Assimilation of Doppler radial winds into a 3D-Var system: Errors and impact of radial velocities on the variational analysis and model forecasts

Transcription

1 QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY Q. J. R. Meteorol. Soc. 134: (2008) Published online in Wiley InterScience ( Assimilation of Doppler radial winds into a 3D-Var system: Errors and impact of radial velocities on the variational analysis and model forecasts Fathalla A. Rihan, a Chris G. Collier, a * Susan P. Ballard b and Sean J. Swarbrick b a School of Environment and Life Sciences, Salford University, Gt. Manchester, UK b Met Office, JCMM, Meteorology Building, Reading University, Reading, UK ABSTRACT: A pre-processing approach to the assimilation of Doppler radar radial wind data into a high resolution Numerical Weather Prediction (NWP) model is described. We discuss the types of errors which might occur in radar radial winds. The construction of simulated high resolution radar data containing such errors is described. Examples of these data are presented and compared with actual radar data from the Chilbolton S-band radar located in central southern England. A new approach to specifying the radial velocity observation error is proposed based upon the radial gradient of the velocity along the pulse volume. The production of super-observations for input to a 3D-Var assimilation system is discussed. Impact of the assimilation of Doppler velocities on the 3D-Var analysis and on the model forecasts, for two case-studies, are shown. Copyright 2008 Royal Meteorological Society KEY WORDS data assimilation; Doppler radial velocity; observation; operator; super-obbing Received 29 March 2007; Revised 4 April 2008; Accepted 1 September Introduction Numerical Weather Prediction (NWP) is considered as an initial/boundary-value problem: given an estimate of the present state of the atmosphere, the model simulates (forecasts) its evolution. Specification of proper initial conditions and boundary conditions for the numerical dynamical models is essential in order to have a well-posed problem and subsequently a good forecast model. (A well-posed initial/boundary problem has a unique solution that depends continuously on the initial/boundary conditions.) The goal of data assimilation is to construct the best possible initial and boundary conditions, known as the analysis, from which to integrate the NWP model forward in time. Over the last thirty years or so, networks of weather radars, providing measurements of radar reflectivity from which rainfall has been estimated, have been established within operational observing systems. Initially the radars, operating at S-band (10 cm) or C-band (5 6 cm) wavelengths, did not have the capability to measure the motion of the targets (mainly hydrometeors but also insects and birds, and for high power systems, refractive index inhomogeneities) towards or away from the radar site. During the last twenty years or so, weather radars having Doppler capability measuring radial motion of the targets have become standard such that now in Europe well over half * Correspondence to: Chris G. Collier, School of Environmental and Life Sciences, Salford University, Gt. Manchester, M5 4WT, UK. E- mail: c.g.collier@salford.ac.uk of the operational radars are Doppler systems (see Collier (2001)). Considerable effort has been, and continues to be, put into the development of nowcasting techniques based upon the extrapolation of radar reflectivity fields aimed at generating forecasts of precipitation up to 3 6 hours ahead (for a review, see Collier (1996) and Krzysztofowicz and Collier (2004)). Whilst such systems have met with some success, particularly when incorporating wind fields from mesoscale numerical models (Golding, 1999), they are not appropriate for forecasting to longer lead times. Improvements to forecasts for these lead times are now being sought through the assimilation into mesoscale models of surface rain rates derived from radar reflectivity using latent heat nudging methods (Macpherson et al., 1996) and variational techniques in which model reflectivity is compared with actual measured reflectivity (Sun and Crook, 2001). More recently, Doppler radar radial winds have also been assimilated into NWP models as vertical wind profiles derived from Velocity Azimuth Display (VAD) analysis (Benjamin et al., 2004) and using variational techniques; see Sun and Crook (1997, 1998), Crook and Sun (2004), Lindskog et al.(2004), Gao (1999, 2006) and Qiu et al. (2006). The models require observations with high spatial and temporal resolution to determine the initial conditions, for which purpose radar data are particularly appealing. However, the resolution of Doppler radar observations is much higher than that of the mesoscale NWP model. Before the assimilation, these data must be pre-processed to be representative of the characteristic scale of the Copyright 2008 Royal Meteorological Society

2 1702 F. A. RIHAN ET AL. model. To reduce the representativeness error and to enable the data to more closely correspond to the model resolutions than do the raw observations, one may spatially average (or interpolate) the raw data to generate the so-called super-observations. In order to assimilate Doppler radial velocity observations, the observation errors which come from several sources must be estimated for inclusion in the variational system. In this paper, we present a new approach to the pre-processing of the Doppler radar wind data. We outline the likely errors in estimates of Doppler radar radial winds, and how they might be represented mathematically. To illustrate the results, we apply the methodology to two case-studies for data collected using the 3 GHz radar situated at Chilbolton in southern England (51.14 N and 1.44 W). The large 25 m antenna affords a beam width of 0.28, using PPI data (Plan Position Indicator, with constant elevation angle) at 300 m range resolution and 0.25 azimuth resolution. We illustrate the structure of radial wind fields for simulated data with realistic errors and compare it with actual observations. We also investigate the impact of assimilation of Doppler radial velocities and their errors on the variational analysis and on the model forecasts. This paper is organized as follows: section 2 describes the sources of errors in radial winds. In section 3, we describe a simulation model which is used to analyse actual radial winds. In section 4, we present a description of the Met Office 3D-Var system for assimilation of Doppler radial winds. In section 5, we discuss the main steps of pre-processing the Doppler radial data that includes data quality control and super-obbing the very high resolution raw data. We describe the representation of the observation errors of radial winds in section 6. The proposed methodology is used to derive the variation of the errors with range. Section 7 provides a discussion of assimilation of radial winds in PPI format and the observation operator. In section 8, we investigate the impact of assimilation of radial velocities on the 3D-Var analysis and on the model forecasts. Section 9 provides the meteorological analysis of the two casestudies. Conclusions and plans are outlined in section Errors in the determination of Doppler radial velocity Targets moving away from or towards a radar produce a Doppler shift between the frequency of the transmitted signal (pulse), and the signal reflected from the targets and received back at the radar. However, ambiguities may arise in these measurements due to range folding and velocity aliasing (see Doviak and Zrnic (1993)). Fortunately, procedures have been developed to minimize these problems (see for example Gong et al. (2003)). Other problems remain, namely the existence of data holes (where there are no targets), and irregular coverage, instrumental noise and sampling errors. Various types of interpolation schemes have been used to fill in data holes and poor coverage (see for example Lin et al. (1993)), although such schemes are unnecessary when three-dimensional assimilation schemes are implemented. However, the impacts of instrumental noise and sampling are more problematic. May et al. (1989) discuss, and assess, a number of techniques used to estimate the Doppler shift in the received signals. The Doppler shift is proportional to the slope of the phase of the autocorrelation function (at zero lag) of the returned signals. An estimator of the shift is the phase at the first lag divided by the value of the lag in time units. This is known as pulse pair processing, and may be improved by averaging more than one value of the phase divided by the lag (poly-pulse pair). An alternative approach is to estimate the Doppler shift directly from the first moment of the Doppler spectra (Doviak and Zrnic, 1993) perhaps using a maximum likelihood estimator (similar to a least squares fit) of the logarithmic spectral signal. A further technique is possible based upon the analysis of the power spectrum, its circular convolution and Fast Fourier Transform (FFT) of the same. Interestingly, it was concluded by May et al. (1989) that the major limitation to the radar performance is the small-scale variability of the wind along the pulse volume. Therefore there is little to be gained by using complicated algorithms to estimate the Doppler shift. The width of the Doppler spectrum, usually assumed to be Gaussian, determines the correlation time of the signal. Therefore, the error in the radial velocity measurements depends on the strength of the returned signal and the spread or width of the Doppler velocity spectrum, which in turn depends mainly on reflectivity and velocity gradients within and across the pulse volume: see Doviak and Zrnic (1993). Sampling errors depend upon the size of the pulse volume corresponding to each data point. The Chilbolton radar in central southern England has a pulse volume of about 300 m by 0.25, and, even though this is relatively small, even smaller-scale wind variability may introduce different sampling errors from measurement point to measurement point. In practice the sampling errors could be weakly correlated from point to point, but only a very small additional error will be introduced if this is ignored. Practically, sampling errors dominate since instrumental errors are usually minimized in operational systems. In the following we outline a system for creating artificial radar radial wind datasets within which different types of error may be included. Figure 1 shows the instrumentally induced effects (with symmetric distribution) and the proposed schematics of the impact upon a Gaussian Doppler spectrum of various effects of strong wind shear (with positive skewness) along the pulse volume. Several of these effects upon the Doppler spectrum may be present in the same radar image, and, in the case of geophysically induced effects, their magnitude may vary with range and azimuth. The height and size of the pulse volumes will increase with increasing distance from the radar.

3 ASSIMILATION OF DOPPLER RADIAL WINDS 1703 observed, then we use the formula v r = u sin θ cos α + v cos θ cos α + w sin α (1) Figure 1. Distribution of wind speed error due to (a) variations of instrumental noise with symmetric distribution, and (b) strong velocity gradient along the pulse volume at a specific range and azimuth with chi-squared distribution (positive skewness). This figure is available in colour online at 3. Comparison of simulated and actual radar winds Air movement varies over time and space. However, Doppler radar allows the measurement of only one (radial) component of the velocity of the targets at a specific range and azimuth. To illustrate the Doppler wind field seen from a single radar we simulate it from a known three-dimensional wind field. The simplest case is to consider a horizontally uniform wind field for both horizontal and vertical (precipitation fall velocity) components. In such a case, if we make measurements of the velocity along circles centred at the radar by azimuthal scanning at a constant elevation angle α (PPI), we get, for a constant distance from the radar, a sinusoidal dependency of the measured radial velocity on the azimuthal angle θ. Assuming that the horizontal wind velocity v h and hydrometeor fall speed w are uniform over the area being to derive the wind velocity and we add simulated errors calculated as Gaussian noise. In Figure 2 we display the simulated data and compare the variation with azimuth at 30 km range of the simulated data and a perfect sine wave. The impact of the simulated errors is to cause the differences between the data (dots) and the no-error sine curve (solid line) shown. Figure 3 (left) displays a real-data case for the PPI scan from the Chilbolton radar on 1 July 2003, and Figure 3 (right) shows the radial velocity versus azimuth angle from the observed data shown. The real observed Doppler winds differ from the simulated fields due to horizontal and vertical variations in the atmosphere caused by boundary layer structure such as the Ekman spiral, turbulence and synoptic scale motions combined with the impact of real measurement and instrumental errors. 4. 3D-Var data assimilation The Met Office implemented a global 3D-Var system (see Lorenc et al. (2000)) in March 1999, a UK 12 km limited-area version in October 1999 (see Macpherson et al. (2002)) and an enlarged European and North Atlantic 20 km version in The global system was extended to 4D-Var in October 2004, a 4 km resolution UK model was introduced operationally in 2005 with 3D-Var, and 4D-Var was introduced operationally for an enlarged domain 12 km limited-area model in This variational analysis system operationally uses conventional observations such as radiosonde, aircraft data, polar-orbiting satellite radiances and surface synoptic observations. The Met Office are extending the system to allow assimilation of radar radial Doppler winds which Figure 2. Artificial radial velocity with Gaussian noise (left), and the data for radial velocity (shown as dots) versus azimuth angle at 30 km range (right). The solid line is the variation without any error. Left panel: y-axis = distance north (km); x-axis = distance east (km); scale in m/s. This figure is available in colour online at

4 1704 F. A. RIHAN ET AL. Figure 3. Observed Doppler radial winds on 1 July 2003 from Chilbolton radar (left), and the data displayed as radial velocity versus azimuth angle at a selected range of 30 km (right). Left panel: y-axis = distance north (km); x-axis = distance east (km). This figure is available in colour online at have recently become available from some of the operational C-band radars in southern England. Here we use the 12 km UK domain 3D-Var system and a reduced area 4 km version for experiments with assimilation of radar radial Doppler winds from a single elevation scan of the S-band Chilbolton research radar. The Met Office 3D-Var system uses an incremental formulation (for a review see, for example, Rihan et al. (2005)) producing a multivariate incremental analysis for pressure, density, horizontal and vertical wind components, potential temperature and specific humidity in the model space, consistent with a non-hydrostatic, height coordinate model. Cloud water and fraction are neglected in the basic system. Under the assumption that the background and observation errors are Gaussian, random and independent of each other and that the increments and model are at the same resolution (as they were in the 12 km model and these experiments), the optimal estimate x a = x b + δx in the analysis space is given by the incremental cost function J(δx) = 1/2δx T B 1 δx + 1/2 (H δx y + H x b ) T E 1 (Hδx y + H x b ), (2) where δx x a x b is the state vector of the analysis increments (the estimated variable is then given by H x b + H δx), x b is the state vector of the background of x that includes the horizontal and vertical wind components, and y is the observation vector that includes the observed radial winds in the observation space. H is the nonlinear observation operator that relates the model variables to the observation variable and a transformation between the different grid meshes, and H is the linear observation operator with elements h ij = H i / x j. B is the background covariance matrix and E is a diagonal matrix of the error covariance in the observations, and in the observation operator; see Lorenc (1997). Miller and Sun (2003), Xu and Gong (2003) and Xu et al. (2006) assumed that the observation error covariance matrix E is diagonal with constant diagonal elements given by the estimated observation error, which was taken as 1 m/s for typical radar observations. We provide later in this paper a different approach to the representation of the errors of observed radial winds. The incremental cost function minimization is performed in a preconditioned control variable space (see Lorenc (1997)). The preconditioned control variables are stream function, velocity potential, the unbalanced part of the hydrostatic pressure and relative humidity, and the space is spectral components of vertical modes. The transformation is achieved by a change of variables which are then multiplied by an operator modelling the inverse of the square root of the background error covariance. This happens in four stages. Firstly, they are transformed in the vertical into uncorrelated vertical error covariance modes. The same vertical modes are used for all vertical columns in the grid. Secondly, the vertical modes are scaled by the square root of their error variance. The vertical part of the background error covariance matrix is represented as the eigenfunctions, E, and eigenvalues, λ. The eigenfunctions form an orthogonal basis set of functions which are used to represent the data. The square root of the eigenvalues, λ, is the square root of the error variance in each mode and is used to scale the vertical modes. Thirdly, a horizontal spectral transform is performed for each vertical mode. In the global version this is a standard Legendre polynomial spectral transform. In the limited-area version a sine transform is performed in both horizontal directions. Fourthly, and finally, the horizontal correlations are removed within each field by dividing by the power spectrum of the (horizontally spectrally) transformed background error correlation. This is done independently for each vertical mode. Note that the power spectrum is the normalized variance in each horizontal wave number, i.e. variance in this wave number/sum of variances in all wave numbers.

5 ASSIMILATION OF DOPPLER RADIAL WINDS 1705 The double-sine transform in the limited-area version of 3D-Var provides a constraint which forces analysis increments to be zero at the boundary for scalar fields and the normal component of the wind. The forecast error covariances are based on 63 days of (T + 24 T + 12) forecast differences for deriving vertical modes. Fixed length-scales are used in a SOAR (Second Order AutoRegressive) function for the horizontal analysis of all variables 130 km for unbalanced pressure, 180 km for stream function, 90 km for relative humidity and 130 km for velocity potential. Here, we assume that the matrix E includes the errors from the observations (original measurements), observation operator, and the super-obbing procedure (Super-obbing is a technique to combine (re-scale) the radar observations, using statistical interpolation, at a larger spatial scale which is compatible with the model; see section 5.2.). For the experiments here the background fields are produced from a short forecast at either 12 or 4 km resolution with T + 3 h window valid at the analysis time. Hourly forecast fields are interpolated to the time and location of the observations. In the next two sections we explain how to pre-process data to make it suitable for the variational assimilation system. 5. Pre-processing of Doppler radial wind data 5.1. Quality control The major steps that may be used in processing radial Doppler wind data before the assimilation are unfolding, interpolation of data from/to a Cartesian grid, removing excessively noisy data, and filtering. For our purposes we do not need to interpolate to a Cartesian grid and are using the raw polar data as our input to the quality control and super-obbing. Data quality control (QC) is a technique, which should be performed for each scan, to remove undesired radar echoes, such as ground clutter and clutter arising from anomalous atmospheric propagation clutter (AP clutter), sea clutter, velocity folding, and excessive noise: see e.g. Alberoni et al. (2003), Gong et al. (2003). Doppler velocity, in the Chilbolton radar, is measured using both horizontally and vertically polarised pulses. This parameter is the component of the target velocity towards the radar (positive velocities are towards the radar) (There are no known biases that need correcting for the dates used in this paper.). With the Chilbolton data used in this paper unfolded radial Doppler velocity is used. The Chilbolton data has the velocity unfolded from its range of +/ 15 m/s. Processing is applied (Robin Hogan, Reading University, personal communication) such that there is removal of noise from target-free areas, and removal of ground clutter. These are set to missing value in the files. The algorithm uses the correlation between nearby pixels to distinguish good signal from noise or clutter. It is effective at removing bad data, although sometimes might remove a little too much good signal. Then the velocities are unfolded. The algorithm chooses a pixel near the radar that it can assume is within the Nyquist folding range, then performs a flood-fill approach to unfold each region of contiguous rainy pixels. It sometimes makes a mistake and a whole region may have its velocity out by a multiple of 30 m/s. This is easy to spot by eye. For the cases reported here the quality control and super-obbing were done prior to ingestion of the data into the Met Office observation processing system and so the standard background checks against model background velocity and buddy checks were not performed Super-obbing radial wind data Doppler radars produce raw radial wind data with high temporal and spatial density. The horizontal resolution of the data is around 300 m (this is too high to be used in the assimilation scheme) whereas the typical resolution of some operational mesoscale NWP models is of the order of several kilometres. To reduce the representativeness error, and match the observations to the horizontal model resolution, one may use spatial averages of the raw data, called super-observations (Figure 4). The desired resolution for the super-observations can be generated by defining parameters (which can be freely chosen) for the range spacing and the angle between the output azimuth gates; see for example Henja and Michelson (1999). As we have mentioned previously, a direct assimilation of PPI data with no vertical interpolation is recommended (see Sun and Crook (2001)). Moreover, assimilation using radar data directly at observation locations avoids interpolation from an irregular radar coordinate system to a regular Cartesian system, which can often be a source of error especially in the presence of data voids (see Rihan et al. (2005)). We have developed a software package, which is based on spatial interpolation in polar space, for processing raw volume data of radial velocity in PPI format to produce super-observations to the required resolutions for the 3D-Var system in the Met Office. The major steps in the processing are interpolation of data in polar coordinates using bilinear approximation in the beam direction and Fourier approximation in the azimuth direction (The errors produced from Fourier approximation for a function with oscillatory behaviour are small compared with other methods of interpolation (as long as there are no missing data).). If the length of a vector x is m, andx has sample interval dx, then the new sample interval for the interpolated vector y, that contains the value of the periodic function x and re-sampled to n equally spaced points, is dy = dx n/m. Some local discontinuities can be seen in the raw data due to radar echoes from non-meteorological targets, such as birds or aliased wind observations. Such discontinuities increase with the range (Figure 4). Since the super-obbing procedure is based on statistical interpolation (at a larger spatial scale which is compatible with the model), the expected error of the super-obbing procedure is regarded as the representative error of

6 1706 F. A. RIHAN ET AL. the super-observations. It is supposed that the error of super-observations is uncorrelated with the background error. The representativeness errors are then accounted for by using the local standard deviation during the interpolation, which increases with the range. Figure 5 shows the standard deviation of the radial velocity errors, due to the super-obbing process, in m/s as a function of range for the 1 July 2003 case. The larger values of the errors as range increases are due to some voids in the raw data. The interpolation method fills in missing data as zero which unfortunately can introduce some values outside the range of the raw data, producing higher errors. Small values produced in data voids and associated with isolated raw data are filtered out by removing values below a threshold. From Figure 4 it can be seen that the super-obbed data are less smooth than the raw data due to this interpolation using zero values in data-free areas, see for example 68 km range and zero azimuth. 6. A representation of errors for observation radial winds In order to optimally assimilate Doppler radar radial velocity observations into an NWP model, it is necessary to know their error covariances in addition to the errors included by the derivation procedure of the superobservations. We assume that the observational errors are uncorrelated in space and time. Under this assumption, the observation error covariance matrix ( E) in the cost function (2) can be reduced to a diagonal matrix. Then the matrix E, in Equation (2), is regarded as a weighting coefficient that reflects the relative precision of the data (measurement uncertainty and representativeness error). The matrix can be expressed as: = diag {σ 2 (ε)} (3) where σ 2 (ε) is the error variance of the radial velocity v r. The most common errors in radar radial winds are (1) the noise in the radial velocity induced by the velocity gradient across the pulse volume with variance σ 2 (ε v ), (2) the instrumental error due to hardware degradation of variance ˆσ 2 (ε i ), and (3) the error due to the superobbing procedure (see Figure 5). Miller and Sun (2003) state that these measurement errors need to be specified so that radar observations can be properly assimilated for NWP. However, they note that the mean radial velocity and spectral width estimators are proportional to the radar wavelength and the time spectral width, and therefore are rather impractical as estimates of the measurement errors. They therefore note a need for error estimators of radial velocity that can be obtained from the measurements themselves Error due to the velocity gradient within and along the pulse volume The local sampling of the radial velocity is employed to approximate the error variance σ 2 v r, since noisy data are usually associated with high values of radial velocity variance. Errors in the original measurements of the radial velocity within each radar pulse volume depend on the strength of the returned signal and the spread (or width) of the Doppler velocity spectrum that depends on the velocity gradients. Since the radar scatterers in the pulse volume move randomly, we assume that the errors of the velocity gradient of the radar backscatterers are given by a normal (Gaussian) distribution, where 1 pdf(ε v ) = e ε v/2 σ 2 dε v. (4) 2π σ The error variance σ 2 is modified by the velocity gradient (which varies with time) along the pulse volume. The variations of this velocity difference along the pulse volume cause the kinetic energy (KE) of the moving scatterers to change. We will assume here for simplicity that this velocity difference is taken in the radial direction only. Here, rate of change in KE = F v r (5) where F is an arbitrary force applied to the scatterers and v r is the velocity difference along the pulse volume. Thus the increase in the kinetic energy in time interval dt, during which the scatterer moves a distance dr, is d(ke) = dw (6) where dw is the work done by the force F in the infinitesimal displacement dr taken as the pulse length and over time dt. Rogers and Tripp (1964) show that the change in the kinetic energy per unit mass can be expressed as d(ke) = σ 2 (v r ) (7) where σ 2 (v r ) is the variance of the mean Doppler velocity, which can be expressed by (see Doviak and Zrnic (1993)) σ 2 (v r ) ={λ/(8 πmt)} ϕ, (8) where λ is the radar wavelength, ϕ is the true spectral width, M is the number of equally spaced pulses, and T is the time between pulses. (The maximum unambiguous (Nyquist) velocity is v nyq = λ/4t.) Nastrom (1997) investigated the factors impacting on the spectral width of Doppler radar measurements. The Doppler spectral width provides information about the spread of Doppler radial velocities being measured by the radar, and is a combination of effects due to turbulence, wind shear and on occasion non-meteorological factors related to the way in which the radar samples. When the radar sampling is at higher elevation angles, spectral width will also contain a component related to the variance in fall speed of the target hydrometeors. For very small bandwidths it was found that the variance was dominated by the effects of wind speed changes along the radar beam. Unfortunately the spectral width, ϕ, ofthe

7 ASSIMILATION OF DOPPLER RADIAL WINDS 1707 Figure 4. An example of the Doppler radar radial raw data (left) and the super-obbed Doppler radial winds (right) at 4 km resolution along the radar beam and 1 azimuth angle resolution. Top left and right panels: y-axis = distance north (km); x-axis = distance east (km). Bottom left and right panels: y-axis = azimuth (deg); x-axis = range (km). This figure is available in colour online at STD (errors due to super-obbing m/s) range(km) range(km) Figure 5. Local super-observation error standard deviation (STD) at 4 km resolution at 12 UTC, 1 July This figure is available in colour online at

8 1708 F. A. RIHAN ET AL. pulse volume observations was not recorded for the cases studied here so a functional form was used assuming that the variance was a function of the beam elevation and the vertical wind shear. The change of radial velocity along the radar beam is largely caused by the vertical wind shear as the beam increases with altitude as the range increases. It is also to be expected that the kinetic energy variation of the scatterers in the pulse volume will increase with increasing range as the radar beam gets wider and the pulse volume greater with increasing range. Here we neglect changes due to beam width and assume a simple formula to express the error variance σ 2 (ε v ) of radial winds in terms of the gradient variance along the pulse volume in a radial direction (from one pulse to another) as follows: σ 2 (ε v ) ={1 exp( v r /v r )} σ 2 (v r ) (9) where v r is the gradient of the radial velocity, measured as a centred difference across the pulse volume in the radial direction (see Bringi and Chandrasekar (2001) (Equation (9) is based upon the equation (5.212) modified by the presence of noise (5.210b) given in Bringi and Chandrasekar (2001). The equation in the book derives from pulse pair processing of the radar signals and assumes that the power spectral density of meteorological echoes is approximated by a Gaussian shape with mean velocity v and standard deviation σ (v).). The term {1 exp( v r /v r )} has been chosen as it makes the value of σ 2 (ε v ) a minimum near the radar. The error in radar radial winds due to the velocity gradient along the pulse volume varies with the range R. Figure 6 (left) displays the local radial velocity errors (m s 1 ) calculated using (9) for the case of 1 July 2003, shown in Figure 3 (left). It indeed shows an increase with range; however in this case it seems sensitive to whether raw or interpolated data extracted at observation locations is used in the calculation, the latter having sharper gradients near data voids (due to use of zeros for missing data). This requires further investigation. Figure 6 (right) shows a proposed S-function for the observation errors as a function of the range which could be used to represent the errors of the radial velocity. Of course there is an element of double-counting arising from the lack of independence of error due to the geometric calculation and the error due to the velocity gradient from one pulse to the next. This may cause an overestimate of the total error used in the variational analysis. It may be possible to reduce the impact of this problem by averaging over a number of pulses, but further work is needed to validate such an approach Error due to hardware degradation Although the instrumental error can have a significant impact on the retrieval, in practice it is difficult to determine how this error varies with time. In this case we assume that the instrumental error does not vary temporally, and take the instrumental error variance as ˆσ 2 (ε i ) assuming there is no hardware degradation with time. Therefore the total error variance of the radial winds is given by σ 2 (ε) = σ 2 (ε v ) +ˆσ 2 (ε i ) = (1 exp ( v r /v r )) σ 2 (v r ) +ˆσ 2 (ε i ) (10) at each (r, θ). In the 3 GHz Chilbolton radar, the measurement accuracy for Doppler radial velocity is about m/s. The instrumental error may not be a function of (r, θ), but is a function of time. In this case, we assume that this error is represented by a skewed distribution such as a chi-squared distribution with probability density function (for ν degrees of freedom) given by pdf v (ε i ) = ε(ν 2)/2 e ε i/2 2 ν/2 Ɣ(ν/2), for ε i [0, ]. (11) Here Ɣ (y) isthegamma function, andε i is considered as the instrumental error. Thus the error variance of the degradation is defined by ˆσ 2 (ε i ) = pdf v (ε i )(ε i ε i ) 2 dε i, (12) where average ε i is the mean value of the instrumental error. 7. Direct assimilation and observation operator of PPI data Due to the poor vertical resolution of radar data, a vertical interpolation of radar data from constant beam elevation levels to model Cartesian levels can result in large errors. For this reason a direct assimilation of PPI data with no vertical interpolation was recommended in Sun and Crook (1998, 2001). However, radar data has higher horizontal resolution than that of the model (the poorest polar radar data has approximately 0.5 km at the farthest range distance). An observation operator must be formulated to map the model variables from model grid into the observation locations such that the distance between the observations and model solution is estimated in the cost function. Thus, we take advantage of the vertical resolution of the model being much better than that of radar data. The observation operation, H e, for mapping (and averaging) the data from the model vertical levels to the elevation angle levels is formulated as v r,e = H e (v r ) = Gv r z/ G z, (13) where v r,e is the radial velocity on an elevation angle level, v r is the model radial velocity, and z is the model vertical grid spacing. The function G = e α2 /2β 2 represents the power gain of the radar beam; β (in radians) is the beam half-width; and α is the distance from the centre of radar beam (in radians). The summation is over the model grid points that lie in a radar beam. Here

9 ASSIMILATION OF DOPPLER RADIAL WINDS 1709 STD (observation error (m/s) Observation error range (km) range (km) Figure 6. Local errors (m/s) of the raw observation radial velocity (not super-obbed) for the 1 July 2003 case (left), and the proposed S-function for the observation errors as a function of the range (right). The errors are measured over all azimuths for each range. We can see the variability of the error increases as the range increases. y-axes = observation error (m/s); x-axes = range (km), from 0 to 90 km. This figure is available in colour online at Figure 7. Impact of assimilation of radar radial winds on the 3D-Var analysis with 12 km (top) and 4 km (bottom) model resolution for the wind speed at level 5 (approx. 420 m), at 12 UTC on 1 July From left to right: the PF (Perturbation Forecast) analysis without radial winds, the PF analysis with radial winds, and the difference. Colour scales for top panels (left to right): m/s; m/s; m/s. Bottom panels (left to right): m/s; m/s; m/s. This figure is available in colour online at

10 1710 F. A. RIHAN ET AL. Figure 8. Impact of assimilation of radar radial winds on the 3D-Var analysis with 12 km (top) and 4 km (bottom) model resolution for the wind speed at level 5 (approx. 420 m), at 12 UTC on 10 July From left to right: the PF analysis without radial winds, the PF analysis with radial winds, and the difference. Colour scales for top panels (left to right): m/s; m/s; 2 4 m/s. Bottom panels (left to right): m/s; 1 15 m/s; 2 12 m/s. This figure is available in colour online at we make a simplification and ignore the beam width and hence function G. We next discuss the observation operator to convert the Cartesian model components to the radial components Observation operator The observation operator here is based on that described by Lindskog et al. (2004). There are two types of observation operator. One is used to interpolate and transfer the radar data from observation locations to the model grids. The second is used to map the model data into the observation locations. In the case of a direct assimilation of radar radial wind at constant elevation angles, which is not a model variable, the observation operator involves: (1) a bilinear interpolation of the NWP model horizontal and vertical wind components u, v and w to the observation location; (2) a projection of the interpolated NWP model horizontal wind, at the point of measurement, towards the radar beam using the formula v h = u sin θ + v cos θ, (14) where θ is the azimuth angle (clockwise from due north). The elevation angle should include a correction which takes account of Earth surface curvature and radar beam refraction (see Lindskog et al. (2004)). Then the third step (3) involves the projection of v h in the slantwise direction of the radar beam as v r = v h cos (α + ϕ) + w sin (α + ϕ), and ϕ = tan 1 {r cos α/(r sin α + d + h)},(15) where α is the elevation angle of the radar beam. The formula for ϕ represents approximately the curvature of the Earth. In the term ϕ, r is the range of observation, d is the effective radius of the Earth and h is the height of the radar above sea level. The effective Earth radius is 1.3 times the actual Earth radius; this compensates for radar beam refraction. Note that the factor of 1.3 is valid only in the case of a standard atmosphere profile. The factor actually varies according to atmospheric conditions and should ideally be calculated from the variation of refractive index with height (see Lindskog et al. (2004)). The effect of this correction term on results is very small.

11 ASSIMILATION OF DOPPLER RADIAL WINDS 1711 Table I. Root-mean-square errors. Case I Case II O-B O-A O-B O-A 4 km model km model Impact of assimilation of Doppler radial winds In this section we investigate the impact of assimilation of radial velocity on the regional 3D-Var analysis and on the model forecasts. Two experiments with the 3D-Var system have been performed at 12 km and 4 km resolution, using the Chilbolton radial winds PPI scan data for two cases: at 12 UTC on 1 July 2003 and 10 July The raw data were averaged to about 4 km resolution and estimated observational errors assigned. The size of the radar radial wind observation error standard deviation assigned was a function of observational error, range and the hardware error magnitude. The magnitude of the total error used was of order 2 6 m/s (larger errors with increasing range), values which are similar to the errors in the radar winds discussed in the review carried out in the EU COST 75 project (see Collier (2001)). This range is similar to the range obtained by adding the errors due to the super-obbing procedure (Figure 5), the error due to radial velocity gradient across the pulse volume (Figure 6) and the hardware error (0.5 m/s). It should be noted that in generating the error due to the super-obbing procedure there is an element of double-counting arising from the lack of independence of the error due to the geometric calculation, and error due to the velocity gradient from one pulse to the next. The results could mean that the magnitude of the total error used in the variational analysis may be overestimated. In this paper we have ignored this by recognising that in using a particular value of the error in the experiment we have introduced a degree of arbitrariness. In each Var experiment analyses were generated using surface, aircraft and radiosonde observations (and also SatWinds in the 4 km experiments), both with and without radial wind observations. Figure 9. Impact of assimilation of Doppler radial winds on the forecast. Atmospheric wind speeds at T + 3(top)andT+ 6 (bottom), level 5 (approx. 420 m), from 12 UTC on 1 July From left to right: the forecast without radial winds, the forecast with radial winds, and the difference. Colour scales for top panels (left to right): 3 21 m/s; 3 21 m/s; 4 11 m/s. Bottom panels (left to right): 1 15 m/s; 1 15 m/s; 5 11 m/s. This figure is available in colour online at

12 1712 F. A. RIHAN ET AL. Figure 10. Impact of assimilation of Doppler radial winds on the forecast. Atmospheric wind speeds at T 1 (top) and T (bottom), level 5 (approx. 420 m), from 12 UTC on 10 July From left to right: the forecast without radial winds, the forecast with radial winds, and the difference. Colour scales for top panels (left to right): 1 17 m/s; 1 17 m/s; 4 7 m/s. Bottom panels (left to right): 1 17 m/s; 1 17 m/s; 5 7 m/s. This figure is available in colour online at Combinations of radial winds (and their errors) and other input data can interact to produce nonlinear results. However, in the cases studied the initial experiments indicated that the radial winds do seem to produce a significant impact on the analysis. Figure 7 shows the impact of the Chilbolton radial winds on the NWP analysis, with 12 and 4 km model resolution. The lefthand panel shows wind speed analysis without radial winds (computed from the analysed u and v increment fields) at level 5 (approx 420 m) for the case of 1200 UTC on 1 July The middle panel is the analysis when radar radial data have been added to the other source of observations, and the right-hand panel is difference between the two analyses, to show the contribution of the radial winds observations to the analysis. Note that the impact of using radial winds is confined to the vicinity of the coverage of the Chilbolton radar located in central southern England. The same analysis was conducted for the 1200 UTC, 10 July 2004 case. Figure 8 shows the same analysis as that of Figure 7. The top panels in this Figure are for the 12 km grid model and the lower panels are for the 4 km grid model. Note the significant impact of assimilation of the radial winds around the location of the radar. Table I shows the root-mean-square (r.m.s.) errors produced at the beginning and end of the 3D- Var minimization for the analyses including the radar Doppler wind data. Results for the two cases (Figures 7 and 8) and two different model/analysis grid sizes (4 km and 12 km) are given. The r.m.s. values for the 4 km model are larger than those of the 12 km model in case II, presumably due to greater detail and variation in the high resolution forecast fields. In case I the 4 km background was an interpolated 12 km resolution forecast and as the background errors were the same in the 12 km and 4 km analyses and it used the same observations, the final analyses are very similar. In case II the 4 km background came from a trial assimilation system at 4 km resolution. R.m.s. errors are produced at the beginning and end of the 3D-Var minimization for the analyses including the radar Doppler wind data, i.e. the fit to the radar Doppler wind super-observations of the background (O-B) and the analysis (O-A). The two case-studies are: 1 July 2003 (Case I) and 10 July 2004 (Case II). Our experiments indicate (from the quantitative results in Table I and qualitative results in the Figures) that it is possible to assimilate the super-obbed Doppler radar radial observations in PPI format into the mesoscale NWP

13 ASSIMILATION OF DOPPLER RADIAL WINDS 1713 Figure 11. Impact of assimilation of Doppler radial winds on the forecast. Atmospheric wind speeds at T + 3(top)andT+ 6, level 5 (approx. 420 m), from 12 UTC on 10 July From left to right: the forecast without radial winds, the forecast with radial winds, and the difference. Colour scales for top panels (left to right): 2 14 m/s; 2 14 m/s; 4 7 m/s. Bottom panels (left to right): 2 14 m/s; 1 15 m/s; 5 7 m/s. This figure is available in colour online at model without producing unreasonable r.m.s. values. The model counterpart is calculated by the observation operator discussed in section 7.1. The results show that the radar data do have an effect on the analysis and on the forecast. However, the results obtained depend strongly on the observation errors that vary with the range. Atmospheric model forecasts for the speeds at T + 3 ( 15 UTC), and T + 6( 18 UTC), level 5 from 12 UTC on 1 July 2003 are displayed in Figure 9. The model forecasts at T 1, T + 0, and at T + 3, T + 6 from 1200 UTC on 10 July 2004 are also shown in Figures 10 and 11. Radial winds appear to have significant impact on both the 3D-Var analysis that provides the model initialization, and then on the model forecasts. As one expects, there is no impact of radial winds at T 1. Indeed, the impact at T + 6 is larger over southeast England than at T + 3. In the case of Figure 9, 1 July 2003, this is consistent with the subsequent development of the line convection which occurred over southern England on this day as the impact is to increase the lowlevel convergence of the air, producing increased ascent. 9. Meteorological analysis of the 1 July 2003 and the 10 July 2004 case-studies The 1 July 2003 case-study is an example of an ana-cold front (line convection) at 13 UTC located from the Isle of Wight towards the northeast, crossing London. The intense core of the front is about 25 km wide narrowing to about 5 km wide over London. Figure 7 shows that both the 12 km and 4 km model analyses give some indication of the front, but neither analysis represents the true spatial scale. The inclusion of the Doppler radial winds actually moves the winds associated with the front further towards the northeast. It is clear that the resolution of the model is not adequately representing the frontal dynamics, and the radial winds are not able to improve the analysis. The forecasts shown in Figure 9 show only minimum impact of the assimilated radial winds except in a few specific locations not associated with the frontal position. The 10 July 2004 case-study is an example of the triggering of convection that developed into three parallel arcs of showers and thunderstorms. An upper-level shortwave trough extended from the southwestern tip of Ireland to south Wales as shown in Figure 12(top). A

14 1714 F. A. RIHAN ET AL. Figure 12. (a: top) Met Office operational surface analysis at 00 UTC 10 July 2004, and the forecast for this time made 24 h earlier. The shaded box shows the area where the case-study of Morcrette et al. (2006) was based. (b: bottom) Rain-rate (mm h 1, see colour code) at 12 UTC on 10 July 2004, derived from the weather network. Pixels are 1, 2 or 5 km square depending on the best resolution available at that location. The diagonal line distinguishes the area of convective outbreaks associated with the short-wave trough (to the northwest of it) from the arcs of showers that developed ahead of it. H and L show the locations of Herstmonceux and Larkhill sounding stations. From Morcrette et al. (2006). This figure is available in colour online at

15 ASSIMILATION OF DOPPLER RADIAL WINDS 1715 detailed analysis of this case has been described by Morcrette et al. (2006). At 1200 UTC the precipitation associated with the trough had reached south Wales extending northeastwards into the Midlands. However, a separate system of multiple north south-oriented bands of convective precipitation had developed to the south and east (Figure 12 (bottom)). The wind fields derived from the 12 km and 4 km model analyses without the Doppler radial winds shown in Figure 8 do not show these two convective bands. However, the assimilation of the Doppler winds does introduce increased wind fields associated with two distinct areas, one over south Wales towards the northeast and the other over the coast of southern England. The latter is some way from the location of the convection over southern England, but the former is located in the region of the convection. The forecast wind fields shown for this case in Figure 11 show the convection from south Wales moving towards the northeast, and the southern band moving along the south coast. However, the impact of the assimilation of the radial winds diminishes with increasing forecast lead time such that at T + 6 the impact is small except in a few isolated locations which may be spurious. 10. Conclusions The mesoscale 3D-Var system has been developed in the Met Office to include the capability of assimilation of Doppler radial velocities. This paper has been concerned primarily with the statistical errors of the radial wind data, the pre-processing of the data before assimilation that includes radial velocity unfolding, and super-obbing the very high resolution raw data to match the model resolutions. The observation operator for the Doppler radial velocity has also been developed and incorporated in the Var system. Examples of radial winds derived from the Chilbolton radar and a simulation model have been considered. The impact of the Doppler radial winds and their errors on the variational assimilation system and on the model forecasts has been investigated. The Met Office 3D-Var system has been run with (and without) Doppler radial velocities (in PPI format) with 12 km and 4 km model resolution. The Met Office Unified Model has also been run with the obtained initializations having a time window of 3 h. A mathematical representation of observation errors for radial winds has been used, and the form of representation has been tested. The S-curve representation seems to be appropriate, and consistent with the super-obbing technique and observations for the particular cases studied. The numerical experiments have suggested that the incorporation of the Doppler radar radial winds into the variational analysis do produce some impact on the analysis and the forecast, but this remains to be tested further before a definite conclusion can be reached. In future work at the Met Office the system will be extended to use multi-elevation scan Doppler winds from the operational radar network both within 3D-Var and 4D-Var at 1.5 km model resolution. Alternative methods of super-obbing, quality control and monitoring will be investigated and implemented within the observation processing system. The super-obbing and observation errors will be investigated for many more cases to see whether the relationships seen here are typical or whether they vary from case to case. The spectral width information will be captured and investigated to provide more information on the observational error. Acknowledgements The work at Salford represented in this paper is sponsored by the National Environmental Research Council (NERC) through the Universities Weather Research Network (UWERN), a part of the NERC Centres for Atmospheric Science (NCAS) (now known as NCAS Weather). The help of Dr Fay Davies (Salford University) in developing of the simulation of radial velocities is acknowledged. The data for the 10 July 2004 case were acquired in part from the NERC CSIP project dataset. References Alberoni PP, Ducrocq V, Gregorič G, Haase G, Holleman I, Lindskog M, Macpherson B, Nuret M, Rossa A Quality and assimilation of radar data for NWP a review. COST-717 Action, 38 pp. Benjamin SG, Dévényi D, Weygandt SS, Brundage KJ, Brown JM, Grell GA, Kim D, Schwartz BE, Smirnova TG, Smith TL, Manikin GS An hourly assimilation forecast cycle: The RUC. Mon. Weather Rev. 132: Bringi VN, Chandrasekar V Polarimetric Doppler weather radar: Principles and applications. Cambridge University Press. 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