Fractal interpolation of rain rate time series

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 109,, doi: /2004jd004717, 2004 Fractal interpolation of rain rate time series Kevin S. Paulson Radio Communications Research Unit, Rutherford Appleton Laboratory, Didcot, UK Received 1 March 2004; revised 27 August 2004; accepted 10 September 2004; published 24 November [1] Meteorological radar databases exist providing rain rate maps over areas with a sampling period of 2 15 min. Such two-dimensional, rain rate map time series would have wide application in the simulation of rain scatter and attenuation of millimeter-wave radio networks, if the sampling period were considerably shorter, i.e., of the order of 10 s or less. However, scanning a large radar at this rate is physically infeasible. This paper investigates a stochastic numerical method to interpolate point rain rate time series to shorter sampling periods while conserving the expected first- and second-order statistics. The proposed method should generally be applicable to the temporal interpolation of radar-derived rain rate maps. The method is based on the experimentally measured simplescaling properties of log rain rate time series. It is tested against 9 gauge years of rapid response drop-counting rain gauge data, with a 10 s integration time, collected in the southern UK. The data are subsampled to yield time series with a 10 s rain rate measurement every 320, 640, and 1280 s. The subsampled time series are then interpolated back to a 10 s sample interval, and the first- and second-order statistics are compared with the original time series. INDEX TERMS: 1854 Hydrology: Precipitation (3354); 3210 Mathematical Geophysics: Modeling; 3250 Mathematical Geophysics: Fractals and multifractals; KEYWORDS: rain, fractal, interpolation Citation: Paulson, K. S. (2004), Fractal interpolation of rain rate time series, J. Geophys. Res., 109,, doi: /2004jd Introduction [2] When meteorological radars are used to scan across near-horizontal planes (PPI scans), they provide nearinstantaneous measurements of radar reflectivity over large areas. Each reflectivity value is a weighted average across a voxel defined by the radar antenna pattern, the angular scan of the radar, and the range gates. Measured single-polarization and dual polarization radar reflectivities may be used to estimate rain rate, either using empirical relationships or theoretical relationships based on parameterized drop size distributions and assumptions about drop shape and terminal velocity, [Goddard and Cherry, 1984]. These radarderived rain rates are averaged over the same voxel as the reflectivity measurement. Large radars, such as the Chilbolton Advanced Meteorological Radar ( ac.uk/camra.htm), can produce near-instantaneous rain rates averaged over voxels with linear dimensions of a few hundred meters within a range up to 100 km. The resulting rain rate maps have been used to model the rain scatter effects on millimeter-wave radio networks [Paulson, 2003] to yield useful results such as diversity performance and path reduction factors. However, as the radars need to be physically rotated to scan across the area being mapped, the period between consecutive maps being produced is usually several minutes and often as long as 15 min. This low Copyright 2004 by the American Geophysical Union /04/2004JD temporal sampling rate severely limits the application of these rain rate map time series. [3] The two-dimensional rain rate maps derived from radar data may be used to estimate the instantaneous rain attenuation on millimeter-wave fixed terrestrial links located within the radar scan. The voxel rain rates are converted to specific attenuation using International Telecommunication Union [2003], and these specific attenuations may be integrated along the link path. If databases of rain rate maps are unbiased samples of the rain experienced by a region, then radar-derived joint distributions of rain attenuation and interference can be calculated for radio links of arbitrary parameters, in radio networks of arbitrary geometry. However, these data are less useful if the rain attenuation or interference time series is being investigated, because of the low temporal resolution. This is frequently the case when fade mitigation techniques are being developed or when complicated dynamic changes occur in the network, such as networks of links utilizing automatic transmit power control. [4] It is often postulated that raindrops act as a passive tracer as they fall through the atmosphere. Each drop s terminal velocity is relative to the local air movement, and so, over scales that are small relative to the size of rain events the rain rate variation has statistics similar to that of the vertical component of air movement. Kolmogorov [1991] has derived expressions for the spectral density of fluid motion from the nonlinear Navier-Stokes equations and predicts that the spectral density would have a power law form with an exponent of 5/3. 1of8

2 Figure 1. Average spectral density of log rain rate time series measured by optical rain gauge averaged over 320 events (solid line). The dash-dotted line indicates a 5/3 slope. [5] Paulson [2002] has shown that log rain rate time series, measured using an optical rain gauge ORG-715, have a spectral density that follows a power law with a 5/3 exponent; see Figure 1. Similar, although more complex, results were found by Crane [1990] and Veneziano et al. [1996]. The two-dimensional spatial variation of log rain rate, measured using meteorological radar, has been shown to have a spectral density function with a 8/3 exponent [Paulson, 2002], for the small number of events examined. This has led to models for the fine-scale variation of log rain rate within rain events being isotropic, homogeneous Gaussian random fields with power law spectral density functions. A log rain rate, power law spectral density with exponent (n + 2/3), where n is the dimension of the random field, has been shown to be a good model of spatial and temporal rain rate variation in widespread, stratiform events. [6] This is equivalent to assuming that the fine-scale variation of log rain rate, within a rain event, can be modeled as a fractional Gaussian noise process with a Hurst coefficient of 1/3, [Hurst, 1951]. Such processes are antipersistent. For rain rate this reflects the processes whereby atmospheric turbulence concentrates raindrops in a volume by depleting the concentration in surrounding volumes. Fractional Gaussian noise processes have simple scaling properties and are a class of fractals. More sophisticated models, such as that by Lovejoy and Schertzer [1992], suggested that rain could be viewed as a turbulent cascade process leading to multifractal rain rate variation. These multifractal models are applicable over wider ranges of scales, including areas of no rain, than the monofractal variation assumed in this paper. Furthermore, interpolation consistent with multifractal rain rate variation is more likely to reproduce the statistics of extreme events with long return times. However, the aim of this research is to produce rain-field time series for radio network simulation where characterization of an average year is more important than reproducing extreme events. [7] Over these infraevent scales, rain rate variation is largely determined by the inertial turbulence of the atmosphere, rather than the far more complex, mesoscale processes leading to the formation and evolution of rain events. Point rain rate time series with a sample period shorter than typical short event duration contain information on event arrival, duration, and an indication of intensity. This paper investigates the interpolation of such time series in a way that is consistent with the fractal variation expected from atmospheric turbulence. The ultimate goal is to develop this algorithm further so that time series of radar-derived rain rate maps can be interpolated to sample periods of the order of 10 s. [8] A large amount of literature exists that investigates the disaggregation of coarsely averaged rain rate data to yield ensembles of consistent, finely scaled data, [Onof et al., 1996, 1998; Mackay et al., 2001; Koutsoyiannis and Onof, 2001; Koutsoyiannis et al., 2003]. These methods are used to produce fine-scale rain rate fields from coarse-scale satellite or radar data, the output of numerical weather prediction software and stochastic pulse models. Disaggregation problems are different from the one investigated in this paper. Radar image time series yield near-instantaneous rain rate measurements, averaged over small voxels, but with sparse temporal sampling. There has been no aggregation of rain rate data between radar scans to act as a constraint on a downscaling process. The process described in this paper could be described as the stochastic interpolation of the missing radar scans. [9] In section 2 a stochastic, numerical method is developed for the interpolation of coarsely sampled, short integration time, point rain rate time series. Section 3 describes the 9 gauge years of rapid response rain gauge data that is used to test the data. A model for the point rain duration statistics is presented in section 4. The interpolation method is tested by subsampling the rain gauge data and then interpolating back to the original sampling interval. The first- and second-order statistics are compared with the original time series. These tests are presented in section Fractal Interpolation [10] The fractional Gaussian noise model of log rain rate assumes that the log rain rate time series are stationary and have a power law spectral density with exponent 5/3. The phases of the Fourier components are uncorrelated and have a uniform distribution. Variation between rain events is parameterized by the mean and variance of log rain rate and the phases of the Fourier components. This model is inadequate for describing the nonstationary start and end of an event where the log rain rate diverges to negative infinity. Furthermore, investigations of strongly convective events suggest higher power law exponents [Féral and Sauvageot, 2002]. However, for much of the duration of many events the model is a good approximation. [11] Voss [1985] devised several algorithms for the simulation of fractional Gaussian fields to produce representations of clouds and landscape in computer graphics. The original algorithm, known as sequential random displacement, refines a sequence by iteratively introducing smallerscale variation consistent with a power law spectral density. 2of8

3 points Y ={Y i ; i =1,...N} halfway between the existing abscissa forming a sequence Z ={X 0, Y 1, X 1,...,Y N, X N } using Y i ¼ 1 2 X ð i 1 þ X i Þþz i z i 2 N 0; s 2 0 : ð1þ Then E(Z) = E(Y) = E(X), and EY 2 ¼ EX 21 2 ½1 þ rð1þšþs2 0 ; ð2þ where r(1) is the lag 1 correlation coefficient, E(X 2 )r(t) = E(X i X i+t ). The mean is conserved while the variance is conserved only if Figure 2. Variance of additive noise s 2 0 = E(X 2 ) 1 2 [1 r(t)] as a function of initial period between samples. In one dimension, at each iteration, one or more points are introduced between existing samples by smooth interpolation followed by the addition of random noise. Consider a series of equispaced samples of a randomly varying signal. If a new sample is introduced by smooth interpolation halfway between the existing samples, then the highest frequency supported by the sampling would have doubled, but the spectral power density at these new frequencies would be near zero. If each new abscissa has added a sample from a zero-mean Gaussian distribution with a variance s 2, then the new part of the spectrum has random phase and near-constant power density. By repeating this procedure while reducing s 2 geometrically at each iteration, a series can be produced with an approximate power law spectral density. [12] Figure 1 indicates that the average power spectral density of log rain rate, within rain events, is well modeled as a power law with exponent 5/3 over temporal scales from 10 s up to several hours. At longer timescales the processes leading to rain rate variation changes from atmospheric turbulence to those processes determining the size and occurrence of fronts and storms. If an event s log rain rate time series is sufficiently finely sampled, then the finer-scale variation, at least down to scales of the order of 10 s, can be introduced using Voss s algorithm. This algorithm can generate an ensemble of different events. [13] If a parameter such as rain rate is sampled at a uniform sampling interval through the duration of a rain event, then the mean and variance of the samples are unbiased estimates of the mean and variance of the continuously varying parameter. If finer-scaled time series are generated by interpolation, then the ensemble average of the mean and variance should be the same as the original time series. In the next paragraph we show that Voss s algorithm conserves the ensemble mean, and the variance of the added Gaussian noise may be chosen to conserve the ensemble variance. [14] Consider a rain event log rain rate time series X = {X i ; i =0,...N}, sampled at even time intervals. The time series is interpolated by the introduction of a new set of s 2 0 ¼ EX2 1 ½ 2 1 rðþ 1 Š: ð3þ This process may be iterated, with the additive random variable being a sample from a Gaussian distribution z 2 N(0, s 2 1 ), and variance will be conserved if s 2 i =2s 2 i+1. [15] Figure 2 illustrates the first iteration Gaussian noise variance required to conserve the variance of a log rain rate time series for different sampling intervals, calculated from the 9 gauge years of data, i.e., s 2 0 derived from equation (3). For further iterations of interpolation, there is a contradiction between the condition to conserve variance and that to produce a sequence with Hurst coefficient of 1/3: s 2 i = 2 2H s 2 i+1. For gauge data a compromise is made, and a higher initial noise variance is used, and these are refined to be consistent with the Hurst coefficient. To some extent this practice is justified by the nature of the data. Rain rate time series derived from drop-counting gauges have artificially inflated high-frequency variation due to the large rain rate quantization. For radar-derived rain rates an alternative strategy is to use the geometric sequence of Voss variances to reproduce the measured Hurst coefficient and to shift and scale the resulting fields to the measured event mean and variance. [16] The proposed interpolation scheme first identifies periods of consecutive, nonzero, rain rate measurements, which are defined as events. The log rain rate time series is calculated for each event, and a sample is added at each end corresponding to a rain rate below the measurement threshold. The Voss interpolation scheme described above is then applied to each event until the desired sample rate is achieved. The interpolated event log rain rate time series is then exponentiated to yield the fine-sampled event rain rate time series. Intervals between zero rain rate measurements in the original time series are interpolated with zeros. 3. Data [17] Data from three Rutherford Appleton Laboratory rapid response drop-counting gauges ( ac.uk/raingauge.htm) are used to estimate rain rates over 10 s intervals. Two gauges are located at Chilbolton Observatory ( N, E): CH001 is situated on the flat roof of a one-story building while CH006 is sited on the ground a short distance away. The third gauge SP001 is situated on the flat roof of a two-story building, 9 km 3of8

4 Table 1. Percentage of Each Year for Which Valid Data Were Collected a Gauge Name CH CH SP a All numbers are percentages. away at Sparsholt ( N, E). Table 1 summarizes the data and their annual availability. [18] Rainwater collected in the funnel of a gauge passes through a sump to a device that produces equally sized drops. These drops are detected optically as they fall to a drain. Each drop in a 10 s period corresponds to a rain rate of 1.43 mm/hr, and this defines the rain rate quantization introduced by the gauge. Each 10 s integration period can start and end with a partially formed drop, and so collected rainwater can be carried over one or more periods. [19] The results of Figure 1 were derived from an optical gauge capable of near-continuous rain rate measurement. Drop-counting or tipping bucket rain gauges carry over collected water from one integration period to another, and this nonlinear transformation of the notional instantaneous rain rate time series leads to spectral densities that are not power laws. Therefore drop-counting gauges cannot be used to measure high-resolution autocovariances, and interpolation schemes based on power law spectral densities are not consistent with drop-counting gauge data. 4. Point Rain Duration Statistics [20] In Paulson s [2004] work these data are used to develop a point rain duration model valid for the region. Power law negative exponential (PLNE) distributions were fitted to annual duration statistics from 12 gauge years of rapid response rain gauge data, for 12 rain rate thresholds between 5 and 60 mm/hr inclusive. Smooth curves were fitted to the PLNE parameters as a function of rain rate. The resulting PLNE model predicts the number of events in an average year where the 10 s averaged rain rate exceeds R for a duration longer than T s and is specified by equations (4) and (5): X 1 NR; ð TÞ ¼ N R i¼1 b e t i= h t i ¼ 10i s; ð4þ t i t 0 where t 0 = 1 s is introduced to make the summation dimensionless and h ffi 2368ðR 2Þ 0:93 s ð5aþ b ffi 1:0 1:438R 0:419 : ð5bþ The parameter h represents an effective duration of events of different intensity while b implies a propensity for more intense events to last longer, also noted by Peters and Christensen [2002]. The power law negative exponential distribution commonly occurs in biased random walks [Ding and Yang, 1995]. [21] The constant N R is calculated for each rain rate threshold so that the duration distribution is consistent with the total time that the rain rate is exceeded in an average year. This model should be a good approximation to the average annual duration distribution for rain rates and durations that are large compared to the quantization of the gauges, i.e., >10 s and >1.43 mm/hr, and for events with average annual occurrences >1. 5. Fractal Interpolation of Rain Gauge Data [22] The fractal interpolation of rain gauge data was tested on the 9 gauge years of rapid response rain gauge data. Each gauge year of data is a time series of 10 s averaged rain rate samples, indexed i = 0, 1, 2,..., Each gauge year time series has been subsampled by retaining every Nth sample where N = 2 n and discarding the rest. A gauge year time series yields N subsampled time series. The jth time series uses samples in the original time series with indices i modulo N = j. In the experiments described below in this section, n 2 {5, 6, 7} corresponding to subsampling to one 10 s rain rate sample every 5 min 20 s, 10 min 40 s, and 21 min 20 s. [23] Each subsampling rate N yields 9 N coarsely sampled time series, each of which spans a gauge year. These time series have been interpolated back to a 10 s sample interval using the Voss-based fractal interpolation scheme described in section 2. First- and second-order statistics are calculated for the original 9 gauge years of rain gauge data and for the subsampled data interpolated back to a 10 s sampling interval. The first-order statistic calculated is the percentage of time that abscissa rain rates are exceeded. Two second-order statistics are calculated: the autocovariance of rain rates, i.e., E[R(t)R(t + t)] E(R) 2, and the rain duration statistics. The rain duration statistics are compared with the statistics for the measured data and the PLNE model. [24] Figures 3, 4, and 5 are for rain rate time series subsampled to 320, 640, and 1280 s sampling intervals, respectively. In each case the subsampled time series is interpolated back to 10 s sampling before the statistics are calculated. Figures 3a, 4a, and 5a compare the percentage of time that abscissa rain rates are exceeded from the measured gauge data (solid) and with the subsampled and interpolated data (dashed). Figures 3b, 4b, and 5b compare the rain rate autocovariance for the measured gauge data (solid) with the subsampled and interpolated data (dashed). Figures 3c, 4c, 5c, 3d, 4d, and 5d illustrate point rain duration statistics from the subsampled and interpolated data (solid). Figures 3c, 4c, and 5c compared these to the rain duration statistics derived from rain gauge data (dashed), while Figures 3d, 4d, and 5d compare with the PLNE model (dashed). From top right to bottom left the rain rate thresholds are 4.3, 7.2, 10.1, 14.4, 20.2, 25.9, 30.2, 34.5, 40.3, 44.6, 50.4, 54.7, 60.4 mm/hr. [25] The variance of the added noise in the first step of Voss s algorithm is the only parameter of the method. An 4of8

5 Figure 3. (a) Rain rate exceedance after interpolation from 320 s subsampling. Solid line is the measured gauge data, and dashed line is the subsampled and interpolated data. (b) Rain rate autocovariance after interpolation from 320 s subsampling. (c) Rain duration statistics after interpolation from 320 s subsampling (solid) compared with original data (dashed). (d) Rain duration statistics after interpolation from 320 s subsampling (solid) compared with PLNE model (dashed). initial approximation was derived using equation (3), illustrated in Figure 2. A single iteration of quasi-newton minimization was used to adjust this parameter to attempt to minimize the difference between the annual rain rate distribution of the interpolated time series, Z f int (R;s 2 0 ), to that of the original data, f meas (R), i.e., min s 2 0 R [ f int (R;s 0 2 ) f meas (R)] 2 dr. This generally changed the parameter by <30%. Figures 3a, 4a, and 5a illustrate the agreement between the measured and interpolated annual rain rate distribution. Figures 3b, 4b, and 5b compare autocovariance of the original and interpolated time series. For subsampling by factors of 32 and 64, the autocovariance differs by a scaling factor. The higher variance of the measured data is due to the gauge rain rate quantization. Figure 5b demonstrates the failure of the method when applied to data subsampled by a factor of 128. The interpolated data underestimate the number of events at all rain rate thresholds and durations. Subsampling to one sample every 20 min misses many events entirely, particularly short and intense convective events. When the initial additive noise variance is chosen in the Voss algorithm to match the first-order statistics, too high a value is chosen to reproduce the incidence of extreme values. The result is too few and too extreme events compared to an average year. [26] Rain duration statistics calculated from the data subsampled by factors 32 and 64 fit both those derived from rain gauge measurements and the PLNE model as well as the latter fit each other. The difference is certainly smaller than that due to year-to-year variation. Measured rain duration statistics suffer systematic errors at low rain rate, because of quantization of the gauge measurements. As the gauge needs to collect enough water to form a drop, low rain rates lead to time series with irregularly occurring minimum rain rate measurements. High rain rates are often inaccurately measured because of the time required for the water caught in the funnel to pass through the drop-forming sump. The lower incidence of high rain rates also leads to greater uncertainty in duration statistics. Similarly, the number of short-duration events is distorted by both the time and rain rate quantization of the gauges. The number of long-duration events is uncertain because of the small numbers measured. Given these uncertainties, the interpolated data fit quite well, particularly at thresholds around the 5of8

6 Figure 4. (a) Rain rate exceedance after interpolation from 640 s subsampling. (b) Rain rate autocovariance after interpolation from 640 s subsampling. (c) Rain duration statistics after interpolation from 640 s subsampling (solid) compared with original data (dashed). (d) Rain duration statistics after interpolation from 640 s subsampling (solid) compared with PLNE model (dashed). 6of8

7 Figure 5. (a) Rain rate exceedance after interpolation from 1280 s subsampling. (b) Rain rate autocovariance after interpolation from 1280 s subsampling. (c) Rain duration statistics after interpolation from 1280 s subsampling (solid) compared with original data (dashed). (d) Rain duration statistics after interpolation from 1280 s subsampling (solid) compared with PLNE model (dashed). 7of8

8 0.01% exceeded rain rate, which is of particular interest in radio spectrum management. 6. Conclusions [27] A method has been developed to interpolate coarsely sampled, rain rate time series of short integration time measurements. The method has been demonstrated on rain gauge data to interpolate a 10 min sampling period time series of 10 s integration time measurements to a time series with a sampling interval of 10 s. The method reproduces the expected first- and second-order statistics. Numerical experiments have demonstrated that the method fails when the sampling period becomes long compared to the duration of short events, somewhere between 10 and 20 min. It is possible to generalize the method to more spatial dimensions to allow the interpolation of time series of meteorological radar images. Notation E( ) expected value. h parameter of PLNE rain duration distribution (s). b parameter of PLNE rain duration distribution. z a sample from a normal distribution z 2 N(m, s 2 ). N(R, T) number of rain events in an average year with rain rate higher than R. (mm/hr) for duration longer than T (s). r(t) autocorrelation for lay t. X, Y, Z log rain rate time series. [28] Acknowledgment. We would like to acknowledge the support of the UK Radiocommunications Agency, in particular Dave Eden and David Bacon, the staff of Chilbolton who acquired the radar data, and other members of the RCRU for their helpful contributions. References Crane, R. K. (1990), Space-time structure of rain rate fields, J. Geophys. Res., 95(D3), Ding, M., and W. Yang (1995), Distribution of the first return times in fractional Brownian motion and its applications in the study of on-off intermittency, Phys. Rev. E, 52(1), Féral, L., and H. Sauvageot (2002), Fractal identification of supercell storms, Geophys. Res. Lett., 29(14), 1686, doi: /2002gl Goddard, J. W. F., and S. M. Cherry (1984), The ability of dual polarization radar (copular linear) to predict rainfall rate and microwave attenuation, Radio Sci., 19(1), Hurst, H. E. (1951), Long term storage capacity of reservoirs, Trans. Am. Soc. Civ. Eng., 116, International Telecommunication Union (2003), Specific attenuation model for rain for use in prediction methods, ITU-R Recomm. P.838-2, Geneva, Switzerland. Kolmogorov, A. N. (1991), The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, Proc.R.Soc. London, Ser. A, 434, Koutsoyiannis, D., and C. Onof (2001), Rainfall disaggregation using adjusting procedures on a Poisson-cluster model, J. Hydrol., 246, Koutsoyiannis, D., C. Onof, and H. S. Wheater (2003), Multivariate rainfall disaggregation at a fine timescale, Water Resour. Res., 39(7), 1173, doi: /2002wr Lovejoy, S., and D. Schertzer (1992), Multifractals and rain, in New Uncertainty Concepts in Hydrology and Water Resources, Int. Hydrol. Ser., edited by Z. W. Kundzewicz, Cambridge Univ. Press, New York. Mackay, N. G., R. E. Chandler, C. Onof, and H. S. Wheater (2001), Spatial disaggregation of rainfall for hydrological modelling, Hydrol. Earth Syst. Sci., 5(2), Onof, C., P. J. Northrop, H. S. Wheater, and V. S. Isham (1996), Spatiotemporal storm structure and scaling property analysis for modeling, J. Geophys. Res., 101(D21), Onof, C., N. Mackay, L. Oh, and H. S. Wheater (1998), An improved rainfall disaggregation technique for GCMs, J. Geophys. Res., 103(D16), 19,577 19,586. Paulson, K. S. (2002), Spatial-temporal statistics of rain rate random fields, Radio Sci., 37(5), 1088, doi: /2001rs Paulson, K. S. (2003), Prediction of diversity statistics on terrestrial microwave links, Radio Sci., 38(3), 1047, doi: /2001rs Paulson, K. S. (2004), High temporal resolution rain duration statistics in the southern UK, Proc. Inst. Electr. Eng., 151(4), Peters, O., and K. Christensen (2002), Rain: Relaxations in the sky, Phys. Rev. E, 66(3), Veneziano, D., R. L. Bras, and J. D. Niemann (1996), Nonlinearity and selfsimilarity of rainfall in time and a stochastic model, J. Geophys. Res., 101(D21), 26,371 26,392. Voss, R. F. (1985), Random fractal forgeries, in NATO Advanced Study Institute on Fundamental Algorithms for Computer Graphics, NATO ASI Ser., Ser. F, Computer and System Sciences, vol. 17, editor R. A. Earnshaw, Springer-Verlag, New York. K. S. Paulson, Radio Communications Research Unit, Rutherford Appleton Laboratory, Chilton, Didcot, OX11 OQX, UK. (k.paulson@rl. ac.uk) 8of8