function of the scatterplot in Fig. 1a. The fluctuations

␦R around the mean regression have a decorrelation time of 1 h. In fact, the decorrelation time of rain rate itself (not shown) is also 1 h! That is, the fluctuations around a mean R–Z relationship have a well-defined temporal structure comparable to the structure of rain itself. The correlation structure of rain rate is well known; see, for example, in Zawadzki (1973). Here we are showing that a similar structure exists in the residuals of relationships between moments. This has important consequences for our understanding and interpretation of remote sensing measurements. Although our data are taken at a point and we show only the existence of temporal structure of fluctuations, we invoke Taylor hypothesis and assume that the same must be true along a line in space. Zawadzki (1973) established the validity of Taylor hypothesis for one moment of the drop size distribution. Here, we assume its validity for other moments and hence for the residuals of a best-fit relationship between moments.

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To analyze properly the time–space structure of these fluctuations, we would need to have a dense and extended network of disdrometers. Or alternatively, we would need remote sensing measurements, in time and space, of a number of DSD parameters sufficient to derive this structure in time and space. Here, we will explore another approach: we will develop a method for the simulation of these fluctuations under the condition that the simulated structure must have all the known statistical properties of the observations. For this we must first decide how many parameters of the DSDs are necessary to describe the variability. It was shown in Lee et al. (2004) that two moments of the DSD capture most of the variability. In section 3 we will consider this question again and make the link between the DSD variability, the concept of scaling normalization, and the time structure of ␦R introduced above. We must also decide whether to treat all fluctuations as a continuum or divide them into categories (associated with different microphysical processes) with a continuum within each category. In reference to Fig. 1b, the question is whether to treat each hour separately, group hours having more similar R–Z relationships, or just consider the change from hour to hour as part of the continuously changing structure. In this particular example there is no compelling reason for not following the latter, simpler, approach. However, because there are situations in which distinct processes may lead to very distinct DSDs, we will include the option of modeling localized structures. Following Smith and Krajewski (1993) and Jordan et al. (2003) the fluctuations in rainfall estimations because of changes in the R–Z relationship are assumed to be multiplicative perturbations. Because of the power law relating rainfall and reflectivity, this is equivalent to assuming that the exponent in the R–Z relationship is fixed and we can model the fluctuations as an adjustment factor: ៣共x兲, R共x兲 ⫽ ␦R共x兲R

共1兲 ៣ b where R(x) ⫽ aZ(x) , for a and b we can take the climatological R–Z parameters (in this paper the climatological Z ⫽ 210R1.47 in Montreal is used), and x is some point in space and in time. Thus, we assume that there is no significant correlation between the fluctuations and Z. It is more convenient to work with the log transformations in decibel units dBR ⫽ 10 log10(R) for nonzero rain rates:

៣ ⬎ 0.1 mm h⫺1 where R, R ⫽0

otherwise.

The parameters required for the modeling of variability of DSDs, as well as the assumptions used here, are discussed in section 4. Section 5 describes the model and the results applied to a case study are described in section 6.

2. Data The J. S. Marshall Radar Observatory of McGill University operates an S-band weather radar that observes radar reflectivity at 5-min intervals, with 1-km resolution over 24 elevation angles. These volume scan data were used to estimate the radar reflectivity at a constant height of 2 km above sea level, the lowest elevation that is relatively free of ground clutter. These data were downscaled to a 1-min and 250-m resolution by subsampling the spatial field and using bilinear interpolation in Lagrangian coordinates to interpolate to the required temporal resolution. This radar is calibrated with disdrometers and polarimetry (Lee and Zawadzki 2006). The observatory also operates a precipitation occurrence sensor system (POSS), which is continuous wave bistatic X-band Doppler radar that can be used to retrieve the DSD for a volume in space near ground (Sheppard 1990; Sheppard and Joe 1994). Every minute the POSS gives an average Doppler spectrum (from about 480 measured Doppler spectra) that is converted into a DSD after taking into account the beam pattern. The POSS has a large sampling volume [V(D) ⫽ 0.3– 190 m3 s⫺1 depending on diameter] and is therefore able to provide estimates of the DSD at a high temporal resolution with minimal undersampling (Lee and Zawadzki 2005). The POSS has been compared with other types of disdrometers and rain gauges (Sheppard and Joe 1994; Campos and Zawadzki 2000; Miriovsky et al. 2004; Lee 2006). The comparison showed that the probability of bigger drops was estimated well as a result of the large sampling volume (Figs. 6–9 of Sheppard and Joe 1994; Fig. A3 of Lee 2006). From measured 1-min DSDs, the rainfall intensity R (mm h⫺1), radar reflectivity Z (mm6 m⫺3 or dBZ in 10 log10Z ) according to the Rayleigh approximation, and nthmoment Mn (m⫺3 mmn) are calculated by the following: R ⫽ 6␲ ⫻ 10⫺4 Z ⫽ M6 ⫽

៣共x 兲 ␦R共x兲共dB兲 ⫽ dBR共x兲 ⫺ dBR

共2兲

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Mn ⫽

冕

冕

冕

Dmax

␷ 共D兲D3N共D兲 dD,

Dmin

Dmax

D6N共D兲 dD,

and

Dmin

Dmax

Dmin

DnN共D兲 dD,

共3兲

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FIG. 2. Time series of (a) R, (b) Z, and (c) ␦R from POSS measurements for the case of 24–25 Sep 2001; ␦R is ៣ ៣ defined as ␦R (dB) ⫽ 10 log10(R) ⫺ 10 log10(R), where R is calculated from Z with a climatological R–Z relationship 1.47 Z ⫽ 210R .

where ␷ (D) (m s⫺1) is the terminal fall velocity according to Gunn and Kinzer (1949) and N(D) (m⫺3 mm⫺1) is the number of drops in a unit volume and given diameter interval dD (mm) centered at D (mm). A storm that occurred on 24–25 September 2001 was selected as a test case because of the long duration of the rainfall (over 24 h) and the range of R–Z relationships that were identified during the period. A low pressure system that was associated with a strong upper-level cutoff low passed through the northwest of the radar site with moderate convective precipitation lines and cells and with following widespread stratiform precipitation. It intensified around southern Quebec and western Ontario. POSS measurements (15-km range and 72° azimuth from the radar site) show a total rainfall accumulation of 28 mm and a few peaks in Z and R associated with convective cells (Figs. 2a,b). The derived R–Z (Z ⫽ 206R1.55) for the whole period is very close to the climatological R–Z relationship (Z ⫽

៣ 210R1.47). However, the deviation ␦R between R and R 1/1.47 ] shows very interesting features (Fig. [⫽(Z/210) 2c). Consider the period from 0000 to 0500 UTC 25 September 2005: the R–Z relationship is very similar to the climatological one, and the values of ␦R are systematically biased high for the next 3 h and biased low for the last 4 h. The coherence of these deviations can be calculated in terms of the qth generalized structure function ␥(␶) for a given time lag ␶:

␥共␶兲 ⫽ 关␦R共t兲 ⫺ ␦R共t ⫹ ␶兲兴 q.

共4兲

The calculated second-order ␥(␶) and autocorrelation ␳(␶) as a function of time lag ␶ are shown in Fig. 3. The decorrelation time is over 1 h and structure function does not reach a sill. As pointed out before, ␦R does not fluctuate as a white noise process but has coherence over 1 h, and this coherence will be considered in the modeling of DSD variability in time and space.

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DSD, the question that should be answered is whether this general DSD can describe the most discernible variability of DSDs. When the ith and jth moments (Mi and Mj) of DSD are used as parameters of interest, the general doublemoment normalized DSDs can be written in the following form (Lee et al. 2004): N共D兲 ⫽ M i共j⫹1兲Ⲑ共j⫺i兲M 共ji⫹1兲Ⲑ共i⫺j兲h共x2兲

共5兲

with x2 ⫽ DM i1Ⲑ共j⫺i兲M j⫺1Ⲑ共j⫺i兲, FIG. 3. The second-order structure function ␥(␶) and autocorrelation ␳ (␶) of ␦R (dB) as a function of time lag ␶. The coherence of structure is over 1 h.

where h(x 2 ) is a generic shape of DSDs. With the definition of the characteristic diameter D⬘m [⫽(Mj /Mi)1/( j⫺i)] and the characteristic number density ], we can rewrite (5) as N⬘0 [⫽M(i j⫹1)/(j⫺i)M(i⫹1)/(i⫺j) j N共D兲 ⫽ N⬘0h共DⲐD⬘m兲.

3. Description of the DSD variability a. Scaling of DSDs The variability of DSDs in both space and time as well as over a range of meteorological situations is a major consideration when using remote sensing techniques to estimate rainfall intensity. The early work by Marshall and Palmer (1948) used the exponential model with varying slopes as a function of rainfall intensity R. It is a fundamental description of the variability of DSDs with a single parameter. That is, the variability of DSDs is solely explained by the change of R. Since then, various methods have been applied by using a variable intercept parameter of an exponential model or different DSD models (gamma, lognormal, generalized gamma models). Recently, the DSD variability has been described by the general doublemoment normalization of DSDs (Lee et al. 2004) that is based on the concept of a scaling law. This method is a general formulation of previously suggested DSD normalization with a single or double moment (Sekhon and Srivastava 1971; Sempere-Torres et al. 1994, 1998; Testud et al. 2001; Illingworth and Blackman 2002; Uijlenhoet et al. 2003a,b) and is based on the widely used power law between moments. An advantage of this general normalization is that a functional form (model) of DSDs is not assumed and that it is a generalization of DSD models that have been proposed previously (Lee et al. 2004). In addition, this scaling form of the DSD is particularly convenient and meaningful when describing microphysical processes, such as aggregation, riming, and equilibrium processes (Field and Heymsfield 2003; Lee 2003; Zawadzki and Lee 2004). Since our ultimate goal is to model the DSDs in space and time using the general double-moment normalized

共6兲

No particular functional form of h(x2) is assumed. For the exponential DSD model [N(D) ⫽ N0e⫺␭D ⫽ M0␭e⫺␭D, where N0 is not a constant], (6) can be rewritten as (Lee et al. 2004) N共D兲 ⫽ N⬘0关⌫共1 ⫹ i兲兴

冓再

⫻ exp ⫺

共j⫹1兲Ⲑ共i⫺j兲

共⫺i⫺1兲Ⲑ共i⫺j兲

关⌫共1 ⫹ j兲兴

关⌫共1 ⫹ i兲兴 关⌫共1 ⫹ j兲兴

冎

1Ⲑ共i⫺j兲

冔

D . D⬘m

共7兲

The single-moment normalized DSD can be derived using (5): N共D兲 ⫽ Mi1⫺共i⫹1兲␤g共x1兲

共8兲

with x1 ⫽ DM i–␤. Again, g(x1) is a generic shape of DSDs with the single-moment normalization. The normalized DSDs h(x2) should have a minimal scatter with respect to an average or generic h(x2) if the variability of the DSD can be modeled using two moments and therefore we can construct DSDs in space and time with two moments and a generic h(x2).

b. Scaling analysis We have applied a scaling normalization of DSDs to evaluate how much of the DSD variability can be explained by one or two moments of DSDs. Figures 4 and 5 show observed DSDs and scaling normalized DSDs with various moments of DSDs for the whole period of the case 24–25 September 2001. As indicated by the large fluctuation, observed DSDs show large dependence on rainfall intensity. The dynamic ranges of the number density N(D) for a given diameter vary from

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FIG. 4. (a) Observed DSDs for the case of 24–25 Sep 2001. (b) Single-moment normalized DSDs with the reference moment Mi ⫽ R (i ≅ 3.67). The gray solid line indicates an average normalized DSD [g(x1)], and ␤ is the scaling exponent. (c) An average normalized DSD [g(x1); gray solid] with its std dev (vertical bars) and exponential adjustment (dashed line). The std dev is calculated at the interval of ⌬x1 ⫽ 0.1 with respect to g(x1). (d), (e) Same as in (b) and (c) but for Mi ⫽ Z (i ⫽ 6).

101 to 105. When the dependence of DSDs on different rainfall intensity or reflectivity is taken into account by the single-moment normalization with R or Z (Figs. 4b,d), the scatter reduces dramatically, but as indicated by the large standard deviation with respect to the average normalized DSD g(x1), significant fluctuation still remains. This is an indication that the overall DSD variability cannot be adequately explained by a single parameter and it reflects the large scatter around the average power-law relationship between any two moments of the distribution. This is related to the various microphysical processes that are responsible for the DSD variability. The structure of ␦R is the direct consequence of the systematic and coherent variability of these microphysical processes. The double-moment normalization of observed DSDs with various combinations of two moments (Mi and Mj) is shown in Fig. 5. The dramatic reduction of the scatter is noticeable. The standard deviation is small (Fig. 5b) but still bigger than the Poisson undersam-

pling uncertainty (not shown here). Nevertheless, a good deal of DSD variability is explained and a welldefined generic h(x2) exists. The h(x2) become slightly steeper when two higher moments (Fig. 5d) are used and vice versa for the two lower moments (Fig. 5e). When two higher moments are used, h(x2) at larger x2 is well defined. When two moments are far apart, the range of small scatters extends to smaller and larger x2. The average h(x2) is close to the normalized exponential DSD in (7). In general, any combinations of two moments provide a comparable scatter around a welldefined h(x2). Thus, to a good first approximation the spatial and temporal distribution of raindrop sizes can be calculated from the spatial and temporal distribution of any two moments. High-resolution fields of radar reflectivity are available from operational radar measurements, so this is an obvious choice for one of the moments that can be used. Thus, we need a stochastic model to generate a second moment, rain rate, that is conditioned on

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FIG. 5. Double-moment normalized DSDs with various combinations of Mi and Mj. Average normalized DSDs h(x2) and their std dev are shown by the gray lines and vertical bars. The std dev is calculated at the interval of ⌬x2 ⫽ 0.2 with respect to h(x2). The exponential h(x2) in (7) is indicated by the dashed line.

the observed Z fields and preserves the observed structure of ␦R. The choice of the second moment is due to practical reason since rain rate can be measured and is needed for hydrological applications. In addition, Z represents bigger drop sizes while R better explains smaller and median drop sizes.

4. Model description Differences between rainfall measured on the ground by means of a rain gauge and rain estimated aloft by means of weather radar include the uncertainty in the R–Z conversion, as well as the sampling errors due to the different spatial and temporal sample volumes and locations of the two observations. The objective in this paper is to build a statistical model of ␦R(x), the departures in space and time from an average R–Z relationship. At this point we will not consider other significant sources of uncertainty. The structure of ␦R(x) due to the uncertainty in R–Z relationships is evaluated by esti-

mating both Z and R from DSDs observed with a disdrometer at the surface and then converting the Z into rainfall using a fixed R–Z relationship, thereby restricting the analysis to errors that arise from fluctuations in the R–Z relationship only. The simulation does not include the discrepancy in Z from radar and disdrometer, which is caused by differences in the height of the measurements and in sample volume. Including these differences could result in excessively large variability in the R fields. The only assumption that we made is that after averaging disdrometric data for 5 min the DSD variability in disdrometric measuring volumes is equivalent to that of radar measuring volumes. The model must be constructed in such a way that makes it possible to generate stochastic realizations of a rain field that are conditioned on radar reflectivity observations if it is to be used with fields of reflectivity that are observed with a radar. It must also describe the fluctuations in space and time in a way that is consistent with the observations, which are limited at present to a

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FIG. 6. Spectral density for the time series of ␦R(t) from POSS measurements for 24 Sep 2001 with a fitted line (solid line), equation, and determination coefficient (␳2).

time series at a point. Therefore, assumptions will have to be made regarding the advection of the ␦R(x) field through the area of interest, the covariance between ␦R(x) and Z(x), and the structure of ␦R(x) in space and time. The covariance between ␦R(x) and Z(x) is assumed to be zero in the interest of simplification [the correlation between ␦R(x) and Z(x) is 0.19 for the case shown in Fig. 2]. It seems reasonable to assume that ␦R(x) and Z(x) are advected together since the physical processes that are driving the perturbations are likely to be embedded in the air mass, which is moving at a rate that is similar to that observed in the Z(x) field. Of course this is a simplification and is unlikely to be correct in cases where the apparent motion of Z(x) is due to development and not advection and in cases where the occurrence of physical rainfall processes are constrained in their horizontal location by topographic influences. The perturbation field is advected using the velocity field of the observed reflectivity data, which is calculated using the optical flow technique of Bowler et al. (2004). The statistical nature of the time series of the perturbations at a point [␦R(t) in Fig. 2] provides some clues regarding the selection of an appropriate model for ␦R(x). If ␦R(t) shows strong evidence of a scaling behavior, then a scaling model for ␦R(x) will be appropriate. POSS data for 24–25 September 2001 were used to estimate Z and R at 1-min time steps over a 24-h period. ៣ Thereafter ␦R(t) was calculated using R ⫽ (Z/210)1/1.47, the climatological R–Z relationship for Montreal. The power spectrum of the time series of the residuals ␦R(t) was calculated using the first 1024 data points in the time series; the resulting power spectrum was averaged in octaves as shown in Fig. 6. The spectrum was found to follow a power-law P ⬀ f ⫺1.04, which is consistent with a scaling model (Menabde et al. 1997).

FIG. 7. Structure functions for q ⫽ 1 and q ⫽ 2, with the power-law model fit (solid line) also shown.

The generalized structure function ␥ ( ␶ ) ⫽ [␦R(t) ⫺ ␦R(t ⫹ ␶)]q was calculated and is shown in Fig. 7. The power-law model

␥ 共␶兲 ⫽ a0␶ ␨ 共q兲

共9兲

is compatible with the scaling hypothesis, at least close to the origin. The generalized structure functions for moments 0.4 ⬍ q ⬍ 2.8 were calculated and the spectra of exponents ␨(q) were estimated as shown in Fig. 8. A straight line fit to ␨(q) explains more than 95% of the variance, so a simple self-affine model with a constant scaling exponent can be used to model the time series, at least near the origin and over a limited range of moments.

5. Modeling the R–Z fluctuations in space and time The scaling nature of ␦R(t) is very similar to that displayed by a time series of rainfall at a point, so the space–time model developed to model rainfall by Seed et al. (1999) was used to generate ␦R(x) after the radar measurement model proposed by Jordan et al. (2003). The space–time model has two components: a model to generate a time series of mean field bias over the domain of the radar reflectivity observations and a space– time multiplicative cascade to model the small-scale fluctuations about this mean.

a. Model of mean field bias A simple broken-line model results from the linear interpolation between equally spaced independent random variables (Mejia et al. 1972). A diagram of a simple broken line is shown in Fig. 9. Seed et al. (2000) developed a multiplicative broken-line model by multiplying N simple broken lines together:

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FIG. 8. Plot of the exponent of the power-law fit to the structure function ␨(q) vs the moment of the structure function q. The fitted line is shown as the solid line with a corresponding equation and determination coefficient (␳2).

FIG. 9. Schematic representation of a simple broken-line process (after Fig. 1 of Mejia et al. 1972).

2 2 ␴ Sk ⫽ ␴S0 q⫺2kHs,

N

Y共t兲 ⫽

兿

exp关␰p共t兲兴,

共10兲

p⫽1

where ␰p(t) is the time series from the pth broken line in the cascade. The ␰p(t) are the values of the pth broken line that are calculated by linear interpolation between the set of n points [nap, ␩(n)], where ␩ is a random, normal, independent, identically distributed variable with mean ⫽ 0 and standard deviation ⫽ ␴ Tp. The variance of each broken line ␴ 2Tp and the spacing between the vertices ap for the pth broken line are given by 2 ␴ Tp ⫽ ␴ 2T 0q⫺pHT and

ap ⫽ A0q⫺p,

共11兲 共12兲

where ␴ 2T 0 is the variance of the broken line at the outer scale A0, q is the fractional change in the spacing of the random numbers between successive broken lines, and HT is the exponent for the change in the variance between successive broken lines in the cascade (0 ⬍ HT ⬍ 1). This model is able to generate fractional Gaussian noise with a spectral density function that follows a power law.

b. Small-scale model The model of Seed et al. (1999) was used to generate the space–time fluctuations in the R–Z relationship about the mean field bias. This model uses a multiplicative bounded lognormal cascade model for the spatial distribution and an autoregressive model for the temporal development in Lagrangian coordinates. A multiplicative cascade field of size (L0, L0) is calculated as the product of N fields of correlated random variables where the variance of the kth level ␴ 2Sk is given by

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共13兲

where q is the ratio of the scales between cascade levels, Hs is the spatial scaling parameter, and ␴ 2S0 is the variance of the weights at level zero. The correlation length Lk of the random field at level is given by Lk ⫽ L0q⫺k.

共14兲

The correlation time ␶Lk in Lagrangian coordinates for level k is assumed to follow a scaling relationship:

␶Lk ⫽ ␶L0q⫺kHST,

共15兲

where HST is the space–time anisotropy exponent, and ␶L0 is the Lagrangian lifetime of a structure at spatial scale L0. The fields in the cascade are updated in Lagrangian coordinates by means of an autoregressive lag 2 [AR(2)] model: Xk,i, j 共t ⫹ 1兲 ⫽ ␾k,1Xk,i, j 共t兲 ⫹ ␾k,2Xk,i, j 共t ⫺ 1兲 ⫹ ␦k, 共16兲 where the model parameters ␾k,1 and ␾k,2 are functions of ␶Lk, and the ␦k have a normal distribution with E 共␦k兲 ⫽ 0 Var共␦k兲 ⫽

and

1 ⫹ ␾k,2 2 关共1 ⫺ ␾k,2兲2 ⫺ ␾k,1 兴 1 ⫺ ␾k,2

共17兲 共18兲

and spatial correlation length Lk. The cascade levels are advected using the advection field estimated from the radar reflectivity field. The complete list of model parameters is found in Table 1. The mean of the time series ␮ was calculated to yield a mean bias of one after conversion from decibel units. The variance of the time series is partitioned into two components: ␴ 2T 0, the variance from the changes in the mean field bias that is

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TABLE 1. List of model parameters. Mean dB␧ Variance at pixel resolution (dB␧)2 Spatial scaling exponent Lagrangian space–time exponent Lagrangian correlation time (min) Correlation time of mean field bias (min) Scaling exponent of mean field bias Variance of mean field bias (dB␧)2

␮ ␴ 2s0 Hs HST ␶L0 T0 HT ␴ 2T0

⫺0.1 1.7 0.15 1.55 30 420 0.1 1.25

modeled using the broken-line model, and ␴ 2s0, the variance due to the small-scale fluctuations. This partitioning was performed by adjusting ␴ 2s0, HS, HST, and ␶L0, so as to reproduce the observed structure function close to the origin, and then attributing the balance of the observed variance to the broken-line model. The starting value for the scaling exponent of the mean field bias, HT, was provided by the analysis of the slope of the scaling function in Fig. 8. Then T0, HT, and ␴ 2T0 were adjusted so as to reproduce the longer range in the structure function.

6. Results for a case study The model was run using the parameters shown in Table 1 and the downscaled radar reflectivity fields. The model output was verified using the second-order structure function of ␦R(t) evaluated at an area of 25 km by 25 km around the site of the POSS. The temporal generalized structure function was calculated for each pixel in the area and the mean and standard deviation of the structure function was calculated from 10 000 structure functions for each lag out to 120 min. The structure function for the POSS together with the mean and standard deviation of the model is shown in Fig. 10. Estimates of the second-order structure function have significant uncertainty, particularly for the longer lag times, but the modeled structure function closely follows the observed structure function for the first 30 min. The structure function from POSS is within one standard deviation of the modeled function for lag times up to 110 min. The mean spectral densities of the input (downscaled) radar rainfall based on the R–Z relationship (thick line) and the model output fields (thin line) are shown in Fig. 11. The power laws are well preserved in the model (simulation). Spectral density from simulated fields shows larger power than that from downscaled radar Z fields because of the added stochastic components. A slight dip is noticed in the spectral density of both the observed and simulated fields at 1-km resolution. This is due to the loss of variance caused by

FIG. 10. Second-order structure functions for POSS (square) and for the model (thick solid line). The structure function for the model is an average of 10 000 structure functions at an area of 25 km ⫻ 25 km around the POSS location. The std dev is shown as thin lines.

the downscaling procedure. Temporal variations of Z, R, and D⬘m from POSS, radar, and simulated fields, as shown in Fig. 12, support the simulation technique by demonstrating their temporal consistency. The mean and variance of D⬘m from the simulation and POSS measurements are comparable (Table 2). The general trend of D⬘m is well simulated, although a systematic bias exists because of the difference between radar Z and POSS Z. This is attributed to the fact that the two measurements have different measurement heights (Zawadzki 1984; Joss and Waldvogel 1990). Since we have simulated only the DSD variability measured from POSS and do not consider the difference between two measurements, a systematic bias does not indicate the failure of the simulation. Spatial distributions of radar Z and simulated R in Fig. 13 illustrate the effects of adding stochastic components that represent the variability of DSDs. We

FIG. 11. Mean spectral density for the input (downscaled) radar and the model (simulated) rainfall fields. The radar reflectivity field was first converted into R using the climatological Z–R relationship and then spectral density was calculated.

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FIG. 12. Time series of Z, R, and D⬘m from POSS, radar, and simulation.

have adjusted the color scale of Z with R–Z relationship in a way that the same color has the same value of R. This procedure reduces the fluctuation of fields that is due to the DSD variability since this transformation does not take into account the scatter (␦R) in R–Z plot. For example, the variances of R and transformed R from Z in Fig. 2 are 9.2 and 8.1 mm2 h⫺2. The structure of simulated R and Z fields is similar at the large scale, but the small-scale fluctuations are more pronounced in R fields because of the DSD variability. This was indicated in the spectral densities in Fig. 11, which show that the simulated field has more variability at the high frequencies. Fields of drop size distributions can be calculated given the R and Z fields and using (5) or (7) with a predetermined h(x) that is derived from actual data or a specific DSD model. In this case, the average h(x) shown in Fig. 5 was used. In addition, two characteristic parameters of DSDs (N⬘0 and D⬘m) can be easily obtained from the two fields as shown in Fig. 13. Spatial distribution of D⬘m is similar to those of simulated R and radar Z. The correlation coefficients for R ⫺ D⬘m and Z ⫺ D⬘m are 0.38 and 0.67, respectively (Table 3). Similar characteristics have been found from POSS measurements and are also shown in

Table 3. The distribution of N⬘0 does not show any coherency with R, Z, or D⬘m fields (the correlation coefficient is less than 0.2), but the fields do have spatial structure. Again these results support the conclusion that the variability of DSDs cannot be fully described with only one parameter, either D⬘m or N⬘0. As shown by Lee et al. (2004), the correlation coefficient between R and N⬘0 is less than 0.3, whereas D⬘m increases with R. Testud et al. (2001) also found a similar trend in convective rain. Thus, our simulation is very consistent with their results, as well as POSS measurements.

7. Conclusions The variability of the drop size distribution in space and time is a key issue in both remote sensing and in

TABLE 2. The mean and variance of D⬘m from POSS measurements and model.

Model POSS

Mean (mm)

Variance (mm2)

1.29 1.34

0.08 0.11

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FIG. 13. Spatial distributions of radar Z, simulated R, derived D⬘m, and N⬘0 at 0100 UTC 25 Sep 2001.

developing an understanding of the microphysical processes that generate rainfall. In this paper, we have shown that two moments are sufficient to capture the bulk, but certainly not all, of DSD variability. We have developed a way of constructing fields of DSDs in space and time that are constrained by actual radar and disdrometric measurements. Fields of one moment (Z or sixth moment) of the DSD are obtained from radar measurements and then these reflectivity fields are converted into deterministic R fields. Last, fields of the other moment (R) are obtained from the combination of the deterministic R fields and stochastic components that are derived from the space–time modeling of the fluctuation with respect to a best-fit R–Z relationship. The fluctuations are assumed to be independent of the Z field. The stochastic model is checked for self-consistency

at a point by comparing the second-order structure functions and characteristic DSD parameters from the model with POSS measurements. Modeled D⬘m is spatially correlated with the fields of Z and simulated R, whereas N⬘0 shows virtually no spatial correlation. This seems to indicate that the most important contributor to DSD variability is N⬘0 and furthermore N⬘0 shows coherent spatial structure. In addition, the time series of D⬘m from POSS measurements and modeling have TABLE 3. Coefficients of determination between DSD parameters estimated from the space–time model simulations and from POSS observations.

Model POSS

N⬘0 ⫺ D⬘m

N⬘0 ⫺ R

N⬘0 ⫺ Z

D⬘m ⫺ R

D⬘m ⫺ Z

0.20 0.28

0.19 0.08

0.02 0.00

0.38 0.44

0.67 0.73

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FIG. 14. (a) One-hour accumulation of radar-derived precipitation using the climatological R–Z relationship. (b) One-hour accumulation taking into account the stochastic component of the R–Z relationship.

similar characteristics, supporting the conclusion that the fields of modeled DSDs are realistic.

8. Discussion With two fields, radar Z and simulated R, the spatial and temporal distribution of DSDs can be derived from the model. The derived DSD fields can be used as a test bed for assessing many remote sensing techniques, such as attenuation correction techniques, radar–rain gauge adjustments, polarimetric rain estimates, etc. Here we will show one example of the application of the simulation. The left-hand side panel of Fig. 14 shows 1-h accumulation of rainfall obtained from radar data using the deterministic R–Z relationship, while the righthand side panel shows the accumulation from the model that incorporates the fluctuations. It is apparent that the latter has greater spatial variability, with higher peaks and deeper valleys, than the former. This is the direct consequence of the time–space coherence of the fluctuations. The effect of time integration and area average is shown as the normalized difference in rain amounts from radar data and simulated fields (Fig. 15). Hourly accumulation at a point (0.25 km ⫻ 0.25 km) shows about 58% difference. It helps in understanding the finding of Wilson and Brandes (1979) that gauge-toradar ratios may remain large even if a good deal of time integration is made. An additional area average over 10 km ⫻ 10 km significantly reduces the difference to about 40%. The difference can be reduced to below

20% when an hourly accumulation is averaged over 30 km ⫻ 30 km. In the same manner, the effect of DSD variability in space and time on radar–rain gauge adjustments can be investigated in a way that the problems of undersampling and drop sorting are fully resolved. In addition, all polarimetric and dual-wavelength parameters, such as differential phase shift, differential reflectivity, and attenuation, can be easily obtained through a scattering

FIG. 15. Normalized difference in rain amount from radar Z and simulated R as functions of different accumulation intervals and average areas for the case in Fig. 14. Five simulations are used to derive these statistics.

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model using derived DSD fields in space and time. With these parameters, several methods of attenuation correction, especially polarimetric attenuation correction, can be tested and the uncertainty of correction methods can be obtained. In addition, the modeling can incorporate the initial errors in radar quantitative precipitation estimates to investigate their importance in precipitation forecasting systems. Since the significant variability in precipitation forecasts is due to evolution of the field or changes in the direction of echo movement, particularly for longer forecast lead times, this variability can be added in the modeling by taking into account its structure. Thus, this is a comprehensive way of generating an ensemble precipitation forecasting. The construction of DSDs presented here utilizes only two moments and a constant h(x) in space and time. As shown in Fig. 5, the scatter around average h(x) is small but nevertheless it exists, suggesting that the shape of h(x) also should be added in the modeling for a better representation of the DSD variability. A possible straightforward extension of the present model is to consider the fluctuations ␦n of a third moment of order n measured with respect to a best fit to (Lee et al. 2004) Mn ⫽ C2,nM 共jn⫺i兲Ⲑ共j⫺i兲M 共ij⫺n兲Ⲑ共j⫺i兲,

共19兲

where i and j could be 3.67 (for rain rate) and 6 (for reflectivity), respectively. It has been recently shown (Szyrmer et al. 2005) that for all practical purposes three moments are sufficient to describe all DSD variability. They also give a three-moment generalization of (19). Currently, the model considers the systematic change of the microphysics for a longer period as a temporally correlated stochastic fluctuation of average fields in the entire domain with no attempt to condition these fluctuations on the changing meteorological situation. This is also true for the small-scale model in space where there could be a systematic change in the microphysics within the area. Thus, the model should be modulated in space and time to include systematic changes of microphysics. Furthermore, the model assumes a spatial structure of fluctuation fields derived from the temporal structure because of the limitation in the measurements. A small network of disdrometers should be beneficial to provide additional necessary information. Acknowledgments. This work was partially supported by a grant from the Canadian Foundation for Climate and Atmospheric Sciences and funds provided by the Canadian Weather Research Program.

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