An Application of the Multi Physics Ensemble Kalman Filter to Typhoon Forecast

Pure Appl Geophys Ó 2013 Springer Basel DOI 10.1007/s00024-013-0681-y Pure and Applied Geophysics An Application of the Multi-Physics Ensemble Kalman Filter to Typhoon Forecast CHANH KIEU,1,3 PHAM THI MINH,2 and HOANG THI MAI2 Abstract—This study examines the roles of the multi-physics approach in accounting for model errors for typhoon forecasts with the local ensemble transform Kalman filter (LETKF) Experiments with forecasts of Typhoon Conson (2010) using the weather research and forecasting (WRF) model show that use of the WRF’s multiple physical parameterization schemes to represent the model uncertainties can help the LETKF provide better forecasts of Typhoon Conson in terms of the forecast errors, the ensemble spread, the root mean square errors, the cross-correlation between mass and wind field as well as the coherent structure of the ensemble spread along the storm center Sensitivity experiments with the WRF model show that the optimum number of the multiphysics ensemble is roughly equal to the number of combinations of different physics schemes assigned in the multi-physics ensemble Additional idealized experiments with the Lorenz 40-variable model to isolate the dual roles of the multi-physics ensemble in correcting model errors and expanding the local ensemble space show that the multi-physics approach appears to be more essential in augmenting the local rank representation of the LETKF algorithm rather than directly accounting for model errors during the early cycles The results in this study suggest that the multi-physics approach is a good option for short-range forecast applications with full physics models in which the spinup of the ensemble Kalman filter may take too long for the ensemble spread to capture efficiently model errors and cross-correlations among model variables Key words: LETKF, ensemble data assimilation, multiphysics ensemble Introduction Evaluating impacts of model internal uncertainties in tropical cyclone (TC) forecasting models is a Laboratory for Weather and Climate Forecasting, Hanoi College of Science, Vietnam National University, Hanoi 10000, Vietnam E-mail: chanhkq@vnu.edu.vn Center for Environmental Fluid Dynamics, Hanoi College of Science, Vietnam National University, Hanoi 10000, Vietnam I M Systems Group, NOAA/NWS/NCEP/EMC, Camp Spring, MA 20746, USA challenging problem While there are many different sources of errors related to observational errors, insufficient vortex initialization, or spurious correlations (see, e.g., DALEY, 1993; ANDERSON, 2007; BAEK et al., 2006; LI et al., 2009), model deficiencies associated with underrepresented physical processes are perhaps the main cause of the forecast errors in TC models In particular, fast sub-grid processes such as turbulence forcing or cloud microphysics are highly variable under the TC extreme wind conditions that inadequate parameterizations of these processes could impact considerably the TC track and intensity forecast skills Numerous studies have shown that a simple change of microphysics scheme or boundary parameterization could result in very different forecasts even with the same initial condition, especially for forecasts of high-impact mesoscale systems such as heavy rainfall or TCs (ZHU, 2005; VICH and ROMERO, 2010; BYUN et al., 2007; LI and PU, 2009; IM et al., 2007; KIEU and ZHANG, 2010; PU, 2011) As a quick illustration, Fig shows four 36-h forecasts of the accumulated rainfall over the Indochina Peninsula, using the weather research and forecasting model (WRF-ARW, V3.2, SKAMARAOCK et al., 2005) These forecasts are initialized with the same boundary and initial condition valid at 1200 UTC 13 July 2010 at which Typhoon (TY) Conson (2010) started to develop rapidly The initial and boundary conditions are taken from the National Center for Environmental Prediction (NCEP) Global Forecast System (GFS) forecast products All model forecasts have similar model configurations except for four different microphysics schemes including the Lin et al., scheme, Kessler scheme, WSM 3-class simple ice scheme, and the WSM 5-class scheme One can notice easily that both the magnitude and distribution of accumulated precipitation differ substantially in the C Kieu et al Pure Appl Geophys four forecasts (Fig 1), especially around and in the eastern part of TY Conson where both the maximum accumulation and the coverage of the rainfall vary widely; for the Kesser and Lin et al., microphysics scheme, the peaked rainfall exceeds 310 mm and is confined mostly along-the-track of TY Conson while the rest have distribution of rainfall extended farther to the east of the Indochina Peninsular area The diverse forecasts due to different microphysics schemes in the above simple example demonstrate one of many severe mesoscale weather systems that are sensitive to forecasting models As seen in Fig 1, the errors associated with such poorly physical representation could suppress any benefit of observational information assimilated into the model no matter how good the initial condition is This Figure Thirty six hours forecasts of the accumulated rainfall using the WRF model with the same initial and boundary condition but with different microphysics schemes including a Kessler microphysics scheme; b WSM 3-class simple ice scheme; c WSM 5-class scheme; and d Lin et al., microphysics scheme All simulations are configured with a single domain of 36-km resolution and initialized at 1200 UTC 12 July 2010 from the GFS global input data and boundary updated every 6-h Multiphysics Ensemble Typhoon Forecast poses some real challenge to any TC model in which a single deterministic forecast could easily provide bias track and intensity information Model errors are thus essential and have to be considered properly in any TC forecasting systems The importance of model errors has been recognized and examined extensively in previous studies (e.g., ANDERSON and ANDERSON, 1999; HOUTEKAMER et al., 2005; WHITAKER and HAMILL, 2002; SZUNYOGH et al., 2008; MENG and ZHANG, 2007; LI et al., 2009; Evensen, 2009) For unbiased systems, ANDERSON and ANDERSON (1999) proposed an approach in which the a priori covariance matrix is enlarged every cycle by a multiplicative factor k [ 1, the so-called multiplicative covariance inflation technique This approach is to some extent equivalent to an assumption that the model error is proportional to the background covariance by a factor of (k - 1), and, thus, bearing all spatial structures of the priori covariance matrix In another approach, MITCHELL and HOUTEKAMER (2000), and HAMILL and WHITAKER (2005) suggested that one can add a random distribution to the posteriori analysis perturbations such that the spread of the ensemble and, consequently, the analysis covariance matrix are augmented The additive and the multiplicative inflation methods have been tested in various assimilation systems and have proven to give encouraging results (see, e.g., HOUTEKAMER et al., 2005; ANDERSON, 2007; WHITAKER and HAMILL, 2002; SZUNYOGH et al., 2008; HUNT et al., 2005; LI et al., 2009) A somewhat different approach was introduced by ZHANG et al (2004) in which modified analysis perturbations are computed by weighting the background and newly-obtained analysis perturbations This technique can be shown to be roughly equivalent to degrading the quality of observations, i.e., to increasing the observation errors, thus inflating the posterior analysis To further enlarge the ensemble spread, FUJITA et al (2007), MENG and ZHANG (2007), KIEU et al (2012) proposed to employ multiple physical parameterizations, which were demonstrated to help improve the performance of the ensemble Kalman filter (EnKF) for TC forecasts In addition to the above approaches for the unbiased models, a number of techniques that deal with the biased models have been also developed (see, e.g., DEE and DA SILVA, 1998; BAEK et al., 2006; DANFORTH and KALNAY, 2008) A review of different model error correction techniques in the presence of the model bias can be found in LI et al (2009) Because the early TC development and subsequent intensification are challenging problems due to the sensitivity of the storm track and the intensity for forecasting models, the case of TY Conson (2010) is chosen in this study to examine the relative importance of the multiple physics (MP) and the multiplicative inflation (MI) in correcting model errors in the EnKF algorithm The MI approach is currently considered as an effective treatment of model errors in many practical applications of the EnKF in regional models While there are several other techniques such as additive inflation or various adaptive versions of the localization that are shown to be valuable (see, e.g., ANDERSON, 2007; BISHOP and HODYSS, 2009; MIYOSHI, 2011), thorough validation of all these inflation techniques is challenging in the context of the regional models with the full primitive equations settings Therefore, this study is limited to examining the performances of the MP and MI approach for the ease of implementation and comparison A variant version of the EnKF, the so-called Local Ensemble Transform Kalman Filter (LETKF), is adopted and implemented in the WRF-ARW model system for our investigation Recent studies have demonstrated that the LETKF is a potential candidate for various real-time global and regional applications (HUNT et al., 2005; SZUNYOGH et al., 2008; MIYOSHI and YAMANE, 2007; MIYOSHI and KUNII, 2012) By far, the MI approach and its related adaptive algorithms are the most common method in accounting for model errors in the LETKF (HUNT et al., 2005; LI et al., 2009; MIYOSHI, 2011) Thus, it is of significance to examine the importance of the MP method against the MI method in forecasting TCs with the LETKF algorithm Because a single case study could be exposed to representativeness errors that make it hard to apply to more general situations, a series of complementary idealized experiments with the Lorenz 40-variable (LORENZ and EMANUEL, 1998) whose model errors are assumed to be represented by a random forcing function will also be conducted Note again that the main objective of this study is to examine how the multi-physics ensemble can help the LETKF algorithm improve the TC forecasts relative to the multiplicative covariance inflation technique C Kieu et al Pure Appl Geophys Therefore, potential representativeness errors arisen from a single case are expected to be of secondary importance for such relative comparison In the next section, a quick overview of the LETKF will be presented Section describes model experiments and data Results are discussed in Sect In Sect 5, some sensitivity experiments with the idealized Lorenz 40-variable model are presented to provide some further information into the dual role of the multi-physics in the EnKF algorithm Discussions and conclusions are given in the final section J wị ẳ k 1ịwT fI Xb ịT ẵXb Xb ịT Xb gw 2ị ỵ Jẵxb ỵ Xb w; LETKF Algorithm Recent studies with LETKF have demonstrated that this ensemble scheme is capable of handling a wide range of scales and observation types (see, e.g., HUNT et al., 2005; SZUNYOGH et al., 2008; LI et al., 2009; KIEU et al., 2012) The main advantage of LETKF is that it allows for the analysis to be computed locally in the space spanned by the forecast ensemble members at each model grid point, which reduces the computational cost and facilitates the parallel computation efficiently The key idea of the LETKF algorithm is to use the background ensemble matrix as a transformation operator from the model space spanned by the grid points within a selected local patch to the ensemble space spanned by the ensemble members, and perform the analysis in this ensemble space at each grid point For a quick summary of the LETKF algorithm, assume that a background ensemble fxbiị : i ẳ 1; .; kg are given, where k is the number of ensemble members (assuming that we are doing analysis at one instant of time, so no time index is written explicitly) Following HUNT et al (2005), an ensemble mean " xb and an ensemble perturbation b matrix X are defined respectively as: k 1X " xb ẳ xbiị k iẳ1 Xb ẳ fxb1ị " xb ; xb2ị " xb ; ; xbðkÞ À " xb g: 1ị Let x ẳ " xb ỵ Xb w, where w is a local vector in the ensemble space, the local cost function to be minimized in the ensemble space is given by: _ where Jẵxb ỵ Xb w is the cost function in the model space If one defines the null space of Xb as _ N = {v|Xbv = 0}, then the cost function J ðwÞ is divided into two parts: one containing the component of w in N (the first term in Eq 2), and the second depending on the components of w that are orthogonal to N By requiring that the mean analysis state _ " a is orthogonal to N such that the cost function J ðwÞ w is minimized, the mean analysis state and its corresponding analysis error covariance matrix in the ensemble space can be found as: _a " a ¼ P ðYb ịT R1 ẵy0 H"xb ị w _a P ẳ ẵk 1ịI ỵ Yb ịT R1 Yb ; ð3Þ ð4Þ b where Y  HðxbðiÞ À "xb Þ is the ensemble matrix of background perturbations valid at the observation locations, and R is the observational error covariance matrix By noting that the analysisa error covariance _ matrix Pa in the model space and P in the ensemble _a space have a simple connection of Pa ¼ Xb P ðXb ÞT , the analysis ensemble perturbation matrix Xa can be chosen as follows: _a Xa ẳ Xb ẵk 1ÞP Š1=2 : ð5Þ The analysis ensemble xa is finally obtained as: _ aiị " a ỵ ẵk 1ịP xaiị ẳ "xb ỵ Xb fw 1=2 g: 6ị Detailed handling of more general nonlinear and synchronous observations in LETKF can be found in HUNT et al (2005) It should be mentioned that the above formation is only valid in the absence of model errors To take into account the model errors, Hunt et al (2005) suggested that a multiplicative factor should be introduced in Eq (4) (specifically, the first factor on the rhs of Eq 4) This simple use of the multiplicative inflation introduces no additional costs to the scheme, and has been shown to be efficient in many applications of the LETKF (e.g., LI et al., 2009; MIYOSHI, 2011; MIYOSHI and KUNII, 2012) Multiphysics Ensemble Typhoon Forecast Experiment Descriptions 3.1 Overview of Typhoon Conson (2010) Conson (2010) was the first typhoon of the 2010 typhoon season in the WPAC basin It originated from a tropical disturbance east of the Philippines around July 2010 The system reached the tropical depression stage on July 11 and intensified quickly into a severe tropical storm around July 12 as it moved westward over favorable environmental conditions By July 13, Conson attained the typhoon stage with the maximum sustained wind *130 km h-1 It weakened substantially after making landfall over the Philippines archipelago but restrengthened when it entered the South China Sea As it approached Vietnam coastal line, it managed to reach the typhoon stage again with the maximum wind of *110 km k-1 right before crossing Hainan Island Despite its fairly straight westward track, it was somewhat surprising to notice that most model guidance, including the GFS forecasts, had strong right bias This caused a lot of confusion to forecasters in Vietnam, as the Vietnam HydroMeteorological Service continued issuing advisories that Conson would bear northward until later July 15 when the consensus forecasts from several guidance models started to converge toward a track that headed to the North of Vietnam Such persistent right biases of the track forecasts appeared to be related to the underestimation of the large-scale steering flow associated with the western Pacific subtropical ridge from the GFS model, which provided the global boundary conditions for most of the regional models As such, all of the downstream model applications that were driving the GFS would suffer from the same biases, leading to inaccurate advisories of Conson’s track and intensity change The impacts of Typhoon Conson were detrimental across several countries including the Philippines, Vietnam, China, and Laos In the Philippines, Conson produced very heavy rains that triggered flooding over a widespread area Seventy-six people were reported to be killed across the area and 72 others are listed as missing Damage was also announced in Vietnam where several people were killed and 17 others were listed as missing In China, at least two people have been killed due to wind-related incidents The total damage over all countries was estimated at more than US $100 million Understanding the source of the bias as well as effectiveness of different data assimilation methods in improving the track forecast of Typhoon Conson is, therefore, of significance for future better preparation and typhoon forecasting 3.2 Ensemble Configuration To examine the performance of the multi-physics LETKF in a realistic setting, the LETKF algorithm is implemented in a non-hydrostatic version of the Weather Research and Forecast (WRF-ARW) model (V3.2, SKAMAROCK et al., 2005) in this study Due to a large number of ensemble experiments conducted, the WRF model is configured with a single domain of 36 km horizontal resolution and initialized with the NCEP/GFS operational analysis The model domain covers an area of 3,700 km 3,700 km with 31 vertical levels, and it is centered in the South China Sea, to the East of Vietnam (Fig 1) The forecasted period spans the lifetime of TY Conson from its earlier depression stage at 1200 UTC 12 July 2010 to its near dissipation at 1200 UTC 15 July 2010 after making landfall over Vietnam To establish some general baseline about the performance of different model error correction techniques, three experiments in which a fixed number of 30 ensemble members are conducted In the first control experiment (NO), a specific set of model physics in the WRF model including (a) the Betts–Miller–Janjic (BMJ) scheme cumulus parameterization scheme (JANJIC, 2000); (b) the Yonsei University planetary boundary layer (PBL) parameterization (HONG et al., 2006); (c) WRF SingleMoment 3-class (WSM3) microphysics scheme (HONG et al., 2004); and (d) the rapid radiative transfer model (RRTM) scheme for both long-wave and short-wave radiations (MLAWER et al., 1997) are applied for all ensemble members with no multiplicative inflation at any analysis cycle This NO experiment serves as a reference to evaluate the effectiveness of the MI and MP approach C Kieu et al Pure Appl Geophys In the second experiment (MP), a spectrum of (1) three microphysics schemes including the Kessler scheme, the Lin et al., scheme, and the WSM3 scheme; (2) two PBL schemes including the YSU and the Mellor–Yamada–Janjic (MYJ) scheme; (3) two cumulus parameterizations schemes including the Kain–Fritsch scheme, and the Betts–Miller–Janjic scheme, and (4) two long-wave radiative schemes including the RRTM and the Geophysical Fluid Dynamics Laboratory (GFDL) longwave scheme are used A total of 24 different combinations of these physical options are formed and assigned to different ensemble members in a sequence of permutations of the above physical options.1 If the number of ensemble members is larger than the number of the combinations, the assignment will be repeated for the next members There is no inflation invoked in the MP experiment such that the increase of the ensemble spread relative to the NO experiment can be attributed to the use of the multiple physics options In the third experiment (MI), a multiplicative inflation factor k = 1.8 is applied to the analysis _a transformed covariance matrix P in Eq (4) The inflation factor is kept constant for all cycles and the same set of model physics as in the NO experiment is used for all members such that the effectiveness of the MI approach in correcting model errors can be compared against that of the MP approach in a transparent way The role of the MI approach is examined further in a number of sensitivity experiments in which the inflation factor varies from 1.0 to 6.5, which appears to be a typical range of the inflation factor for the TC environment as shown in MIYOSHI and KUNII (2012) These sensitivity experiments are expected to provide some information about the significance of the adaptive multiplicative inflation in the LETKF Likewise, the effectiveness of the MP approach can be investigated by varying the number of ensemble members from 10 to 50 in several additional sensitivity experiments See Table for the list of experiments Since there are no ensemble backgrounds for the first analysis cycle, cold-start background ensemble members are first initialized by adding a random noise with standard deviations of m s-1 for wind, K for temperature, and 10-3 kg kg-1 for specific humidity into the GFS data 12-h earlier, i.e., at 0000 UTC 12 July The short 12-h forecasts of the cold-started members from 0000 UTC to 1200 UTC 12 are then used as the initial background for the first analysis cycle at 1200 UTC 12 In the WRF-ARW model, each physical parameterization option has a designated number So, the 24 combinations of the physical schemes are simply permutations of these options, which are of the form (i, j, k, l) in which i [ [1, 2, 3] is for microphysical scheme, j [ [2, 3] for the PBL schemes, k [ [1, 2] for the cumulus schemes, and l [ [1, 2] for longwave radiation schemes 3.3 Observation Data To create observational data, a hypothetical truth is formed by using the NCEP Final Operational Global Analysis (FNL) dataset during the period that encompasses the whole lifecycle of TY Conson (2010) Such FNL data represents roughly the true state of the atmosphere and will be considered in this study as a reference for later comparison After the truth is obtained, bogused observations are generated every 6-h by assigning a random noise of a standard deviation of m s-1 for the horizontal wind components, K for temperature, and 10-3 kg kg-1 for the specific humidity to the truth at all of the grid points from surface up to level z = 13 km These bogus observational data points are generated in the forms of radiosonde columns in the LITTLE_R.2 format; any data levels that are below the terrain height will be eliminated As a step to orient the WRF-LETKF system to be consistent with the WRF assimilation system (WRFDA), all of the bogus observations are assigned observational errors that are based on the error statistics imposed by the quality control component of the WRFDA system Of the seven prognostic variables in the WRFARW model including the horizontal wind components, vertical velocity w, perturbation potential temperature, perturbation geopotential, surface pressure, and water vapor, then there are the variables that are directly assimilated by the LETKF are the LITTLE_R is a legacy data format that was developed earlier to ingest observational data for the MM5 model This format is adopted in the WRF-ARW as part of continued support for different observational format inputs More information about the LITTLE_R format can be found at: http://www.mmm.ucar edu/mm5/mm5v3/little_rv3.html Multiphysics Ensemble Typhoon Forecast Table List of experiments with the WRF-LETKF configuration Experiment Description NO MP MI MP–MI MI-sensitivity MP-sensitivity A single set of model physics; no inflation; 30 ensemble members Combination of 24 physics options; no inflation; 30 ensemble members A single set of model physics as in the NO experiment; inflation factor k = 1.8; 30 ensemble members A combination of 24 physics options; inflation factor k = 1.8; 30 ensemble members A single set of model physics as in the NO experiment; inflation factor k varying from 1.0 to 6.5; 30 members A combination of 24 physics options; no inflation; ensemble members varying from 10 to 50 horizontal (u and v) winds, the potential temperature, and the relative humidity The other three prognostics variables are updated every cycle via cross-correlations with the observed variables The degree to which such prognostic variables can update observational information depends essentially on how the ensemble cross-correlations are represented, and can be used to evaluate the effectiveness of different model error correction methods To minimize the impacts of the homogeneous covariance localization, the observational network is designed in such a way that observations are given at all model grid points for all cycles This partly removes the need of a spatially adaptive localization, albeit the scale of the localization still needs to be tuned at the initial time for the best performance For the LETKF, a local volume of 11 11 grid points in horizontal direction and vertical extension of 0.2 (in r-coordinate) is fixed, and the localization scale is chosen to be 700 km and kept constant in time 3.4 Boundary Condition To ensure that each member has its own lateral boundary condition consistent with its updated analysis, the WRFDA boundary routine is used to generate boundaries for each ensemble member after the ensemble analysis step is finished for every cycle Because the GFS forecasts are outputted every 6-h, the lateral boundary conditions are updated at the same temporal period The roles of the lateral boundaries in regional models should be especially highlighted as our various experiments with different types of boundary conditions show that idealized boundaries tend to be detrimental to the TY development (not shown) E.g., use of the open boundaries destroys the coherent structure and development of TY Conson after just two or three cycles This is because the convective instabilities needed to fuel the TY growth appear to be radiated away from the domain As a result, the storm dissipates and the ensemble collapses (i.e., ensemble spread approaches zero) and drifts away from the truth quickly, thus reducing the capability of the ensemble filter in assimilating new observation Results 4.1 Control Experiments As a preliminary illustration of the performance of the MI and MP approach in the forecasts of TY Conson, Fig shows the track forecast errors averaged from 1200 UTC 12–1200 UTC 13 and the time series of the storm intensity valid at 1200 UTC 12 Note that this 12-h period rather than the entire forecast period is selected for the averaging shown in Fig because the TC track and intensity forecasts require tracking the point-like storm centers for which a systematical track bias can be carried over the next cycles if it is not corrected for every cycle Unless a vortex initialization scheme is used to recorrect the vortex center (or the initial condition is continuously updated from the GFS analysis), the storm center will deviate gradually away from the best track, leading to artificial accumulated track and intensity errors for the later cycles Since we have no bogus vortex component implemented in this work, any analysis of track or intensity forecast errors is limited to the first several cycles during which the storm centers are located at the similar position as the truth location from the FNL dataset C Kieu et al Pure Appl Geophys Figure a Comparison of the track forecast errors between experiments with no model error correction (dark gray), the multiplicative inflation factor of 1.8 (medium gray), multiple physics (light gray) b time series of the minimum sea level pressure for the MP experiment, and c similar to b but for the MI experiment valid at 1200 UTC 15 JUL 2010 All experiments have 30 ensemble members One first sees in Fig that although the track errors in the MI experiment not show much improvement with respect to the NO experiment, the MP experiment exhibits a noticeably better track forecast with the track errors reduced about 20 % at 36-h lead time and longer during the selected period This is because the broad spectrum of the physics schemes in the MP experiment helps generate a range of storms with different intensity, which interact differently with the steering environment (Fig 2b, c) Multiphysics Ensemble Typhoon Forecast Unlike the MI ensemble in which all members show a similar intensity with very minimum spread for the entire forecasted period, the MP ensemble possesses a much larger intensity spread with half of the members showing higher intensity while the other half possessing much weaker intensity Such bifurcation of the intensity seen in the MP ensemble is attributed to the fact that the members with the Kain– Fritsch cumulus scheme tend to produce storms with much stronger intensity than those with the BMJ scheme This is not limited to the selected cycle but in fact observed very consistently in all experiments conducted thus far with our WRF-LETKF system Because of their stronger intensity and more welldefined circulation, members with the Kain–Fristch cumulus scheme are subject to larger northward bias under the strong influence of the subtropical ridge over Mainland China In contrast, the ensemble members with the BMJ cumulus scheme have weaker intensity and not seem to be influenced much by this northward steering, leading to an overall larger ensemble spread and thus smaller ensemble mean track errors as shown in Fig Of course, any analysis of the storm intensity at the 36-km resolution should be cautioned as this coarse resolution may not capture reliably the magnitude of the maximum surface wind at any stage However, the mesoscale characteristics are often sufficiently represented in the model for the storms to experience a different response to the large-scale steering flow under different parameterizations This explains the different performance in the track and intensity errors seen in Fig Because the point-like metrics based on the TY track and intensity errors are rather sensitive to the model resolution and could be subject to representativeness errors, Fig compares further the total errors between the MP and MI experiment Here, the total errors are defined as the volume-averaged energy root mean squared errors (EME) as follows: Cp EME ẳ U U ỵ V V ỵ " T T ị1=2 ; T ð7Þ where U, V are the zonal and meridional wind components, respectively, T is the temperature (in K unit), the prime denotes the differences between the analysis and the truth valid at the same time, Cp is the constant pressure heat capacity, T" = 273 K is the reference temperature, and the average is taken over the entire model grid mesh Consistent with the track errors, the MP has overall better performance during the entire forecast period with the average EME error *0.2 m s-1 as compared to 1.2 and 0.8 m s-1 in the NO and MI experiment, respectively In particular, the MP EME errors reduce *80 % after just three cycles and maintain a steady small magnitude afterward Except for the first cold-started cycle, the smaller EME errors in the MP experiment are observed for a range of ensemble members from 20 to 50 Figure Comparison of the volume-averaged energy root mean squared errors (EME) with no model error correction (dark gray), a fixed multiplicative inflation factor k = 1.8 (medium gray), and multiple physics (light gray) C Kieu et al Pure Appl Geophys regardless of the domain size or the model resolution (cf also Fig 9) The outperformance of the MP method during the entire period, especially for the few early cycles, can be understood if one notes that convective- to meso-scale instabilities often develop rapidly during the incipient phase of model integration (TOTH and KALNAY, 1997) Such development of the uncertainties related to the instabilities at the convective scale depends sensitively on the different parameterization schemes used by the model With a diverse set of physics schemes, the MP ensemble could generate a large spread quickly after 12 h into integration, allowing for the model errors associated with physical representations to be captured more efficiently To investigate the MP and MI ensemble spread more explicitly, Figs and show a series of horizontal distributions of the ensemble spreads, which are defined as the standard deviation of the wind speed with respect to the ensemble mean, in the MI and MP experiment Despite its fairly organized structure in both coverage and amplitude, the MI spread is in general small even near the storm center where uncertainties are supposed to be the most significant at the convective scale Because of the inflation, it is seen that the analysis increments still manage to capture some of the new observational information at each cycle Apparently, the inflation is essential to allow for such analysis to be updated despite the small ensemble spread As long as the spread does not completely collapse, the inflated covariance always enhances the analysis perturbations such that new observational information is updated over the area where the spread is significant Similar analyses of the ensemble spread versus analysis update for the NO experiment confirm that without the inflation, the analysis increments are indeed very small and virtually negligible over the entire domain after three cycles (not shown) Although the MI analysis increments exhibit some update of new observational information for every cycle, it should be noted that the percentage of the analysis increments relative to these increments diminishes quickly in time in the MI experiment (Fig 4e–h) Indeed, examining the observational increments, shows that these increments keep growing in time because the model state is drifting away from the truth This is anticipated as the MI analysis increments could not assimilate fully new observational information at each cycle due to the small ensemble spread even after inflated (Fig 4) As a result, the difference between the analyzed and the true states is accumulated every cycle, leading to a growing deviation of the model state from the true atmosphere toward the end of the forecasted period Unlike the MI spread, the MP spread shows much more signal with time in both horizontal and vertical cross sections (Figs 5, 6) Specifically, the MP spread is more organized and could maintain well the structure of the uncertainties along the track of the storm Inspecting the regions of strong wind convergence reveals that the MP spread is more representative in the first five cycles during which model uncertainties related to convective instabilities develop rapidly, leading to a consistent and coherent structure of the spread The clustering of the MP spread along the convergent areas seen in Fig implies that new observations are most updated within these large spread areas but minimally used in other areas where the ensemble spread is otherwise small The characteristically large concentration of the MP spread along the convergent zones does not seem to depend on the number of ensemble members as this is seen for all ranges of the number of ensemble members The difference in the MP and MI spread is even more apparent in the vertical cross sections (Fig 6) While the MI spread is confined mostly at the middle levels, we observe that the MP spread could capture model uncertainties related to the TC physics from the surface up to 200 hPa Comparison of the analysis increments in the MI and MP experiment shows that the MI method tends to be slightly more efficient at the upper levels where the magnitude of the MI and MP ensemble spread is somewhat similar, but it is less efficient than the MP approach from the surface to *300 hPa where the MI spread is not sufficient even after inflated Note that the performance of the MP approach is somewhat degraded toward the end of the forecast because the saturation of the ensemble spread affects the effectiveness of the LETKF Therefore, the LETKF is no longer capable of updating new observations efficiently and the EME errors start to grow afterward (Fig 3) Multiphysics Ensemble Typhoon Forecast Figure Plane views at 900 hPa of a–d the ensemble spread with the inflation factor of 1.8 (shaded, m s-1) and zonal wind analysis increments -1 (contoured at interval of 0.5 m s ); and e–h rms wind speed errors (shaded) and observed zonal wind increments (contoured at interval of 0.5 m s-1) valid at 0000 UTC 13, 1200 UTC 13, 1200 UTC 14, and 1200 UTC 15 Superimposed are storm-relative flows C Kieu et al Figure Similar to Fig but for the multi-physics approach with no inflation Pure Appl Geophys Multiphysics Ensemble Typhoon Forecast Figure Vertical cross sections of the ensemble spread (shaded, m s-1) and zonal wind analysis increments (contoured at interval of 0.5 m s-1) for a– d the multi-physics approach; and e–h the inflation approach with a fixed inflation factor k = 1.8 valid at 0000 UTC 13, 1200 UTC 13, 1200 UTC 14, and 1200 UTC 15 C Kieu et al Pure Appl Geophys The benefit of the MP method is also notable when the cross-correlation between assimilated variables and other prognostics variables is analyzed Figure compares the horizontal distributions of the cross-correlation between the geopotential height and the zonal wind for the MP and MI experiment Although both the MP and MI can update geopotential perturbations along the track of Conson, there is little coherent correlation between the mass and the wind fields in the MI experiment; most of the geopotential increments in the MI experiment are due to the inflated correlation near the storm center In contrast, the MP ensemble exhibits well the crosscorrelation between the mass and wind fields not only in the vicinity of the storm but also in the other parts of the domain as well This indicates that the MP approach is capable of enhancing the cross correlation among variables and, therefore, spreading the observational information well to other variables This improves the LETKF performance as compared to the covariance inflation Note in particular that the EnKF-based schemes tend to underestimate the geostrophic constraint if the mass and wind field are assimilated simultaneously due to the negative impact of the covariance localization (LORENC, 2003) This is most apparent for the ensemble with a small number of members In this regard, the capability of the MP approach in enhancing the cross-correction between the mass and the wind field is encouraging as it helps alleviate potential unbalances related to localization small-scale features tend to be dissipated in terms of gravity waves and contribute little to the storm-scale development at the short range scale Therefore, model integration can be maintained as long as the small noises associated with the inflated covariance not cause any computational instability Of course the exact upper limit for the inflation factor is not known and may be casedependent The study by MIYOSHI and KUNII (2012) showed that the maximum value of their adaptive inflation is around 4.0 However, our several sensitivity experiments show that the WRF model is generally stable until k [ 6.5 in all of the MI experiments Note again that because the observations are given at all model grid points for all cycles, the negative impacts of the homogeneous covariance localization are minimized and hereinafter not examined Figure shows the EME errors for the MP and MI experiment in which k varies from 1.0 to 6.5 Overall, the larger the inflation factor is, the better the performance of the MI approach While the MP errors are consistently smaller than the MI errors for all the range of the inflation factors, it is of interest to notice that the MI errors start to decrease after 0000 UTC 14 for k [ 4.5 For k = 6.5, the MI errors are somewhat comparable to the MP errors near the end of the forecast period That the MI has better performance for a larger inflation factor is reasonable because the analysis covariance is increasingly inflated every cycle no matter how small the ensemble spread is This leads to augmented analysis perturbations that are capable of updating more observational information with time in the experiments with larger inflation factors Because the MI has the best performance with the largest inflation k = 6.5, Fig compares the performance of several MP experiments in which the ensemble members vary from ten to 50 with respect to the MI approach Except for the ensembles with ten and 15 members, the MP shows better skill than the best-tuned MI even with the number of the ensemble members as small as 20 In general, more ensemble members result in better performance in the MP experiments However, for the number of ensemble members [30, we notice that the MP appears to reach some limit at which the EME errors not seem to be reduced further This could be related to the fact that the maximum number of different physics 4.2 Sensitivity Experiments Recent studies of the adaptive inflation have demonstrated a number of benefits with the adaptive inflation and covariance localization that could follow the error propagation instead of being spatially and temporally homogeneous In this section additional sensitivity experiments are conducted in which the inflation factor in the MI experiment is allowed to vary within a wide range of values from k = to k = 6.5 such that the overall performance of the MI approach can be evaluated systematically when compared to the MP method That the inflation factor can be as large as 6.5 in the WRFLETKF system is probably because the real atmosphere is resilient to small scale variations; any unbalanced Multiphysics Ensemble Typhoon Forecast Figure Similar to Fig but for the cross-correlation between geopotential height perturbation and the zonal wind (shaded) and geopotential height increments (contoured, m) for a–d the multi-physics approach; and e–h the covariance inflation approach with k = 1.8 C Kieu et al Pure Appl Geophys Figure The EME errors for sensitivity experiments in which the inflation factor k is set equal to 1.8 (dark gray), 3.5 (medium gray), 4.5 (light gray), and 6.5 (striped) Superimposed also are the EME errors in the MP experiment (black) Figure EME errors for sensitivity experiments with the MP approach in which the number of the ensemble members is set equal to 10 (times), 15 (triangle), 20 (square), 25 (circle), 30 (open circle), 35 (asterisk), 40 (plus), and 50 (diamond) Superimposed also are the EME errors for the MI experiment with k = 6.5 and 30 members (dashed); and the MP–MI experiment with k = 6.5 and 30 members (dot-dashed) combinations used in the MP experiments is 24.3 As the number of members is greater than 24, the physics options are repeated Because the spread among members that share the same physics options is generally small, the additional members does not help to increase the ensemble spread further As such, the optimal number of ensemble members should be around 25 as seen in Fig Recall that the combination of the physics schemes used in the MP experiment include two PBL schemes, two radiation schemes, two cumulus parameterization schemes and three microphysics schemes With the better performance of the MI approach for large inflation factors as well as the degrade trend of the MP approach toward the end of the forecast period, one may suspect that combination of the MP and MI would help improve the overall skill of the forecast To verify this possibility, another experiment (MP–MI) is conducted in which both multiplicative inflation with k = 6.5 and multiple microphysics options are used to examine the performance of the LETKF It is of interest to find that while the EME errors during the later cycles show a general smaller magnitude as compared to those in either the MP or MI experiment, the MI-MP Multiphysics Ensemble Typhoon Forecast Figure 10 Vertical profiles of the mean ensemble spread (solid) and absolute errors (dash) with respect to the FNL analysis of the horizontal wind components that are averaged over the entire forecast period for a the MI experiment with inflation factor k = 1.8; b the MP experiment with no inflation, and c combination of inflation and multi-physics C Kieu et al Pure Appl Geophys combination actually has higher EME errors than a simple use of the MP during the first four cycles (Fig 9) This is likely because the early spread associated with the multi-physics options is sufficiently broad that further inflating the priori covariance matrix actually degrades the background ensemble to an extent that a new observation is not used optimally It is observed furthermore that the MP–MI combination gives the best results only for the number of ensemble members less than 30 As the number of ensemble members increases, such combination does not give better results than either simple use of multi-physics for the first 18-h or MI at the later 18-h Additional examination of the time-averaged vertical profiles of the relative difference between the ensemble spread and absolute errors in the MI, MP, and MP–MI experiments shows that the most significant reductions of the EME errors in the MP– MI experiment are below 500 hPa, whereas the MP– MI ensemble spread exhibits a slight decrease (Fig 10) If the ratio of the ensemble spread over the errors is used to characterize the performance of different approaches, one sees in Fig 10 that the benefit of using a combination of the multi-physics and covariance inflation is minimal during the entire period The large improvement in the ratio of the ensemble spread over the errors between the MI and MP experiment and the insignificant difference between the MP and MP–MI experiment imply that it is the diverse physics schemes in the WRF model that help the LETKF become more efficient in capturing observational information and maintain a reasonable ensemble spread during the forecast period This suggests that a multi-physics approach is a good option for the short-range mesoscale forecasts in which the spinup of the LETKF may take too long for the ensemble spread to efficiently span the spectrum of instability model is dual On the one hand, these techniques help take into account model errors by either inflating the posteriori covariance matrix by a factor k or enlarging the covariance matrix via different representations of physical processes On the other hand, they are important to augment the performance of the EnKF due to the inherent dependence of the EnKF on the local rank of the local ensemble space.4 These two different roles are hard to separate in the full physics models due to a single use of the ensemble to represent both the model states and model errors As mentioned by LORENC (2003), techniques for correcting model errors are not merely restricted to the EnKF but they should be applied to the full rank Kalman filter (KF) as well because the objective of these techniques is to parameterize the model errors Since the performance of the MP and MI approach in the WRF-LETKF system depends sensitively on the background ensemble spread and could possess representativeness errors associated with a single case study, it is not clear to see their roles in correcting model errors In this section, the relative importance of the MP and MI method in correcting model errors is examined in further details with the full rank KF, using the idealized Lorenz 40-variable (L40) model The use of the full rank KF eliminates the local ensemble rank issues associated with the EnKF, thus isolating the role of the MI or MP method in accounting for model errors While the MI approach can be applied straightforwardly to the KF, the implementation of the MP approach requires some additional expressions for the full rank formulation Note again that since the role of the multi-physics approach in the WRF-LETKF system is to provide some kind of random forcing with some schemes overestimating while other schemes underestimating the true model forcing (i.e., unbiased model errors), it is desired that this type of random forcing could be carried over to the L40 model in the idealized experiments as well To express this more explicitly, we first write the L40 model equation in the form: Idealized Experiment 5.1 Formulation The role of the multiplicative inflation or the multiple physics technique in the LETKF algorithm as demonstrated in the previous section with the WRF For the autonomous dynamical systems with a constant forcing, it is possible to design two different ensembles; one represents the initial uncertainties and the other represents the model internal error See the next section for one sample Multiphysics Ensemble Typhoon Forecast dx ẳ Mxị ỵ F; dt ð8Þ where x is the 40-dimension state vector, M is a 40 40 nonlinear matrix, and F is a vector of forcing To mimic the role of the multi-physics, we assume that F is a random vector, i.e., F = 8.0 I ? dF, where dF is a random component represented by a Gaussian distribution with amplitude of and null mean, the exact solution of Eq (8) is given by (see Appendix 1): 93 t