Scales in Hydrology and Water Management /Echelles en hydrologie et gestion de l'eau (IAHS Publ. 287, 2004)

137

Modelling and forecasting rainfall in space and time

ALAN SEED Bureau of Meteorology Research Centre, GPO Box 1289K, Melbourne, Victoria 3001, Australia

a . seed(5 ibom.gov .au

Abstract The distribution of rainfall in space and time is well known to be both variable and scale dependent. Models based on scale invariance have been shown to provide parsimonious descriptors of the distribution of rainfall over a wide range of scales in both time and space. This paper presents some of the methods that are used to describe the scaling behaviour of rainfall. The scaling nature of rainfall affects both the ability to forecast rainfall and the selection of stochastic methods to generate plausible fields of rainfall. Examples of methods that exploit the space and time scaling of rainfall to produce short duration rainfall forecasts and stochastic simulations are provided. K e y w o r d s forecas t ing; m o d e l l i n g ; rainfal l ; sca l ing ; s tochas t i c

Modélisation et prévision de la pluie dans l'espace et dans le temps Résumé II est bien connu que la distribution de la pluie dans l'espace et dans le temps est variable et dépend de l'échelle d'observation. Les modèles fondés sur l'invariance d'échelle ont montré leur capacité à fournir des descripteurs concis de la distribution de la pluie sur une grande gamme d'échelles de temps et d'espace. Ce papier présente certaines des méthodes utilisées pour décrire l'invariance d'échelle de la pluie. Cette scalance est importante aussi bien en ce qui concerne la possibilité de prévoir la pluie que pour le choix de méthodes stochastiques de génération de champs de pluie plausibles. On présentera des exemples de méthodes s'appuyant sur la scalance de la pluie dans le temps et dans l'espace pour réaliser des prévisions à court terme et des simulations aléatoires. M o t s c lefs p lu ie ; s ca l ance ; p rév i s ion ; modé l i s a t i on ; a léa to i re

INTRODUCTION

Rainfall is the result of complex atmospheric phenomena and therefore is itself complex, displaying both variability and intermittency over a wide range of scales. The statistical properties of rainfall in space and time are highly dependent on the scale of the accumulation or average, making it an "intriguing and challenging process to study" (Foufoula-Georgiou, 1996). In general, there are two generic types of rainfall models, viz. those based on clustered point process models using the notion of rain cells (Cox & Isham, 1996) and those based on treating the rain field as a scaling or muitifractai process. Scaling models of rainfall provide "attractive and parsimonious" (Foufoula-Georgiou, 1996) representations that are able to generate rain fields that are able to reproduce the observed relationship between the statistics of precipitation amounts at different spatial and temporal scales (Rodriguez-Iturbe et al., 1998). While some argue that rain fields (or more accurately the fluctuations of rain) are muitifractai in a fundamental sense, e.g. Lovejoy & Schertzer (1995), it is sufficient to note that they are able to reproduce a reasonable approximation of key observed statistical

138 Alan Seed

characteristics over the range of scales usually found in hydrology, and are therefore useful in modelling and simulating rain fields for hydrological applications. This paper starts with a very short outline of the theory that underpins the analytical methods, including useful references to more detailed treatments. Thereafter, the use of scaling models to characterize the statistical nature of rainfall, to generate plausible random fields, and to produce conditional simulations, will be discussed and conclusions will be drawn. This paper is not an encyclopaedic treatment of the subject, but rather an introduction to scaling analysis of rainfall and a presentation of some models that the author has found useful.

BASIC THEORY

The structure of the fluctuations in a homogeneous stationary random field X(t) can be characterized through the covariance and correlation functions:

Cov[X(t), X(t')] = B(t, t') = E[X(t)X(t')] - m(t)m(t') (1 )

PCt.O = ^ r (2) 0(t)c(t )

where m(t) and m{t') are the means and cr(f) and a 2 (f) are the variances of the n-dimensional random fieldX(t) at locations t = (ti,...,t2) and f = (t'i,...,t' 2) (Vanmarcke, 1983).

If the variances exist, the field is defined as homogeneous (Vanmarcke, 1983) or as second-order (or weak or wide-sense) stationary (Cresse, 1993) if:

m(t) = \i, for all t e D, and (3)

B(t, t') = B(t -1') = 5 ( T ) , for all t G D (4)

The rate at which the variance of a field decreases under increased local averaging is called the variance function. For a two-dimensional field, if XA(t\,t2) is the local average (calculated over the rectangular area A = T\Ti, centred on \\,ti) of the random field JY(ti,t2), then the variance function, y{T\,T2), is defined as:

y(T]T2) = Var[XA]/o2 (5)

If the second order moment of the correlation function is finite then in the limit:

Y(71,r 2) = ~ , 7,,r, large (6) A

In this case a is the correlation area (or scale of fluctuation for one-dimensional processes) and is equal to the integral of the con-elation function (Vanmarcke, 1983):

« = £, £~p(-x:1,T2)dTjdx2 (7)

The variance function can be approximated by a power law:

7 ( 0 - r * (8)

where / is now the length scale. Vanmarcke (1983, p. 206) shows that h converges

Modelling and forecasting rainfall in space and time 139

quite slowly to the dimension of the field, D, and that the value will depend on the range of scales approximated by the power law and the underlying scale of fluctuation.

Tools to describe and model (Gaussian) homogeneous random fields are well developed, but rainfall fields are nonhomogeneous due to spatial variability in the mean and variance of the field and second order statistics. One approach is to transform the measured field into a homogeneous field by standardizing the random field with spatially distributed estimates of the mean and variance and by using the logarithmic transformation. Vanmarcke (1983, p. 183) states:

It is difficult to generalise the manner in which random field non-homogeneity should he modelled. Stochastic model building is always an art as much as a science, and artistry tends to dominate when non-homogeneity is involved.

The case where the field is a stationary process with a very long memory (correlation function has an infinite second moment) can be modelled as a self-similar process. A random field will display self similarity if, and only if, the power spectrum P(f) obeys the power law:

^ ( / ) « = / - p (9) The scaling can be characterized by spectra of exponents; the spectra for the scaling of the moments, K(p), is defined as:

(Ffix))*!-*^ (10)

where F? (x) denotes the average of the field over the interval / centred at x, and < > denotes ensemble average, or the spectra for the scaling of the generalized structure function, Ç(q), defined as:

'\F(x + l)-F{xf \ o c / * / ) (11)

Scaling can also be expressed in terms of the scaling of the probability distributions for the field averaged over scales /:

Pr(F, > / 7 )« r e W (12) where c(y) is the spectrum of scaling exponents for the probability distribution, and ~ represents equality to within slowly varying functions of / (Lovejoy & Schertzer, 1995). The exceedence probabilities of multifractals have asymptotically hyperbolic tails so that Pr(R > r) <* r~q" (Mandelbrot, 1974). Here qcr is a critical exponent indicating the moment beyond which moments are divergent, and K(q) becomes a straight line (Harris et ah, 1996).

It can be shown that fields are multiscaling as defined by equation (10) if the slope of the power spectrum, P, is less than D, the dimension of the support (D = 1 for time series, 2 for spatial fields). In the case for p > D, the scaling is characterized through the generalized structure function defined by equation (11) and is said to be multiaffine. Moment scaling analysis can only be performed on the field with P > D after fractional differentiation (Lovejoy & Schertzer, 1995).

Estimation of K(q) involves calculating the qth moment of the field at the smallest scale and then averaging this field over successively larger scales. The K(q) curve is

140 Alan Seed

only defined over a finite range of uniform scaling and care must be taken to verify that the field does in fact scale over a useful range. The analytical form for K(q) can be derived theoretically for specific cascade constructions, e.g. the Universal Multifractals of Lovejoy & Schertzer (1995), or Bounded Lognormal Cascades (Menabde, 1998).

The advantage in these approaches is that they provide a convenient framework for the characterization of highly nonlinear fields over a wide range of scales by means of a small number of parameters. For example, C\, the derivative of K(q) at q = 1, is a measure of the intermittency of the field (the co-dimension of the set of points that are greater than the mean). If C\ is close to zero the field behaves as a field of white or llf noise which fills the embedding space, whereas if C\ is close to D then the field is intermittent and the flux is concentrated on a small fraction of the area. A good example of the use of multiscale statistical properties to characterize rainfall is found in Harris et al. (2001).

Most of the theory and models for scaling invariance are based on the random multiplicative cascades, which arose in the theory of turbulence. The random multiplic­ative cascade can be constructed as a discrete cascade, e.g. Over & Gupta (1996), Menabde (1998), or as a continuous cascade, e.g. the Universal Multifractals of Lovejoy & Schertzer (1995). An alternative approach is to use the wavelet decomposition, e.g. Perica & Foufoula-Georgiou (1996). Substantial reviews of the models and related theory can be found in Lovejoy & Schertzer (1995) and Foufoula-Georgiou (1996).

A discrete random cascade is constructed as follows: at an arbitrary «-th step in the process, a square of size /„ = L2~" with mean areal rainfall F„ is divided into four equal squares, each with size 1„I2 and value F„+\ = W(i)F„ where the W{i) are random numbers drawn from some generator. If the weights are selected subject to the

4

constraint that "Y_lW{i) = \, then the cascade is said to be micro-canonical and the 1=1

resulting field will have simple scaling characteristics. A less restrictive condition is to set the expectation of the generator equal to 1, the

cascade is said to be canonical and the resulting field will be multiscaling if the weights are independent and identically distributed for all levels of the cascade, or multiaffine if the width of the generator is reduced with scale according to a power law (Menabde, 1998).

HOMOGENEITY OF RAINFALL

The assumption that statistical homogeneity (i.e. the first and second order statistics of a field are independent of location) is implicit in both the classical Gaussian and scaling analysis of rainfall, so it is interesting to evaluate the range in scales over which the first and second order statistics can be considered to be independent of location within the field, and whether they are independent of the meteorology generating the field. Figure 1 shows the spatial distribution of the mean annual rainfall over Victoria, Australia. It is clear that there is a large east-west gradient of mean annual rainfall in Melbourne, which makes is very unlikely that the spatial statistics within an area of 200 Ion centred on Melbourne will be homogeneous due to the effect of the topography in the east.

Modelling and forecasting rainfall in space and time 141

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Fig. 1 Mean annual rainfall for Victoria. The east-west extent of the map is approximately 750 km.

The scaling characteristics of rainfall are also non-stationary in both space and time. Harris et al. (1996) used the measures [3, C;, and qcr to characterize the statistical nature of time series of precipitation at a point as a function of location in a mountain range and hence orographic enhancement, A network of raingauges with 15-second resolution was installed on the west coast of the South Island of New Zealand. The analysis showed a systematic trend of decreasing intermittency and extreme values with increasing altitude (or proximity to the crest of the mountain). For example, (3 decreased from a value of 1.5 at the coast to 0.95 at the main divide pointing to a relative reduction in the power of low frequency structures at the main divide. C/, the measure of intermittency, decreased from the coast to the main divide as did qcr, a measure of the width of the distribution.

While there is some validity in the observation that the number of classes in a meteorological classification scheme expands to the number of cases in the study, it has been very helpful to classify rainfall as observed by radar into regions of convective and stratiform rain. House (1997) defines stratiform rainfall as caused by the set of physical processes that are dominant when the upward air motion is small relative to the fall speed of the hydrometeors, resulting in rainfall that is more variable in the vertical than the horizontal. Convective rainfall on the other hand is defined as rainfall that is associated with regions where the upward air motion is large relative to the fall speed of the hydrometeors, resulting in rainfall that is as variable in the horizontal as in the vertical dimension.

Average Rainfall - Annual M i l l imet res

142 Alan Seed

Rainfall is a three-dimensional process and it is important to understand the variability of rainfall in the vertical relative to variability in the horizontal when using radar estimations at some height aloft to represent rainfall on the ground. The variogram for the vertical profile of reflectivity, y(h), can be calculated by first estimating the mean vertical profile of reflectivity for all bins that are within a 50 km range of the radar and have a reflectivity that exceeded 15 dBZ and then by using:

7(A) = ([(z(A, ) - u( /7, ) ) - (z(h2 ) - ii(h2 )f I (13) where \i(h) is the conditional mean reflectivity and z(h) is the reflectivity of a polar bin at height h above the radar. Figure 2 shows the variograms for radar reflectivity in the vertical and horizontal dimensions for convective and stratiform rainfalls measured in Sydney. It is clear that there is significant variability in the vertical dimension, and that the extent of the vertical-horizontal anisotropy present in rainfall is dependent on the physics that dominate the rainfall production process.

Stratiform and convective rainfall is usually present in the same rain field (House, 1997) but there are occasions where the field is predominantly stratiform or convective and it is instructive to use these cases to estimate the scaling characteristics for stratiform and widespread rainfall. Chumchean (2004) classified over 800 fields of hourly radar rainfall accumulations into stratiform and convective rainfall based on the vertical profile of radar reflectivity and found systematic differences in the slope of the power spectrum at the low-frequency end of the power spectrum as shown in Fig. 3. Nagata (2001) used the scaling parameters P and Cj to classify fields of hourly rainfall accumulations into stratiform and convective rainfall classes.

This implies that the scaling nature of the rainfall field is a function of the meteorology that generates the rainfall and therefore the fields have scaling parameters that are non-stationary in both space and time. This is a major problem with verifi­cation of a multifractal or multiaffine behaviour in rainfall data sets using equations (10) or (11) since the analysis techniques require long sequences of statistically homogeneous rainfall data. Long sequences of rainfall data may not be scaling over the entire period and usually include sequences from several different rainfall processes, each with significantly different statistical and scaling properties.

Rainfall fields are also prone to measurement errors and instrumentation artefacts and great care needs to be taken to eliminate as much corrupt data as possible prior to

Modelling and forecasting rainfall in space and time 143

25

2 20

* Convect ive

• Stratiform

a

I 15 o a

re 10

0

-23 -22 -21 -20 -19 -18 -17 -16 -15

10Log(frequency(1/km))

Fig. 3 Average power spectra for hourly widespread and convective rainfall over Sydney, Australia, using 256 x 256 Ion fields of 1-km resolution radar rainfall estimations (Chumchean, 2004).

analysis (Krajewski et al, 1996). This is particularly true when using radar rainfall data to investigate the scaling of the moments or the probability distribution. A review of the impact of measurement noise and non-stationarity on scaling techniques is given by Harris et al. (1998) and robust techniques for estimating the variogram (the second order generalized structure function) can be found in Cressie (1993). Gebremichael et al. (2004) assessed the ability of radar data to characterize the small-scale variability of a rain field that was sampled by both a high-resolution radar and a raingauge network and found that the radar under-estimated the con-elation at very short range (less than 5 km) which is significant since the fractal dimension can be inferred from the behaviour of the variogram at the origin (Cressie, 1993).

MODELLING RAINFALL IN SPACE AND TIME

Theories of space-time rainfall based on scaling ideas have only recently emerged (e.g. Bell, 1987; Over & Gupta, 1996; Marsan et al, 1996; Lebel et al, 1998; Deidda et al, 1999; Seed et al, 1999; Pegram & Clothier, 2001). A general feature of atmospheric turbulence is that for a feature with a given size /, there is a typical lifetime or correlation length T i ° c lx'H (Marsan et al, 1996; Venugopal et al, 1999), leading to anisotropic scaling in space and time. This space-time model is quite different to the usual meteorological phenomenology, which assumes that there is a hierarchy of qualitatively different dynamical mechanisms that depend on scale (Schertzer et al, 1997). This complicates space-time models since the scaling in space is different from that in time and the anisotropy is difficult to quantify since it must be estimated in Lagrangian rather than Eulerian coordinates.

A further complication arises from the fact that cascade models are essentially event-based models since rainfall typically does not scale convincingly beyond a couple of days (Fabry, 1996), or that there is a break in the scaling relationship at about one day (Deidda et al, 1999). Pegram & Clothier (2001) model the arrival and duration of the events as an alternating renewal process and then disaggregate each

144 Alan Seed

event into a sequence of rain fields. The disaggregation is done by using an anisotropic power law spectral filter on a cube (two-dimensional space and time) of Gaussian noise so as to achieve the observed correlations in space and time, and then exponentiating to obtain a time series of correlated lognormal fields. The time series of mean areal rainfall over the entire field is modelled by first generating a time series of the wetted area ratio based on an autoregressive lag 6, AR(6), model, and then using a derived relationship between the mean and the wetted area ratio to estimate the mean and variance of the field at a particular time step. Each field in the time series of the event is then renormalized so as to produce the correct field mean and variance.

Seed et al. (1999) developed a stochastic space and time event model that consists of three components viz. a broken line model (Seed et al., 2000) to generate the time series of mean areal rainfall over the domain of the cascade, a model based on a bounded lognormal cascade to generate the spatial pattern, and an autoregressive lag 2 (AR(2), equation (17)) model for the Lagrangian temporal evolution of the cascade weights. The field advection is modelled as a single vector that is held constant for the duration of the event. The model is essentially a down-scaling model and therefore a model of the temporal distribution of the mean areal rainfall at the outer scale of the cascade is required to drive the process. Rainfall typically has a power spectrum where the slope (3 is greater than the dimension of the data and data are distributed approximately as lognormal (Crane, 1990), therefore a multiaffine model, like the bounded lognormal model, is appropriate for distributing the large-scale mean areal rainfall onto a field. It is possible to manage the space-time anisotropy by generating a cascade in space and time, but this places limits on the duration of the period that can be simulated. Therefore, the temporal evolution of the space-time field was modelled as an autoregressive process which is causal (the present only depends on the past) and is therefore appropriate in this context.

The construction of the model is as follows. The model domain, Lo, is decomposed into a cascade of N levels. Level k in the cascade represents the scale Lk = q'kCo; q is the ratio of the scales for successive levels in the cascade such that 0 < q < 2. Co is the correlation length of the outer scale of the cascade and is set such that Co < L0. Each level in the cascade is generated by smoothing a field of independent N(0,l) noise by means of a notch filter centred on the scale for that level in the cascade and then re-normalizing such that the field has a mean of zero and variance of one. The levels in the cascade are updated in Lagrangian coordinates by means of an autoregressive lag 2 model (equation (17)) where the temporal correlation length for level k, T/ f, is assumed to follow a scaling relationship:

**=<r t aT„ (i3) where a is the space-time anisotropy exponent, and To is the lifetime of a structure at scale Co.

The model requires the gross statistics for the event (duration, mean intensity, variance at the small and large scale, and advection velocity) and scaling descriptors that control the space and time correlation structure of the field. The scaling para­meters, variance of the rain field, and advection velocity can be estimated using radar data. The model generates fields of instantaneous rainfall intensity at 1-km, 5-minute resolution which are then integrated in space and time for use in hydrological models.

Modelling and forecasting rainfall in space and time 145

The model has been used to generate stochastic 24-h design storms for an urban sewer system. The historical record was searched for the storms with mean areal 24-h rainfall that was close to the 1 in 5 year design storm depth. Synoptic charts were used to classify these storms into a set of six characteristic storm types and then the model parameters were estimated for storms that had radar data available. Figures 4-7 and Table 1 show the verification results for one such storm type. The temporal correlation function is a stringent test of the model since it is not modelled directly and depends on the temporal Lagrangian development, spatial correlation and advection velocity being correct.

SCALE DEPENDENT FORECASTING

The characterization of a rain field as an ensemble of structures of scales between the single pixel and the outer scale of the field, characterized by dynamic scaling where the life time of a structure is a power law of its scale, leads to an intuitive picture of the predictability of a rain field as a function of scale (Zawadzki et al, 1994; Marsan, 1996; Germann & Zawadzki, 2002). Since there is no information in the current image about

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Fig. 4 Spatial correlation functions for measured 10 February 1994 rain fields and the pre-frontal trough design storm.

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Fig. 5 Temporal correlation functions for measured 10 February 1994 rain fields and the pre-frontal trough design storm.

10 20 30 4 0

146 Alan Seed

'4

5-minute average 60-minute average 10 20 30 km

Fig. 6 Typical 5-min and 60-min average rainfall intensity fields from the pre-frontal trough design storm. The images cover a 256 x 256 km domain with a resolution of 2 km, 5 min.

Rain depth mm 0.1 2.0 4.0 8.0 8.0

j 10.0 i 15.0 j 20.0

to 10 20 30 | km |

Fig. 7 A 12-h accumulation of the synthetic data for a pre-frontal trough situation.

Table 1 A summary of the point and areal statistics for hourly rainfall accumulations of the measured rainfall and three simulations.

Gauge Simulationl Simulation 2 Simulation 3 Mean (mm h"1) 2.32 2.32 2.32 2.32 Point a (mm If 1) 2.09 2.71 2.00 2.73 Areal o (mm h"') 1.48 2.27 1.28 2.04 Point Lag 1 correlation 0.49 0.58 0.30 0.50 Areal Lag 1 correlation 0.76 0.78 0.58 0.70 Point maximum (mm h"') 14 20 16 21 Areal maximum (mm h"1) 5.4 9.2 5.1 8.2

a structure after a period that is greater than its lifetime, the optimal (minimum mean square error) strategy is to smooth structures once their lifetime has been exceeded. Dynamic scaling implies that one needs to analyse the field on increasingly large scales in

Modelling and forecasting rainfall in space and time 147

order to predict further into the friture. These ideas have been exploited by Seed (2003) who used Fourier techniques to decompose the field into spectral components, and by Germann & Zawadzki (2002) and Turner et al. (2004) who used wavelet transformations.

Turner et al. (2004) used a wavelet transformation to study the predictability of a rain field as a function of scale. The predictability limit of the fieldl at a one-hour lead-time was evaluated in real time and a linear relationship was then used to determine the predictability limits at the other lead times in the forecast. The Wavelet transform coefficients were multiplied by a weighting factor that represented trie predictability of that particular scale at that particular lead-time. Seed (2003) calculated the cascade of random fields from a measured rain field (a quantitative radar rainfall estimation) by first transforming the field into radar reflectivity in dBZ units and then using notch filters in the frequency domain to disaggregate the ensemble of scales that are present in the field into a hierarchy of levels in a cascade that will then sum back into the observed dBZ field.

The temporal development of each level in the cascade is modelled using an AR(2) model for each level in the cascade. The parameters for the hierarchy of AR(2) models are calculated at each time step by using the most recent estimates of the Lagrangian lag 1 and 2 autocorrelations to solve the Yule-Walker equations, applying heuristic rules to maintain stationarity. This has the effect of making the model quite adaptive and generic, learning the characteristics of each rainfall event as it unfolds.

A field (size LxL pixels) of decibels radar reflectivity can be decomposed into an additive cascade structure:

dBZu(t) ^X^jit), i = l,...,L, j = l,...,L, L = T (14) k-\

where the k-i\i level in the cascade Xifi) represents (in the spatial not frequency domain) the variability in the original field with frequencies within the range l/X2" ( / c + 1 ) pixel"1 at time t. The dBZ field is transformed into the spectral domain by means of an FFT transfonnation. Thereafter each level Xift) of the cascade is calculated by using a band-pass filter based on a Gaussian window that passes the appropriate frequencies and calculating the inverse so as to transform each Fourier component of the field back into the spatial domain. Finally, the cascade levels are normalized for convenience using:

XkjjW-V-klf)

MO where [lift) and Ok(t) are the mean and standard deviation of the /c-th level, respectively, and are assumed to be constant throughout the forecast period and over the field domain. The Zk{t) fields are referred to as "levels" in the cascade and represent the Fourier components transformed into the spatial domain and then normalized to have zero mean and unit variance. Figures 8 and 9 show a rain field and the first three levels of the spectral decomposition of the field.

The calculation of the advection vectors is based on the optical flow technique, which assumes Taylor's hypothesis and solves:

ddBZ ddBZ ddBZ = v + v„ (16)

dt dx * dy y V '

Zkjj(o= tjj:.\.r ( is)

148 Alan Seed

h .

Data at 21-APR-2001 06:01 :OOUTC

Fig. 8 Field of radar reflectivity. The field is from Melbourne, Australia, and is a 256 x 256 km image with 1-km resolution.

m

2 5 6 - 126 km 1 2 8 - 6 4 km ft 6 4 - 3 2 km

Fig. 9 The first three spectral components of the field in Fig. 8.

(Thévenaz, 1990; Bowler et al, 2004a), where dBZ is the radar reflectivity field (not a cascade level) and vx, vy are the x and y velocity components. This is done by using finite differences to estimate the gradients, and then using least squares to estimate yv, vy over the sub-area. All levels in the cascade are assumed to be moving with the same advection field.

The Lagrangian development of each field in the cascade is assumed to be able to be modelled by an AR(2) model:

ZkJJ (/ + » + l) = (t)ZkJJ {t + n) + ^k2 (t)ZkJJ (t + n -1) + e k J J (t + n + l) (17)

The 8i(r) fields are a cascade of stochastic noise and represent the new development that takes place in the rain field during the forecast period. The noise fields have the following properties (Salas et al., 1980):

E[ek(t)] = Q (18)

Var(ek(t)) = j + ^ 2 ^ [(l-* k , 2(t)f -^(t)2] (19)

Modelling andforecasting rainfall in space and time 149

Finally, the output forecast field at time t + n + 1 is calculated using:

dBZtJ (t + n +1) = £ | i , (0 + ak (t)ZkJJ (t + n +1) (20) lc=\

assuming that u./((r) and Gift) are constant during the forecast period and are estimated at time t, the time that the forecasts are made, and transformed back into a rainfall field.

In effect, the forecast is a deterministic forecast based on advection blended with stochastic noise, which progressively dominates the larger scales as the features in the various levels in the cascade perish. The e/ft) fields are correlated noise and are generated by smoothing a normally distributed independent random field of N(0,\) with a notch filter. This model can be used to generate an ensemble of stochastic nowcasts where each ensemble member is equally likely. This model is able to estimate the probability distribution of forecast rainfall over an area, e.g. a catchment, or along a line, e.g. a flight path for aircraft.

The model can also be ran in a deterministic mode by not adding the noise term to the forecast; in this case the forecast becomes progressively smoother as the details in the analysed field perish. The forecast fields are normalized so as to preserve the mean conditional rain rate and the raining area. A verification of an early version of this model can be found in Ebert et al. (2004). Examples of the deterministic and stochastic nowcasts are shown in Figs 10 and 11. Note that the 60-minute forecast has less small-scale detail than the 30-minute forecast in Fig. 10.

MERGING NOWCASTS WITH NWP FORECASTS

Bowler et al. (2004b) describes the development of STochastic Ensemble Prediction System (STEPS) that merges stochastic rainfall nowcasts with a stochastic down-scaling model for NWP rainfall forecasts. This model is able to generate a set of stochastic forecasts out to 6 or 12 hours. The NWP rainfall field is decomposed into its spectral components and then the skill of the model is evaluated in real time by calculating the fraction of the variance that the model is able to explain for each level in the cascade. The AR(2) models for each level in the observed cascade are used to calculate the fraction of the variance that is explained by a deterministic nowcast, thereby enabling the nowcast cascade to be merged with the NWP forecast at a rate that is dependent on both scale and lead time. The detail in the rain field that exists below the scale for deterministic forecasts, or beyond the effective scale of the NWP forecasts, is modelled as stochastic noise. This is generated using the same space-time model as in the previous section, and blended at each level in the cascade with the merged deterministic forecasts so as to generate the ensemble of stochastic forecasts. The advection field analysed from the radar data is merged with that analysed from the set of NWP rainfall forecast fields and is used to advect the stochastic noise through the forecast period. This advection field is perturbed for each member in the ensemble.

CONCLUSIONS

The scaling exponents are powerful descriptors of the space-time variability of rainfall. However, all current analysis methods require very large samples of data. Problems

150 Alan Seed

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arise therefore from non-stationarity in the data and contamination with measurement noise. The lack of robust methods to evaluate either the scaling of the moments or the generalized structure function for rainfall is a major problem that requires further research.

The moment scaling methods of double trace moments by Lovejoy & Schertzer (1995) are attractive in that they rely on successive averaging over larger spatial scales, thereby reducing the impact of small-scale noise, but have unsolved issues regarding the treatment of the zero values. The generalized structure function is difficult to estimate, particularly for high moments and in the presence of noise, and this sampling variability needs to be understood in order to assess the statistical significance of the difference between two structure functions.

Scaling models of rainfall have been shown to be useful in generating both stochastic data for simulations and forecasts that respect the predictability limits in the data. All current models assume that the scaling is homogeneous in both space and time during an event. This is most unlikely to be the case, but very little work has been done on either quantifying the variability of the scaling descriptors in space and time, or relating them to meteorology, and no work has been published on models that can generate spatially inhomogeneous scaling fields of rainfall.

Modelling andforecasting rainfall in space and lime 151

The spatial organization of rainfall accumulations on the ground is dependent on both the nature of the instantaneous rain field and the velocity with which the field is advected over the area. To some extent, the accuracy of small-scale precipitation forecasts is limited by the predictability of the advection field at small-scales. The predictability and variability of field advection both during an event and from one event to the next has yet to receive adequate attention, and the predictability of the advection vector as a function of scale has yet to receive any attention.

Scaling models of rainfall are able to provide powerful representations of rain fields in both space and time. These models are generally parsimonious and robust, capturing many of the features in rainfall that are important to hydrology. The theory is still very young and therefore a standardized argot has yet to emerge, making it difficult to relate the research from one group to that from another. While the jury is still out on the question of whether rain fields are muitifractai in a fundamental sense, muitifractai models are still useful representations of the measurements and worthy of further development.

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