Attenuation Correction and Direct Assimilation of Attenuated Radar Reflectivity Data using Ensemble Kalman Filter: Tests with Simulated Data

Transcription

1 Attenuation Correction and Direct Assimilation of Attenuated Radar Reflectivity Data using Ensemble Kalman Filter: Tests with Simulated Data Ming Xue 1,2, Mingjing Tong 1 and Guifu Zhang 2 1 Center for Analysis and Prediction of Storms and 2 School of Meteorology University of Oklahoma Norman, Oklahoma 73072, USA Submitted to Geophysical Research Letters September, 2007 * Corresponding Author Address: Dr. Ming Xue, School of Meteorology, University of Oklahoma, NWC Suite David Boren Blvd, Norman OK mxue@ou.edu 1

2 Abstract A new approach to dealing with attenuated radar reflectivity data in the data assimilation process is proposed and tested with simulated data and the ensemble squareroot Kalman filter technique. Different from the traditional approach that corrects the attenuation in the observation space first and then assimilates the data into numerical models, we build attenuation correction into the data assimilation system by calculating the expected attenuation within the forward observation operators. Doing so takes advantage of the knowledge about the hydrometeor species and their DSDs within the model and data assimilation systems. Tests with simulated X-band reflectivity data for a supercell storm show that our attenuation procedure is very effective the analysis results are almost as good as those from un-attenuated data. The analysis errors are very large if no attenuation correction is applied. The effect of attenuation and its correction when radial velocity data are assimilated as well is also discussed. 2

3 1. Introduction Compared to longer-wavelength S-band weather radars, X-band radars have smaller antennas, therefore lower construction costs, and provide relatively higherresolution measurements. X-band radars are also more suitable for airborne deployment and can be more cost-effectively deployed in high-density networks. The latter include the experimental X-band networks of CASA (Center for Collaborative Adaptive Sensing of the Atmosphere, McLaughlin et al. 2007). The Observing System Simulation Experiments (OSSEs) of Xue et al. (2006, hereafter XTD06) using simulated data have shown that the assimilation of additional data from CASA-type radars improves the analysis of a supercell storm. However, in that study, simulated radar data were assumed un-attenuated. Compared to S-band, attenuation poses an additional challenge at X-band, especially when mixed phases exist and/or when polarimetric information is not available (see discussion in Bringi and Chandrasekar 2000). The presence of wet ice particles, such as melting or water-coated hail or graupel, can further complicate the attenuation correction problem. Existing attenuation correction techniques are all applied in the observation space, i.e., based on direct observation data. Such an approach usually requires prior assumptions about the hydrometeor species as well as their drop/particle size distributions (DSDs); it is difficult to quantitatively consider the amount of individual species with such an approach. This current study explores an alterative approach in which attenuation correction is built into the forward observation operators of radar reflectivity measurement, and the attenuation correction is performed simultaneously with the state estimation through a 3

4 data assimilation process. With this approach, attenuated radar measurements are directly assimilated. The ensemble Kalman filter (EnKF) (Evensen 1994) is particularly useful for radar data assimilation, because flow-dependent background error covariances can be estimated from the forecast ensemble and used to retrieve state variables not directly observed. Tong and Xue (2005, TX05 hereafter) show that EnKF is able to effectively assimilate reflectivity data in the presence of mixed phase microphysics. The EnKF method is therefore chosen as the assimilation tool in this study. The attenuation equations together with the equations for equivalent reflectivity factor Z e and attenuation coefficient k as functions of hydrometeor state are described in section 2. The experimental setup and data assimilation configurations are given in section 3. The results of the OSSEs with and without attenuation correction are compared in section 4 and a summary is given in section Reflectivity and attenuation equations In this section, we present the equations for X-band reflectivity and attenuation, which form part of the forward observation operator for radar reflectivity data. The operator is used in both radar data simulation and assimilation. First, the relations between Z e, k and the state of hydrometeors are derived based on assumed drop size distribution (DSD). These relations, combined with the radar emulator introduced in XTD06, form the complete forward observation operator. a. The attenuation equation The measured equivalent reflectivity factor in the presence of attenuation at a given range r can be expressed as 4

5 Z () r = Z () r A() r, (1) m e where Ze( r ) is the equivalent reflectivity factor before attenuation, r Ar ( ) = exp[ 0.46 ksds ( ) ] is the two-way path-integrated attenuation (PIA) factor for 0 equivalent reflectivity, and k is the attenuation coefficient (db km -1 ). The attenuated reflectivity in dbz can be obtained by applying 10log 10 ( ) to Eq. (1), so that ( ) ( ) 0 r Z r = Z r 2 k( s) ds, (2) where Z() r and Z () r are reflectivity in dbz before and after attenuation. It can be seen that the path-integrated attenuation (PIA) in db, i.e., PIA = 10log 10 Ar ( ), is equal to twice the integral of k between range 0 and r, reflecting the effect of two-way attenuation. For the purpose of data assimilation, the attenuation correction can be achieved by including Eq. (2) as part of the observation operator for reflectivity, since it is Z that is observed. The radar reflectivity factor Z e and attenuation coefficient k are directly related to the state of hydrometeors. In the following, we will derive the relations between Z e, k and the water mass content of hydrometeors. b. Z e -W and k-w relations for X-band wavelength The Z e and k are linked to hydrometeor mass content (W, in mass per unit volume of air) through DSD. To be consistent with the DSD assumption in the microphysics scheme used by our forecast model, which is a 5-class (cloud water, rain, cloud ice, snow and hail/graupel) single-moment scheme after Lin et al. (1983, hereafter LFO83), the DSDs of rain, snow and hail/graupel are also assumed to have an exponential form: N(D) = N 0 exp( ΛD), (3) 5

6 where N 0 is the intercept parameter and Λ is the slope parameter. The intercept N 0 is a fixed constant, and the slope parameter is then uniquely linked to W ( = ρ q, where ρ a α is air density and q is the mixing ratio) given the assumptions on the DSD and hydrometeor density. The hydrometeor content and radar variables are represented by weighted integrals over the DSDs as follows: W = π ρ Z e D 3 N(D)dD 4 λ = σ ( ) ( ) 5 2 b DNDdD π K W k = σ e( D) N( D) dd 3 [g m ], (4) 6 3 [mm m ], (5) 1 [db km ], (6) where ρ is the density of hydrometeors, K W = ε r 1 ε r + 2 is the dielectric factor of water, σ b is the backscattering radar cross section and σ e the extinction cross section for hydrometeor particles. The cross sections are calculated using Mie theory. In practice, within data assimilation, we want to avoid direct Mie scattering calculation and integration over DSD for efficiency reason. We derive relatively simple parameterized relations between model predicted W with Z e and k. The procedure is as follows. 1) Use (3) in (4) and solve for Λ as a function of W, yielding the slope parameter Λ= N 0πρ W 1/4. 2) Let W vary in its possible range and calculate Λ then the corresponding Z e and k using (5) and (6). 6

7 3) Performing least-square-fitting to the data for Z e -W and k-w in log domain, leading to power-law relations Z b Z bk e = aw Z and k akw =. (7) The above procedure is applied to all hydrometeor species and the results are given below. 1) Rainwater The backscattering and extinction cross sections of raindrops are calculated at temperature of 10 o C, with the dielectric constant of (55.43, 37.85), a complex number. Reflectivity and attenuation are calculated with the rain intercept parameter N 0r = m -4 (r is for rain), the data are then fitted to the power-law relations to give Z er = W in mm 6 m -3, (8) r k r = 0.319W in db km -1. (9) 1.38 r 2) Dry snow and hail The calculation and fitting procedures are the same as those for rain except for the use of the corresponding intercept parameters and densities. For snow, N 0s = m -4, ρ s = 0.1 g cm -3 (s is for snow), and for hail N 0h = m -4, ρ h = g cm -3 (h is for hail), all are default values used in LFO83. The resultant formulas are Z es = W in mm 6 m -3, (10) s k s = W in db km -1, (11) 1.28 s Z eh = W in mm 6 m -3, (12) h k = 0.159W in db km -1. (13) h 1.64 h 7

8 3) Melting snow and hail The melting ice model of Jung et al. (2007) is used to derive the formulas for melting snow and hail. Since Mie theory for hydrometeor scattering is needed at the X- band, the coefficients are derived as functions of percentage of melting. The melting percentage f w is calculated following Eqs. (2) and (3) of Jung et al. (2007). The density of melting snow is also diagnosed from the melting percentage f w (Eq. (4) of Jung et al. 2007). Using the same procedure as used above, we obtained the coefficients for the power-law relations. For snow: a = ( f 5.58 f ) 10, (14) 2 5 Zs ws ws b = f f, (15) 2 Zs ws ws a = f 50.5 f f, (16) 2 3 ks ws ws ws b = f f 1.24 f, (17) 2 3 ks ws ws ws and for hail: a = ( f 5.98 f ) 10, (18) 2 5 Zh wh wh b = f f, (19) 2 Zh wh wh a = f f f, (20) 2 3 kh wh wh wh b = f f f. (21) 2 3 kh wh wh wh 3. Experiment Setup The 20 May 1977 Del City, Oklahoma supercell storm (Ray et al. 1981) is simulated using the ARPS model (Xue et al. 2001) to serve as the truth for OSSEs. The model domain is km 3 in size, and the horizontal and vertical grid spacings 8

9 are 2 and 0.5 km, respectively. The radar is located at the southwest corner of the model domain. The same simulation configuration is used in TX05. The simulation of radar data follows XTD06 by using a Gaussian power weighting function in the vertical for observations simulated on radar elevation levels. The X-band radar is assumed to operate in the standard WSR-88D precipitation scan mode, having 14 elevations with one volume scan every 5 minutes and a 1 beam width. The radar range is large enough to cover the entire storm. Gaussian-distributed errors of zero mean and 1 m s -1 and 2 dbz standard deviations, respectively, are added to simulated radial velocity (V r ) and reflectivity (Z) from the truth simulation data. These values are also used to specify the observation error variances in the assimilation except for two experiments to be discussed later. The data assimilation algorithm is based on the ensemble square-root Kalman filter (EnSRF) of Whitaker and Hamill (2002), and the filter configurations follow the CNTL experiment of Tong and Xue (2007) exactly. Briefly, spatially smoothed perturbations are added to the first guess of the ensemble-mean initial condition that is horizontally homogeneous as defined by the Del City sounding. The initial ensemble forecast starts at 20 min of the simulated supercell storm. Radar observation volumes are assimilated every 5 min, starting at 25 min until 100 min. Two sets of experiment are performed, one assimilates Z data only and the other assimilates both V r and Z. Forty ensemble members are used, following XTD06. The analyzed state variables include velocity components u, v, w, potential temperature θ, pressure p, and mixing ratios of water vapor q v, cloud water q c, rainwater q r, cloud q i, snow q s, and hail q h. Covariance localization radius used is 6 km and no covariance inflation is applied. Both sets of experiments contain four runs; their names start with NA, NAC, 9

10 NACLE and AC, and end with ZV or Z (e.g., NAZV). NA stands for no attenuation, which assumes that the radar data are not attenuated at all and accordingly no attenuation correction is applied. Attenuated radar data are assimilated in experiments whose names start with NAC, NACLE and AC, but attenuation correction, as outlined in previous sections, is only applied in AC cases. In experiments starting with NACLE, a larger error variance of (10 dbz) 2 is specified for the Z data, reflecting the fact that the data contain attenuation-related error. 4. Results of Experiments Fig. 1 shows the simulated radar reflectivity on the 4.3 elevation with and without attenuation at 70 and 100 min of model time. In the unattenuated fields shown in the left panels of Fig. 1, high reflectivity (Z > 45 dbz) is found at 3 to 4 km above ground, which is mainly associated with high mixing ratios of rainwater and hail, including melting hail. The most significant attenuation occurs behind the high reflectivity region opposite of the radar (located at the southwest corner of the grid). As can be seen in the right panels of Fig. 1, the Z behind, or to the northeast of, the precipitation core of the right moving cell (near the center of domain) is completely attenuated, resulting in a wedge of no reflectivity. The maximum reflectivity in the core region is reduced by about 10 dbz. Such pattern and magnitude of attenuation appear realistic. Fig. 2 compares the root-mean square (RMS) errors of ensemble mean analyses of all 8 experiments, including the set (NAZ, ACZ, NACZ, and NACLEZ) that assimilates Z data only (gray curves) and the set (NAZV, ACZV, NACZV, and NACLEZV) that assimilates both Z and V r (black curves). Similar to the results of our 10

11 earlier studies (TX05 and XTD06), with no attenuation, the ensemble mean analysis RMS errors of NAZ and NAZV (thick gray and thick black curves, respectively) are very low during the later cycles for all state variables. For example, the errors of u and v are below 1 m s -1, that of w is below 0.5 m s -1 while those of hydrometeors are close to or below 0.05 g kg -1. Between them, the errors of NAVZ, with the help of V r data, are consistently lower, and more so in the earlier cycles. When our attenuation correction procedure is applied in ACZ and ACZV, the error levels of all variables (thick dashed gray and thick dashed black curves, respectively, in Fig. 2) during the intermediate and later cycles are very close to those of the corresponding no attenuation cases, indicating that our attenuation correction procedure is very effective. There is more difference in the early cycles between the attenuation correction and no attenuation cases, because the state estimation is not very good yet therefore attenuation calculation based on the estimated state is not very accurate at the stage. It is interesting to note that if attenuated data are assimilated as if they were not attenuated, as in NACZ and NACZV, the analysis errors (thick dashed gray and thick dashed black curves, respectively) are rather large, especially during the later data assimilation cycles when attenuation is more severe with large hydrometeor production in the storm system. The errors of u and v remain above 1.5 m s -1 throughout the period and are significantly above 2 m s -1 at the end of assimilation. The errors of hydrometeor fields are many times larger than those of corresponding no attenuation or attenuation correction cases. It is interesting to note that the errors of NACZV are only slightly lower in general than those of NACZ, despite the inclusion of quality V r data. This indicates a 11

12 significant negative impact of the attenuated Z data when no correction is applied, even when quality V r data are available. Obviously, attenuation correction is very important. In NACZ and NACZV, the error variance specified in EnKF for the attenuated reflectivity data is still the low value of (2 dbz) 2. In NACLEZ and NACLEZV, this error variance is increased to the more appropriate (10 dbz) 2 to reflect the larger errors due to attenuation. It turns out that the analysis of NACLEZ (thin gray solid curves) are significantly worse than that of NACZ (thin gray dashed curves), and even more so in the later cycles. This is apparently because when only Z is assimilated, specifying a rather larger error variance for the Z data further decreases the constraint of the observations imposed on the model solution, resulting in worse storm analysis. When a larger error variance of (10 dbz) 2 is specified for the Z data in NACLEZV (thin black solid curves), the analysis is noticeably improved over that of NACZV (thin black dashed curves) during the later cycles, rather than becoming worse as in the Z only case of NACLEZ. This is because in this case, Z data get a reduced weight in the assimilation, allowing quality V r data to have a larger positive impact. The above findings are further corroborated by the comparison of analyzed lowlevel model fields (of cold pool perturbation potential temperature θ', reflectivity Z and perturbation wind vectors) shown in Fig. 3 for the end time of assimilation (100 min). Immediately clear is that the analyses of ACZ and ACZV (d and h) with attenuation correction are very close to the truth (a), while those of NACZ and NACZV (b and f) are similarly poor, with the reflectivity patterns look similar to that of attenuated truth in (e). There are clear differences in the analyzed perturbation wind fields of these two runs from that of truth, although that of NACZV is better due to the inclusion of V r data. 12

13 When a large error variance is specified for Z data in NACLEZV (g), the analysis is much better than that of NACZV, due to the increased impact of V r data. In fact, despite the use of attenuated Z data without correction in NACLEZV, the analyzed Z field looks closer to the truth in (a) than to the attenuated truth in (e). The analysis of NACLEZ is the worst among all experiments, where the impact of available attenuated Z data is further reduced by large specified error. The analyzed cold pool is the weakest in this case (c) while that of NACLEVZ is rather good (g). 5. Summary In this article, we introduced a new approach to dealing with attenuated radar reflectivity data and tested it with simulated data and the ensemble square-root filter data assimilation technique. Different from the traditional approach that performs attenuation correction in the observation space and then assimilates the data into NWP models, we build the attenuation effect into the data assimilation system by calculating the expected attenuation within the forward observation operators. Doing so allows us to take advantage of the knowledge about the hydrometeor species and their DSDs within the model and data assimilation systems. As the model state estimation improves through the data assimilation cycles, the estimate and correction of attenuation also improve. Our simulated data show total attenuation of X-band reflectivity behind the precipitation core of a supercell storm. It is shown that without attenuation correction, the analyzed storm and precipitation core are much weaker. The RMS errors of the analyzed model fields are 4 to 10 times larger than those of the corresponding no attenuation case, depending on the fields. With our attenuation correction procedure, the quality of storm analysis is almost as good as the corresponding no attenuation case, proving the 13

14 effectiveness of the method. It is also shown that when quality radial velocity data are assimilated at the same time, the attenuation in reflectivity data still has similar negative impact on the analysis if no attenuation correction is applied. The negative impact is reduced when the Z data is weighted less by specifying a larger error variance for the data and when V r data are also assimilated, but the negative impact is further increased when only Z data are available. Acknowledgement: This work was primarily supported by NSF grants EEC , ATM and ATM Computations were performed at the Pittsburgh Supercomputing Center. 14

15 References Bringi, V. N. and V. Chandrasekar, 2000: Polarimetric Doppler Weather Radar. Cambridge, 636 pp. Evensen, G., 1994: Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J. Geophys. Res., 99( C5), Jung, Y., G. Zhang, and M. Xue, 2007: Assimilation of simulated polarimetric radar data for a convective storm using ensemble Kalman filter. Part I: Observation operators for reflectivity and polarimetric variables. Mon. Wea. Rev., Accepted. Lin, Y.-L., R. D. Farley, and H. D. Orville, 1983: Bulk parameterization of the snow field in a cloud model. J. Climate Appl. Meteor., 22, McLaughlin, D., E. Knapp, Y. Wang, and V. Chandrasakar, 2007: Short wavelength technology and the potential for distributed networks of small radar systems. IEE Radar 2007 Conference Digest. Ray, P. S., B. Johnson, K. W. Johnson, J. S. Bradberry, J. J. Stephens, K. K. Wagner, R. B. Wilhelmson, and J. B. Klemp, 1981: The morphology of severe tornadic storms on 20 May J. Atmos. Sci., 38, Tong, M. and M. Xue, 2005: Ensemble Kalman filter assimilation of Doppler radar data with a compressible nonhydrostatic model: OSS Experiments. Mon. Wea. Rev., 133, Tong, M. and M. Xue, 2007: Simultaneous estimation of microphysical parameters and atmospheric state with radar data and ensemble square-root Kalman filter. Part I: Sensitivity analysis and parameter identifiability Mon. Wea. Rev., Accepted. 15

16 Whitaker, J. S. and T. M. Hamill, 2002: Ensemble data assimilation without perturbed observations. Mon. Wea. Rev., 130, Xue, M., M. Tong, and K. K. Droegemeier, 2006: An OSSE framework based on the ensemble square-root Kalman filter for evaluating impact of data from radar networks on thunderstorm analysis and forecast. J. Atmos. Ocean Tech., 23, Xue, M., K. K. Droegemeier, V. Wong, A. Shapiro, K. Brewster, F. Carr, D. Weber, Y. Liu, and D.-H. Wang, 2001: The Advanced Regional Prediction System (ARPS) - A multiscale nonhydrostatic atmospheric simulation and prediction tool. Part II: Model physics and applications. Meteor. Atmos. Phy., 76,

17 Figure Captions Fig. 1. Simulated Z observations at 70 (upper panels) and 100 (lower panels) minutes of model time at the 4.3 elevation level, without (left panels) and with (right panels) attenuation. Fig. 2. Ensemble mean analysis RMS errors averaged over points where true Z is greater than 10 dbz for (a) u, (b) v, (c) w, (d) θ, (e) q v, (f) q c, (g) q r, (h) q i, (i) q s, and (j) q h, for experiments NAZ (thin solid black), NACZ (thin gray), NACLEZ (thin gray dashed) and ACZ (thin black dashed), and experiments NAZV (thick solid black), ACZV (thick black dashed), NACZV (thick solid gray), and NACLEZV (thick gray dashed). Units are shown in the plots. Fig. 3. Perturbation wind (vectors; m s -1 ), perturbation θ (thick black lines for 0 K and thin-dashed contours at 0.5 K intervals) and computed Z (thin solid contours and shading at intervals of 5 dbz) at z = 250 m of truth simulation (a), attenuated truth (e), and ensemble mean analyses from experiments labeled in the figure, at 100 min or the end of assimilation cycles. 17

18 Z(dBZ) MIN= 0.00 MAX= Z(dBZ) MIN= 0.00 MAX= y (km) (a) Z(dBZ) MIN= 0.00 MAX= (b) Z(dBZ) MIN= 0.00 MAX= y (km) (c) x (km) (d) x (km) Fig. 1. Simulated Z observations at 70 (upper panels) and 100 (lower panels) minutes of model time at the 4.3 elevation level, without (left panels) and with (right panels) attenuation. 18

19 u (m/s) 5.0 v (m/s) w (m/s) θ (K) qv (g/kg) a) b) c) 2.0 d) 1.0 e) RMS error qc (g/kg) qr (g/kg) qi (g/kg) qs (g/kg) qh (g/kg) f) g) h) i) j) RMS error time (min) time (min) time (min) time (min) time (min) Fig. 2. Ensemble mean analysis RMS errors averaged over points where true Z is greater than 10 dbz for (a) u, (b) v, (c) w, (d) θ, (e) q v, (f) q c, (g) q r, (h) q i, (i) q s, and (j) q h, for experiments NAZ (thick solid gray), ACZ (thick dashed gray), NACZ (thin dashed gray) and NACLEZ (thin solid gray), and experiments NAZV (thick solid black), ACZV (thick dashed black), NACZV (thin dashed black) and NACLEZV (thin solid black). Units are shown in the plots. 19

20 64 48 Truth NACZ NACLEZ ACZ y (km) (a) (b) (c) Attenuated truth NACZV NACLEZV ACZV (d) y (km) (e) x (km) (f) x (km) (g) x (km) (h) x (km) Fig. 3. Perturbation wind (vectors; m s -1 ), perturbation θ (thick black lines for 0 K and thin-dashed contours at 0.5 K intervals) and computed Z (thin solid contours and shading at intervals of 5 dbz) at z = 250 m of truth simulation (a), attenuated truth (e), and ensemble mean analyses from experiments labeled in the figure, at 100 min or the end of assimilation cycles. 20