The retrieval error analysis of atmospheric temperature profile from Satellite Data

Transcription

1 The retreval error analyss of atmospherc temperature profle from Satellte Data HUANG Jng 1, QIU Chongjan 1 and MA Gang 1 College of Atmospherc Scences, Lanzhou Unversty, Chna Natonal Satellte Meteorologcal Center, Chna Meteorologcal Admnstraton, Bejng 181 Abstract A theoretcal analyss s performed to evaluate the retreval precson of atmospherc temperature profles obtaned wth the HIRS/3 data. Ths method s based on sngular value decomposton and emprcal orthogonal functon technology. The theoretcal results showed that the absolute errors n low lattude were less than that n the mddle-hgh lattude and the geography dstrbuton of relatve errors s opposte to absolute errors n general. For vertcal dstrbuton, the errors are bggsh n the upper and lower atmosphere;lesser from 5hpa to 85hpa. The sensbltes of retreval to observaton are the lowest n mddle atmosphere. Introducton The meteorologcal satellte can detect the global. The functons of satellte data used n weather and clmate predcton are always regarded very mportant. Recently, hgh performance vertcal detectors are put nto the launched meteorologcal satellte one after the other. It has been proved that usng three- and four-dmenson assmlaton method can assmlate the satellte radaton data drectly and mprove the global weather forecast dstnctly. But as an nformaton source of atmospherc parameter, the nformaton provded by satellte data s narrow. In other words, the retreval precson of temperature and humdty s lmted. So the estmaton of retreval precson s mportant to the exerton of retreval results and the assmlaton of satellte data. Theoretcally speakng, retreval errors come from three aspects: radance observaton error, vertcal resoluton error and error of the fast radatve transfer model. Many scholars have theoretcally analyzed the retreval vertcal resoluton and errors on temperature and humdty profles snce 197s (Huang et al., 199; Rodgers, 1988; Thompson et al., 1986). These analyses were based on radatve transfer equaton (RTE). Because the retreval errors n theory vary wth regons and seasons, a more detaled mage of error dstrbuton s needed. In ths paper, an analytc method of retreval error s proposed. It s based on generalzed lnear nverses theory. The retreved atmospherc parameter modal can be separated under the hypothess of RTE S lnearzaton. At the same tme, the basal vertcal structure of temperature can be pcked-up by usng EOF technque. The method of error analyss On the condton of lnear approxmaton, the ssue of retrevng temperature profle can be come down to calculate the ntegral equaton: p s δi = K ( p) δt( p) dp (1) ν ν where I v s the radaton dose reached satellte nductor at frequency v, T( p ) s the ar 771

2 temperature at pressure p, p s the surface press, K ( p) s varaton kernel functon. ν ν ν s δ I = I I, δ T = T T,superscrpt denote reference value. For practcal applcaton, when the number of avalable observaton channels s N and the layers of the profle to be retreved s set to M, N lnear equatons can be constructed and expressed by: δi=kδ T. As mentoned n ntroducton, the characters of kernel functon often make the retreval ll-posed and cause the soluton nonunque or senstve to the observaton errors. The propertes of the soluton can be examned by generalzed nverse theory (Wggns, 197). Ths theory s based on sngular vector decomposton (SVD). From the theory, matrx K can be decompounded as: T K=UpΛpV p, () where Λ p s P P matrx contanng P nonzero egenvalues along the dagonal; U p, Vp s constructed by the left vectors u and rght vectors v, respectvely. T T relaton u u = δ and v v δ ν u, v satsfy the orthogonal j, j j =, j. The decomposton mples that δt s projecton n V p have relatonshp wth the observaton data and can be retreved from observaton. Whle the projecton of δt n subspace V, s ndependent of the observaton. That s the so-called resoluton error. Assume the observaton errors are ndependent and have the same varanceσ d. Then the varance of observaton-caused error s obtaned by: σ d - T σ rd = tr( VpΛ PV p ) = Rσ d, (3) M where tr(a) s the trace of matrx, R s error amplfcaton factor whch s manly determned by the mnmum of egenvalueλ. In actual calculaton, λ s set to zero f λ s less than a specfed p value (Chou, 1986; Wunsch, 1978). Another prmary source of retreval error s the resoluton error. In order to analyss t, EOF technque s used. Consder a tme seres of temperature profles represented by a L M matrx A. T T T Solve A Aq = q γ, we get M egenvalues γ and egenvectors q. δtl can be expanded accordng to the orthogonal prmary functon. After the projecton of q n space V p had been counted, the total resoluton error can be obtaned by: where σ M M P sk, = c ek, = c ( q ajvj) = 1 = 1 j= 1, (4) c can be drectly calculated from sample data. When the model error ddn t be consdered, retreval error σ rk, s the square root of the sum of observaton-caused error, resoluton error and truncaton error of EOF. The retreval error analyss Utlzng the error estmate method gven n last secton, the global retreval error dstrbuton of temperature profle can be calculated wth HIRS/3 data. The ar parameter samples are the 1 1 NCEP data n and The reference values are the average temperature profles at each spot. The vertcal structure of temperature ncrement n most regons can be represented by the frst 7 truncated EOF vectors. Optmum truncaton order n SVD It s seen from last secton that resoluton errors decrease when the truncaton order n SVD 77

3 ncreases, whle the observaton-caused errors ncrease. A good truncate order should be selected by consderng both stablty and resoluton. When the observaton errorσ and atmosphere parameter samples were gven, the optmum truncate order can be confrmed accordng to the least error n whole layer. Fg.1 s the dstrbuton of optmum truncaton order obtaned wth HIRS/3 data n and. Here the assumpton that brghtness temperature error s.5k s appled. It was shown n fg.1 that P s small near Tbetan Plateau, low lattude regon and some regons of Antarctca. In, P s on some equator regons of Indan Ocean. In, the regons correspondng to small P extend. In most regons of mddle-low lattude of Northern Hemsphere P s smaller than or equal to 4. In Southern Hemsphere, the values of P are smlar to that n. The magntude of P represents the nformaton of temperature profles retaned n the satellte data. The larger P, the more nformaton can be retreved from satellte data. d Fg The optmum truncaton order P n and In fact, the dstrbuton of optmum truncaton order s dependent on the dstrbuton of temperature mean square devaton. Fg. s the dstrbuton of averaged mean square devaton of temperature over the whole layer. It shows that the value n low lattude regon has mnmum and t s larger n wnter than that n summer n md-hgh lattude regon (Northern Hemsphere n and Southern Hemsphere n y).ths explans that hgh (or low) mean square devaton of temperature s correspondng to hgh (or low) optmum truncaton order. 6E 1E 18 1W 6W 6E 1E 18 1W 6W Fg. The rms (K) of temperature n whole level n and Result analyss The retreval errors can be calculated accordng to the optmal truncaton order n each net pont. Fgs.3 s the retreval errors at 5hpa, 85hpa n and. At 5hpa, the hgh error regons n northern hemsphere le n the regon from north Pacfc to North Amercan. 773

4 Qngha-Xzang Plateau s also a hgh error zone; the error s about.k currently. The hgh error zone n the southern hemsphere s stll n the hgh lattude regon, the error s about.k. The error n low lattude s less than 1.K. Error dstrbuton n s smlar to. But the values are less than, about.5k. The retreval errors n 85hpa te to landform obvously. Whether or, the errors n manland are hgher than that of ocean regons clearly. The errors n manland are about.k () or 1.5K (y). The errors n ocean regon are about.5-1.5k. 5hpa 5hpa 6E 1E 18 1W 6W 6E 1E 18 1W 6W 85hpa 85hpa 6E 1E 18 1W 6W 6E 1E 18 1W 6W Fg.3 The absolute errors at 5hpa and 8hpa pressure(hpa) pressure(hpa) relatve error relatve error 6 o N 3 o N Fg. 4 Mean relatve error at 6 N and 3 N The total concluson ganed from above analyss s that the retreval errors at ocean and Torrd Zone were lesser. But that ddn t represent the retreval ablty, because the temperature 774

5 tme varablty n these regons s lesser. In order to show t, the relatve errors that are defned as the quotent of absolute errors and the rms of temperature have been calculated. Fgs.4 s the vertcal profles of relatve errors averaged n two lattudes. The results showed that the character of vertcal dstrbuton at 6 N, 3 N s smlar n and. That s, the relatve errors are small n mddle atmosphere; hgh n upper and lower atmosphere. The character of concrete values s that the hgher (lower) lattude s correspondng to smaller (bgger) relatve error. The varaton trend at lattude s dfferent to absolute errors completely. The comparson of relatve error n and showed that the value n was smaller than that n. Especally the dfference between and at 3 N s relatvely bg. These dstrbuton characters of error n lattude relate to the tme varablty of the atmospherc temperature. Summary and dscusson The method of countng resoluton errors and observaton-caused errors wth satellte data s based on generalzed lnear nverses theory. When the observaton error s gven, the retreval errors at each alttude can be calculated easly. The man modal of atmospherc vertcal structure can be ganed by EOF. Usng NCEP temperature data as the samples, ths method can calculate the optmal truncaton order n generalzed lnear nverses and calculate the error dstrbuton of temperature profle at each equpressure surface wth HIRS/3 data n and. The basc characters are as follows: (1)The optmum truncaton order decdes the effectve nformaton offered by satellte measurements. The values obtaned wth HIRS/3 data are between 3 and 7. The results show that the optmum truncaton order s small n low lattude regons and hgh n md-hgh lattude areas. Ths character relates to the tme varablty of temperature profle. ()For vertcal dstrbuton, the retreval errors of temperature profle are more at upper and lower atmosphere. The errors between 4hpa and 85hpa are relatvely small. (3)For geography dstrbuton, the absolute errors n mddle-hgh lattude regon, especally n the northern hemsphere, are the most; n equator zone are the least. Whle the dstrbuton character of relatve errors s reverse to absolute errors. Ths ndcates that satellte data offered more effectve nformaton n the regons of bggsh temperature varablty. References Chou Jfan, 1986: Long-perod numercal weather forecast. Meteorology publshng company, Bejng, Huang, H. L., W. L. Smth, and H. M. Woolf 199: Vertcal resoluton and accuracy of atmospherc nfrared soundng spectrometers. Journal of Appled Meteorology, 31, Rodgers, C.D., 1988: A general error analyss for profle retreval. Advances n Remote Sensng Retreval Methods, A. Deepak Publshng, Thompson, O.E. 1991: Regularzng the satellte temperature-retreval problem through 775

6 sngular-value decomposton of the radatve transfer physcs. Mon. Wea. Rev., 1, Thompson, O.E., D. D. Dazlch, and Yu-Ta Hou 1986: The ll-posed nature of the satellte temperature retreval problem and the lmts of retrevablty. Journal of Atmospherc and Oceanc Technology, 3, Wggns, R. A. 197: The general lnear nverse problem: mplcaton of surface waves and free oscllatons for Earth structure. Revews of Geophyscs and Space Scences, 1, Wunsch, C., 1978: The North Atlantc general crculaton west of 5 W determned by nverse methods. Revews of Geophyscs and Space Physcs, 16(4),

7