Improving the Efficiency of a Digital Filtering Scheme for Diabatic Initialization

1976 MONTHLY WEATHER REVIEW VOLUME 15 Improving the Efficiency of a Digital Filtering Scheme for Diabatic Initialization PETER LYNCH Met Éireann, Dublin, Ireland DOMINIQUE GIARD CNRM/GMAP, Météo-France, Touloue, France VLADIMIR IVANOVICI Romanian Intitute of Meteorology and Hydrology, Bucharet, Romania 4 April 1996 and December 1996 ABSTRACT A method of diabatic initialization i decribed in which a nonrecurive digital filter i applied to both the backward (adiabatic) and forward (diabatic) tep. It ha clear advantage over previouly propoed cheme: the initialization require ignificantly le computation time and the reulting change in the analyzed field are conitently maller. Depite thee maller increment, the uppreion of puriou high-frequency noie i at leat a effective a for the earlier cheme. 1. Introduction Digital filtering ha been hown to be an effective mean of initializing data for numerical weather prediction. In Lynch and Huang (199, hereafter LH9) a imple filter wa applied to a equence of value centered on the initial time t 0; thee value were generated by two hort adiabatic model integration, one forward and one backward from t 0. It wa found that a total pan of 6 h wa required for effective elimination of high-frequency noie. Huang and Lynch (1993, hereafter HL93) howed, by mean of an optimal filter, that equally effective noie control could be achieved with a pan of only 3 h. They alo decribed a mean of incorporating diabatic effect: the backward adiabatic integration (of length 1.5 h) i followed by a diabatic forecat of twice the length, and the value generated by the diabatic integration are proceed by the filter. In Lynch and Huang (1994, hereafter LH94), recurive filter were applied to the initialization problem. Several cheme were invetigated; in one, deignated RAD recurive adiabatic-diabatic in LH94, the backward adiabatic integration wa proceed with a recurive filter, and the terminal output value wa ued to initiate a forward diabatic integration, to which re- Correponding author addre: Dr. Peter Lynch, Met Éireann, Dublin 9, Ireland. E-mail: plynch@irmet.ie curive filtering wa again applied. The application of filtering to both the revere and forward integration i alo poible uing a nonrecurive filter. In thi note we examine the advantage of thi idea.. A imple conceptual model Let u conider a function of time x(t) governed by the ocillation equation dx i 0 x 0 (1) dt with the initial value x(0). The olution i x(0) exp(i 0 t). Suppoe thi olution i filtered by convolution with an impule repone function h(t): x (t) h x(t) h()x(t ) d. Subtituting the olution x(0) exp(i 0 t) into the integral, it follow immediately that x (t) H( 0 )x(t), where the frequency repone function H() i the Fourier tranform of h(t). Now let u ue a value of the filtered olution y(0) x (0) a an initial value for a olution of the inhomogeneou equation dy i y f(t). () 0 dt 1997 American Meteorological Society

AUGUST 1997 NOTES AND CORRESPONDENCE 1977 FIG. 1. Schematic illutration of original and modified filtering procedure. Thin line indicate backward adiabatic integration; thick line indicate forward diabatic integration. (a) Scheme of Huang and Lynch (1993): adiabatic tep i not filtered. (b) Modified cheme: both tep are filtered. The olution i eaily obtained by Laplace tranformation or otherwie. For inuoidal forcing, f(t) f 0 exp(i 1 t)( 0 1 ), it may be written ] 0[ exp(i0t) exp(i1t) y(t) y(0) exp(i0t) f. (3) i( ) If thi i filtered, the reult i ] 0[ i0t i1t H( 0)e H( 1)e ȳ(t) [H( 0)] x(t) f, (4) i( ) wherea tarting from the unfiltered value x(0), a in HL93, it would be ] 0[ i0t i1t H( 0)e H( 1)e ȳ(t) [H( 0)]x(t) f. (5) i( ) Let u conider that (1) correpond to the adiabatic model and () to the diabatic model, with phyical forcing repreented by f(t). Suppoe now that the adiabatic model i integrated and the olution x(t) i filtered to give x (t). Thi i ued to define initial data y(0) for an integration of the diabatic model (). The reulting olution y(t) i again filtered, producing ȳ(t). Then (4) how that the adiabatic part of the olution i effectively filtered twice, while the diabatic part i filtered once. If the amplitude f 0 of the forcing i ufficiently mall, a ingle filtering of the diabatic component hould be adequate to reduce it high frequency component to an acceptable level. The initialization cheme decribed in HL93 i illutrated chematically in Fig. 1a. The thin line repreent the revere adiabatic integration of duration T. Thi i not filtered; it terminal value, depicted by an open circle, i ued to initiate a forward diabatic integration (the thick line) that i then ubjected to filtering. Thi integration mut be of duration T to enure that the output of the ymmetric filter i valid at t 0. The modified cheme, to be analyzed here, i illutrated in Fig. 1b. The revere adiabatic integration (thin curve) i filtered, yielding an output valid at t T/. Thi value, depicted by the black pot, i ued to initiate a forward diabatic integration (thick curve) of duration T, centered on the initial time t 0. Conidering the adiabatic component of the olution, we ee that the HL93 cheme involve a ingle application of a filter of pan T; the modified cheme involve two application of a filter of pan T. Comparion between the cheme can be made by conidering the relationhip between the magnitude of the repone function, H T (), of the filter with the longer time pan and the quared repone function H T () of the filter with the horter time pan. The diabatic initialization cheme apply filtering to trajectorie that do not correpond, at time t 0, to the original analyi. Let u uppoe that both 0 and 1 are low frequencie, o that we may aume H( 0 ) H( 1 ) 1. Thu, the filtering doe not ubtantially affect thee component. The reult of a backward adiabatic integration of length T followed by a forward diabatic integration of the ame length may eaily be deduced: integrate (1) backward from x(0) to time t T to get x(t) x(0) exp(i 0 T); ue thi value to initiate a forward integration of () from t Tto t 0 to obtain 0[ ] exp(i0t) exp(i1t) y(0) x(0) f. (6) i( ) For hort time pan ( 0 T and 1 T mall), auming the filter repone i unity, we have ȳ(0) x(0) f 0 T. (7) Thi differ from the original value x(0) by the diabatic dicrepancy f 0 T, which i proportional to the amplitude f 0 of the diabatic forcing and alo to the time pan T. Since the pan of the new cheme i typically half that of the old cheme, the effect of thi dicrepancy i reduced. Of coure, the problem of diabatic dicrepancy diappear when uing a finalization or launch technique a decribed in Fillion et al. (1995) and LH94, repectively; but then the filtered field are not applicable at the initial time. 3. Some example of filter The filter ued in HL93 wa an optimal filter having pa-band and top-band edge with period p 15 h and 3 h (recall that the period i given by / t/). The total pan wa T 3 h and the time tep wa t 360. A hown in Lynch (1997), the Dolph Chebyhev filter with a top-band edge 3 h and the ame time pan (3 h) i a imple optimal filter with virtually the ame frequency repone. We will

1978 MONTHLY WEATHER REVIEW VOLUME 15 FIG.. Frequency repone (db) for Dolph filter with ripple ratio r 0.1. Solid line: filter order N M 1 37; 0 log H() plotted. Dahed line: filter order N M 1 19; 0 log H() plotted. FIG. 3. Amplitude repone for Dolph filter a in Fig. above. Solid line: filter order N M 1 37; H() plotted. Dahed line: filter order N M 1 19; H() plotted. therefore ue imple Dolph Chebyhev filter for the comparion below. We chooe a time tep t 300 o that M T/t 18 i an even number. Let u recall that the frequency repone of a Dolph Chebyhev filter with a top-band edge frequency i co H() rt M, (8) co where T M (x) i a Chebyhev polynomial of order M and the total time pan i T Mt. The ripple ratio r meaure the maximum repone in the top band ( ) and i determined by requiring H(0) 1: 1 1 1 1 TM coh M coh (9) r co co (ee Lynch 1997). In Fig., two repone function are hown. The ordinate i the filter attenuation, 0 logh() (db). The olid curve i the repone of a Dolph Chebyhev filter (denote it H T ) of total pan T 3 h; the topband edge i 3 h and M 18. The dahed curve i the quared repone of a Dolph Chebyhev filter ( H T ) of total pan T 1.5 h; thu, M 9. The topband edge.5 h ha been choen by experiment o that the total attenuation in the top band wa about the ame in both cae: 1.3 db for the filter with the longer pan and 1. db for the quared repone of the filter with the horter time pan. A more ytematic mean of chooing will be decribed below. The amplitude of the frequency repone of the two filter are hown in Fig. 3 with the axi expanded for clarity: the repone are very imilar, but the pa band of the quared hort filter i omewhat broader. If a emi-lagrangian advection cheme i ued, relatively long time tep are poible. We conider a comparion between filter H T and H T having the ame top band and total pan a above, but with a time tep t 900. Thu, M 6 for H T and M 3 for H T. The frequency repone are hown in Fig. 4. Once again, FIG. 4. Frequency repone (db) for Dolph filter with ripple ratio r 0.1. Solid line: filter order N M 1 13; 0 log H() plotted. Dahed line: filter order N M 1 7; 0 log H() plotted.

AUGUST 1997 NOTES AND CORRESPONDENCE 1979 the attenuation in the top band i imilar for both filter (1.6 db in each cae) and the pa band of the quared hort filter i lightly broader. The repone in the pa band for filter with the longer and horter time tep were compared (by recaling the axi) and were found to be very imilar: thu, effective frequency dicrimination i achievable with the longer time tep. 4. Chooing the filter parameter Method of enuring that ingly and doubly applied filter have the ame level of damping for high frequencie will now be decribed. Let unprimed and primed quantitie refer to the filter with the longer and horter time pan, repectively. Firt, let u aume that the horter pan i exactly one-half of the longer one. We now determine the parameter for a filter with one-half of the pan, having a ripple ratio r r, o that it quared repone will have the ame high frequency attenuation a the longer filter. It top-band edge i determined by inverion of (9) with appropriate parameter value: [ ] 1 1 1 1 co coh coh. (10) M r It wa found in the above example that wa omewhat larger than. It mut be acertained in each cae whether i mall enough to provide the required frequency electivity. Numerical tet uing (10) revealed that, in ome cae, the widening of the top band for the horter filter wa unacceptable. An alternative approach, which enure that both filter have the ame top-band edge, will now be decribed. Firt we note that, if and r are pecified, the required filter pan can be deduced from (9): we have T Mt with 1 coh (1/r) M. 1 coh 1 co Noting that, for mall, coh 1 (1/co ), amay be eaily verified by tandard Taylor expanion, and recalling that t/, we find that 1 1 1 1 T coh, (11) r provided K or t K, which i alway true in practice. Now aume that the longer and horter pan are T Mt and T Mt, repectively, and that both filter have the ame top-band edge. The repone function are co H() rt M, co where r 1/T M (1/co /) for the longer filter, and co H() rt M, co where r 1/T M (1/co /) for the horter one. Since the filter with the horter pan i to be applied twice, we require r rto enure that the top-band attenuation i equivalent in both cae. Thu, from (11) we have 1 1 1 T coh. (1) r Taking typical value 3 h and r 0.05 we find from (11) that the filter applied only once mut have a pan at leat equal to 3.5 h, wherea (1) implie that the filter applied twice achieve the ame high-frequency damping with a pan of only.08 h. Thi lead to a ubtantial reduction in computation time. 5. Application to a forecat model The new initialization cheme ha been evaluated by application to the limited-area pectral model ALADIN (partially decribed in Bubnová et al. 1995). Thi model i run quai-operationally at Météo-France and coupled with the global pectral NWP model ARPEGE. Horizontal repreentation of the variable i achieved by double Fourier erie. A nonhydrotatic verion of the model ha been developed, but the reult preented below are for the hydrotatic verion (application to the nonhydrotatic verion would imply involve filtering of the two additional prognotic variable). The gridize here i approximately 18.3 km and there are 7 vertical level. The time tep i t 450. It wa found to be neceary to ue a maller time tep, t 5, for the backward adiabatic integration to enure tability. The geographical area covered may be een later in Fig. 7. ALADIN doe not have it own analyi cheme. The initialized analyi produced by the global pectral model ARPEGE i tranformed to the reolution required for ALADIN. Thi tranformation introduce puriou noie that mut be removed by initialization on the limited domain. Four forecat were performed tarting from analye valid at 0000 UTC 1 January 1996. One wa from the tranformed analyi without any ubequent initialization. The other three forecat followed digital filtering initialization in three verion. 1) A Lanczo filter with pan 6 h and a cutoff c 6h.

1980 MONTHLY WEATHER REVIEW VOLUME 15 FIG. 5. The frequency repone of the filter ued in the three cheme decribed in the text. Solid line cheme 1, Lanczo filter applied once; dahed line cheme, Dolph Chebyhev filter applied once; dotted line cheme 3, Dolph Chebyhev filter applied twice. Note that for cheme 3 the quare of the repone function i plotted. FIG. 6. Evolution of the mean abolute urface preure tendency for the firt ix forecat hour, tarting from uninitialized data (olid) and from data filtered with cheme 1 (dotted), (dahed), and 3 (dot dahed). A backward integration of 3 h wa followed by a 6-h diabatic forecat. Filtering wa applied once only, to the forward integration. ) A Dolph Chebyhev filter with pan 4.5 h and topband edge 3 h. Filtering wa again applied once only, to the forward integration. 3) A Dolph Chebyhev filter with pan.5 h and topband edge 3 h. Filtering wa applied twice, to both the backward and forward integration. Thi i the new cheme. The frequency repone of the filter ued in the three cheme are hown in Fig. 5. Note that for the new cheme it i the quare of the repone function that i plotted ince the filter i applied twice. The Lanczo filter i preciely that ued in LH9 and i known to be effective. Note that the abcia i the period. Iti clear that the repone of the three filtering cheme are very imilar. The time pan ued here are longer than thoe conidered in ection 3 and allow a better damping of high frequencie, with a ripple ratio lower than 0.05 intead of 0.10 for Dolph Chebyhev filter. At the ame time, the attenuation of period longer than 1 h remain mall. Note that the contraint on the filter are not a trong in thi cae a might be required for a global model (a dicued in Fillion et al. 1995) or a limited-area model with it own aimilation cycle. The evolution of the abolute tendency of urface preure averaged over the forecat domain i hown in Fig. 6. The exceive noie preent in the uninitialized forecat (olid curve) i uccefully removed by all the filtering cheme, and they appear to be equally effective in thi regard. There i a mall amount of reidual noie but it i not of ufficient amplitude to caue concern. The average abolute urface preure tendency i an indicator of the level of noie in the external gravity wave component. It i alo neceary to tudy the impact of filtering on puriou internal gravity wave energy. For thi purpoe, the 500-hPa vertical velocity at the initial time wa examined for the four forecat (reult not hown). It wa found that large amplitude, mall-cale noie in the uninitialized run, particularly in the region of the Alp, wa effectively removed by all the filtering cheme. The change induced by the three filtering cheme were examined and compared. It wa found that the new cheme conitently led to maller change than either of the cheme involving ingle filtering. The impact on the 500-hPa temperature analyi i hown in Fig. 7. Figure 7a,b,c how, repectively, the increment due to filtering cheme 1,, and 3. It i clear that the change brought about by the new cheme are ignificantly maller than thoe of the alternative. Notwithtanding thi, the noie reduction i equally effective. It i reaonable to acribe the reduction in initialization increment for the new cheme to the reduced diabatic dicrepancy aociated with the horter pan ince, a we have een, the filter frequency repone i virtually identical for the three cheme. The aving in computation time for the new cheme may eaily be etimated. Let t B and c B be the time tep and cot per time tep for the backward adiabatic run, t and c the correponding value for the forward diabatic run, and T the total time pan. We may uppoe t B t/ and c B c/. Then the cot of the old cheme (ingle filtering) i

AUGUST 1997 NOTES AND CORRESPONDENCE 1981 FIG. 7. Impact on the 500-hPa temperature analyi: (a), (b), and (c) how the increment due to filtering cheme 1,, and 3. The contour interval i 0. K. T T 3Tc C cb c. tb t t If the filter pan of the new cheme i half a long, the cot i T T Tc C cb c, tb t t which reult in a aving of 33%. Thi wa the cae in the above example: the computation time (econd of

198 MONTHLY WEATHER REVIEW VOLUME 15 CPU uage) for cheme 1 wa 19 ; for cheme it wa 955 ; and for cheme 3 it wa 66. The reduction i of practical benefit in an operational context. The above reult demontrate that a ignificant gain in efficiency can be made by mean of the new cheme. Thi cheme reult in maller increment to the initial field but i equally effective in reducing high frequency noie in the forecat. The new cheme with Dolph Chebyhev filter ha been implemented in the operational ALADIN-LACE uite (focuing on central and eatern Europe) and in the preoperational ALADIN- FRANCE uite at Météo-France. It ha alo been implemented in the global ytem ARPEGE, in the framework of incremental initialization, o a to further reduce the diabatic dicrepancy and preerve tidal mode (a uggeted in LH94). In order to reaonably damp the large-cale gravity component introduced by analyi, a more elective Dolph Chebyhev filter ha been ued (with 5 h a the top-band edge). REFERENCES Bubnová, R., G. Hello, P. Bénard, and J.-F. Geleyn, 1995: Integration of the fully elatic equation cat in the hydrotatic preure terrain-following coordinate in the framework of the ARPE- GE/Aladin NWP ytem. Mon. Wea. Rev., 13, 515 535. Fillion, L., H. L. Mitchell, H. Richie, and A. Staniforth, 1995: The impact of a digital filter finalization technique in a global data aimilation ytem. Tellu, 47A, 304-33. Huang, X.-Y., and P. Lynch, 1993: Diabatic digital-filtering initialization: Application to the HIRLAM model. Mon. Wea. Rev., 11, 589 603. Lynch, P., 1997: The Dolph Chebyhev window: A imple optimal filter. Mon. Wea. Rev., 15, 655 660., and X.-Y. Huang, 199: Initialization of the HIRLAM model uing a digital filter. Mon. Wea. Rev., 10, 1019 1034., and, 1994: Diabatic initialization uing recurive filter Tellu, 46A, 583 597.