NWP SAF Satellite Application Facility for Numerical Weather Prediction

Transcription

1 Satellte Applcaton Faclty for Numercal Weather Predcton Document NWPSAF-MF-UD-5 Verson 1.3 October 11 ANNEX OF SCENTFC DESCPTON AAPP NAVATON Anne Marsoun (Météo-France) Pascal Brunel (Météo-France)

2 5_Navgaton.doc Date : October 11 Ths documentaton was developed wthn the context of the EUMETSAT Satellte Applcaton Faclty on Numercal Weather Predcton (NWP SAF), under the Cooperaton Agreement dated 1st December 3, between EUMETSAT and the Met Offce, UK, by one or more partners wthn the NWP SAF. The partners n the NWP SAF are the Met Offce, ECMWF, KNM and Météo France. Copyrght 6, EUMETSAT, All ghts eserved. Change record Verson Date Author / changed by emarks 1. June 4 A. Marsoun, Orgnal P. Brunel Aprl 5 N. Atknson Add secton 8 1. June 6 N. Atknson Change ttle and ntroducton text for AAPP v6 1.3 Oct 11 N. Atknson Update logos, for AAPP v7 Page / 9

3 TABLE OF CONTENTS 5_Navgaton.doc Date : October ntroducton Defntons, unts, conventons eference frames and converson matrx reenwch reference frame ( ) Local reference frame of an observaton pont ( L ) Nomnal atttude reference frame ( A ) Local normal pontng mode ( A ) Yaw steerng mode ( A1 ) eocentrc mode ( A ) Velocty calculatons Spacecraft-fxed reference frame ( S ) nstrument reference frame ( ) ntermedate reference frames Scannng geometry Prncples Scannng n a plane Concal scannng AS scannng Earth locaton of the vewed pont Satellte and sun vewng angles Footprnt calculaton Prncples Footprnt for a radometer scannng n a plane Footprnt for concal scannng AS footprnt Orbt predcton usng -lne elements esults of a 6-month comparson Concluson... 7 Page 3 / 9

4 FUES 5_Navgaton.doc Date : October 11 Fgure -1: vector lyng n a cone... 5 Fgure 3-1: and L reference frames... 8 Fgure 3-: zenth and azmuth angles... 8 Fgure 3-3: A reference frame... 1 Fgure 3-4:, 1 and reference frames Fgure 3-5: 1 to 3 converson Fgure 8-1 : ANA NOAA-17 wth TBUS... 8 Fgure 8- : ANA NOAA-17 wth TLE NTODUCTON Versons 1 to 5 of the ATOVS and AVH Processng Package (AAPP) support the processng of drect readout HPT data from the NOAA polar orbter satelltes. To prepare for the launch of METOP (n 6), the AAPP navgaton modules were revsed n AAPP v5. Ths document detals the navgaton method that s used n AAPP v5 and subsequent versons of AAPP. The basc algorthm s the same as n the earler AAPP versons and has been presented n: [1] Brunel P. and Marsoun A.,, Operatonal AVH navgaton results, nternatonal Journal of emote Sensng, Vol. 1, No. 5, [] osborough.w., Baldwn D. and Emery W., 1994, Precse AVH mage Navgaton, EEE Transactons on eoscence and emote Sensng, Vol. 3, No. 3, May 1994, The new verson dffers on the followng ponts: several modes of the satellte atttude can be used, several scannng geometry are consdered (plane, conc, AS) the software s more modular and a new satellte or nstrument can be added easly Ths text gves a detaled descrpton of the equatons used n AAPP verson 5 and subsequent versons, and ndcates the names of the correspondng AAPP subroutnes. AAPP contans subroutnes that are specfc to AVH navgaton: nvloc and nvlorb, to convert the lattude and longtude of the vewed pxel nto tme and scannng angle, estmatt and estmrp, to estmate the atttude error usng landmarks true postons and observed vewng vectors. These routnes have not been revsted but only modfed concernng the calls to other subroutnes. They are not consdered here. The navgaton subroutnes that have not been changed are also gnored n ths text. Page 4 / 9

5 5_Navgaton.doc Date : October 11. DEFNTONS, UNTS, CONVENTONS The equatons presented n ths document assume the followng unts: angle n radans, tme n seconds and dstance n klometers. A vector s wrtten V, SM or, f t s a unt vector, v, sm. A matrx s wrtten M. The components of a vector v on the X, Y, Z axs of a reference frame are v(1), v(), v(3), n order to be close to the notatons of the FOTAN code. n the formula w = u x v, x ndcates a vector product, whch corresponds to : w( 1) = u() v(3) u(3) v() w( ) = u(3) v(1) u(1) v(3) w( 3) = u(1) v() u() v(1) Defnng a cone A cone s used for several applcatons: concal scannng, footprnt calculaton and AS vewng geometry. So, a few defntons relatve to a cone are gven here. A cone s defned by an axs and a half angle, δ. Usng a reference frame where the center s at the cone apex, the X-axs s algned wth the cone axs and Y- and Z - axs are normal to the cone axs, any vector V lyng n the cone can be wrtten: cos( δ ) V = sn( δ ) cos( φ) sn( δ ) sn( φ) where φ s the angle between the plane (X,Y) and the plane (X,V), varyng from to π. Ths angle has no specfc name (unlke the half angle). As t s smlar to an azmuth, ths name wll be used n the document. X δ Y φ Z Fgure -1: vector lyng n a cone Page 5 / 9

6 5_Navgaton.doc Date : October 11 eference frames and converson matrx The reference frames used for navgaton have a physcal meanng and each axs drecton can be precsely defned. However, ts orentaton can be chosen arbtrarly, as long as the reference frame s drect. The orentatons already used n AAPP have been kept; they correspond to those of the NOAA documents. The converson matrx M old_to_new from a reference frame old to a reference frame new s the matrx that converts the coordnates n old of a vector V nto ts coordnates n new V new = M old_to_new x V old The new unt vectors n old are the columns of M new_to_old the lnes of M old_to_new For all converson matrces used here: M -1 =M T f new s obtaned from old by a rotaton of angle ϕ about one axs of old, the converson matrces between new and old are gven by two of the sx formulas: M new_to_old : M old_to_new : rotaton about X axs rotaton about Y axs rotaton about Z-axs 1 cos ( ϕ ) sn( ϕ ) ( ϕ ) sn( ϕ) cos( ϕ) sn( ϕ) ( ) 1 sn( ϕ) cos ϕ ( ) ( ) sn( ϕ) c os( ϕ) sn ϕ cos ϕ 1 ( ϕ ) sn( ϕ ) cos( ϕ) sn( ϕ) ( ) 1 sn( ϕ) cos ϕ ( ) ( ) sn ϕ cos ϕ cos cos( ϕ ) sn( ϕ) sn( ϕ) c os( ϕ) 1 1 Page 6 / 9

7 5_Navgaton.doc Date : October EFEENCE FAMES AND CONVESON MATX 3.1. EENWCH EFEENCE FAME ( ) s an Earth fxed reference frame. The center s at the Earth s gravty center, O. X s the lne from the Earth s gravty center to the ntersecton between the equator and the reenwch merdan, whch defnes the postve drecton. Z s the polar axs, postve toward the North Pole. Y s normal to X and Z (n the equatoral plane, toward longtude 9 E). The followng relatons are useful: wth ( lac ) cos( lo) OM = cos( lac ) sn( lo) sn( la ) C = 1 e P cos ( la M : a pont on the Earth s surface C ) tan( la e C = 1 ) = tan( la P E ) lo to the plane (polar axs, merdan of M) la : geographcal lattude of M,.e. angle from the equatoral plane to the normal to the Earth ellpsod la C : geocentrc lattude of M,.e. angle from the equatoral plane to OM : Earth s radus at M E, P : Earth s equatoral and polar radus; AAPP uses the reference ellpsod : longtude of M,.e. angle from the plane (polar axs, reenwch merdan) S 8: E = km flattenng factor f=1 / P = (1 - f) E = km P E Page 7 / 9

8 5_Navgaton.doc Date : October 11 North Pole Z Z L reenwch merdan O lo parallel M la la C equator merdan X L Y L Y toward sun or satellte north Az θ Z L M Y L east X L south X Fgure 3-1: and L reference frames Fgure 3-: zenth and azmuth angles Page 8 / 9

9 5_Navgaton.doc Date : October LOCAL EFEENCE FAME OF AN OBSEVATON PONT ( L ) L s an Earth fxed reference frame. The center s at the observaton pont M. Z L s the local vertcal (.e. the normal to the Earth ellpsod), postve upward X L s normal to Z L and n the half plane that contans the merdan of the pont, postve toward south. Y L s normal to X L and Z L. L s deduced from by two rotatons rotaton of an angle lo about the axs Z, whch gves an ntermedate reference frame rotaton of an angle π/ la about the Y-axs of the ntermedate reference frame So, the converson matrx L from to L s: L = L = L = sn sn cos ( π / la ) sn( π / la ) 1 ( ) ( ) π / la cos π / la ( la ) cos( la ) 1 ( ) ( ) la sn la sn( la ) cos( lo) sn( lo) cos( la ) cos( lo) sn( la cos( la x ( lo) cos sn( lo) ) sn( lo) cos( lo) ) sn( lo) x ( lo) cos sn( lo) sn( lo) cos( lo) cos( la ) sn( la ) 1 sn( lo) cos( lo) L s used only to calculate the vewng angle of an object, sun or satellte, S, seen from an observaton pont on Earth, M: the zenth angle, θ, whch s the angle between the local vertcal and the pont-to-object vector, vares from to 18 degrees. the azmuth angle, Az, whch s the angle between a vertcal half plane contanng the north drecton (.e. opposte to X L ) and the pont-to-object vector (see Fgure 3-); ths angle vares from 18 to 18 degrees, s at true north and postve toward east. The azmuth angle defnton s the one of the NOAA level 1b. Due to L axs orentaton, Az s the angle from the pont-to-object vector to the vertcal half plane contanng the north drecton. The vector MS can be expressed n L : sn( θ ) cos( π Az) MS = D sn( θ ) sn( π Az) cos( θ ) = D sn( θ ) cos( Az) sn( θ ) sn( Az) cos( θ ) where D s the dstance from the observaton pont to the object. 1 Page 9 / 9

10 5_Navgaton.doc Date : October 11 The converson matrx L s calculated n two subroutnes: zenaz, the general routne to calculate the vewng angles and ntrack, whch s adapted to the case of a trackng staton NOMNAL ATTTUDE EFEENCE FAME ( A ) A represents to the nomnal orentaton of the spacecraft,.e. the atttude that the spacecraft would have f the atttude control system was perfect. The precse defnton of ths frame depends on the current atttude mode, local normal pontng for NOAA POES and yaw steerng for METOP. However, the varous atttude modes of the polar orbter satelltes are not so dfferent and a general defnton can be gven for A : The center s at the satellte gravty center. X A s close to the local terrestral vertcal, postve toward the Earth. Y A s nearly collnear to the satellte velocty, opposte n drecton. Z A s nearly orthogonal to the orbt plane, postve n the drecton of the angular momentum vector of the orbt.e toward the left of the trajectory. The followng sectons precsely defne A and calculate the converson matrx T from A to, for several atttude modes. The matrx T s obtaned by calculatng u, v, w, the unt vectors of A n. T = u(1) u() u(3) v(1) v() v(3) w(1) w() w(3) left Z A Y A X A satellte velocty Earth's center Fgure 3-3: A reference frame Local normal pontng mode ( A ) A s defned as follows: X A s algned wth the local terrestral vertcal (.e. the normal to the Earth ellpsod passng by the satellte), postve toward the Earth. Z A s normal to X A and to the satellte velocty; t s postve n the drecton of the angular momentum vector of the orbt. Y A completes the orthogonal system The vector u s calculated wth the satellte longtude, lon, and geographcal lattude lat u = cos( lat ) cos( lon) cos( lat ) sn( lon) sn( lat ) The vector w s normal to u and to the satellte velocty vector V. Ths vector s the absolute velocty or the velocty n an nertal reference frame (as usual orbt predcton). w = u x V / u x V And fnally: v = w x u Page 1 / 9

11 5_Navgaton.doc Date : October Yaw steerng mode ( A1 ) A and A1 have the same X-axs and they dffer on the Y- and Z-axs: X A1 s algned wth the local terrestral vertcal (.e. the normal to the Earth ellpsod passng by the satellte), postve toward the Earth. Z A1 s normal to X A1 and to the satellte velocty relatve to the Earth,.e. the velocty n ; t s postve n the drecton of the angular momentum vector of the orbt. Y A1 completes the orthogonal system The calculatons are the same as n except for the vector w s normal to u and to the satellte velocty vector, Vr, n the reference frame. w = - u x Vr / u x Vr Vr s a relatve velocty vector, obtaned as follows: Vr = V - ϖ E x OS Vr ( 1) = V (1) + ω OS() Vr ( ) = V () ω OS(1) Vr (3) = V wth ϖ E : angular velocty vector of the Earth OS (3) ϖ E = rad/s (.e rad/day ) : vector from Earth's gravty centre to satellte E E eocentrc mode ( A ) A s defned as follows: X A s algned wth the lne from the Earth center to the satellte center,.e. the geocentrc vertcal, postve toward the Earth. Z A s normal to X A and to the satellte velocty,.e. orthogonal to the orbt plane; t s postve n the drecton of the angular momentum vector of the orbt. Y A completes the orthogonal system The calculatons are smpler: u drectly depends on the satellte Cartesan coordnates and w s normal to u and to the satellte velocty vector V. u= - OS / OS w = - u x V / u x V v = w x u Velocty calculatons t should be noted that the velocty vectors stored n the satpos fles are relatve velocty vectors n. They have been calculated by one of the subroutnes pvtogrw, pv5togrw or pvtegrw. The matrx T s calculated by the subroutne snagre, whch use a satpos velocty vector as nput. So, snagre does not convert V nto Vr but Vr nto V, when needed (NOAA operatonal mode and geocentrc mode): Page 11 / 9

12 5_Navgaton.doc Date : October 11 V ( 1) = Vr (1) ω OS() V ( ) = Vr () + ω OS(1) V (3) = Vr (3) E E 3.4. SPACECAFT-FXED EFEENCE FAME ( S ) S s defned as follows: The center s at the satellte gravty center. X S s the vertcal axs of the spacecraft. Y S s the longtudnal axs of the spacecraft. Z S s the transversal axs of the spacecraft. These axes are defned precsely wth respect to some elements of the spacecraft structure, ther postve drecton beng consstent wth the spacecraft nomnal orentaton. S should be perfectly algned wth A, but t dffers from t by three small rotaton angles: yaw about X A, roll about Y A and ptch about the Z A. Usng the small angle approxmaton, the converson matrx between S and A are gven by smple formulas. converson from S to A converson from A to S A = 1 p r p r 1 y y 1 A T = where y, r, and p are the yaw, roll and ptch angles. 1 p r p r 1 y y 1 The sgn conventons are the followng: yaw > : the spacecraft longtudnal axs s orentated toward the rght of the trajectory roll > : the spacecraft vertcal axs s on the rght of the sub-track ptch > : the spacecraft vertcal axs s behnd the sub-pont The atttude matrx A s calculated n the subroutne calatt 3.5. NSTUMENT EFEENCE FAME ( ) Each nstrument s mounted on the spacecraft but some msalgnments may occur. So, the nstrumentfxed reference frame, whch should be perfectly algned wth S, may dffer slghtly from S t by three small msalgnment angles. s defned as follows: The center s at the satellte gravty center. X s nearly algned wth X s. Y s nearly algned wth Y s. Z s nearly algned wth Z s. s obtaned from s by 3 rotatons: n yaw about X S, n roll about Y S and n ptch about Z S. Page 1 / 9

13 5_Navgaton.doc Date : October 11 The converson matrces are the followng: converson from to S converson from S to D = 1 p r p 1 y r y 1 D T = 1 p r p 1 y where y, r and p are the msalgnment angles n yaw, roll and ptch. y r NTEMEDATE EFEENCE FAMES Addtonal reference frames are used for some nstruments. They are defned below. 1, useful for an nstrument nclned forward or backward of the satellte nadr, s deduced from by a rotaton of an angle β about the nstrument transversal axs Z : The center s at the satellte gravty center. X 1 s obtaned from X by a rotaton of β about Z. Y 1 s obtaned from Y by a rotaton of β about Z Z 1 = Z. The converson matrx from 1 to s; M 1 = ( β ) sn( β ) sn( β ) c os( β ) 1 β s counted postvely accordng to the drect orentaton of, whch means: β > X 1 looks backwards of the satellte sub-pont (same conventon as for the roll), useful to defne AS scannng and to calculate the nstrument footprnt, has ts X-axs algned wth the vewng drecton. s deduced from 1 by a rotaton of an angle -α about axs Y 1 (the negatve sgn s due to the scannng angle defnton, see 4. for detals): The center s at the satellte gravty center. X s obtaned from Z 1 by a rotaton of -α about Y 1. Y = Y 1. Z s obtaned from Z 1 by a rotaton of -α about Y 1 The converson matrx from to 1 s: M = cos sn ( α ) sn( α ) 1 ( ) ( ) α cos α = sn ( α ) sn( α ) 1 ( ) ( ) α cos α α s counted postvely accordng to the drect orentaton of 1, whch means: α> X looks on the left of the satellte trajectory Page 13 / 9

14 5_Navgaton.doc Date : October 11 3, useful to calculate the nstrument footprnt n case of concal scannng, has ts X-axs algned wth the vewng drecton. 3 s deduced from 1 by two rotatons rotaton of an angle φ about axs X 1, whch gves the ntermedate reference frame 13 rotaton of an angle δ about axs Z 13, whch gves the reference frame 3 The converson matrx from 3 to 1 s: 1 M 3 = M 13_to_1 x M 3_to_13 = cos( φ) sn( φ) sn( φ) ( ) cos φ cos( δ ) sn( δ ) M 3 = sn( δ ) cos( φ) cos( δ ) cos( φ) sn( φ) ( ) sn( δ ) sn( φ) cos( δ ) sn( φ) cos φ x ( δ ) sn( δ ) sn( δ ) c os( δ ) 1 Z left Z =Z 1 α β Y 1 =Y α Y backwards Z X β left Z 1 α Y 1 =Y X Earth X 1 Z =Z 1 β Y Y 1 =Y backwards X α X 1 Earth X β Earth Fgure 3-4:, 1 and reference frames. X 1 (n red) s rotated by an angle β about the axs Z to obtan 1 (n black), whch s rotated by an angle -α about the axs Y 1 to obtan (n blue). Page 14 / 9

15 5_Navgaton.doc Date : October 11 Z 13 =Z 3 φ Y 3 left X 1 =X 13 Z 1 Z 13 =Z 3 Y 3 δ Y 13 Z 13 =Z 3 left φ Z 1 Y 13 δ δ φ Y 1 Y 13 backwards φ Y 1 backwards X 3 δ X 3 Earth X 1 =X 13 Earth X 1 =X 13 Fgure 3-5: 1 to 3 converson. 1 (n red) s rotated by an angle φ about the axs X 1 to obtan 13 (n black), whch s rotated by an angle δ about the axs Z 13 to obtan 3 (n blue). n Fgure 3-4 and Fgure 3-5, the words left and backwards refer to the satellte trajectory and ndcate only approxmate drectons. Page 15 / 9

16 4. SCANNN EOMETY 4.1. PNCPLES 5_Navgaton.doc Date : October 11 The vewng drecton of the FOV center (FOV = nstantaneous Feld of Vew) can be calculated n the spacecraft-fxed reference frame S n three stages: a) the lne and pxel numbers are converted nto a tme and an angle, scannng angle or scannng azmuth, b) the tme and angle gves the unt vector of the drecton satellte-toward-vewed-pont (sm) n the reference frame, c) the unt vector s converted from to S, taken nto account the nstrument msalgnments. Stages a) and b) depend on the scannng geometry and are descrbed n the followng paragraphs. Stage c) s consdered as ndependent of the scannng geometry and made as follows: 1 sm S = D sm = p r sm sm p 1 y r y 1 sm ( 1) = sm (1) + p () (3) sm r sm ) = p sm (1) + sm () + y sm (3) S ( S ( 3) = r (1) () S sm y sm sm sm + There s not a unque subroutne assocated to each subsecton of secton 4. lptovewvect do the calculatons presented n 4., 4.3 and 4.1 lptovewvect_as do the calculatons presented n 4.4 and 4.1 AS has been consdered separately snce four pxels are obtaned smultaneously, as explaned n 4.4. (3) Page 16 / 9

17 5_Navgaton.doc Date : October SCANNN N A PLANE Most radometers scannng n a plane are cross-track scanners,.e. they scan perpendcular to the drecton of movement of the satellte. The scannng plane may also be tlted backward or forward, by an angle β as defned n 3.6. The lne and pxel numbers are converted nto tme and scannng angle as follows: t = t + l l ) T + t + ( p 1) T ( α = p p ) α ( stepl step offset wth l, l : lne number and reference lne number t, t : start tmes of the lne numbers l and l p p : pxel number stepp : pxel number at sub-track,.e. (real) pxel number when the vewng drecton s algned wth X 1 α : scannng angle, defned as the angle from the vewng drecton to the axs Z 1 counted postvely accordng to the drect orentaton of 1 α step T stepl T stepp t offset α > on the left of the satellte trajectory : center to center FOV step angle (FOV = Feld of Vew), as the angle α ncreases from rght to left α step > for a radometer scannng from rght to left : full scan perod,.e. the tme nterval between two consecutve lnes : step tme / FOV,.e. the tme nterval between two consecutve pxels : tme nterval between the start tme and the frst pxel of the lne (ntal offset tme from TP to start of ntegraton perod, for NOAA) The above equatons and defntons are based on the NOAA satellte scannng nstrument parameters. However they are general enough to be appled to other nstruments. For NOAA nstrument, the tme s obtaned through the TP clock (TP=TOS nformaton Processor) onboard the spacecraft. The vewng drecton unt vector, sm, s n the scannng plane (Z 1,X 1 ) and ts angle wth the axs Z 1 s π/-α: ( π / α ) sn sm 1 = ( ) = cos π / α ( α ) ( ) sn α For a cross-track scanner, β=, the above formula drectly gves vewng vector n. n the general case, the vewng vector s converted from 1 to wth the matrx M 1 defned n 3.6: sm = M 1 sm 1 = ( β ) sn( β ) sn( β ) cos( β ) x 1 ( α ) ( β ) ( ) = sn( β ) sn α sn cos( α ) cos( α ) ( ) α Page 17 / 9

18 4.3. CONCAL SCANNN 5_Navgaton.doc Date : October 11 The radometer vewng drecton les n a cone and rotates about the cone axs at a constant speed. The cone axs s n the nstrument longtudnal plane, algned wth the vertcal axs or nclned forward or backward, by an angle β (as defned n 3.6). t corresponds to the axs X 1 of 1. A lne corresponds to a lmted part of the cone, forward or backward of the cone axs. The lne number s converted nto tme as n 4. and the pxel number s converted nto scannng azmuth by the followng equatons: γ = p p ) γ φ = γ ( step f backward scannng (of the cone axs) or φ = γ + π f forward scannng (of the cone axs) wth p : pxel number p : pxel number at sub-track,.e. (real) pxel number when the vewng drecton corresponds to X 1, f backward scannng, or to -X 1, f forward scannng γ : scannng angle, defned as the angle from the vewng drecton at p to the vewng drecton, counted postvely accordng to the drect orentaton of 1 φ : scannng azmuth,.e. angle from the axs Y 1 to the vewng drecton counted postvely accordng to the drect orentaton of 1 ([, π] on the left of the satellte trajectory and [π, π] on the rght) : center to center FOV step angle γ step Wth the angle γ and pxel p defntons, γ and γ step have the followng sgns wth respect to the satellte trajectory: f backward scannng γ > on the left f forward scannng γ > on the rght γ step > scannng from rght to left γ step > scannng from left to rght The scannng azmuth and half angle, δ, gve the vewng vector n 1 (as presented n ): cos( δ ) sm 1 = sn( δ ) cos( φ) sn( δ ) sn( φ) f the cone axs s algned wth the nstrument vertcal, β=, the above formula drectly gves vewng vector n. n the general case, the vewng vector s converted from 1 to wth the matrx M 1 defned n 3.6: sm = M 1 sm 1 = sm = ( β ) cos sn( β ) cos ( β ) sn( β ) ( δ ) ( δ ) sn( β ) cos( β ) x 1 sn( β ) sn( δ ) cos( φ) + cos( β ) sn( δ ) cos( φ) sn( δ ) sn( φ) cos( δ ) sn( δ ) cos( φ) sn( δ ) sn( φ) Page 18 / 9

19 4.4. AS SCANNN 5_Navgaton.doc Date : October 11 AS nstrument, nfrared Atmospherc Soundng nterferometer, s a sounder coupled to an ntegrated magng subsystem (S). AS feld of vew contans 4 sounder pxels and an S mage of 64 by 64 pxels. The operatonal AS navgaton wll be based on a co-regstraton of S and AVH, so the AVH Earth locaton wll be converted nto an AS Earth locaton. Such a method has been aded because AS s attached on METOP platform va a vbraton dampng mechansm. Ths secton s restrcted to AS sounder and smply apples the geometrcal laws of the scannng, producng a draft navgaton. AS scannng can be summarzed as follows a so-called cal axs moves n a plane normal to the spacecraft longtudnal axs, a vew corresponds to 4 pxels, at a gven angular dstance, ζ, of the cal axs. A vew s consdered as nstantaneous. ts lne and pxel numbers are converted nto tme of the vew and scannng angle of the cal axs, α, wth the equatons of secton 4.. Then the cal axs unt vector, so, s gven n by: so = ( α ) ( ) sn α The four pxels le n a cone. The cone axs s the cal axs, so, and the half angle s the small angle ξ. so s algned wth X axs. So, the vewng drecton of each pxel M s obtaned n by: sm = cos( ζ ) sn( ζ ) cos( φ ) sn( ζ ) sn( φ ) = 1 ζ cos( φ ) ζ sn( φ ) The AS scan azmuth values are dctated by the drect orentaton of and the numberng of the AS pxels: φ 1 = 7π/4 φ = 5π/4 φ 3 = 3π/4 φ 4 = π/4 M 1 on the rght and backward of the cal axs M on the rght and forward of the cal axs M 3 on the left and forward of the cal axs M 4 on the left and backward of the cal axs The scannng plane s normal to the spacecraft longtudnal axs, 1 = and the vewng vector s converted from to wth the matrx M (defned n 3.6) where the scannng angle concerns the cal axs: sm = M sm α sm = sn α ( ) sn( α ) ( ) ( ) cos α 1 1 cos( φ ) ζ sn( φ ) ζ = ( α ) ζ sn( φ ) sn( α ζ cos( φ) sn( α ) + ζ sn( φ ) cos( α ) ) Page 19 / 9

20 5_Navgaton.doc Date : October EATH LOCATON OF THE VEWED PONT Earth locaton of the vewed pxel s the calculaton of the FOV center lattude and longtude or, whch s equvalent, of ts cartesan coordnates n the reference frame. The unt vector of vewng drecton, sm, s known n the reference frame s through the equatons presented n secton 4. sm, s converted from S to by: sm = T T A T sm S The vector OS, Earth's gravty centre to satellte, s known n. The vector OM, Earth's gravty centre to vewed pont, s gven by: OM = OS + D sm where D, dstance between the satellte and the vewed pont, s the only unknown. The vewed pont s n the Earth's ellpsod, whch gves a second order equaton n D: OM (1) E OM () + E OM (3) + P = 1 a D + b D + c = wth a = sm(1) b = OS(1) c = OS + sm() sm(1) + sm(3) + OS() E P sm() E ( 1) + OS() + OS(3) E P + OS(3) sm(3) E and P are respectvely the terrestral equatoral and polar radus E P D s the smallest soluton of the equaton (the other soluton corresponds to the pont where the vewng drecton goes out of the Earth). As the varable b s negatve, D s gven by: b D = b a a c Wth D, the vector OM can be calculated and ts cartesan coordnates are converted nto lattude and longtude. The calculatons presented n ths secton are done n the subroutne earthpx Page / 9

21 5_Navgaton.doc Date : October SATELLTE AND SUN VEWN ANLES The object, satellte or sun, and the pont M are known n. The vector MS, pont to object, s calculated n then converted to L wth the converson matrx L (defned n 3.): MS = OS - OM MS L = L MS As ndcated n 3., MS L s gven by: sn( θ ) sn( π Az) MS = D sn( θ ) cos( π Az) cos( θ ) where θ s the zenth angle, Az the azmuth angle and D the dstance pont-to-object. So, the zenth and azmuth angles are calculated by: MS(3) = θ arcos MS Az = π - datan ( MS ( ), MS(1) ) L L L These calculatons are done n the subroutne zenaz, for the general case, and by the subroutne trackang, for the case of a trackng staton. Page 1 / 9

22 5_Navgaton.doc Date : October FOOTPNT CALCULATON 7.1. PNCPLES For a sounder havng a rather large FOV, t may be useful to locate not only the FOV center but also the footprnt,.e. the surface on Earth correspondng to the FOV. Ths secton presents an exact calculaton of the footprnt. Such a method s probably over complcated for most applcatons but can be used as reference to test approxmate formulas. The FOV s generally delmted by a cone. The cone axs s the vewng drecton of the pxel center and ts half angle, ε, s a small angle. ε s equal to half of the FOV wdth, whch s the parameter commonly used to descrbe the nstrument characterstcs. There s not a unque subroutne assocated to each subsecton of secton 7: footprnt calculates a footprnt n, as presented n 7.1 and 7. footprnt_as calculates the four AS footprnts n, as presented n 7.3 contour_sondeur_avhrr calculates a footprnt n geographcal coordnates then n AVH coordnates (call footprnt and other routnes) ellpse_sondeur_avhrr calculates a footprnt n AVH coordnates, accordng to the ellpse approxmaton presented n 7. (call footprnt and other routnes) contour_as_avhrr calculates the four AS footprnts n AVH coordnates (call footprnt_as and other routnes) 7.. FOOTPNT FO A ADOMETE SCANNN N A PLANE The vewng drecton of the pxel center, whch s the cone axs, s algned wth X axs. Any vewng drecton, sp, lyng n the cone s gven n by: 1 sp = ε cos( ψ ) wth ψ varyng from to π ε sn( ψ ) The vewng vector s converted from to 1 wth the matrx M : sp 1 = M sp = sn ( α ) sn( α ) 1 ( ) ( ) α cos α 1 x ε cos( ψ ) ε sn( ψ ) Page / 9

23 5_Navgaton.doc Date : October 11 sp 1 = ( α ) ε sn( ψ ) sn( α) ε cos( ψ ) sn( α) + ε sn( ψ ) cos( α ) ψ π For a cross-track scanner, β=, the above formula drectly gves vewng vector n. n the general case, the vewng vector s converted from 1 to wth the matrx M 1 defned n 3.6: sp = M 1 sp 1 = ( β ) sn( β ) sn( β ) cos( β ) ( α ) ε sn( ψ ) sn( α) x ε cos( ψ ) 1 sn( α) + ε sn( ψ ) cos( α ) sp = ( β ) [cos( α ) ε sn( ψ ) sn( α)] sn( β ) ε cos( ψ ) sn( β ) [cos( α ) ε sn( ψ ) sn( α)] + cos( β ) ε cos( ψ ) sn( α) + ε sn( ψ ) cos( α ) ψ π Earth locaton of any pont on the FOV border (.e. the pont correspondng to sp) s the same as Earth locaton of the FOV center and t follows equatons of secton 5. f the objectve s to obtan the footprnt n AVH coordnates, the geographcal coordnates are then converted nto AVH lne and pxel numbers. An ellpse s a good approxmaton of the footprnt. The sem-axs n lne s the dstance between the FOV center, M, (already calculated n 4.) and the pont Q at ψ=: sq = ( β ) cos sn( β ) cos sn ( α ) ε sn( β ) ( α ) + ε cos( β ) ( ) α sm = ( β ) sn( β ) sn cos( α ) cos( α ) ( ) α The sem-axs n pxel s the dstance between the FOV center and the pont Q at ψ=π/: However, f the objectve s to obtan the footprnt n AVH coordnates, there s a very smple formula: ε apx = n number of AVH pxels α (AH) step 7.3. FOOTPNT FO CONCAL SCANNN The pxel center vewng drecton, whch s the cone axs, s algned wth the X 3 axs. of the reference frame 3 defned n 3.6. Any vewng drecton, sp, lyng n the cone s gven n 3 by: 1 sp 3 = ε cos( ψ ) ψ π ε sn( ψ ) The vewng vector s converted from 3 to 1 wth the matrx M 3 : sp 1 = M 3 sp 3 Page 3 / 9

24 5_Navgaton.doc Date : October 11 cos( δ ) sp 1 = sn( δ ) cos( φ) sn( δ ) sn( φ) sn( δ ) cos( δ ) cos( φ) cos( δ ) sn( φ) sn( φ) ( ) cos φ x 1 ε cos( ψ ) ε sn( ψ ) cos( δ ) ε cos( ψ ) sn( δ ) [ ] sn( δ ) sn( φ) + ε cos( ψ ) sn( φ) cos( δ ) + sn( ψ ) cos( φ) sp 1 = sn( δ ) cos( φ) + ε [ cos( ψ ) cos( φ) cos( δ ) sn( ψ ) sn( φ) ] ψ π f the cone axs s algned wth the nstrument vertcal, β=, the above formula drectly gves vewng vector n. n the general case, the vewng vector s converted from 1 to wth the matrx M 1 defned n 3.6: ( ) β sp = M 1 sp 1 = sn( β ) sn( β ) cos( β ) x sp 1 1 sp ( 1) = cos( β ) sp 1 (1) sn( β ) sp () 1 sp ( 1) = sn( β ) sp 1 (1) + cos( β ) sp () 1 sp 3) = (3) ( 1 sp 7.4. AS FOOTPNT The FOV of each AS pxel s delmted by a cone. The cone axs s the vewng drecton of the pxel center and ts half angle, ε, s a small angle. The four AS pxels le n a cone whose axs s the cal axs, algned wth the axs X of. We now ntroduce a reference frame 4, whch s smlar to the reference frame 3 used for the concal scannng (defned n 3.6). 4 has ts X-axs, algned wth the vewng drecton of one AS pxel M and s deduced from by two rotatons: rotaton of an angle φ about axs X, whch gves an ntermedate reference frame rotaton of an angle ζ about the Z-axs of the ntermedate reference frame. The converson matrx from 4 to, M 4, s smlar to the matrx M 3 (defned n 3.6), where the angles δ and φ are replaced respectvely by ζ and φ and ζ s a small angle: M 4 = 1 ζ cos( φ ) ζ sn( φ ) ζ cos( φ ) sn( φ ) sn( φ ) ( ) cos φ Any vewng drecton, sp, lyng n the cone that corresponds to the footprnt of the AS pxel M, s gven n 4 by: sp 4 = 1 ε cos( ψ ) ψ π ε sn( ψ ) Page 4 / 9

25 5_Navgaton.doc Date : October 11 The vewng vector s converted from 4 to wth the matrx M 4 : 1 sp = M 4 sp 4 = ζ cos( φ ) ζ sn( φ ) ζ cos( φ ) sn( φ ) sn( φ ) ( ) cos φ x 1 ε cos( ψ ) ε sn( ψ ) As ε and ζ are both small angles, only the frst order terms n ε or ζ are kept: 1 [ ] ζ sn( φ ) + ε cos( ψ ) sn( φ ) + sn( ψ ) cos( φ ) sp = cos( φ ) + ε [ cos( ψ ) cos( φ ) sn( ψ ) sn( φ )] ζ = 1 ζ cos( φ ) + ε cos( φ + ψ ) ζ sn( φ ) + ε sn( φ + ψ ) Then t s converted from then to wth the matrx M : sp = M sp = p = sn sn ( α ) sn( α ) ( ) ( ) α cos α 1 ( α ) sn( α ) [ ζ sn( φ ) + ε sn( φ + ψ )] ( ) ( ) [ ] α + cos α ζ sn( φ ) + ε sn( φ + ψ ) ζ cos( φ ) + ε cos( φ + ψ ) x 1 ζ cos( φ ) + ε cos( φ + ψ ) ζ sn( φ ) + ε sn( φ + ψ ) ψ π Page 5 / 9

26 5_Navgaton.doc Date : October OBT PEDCTON USN -LNE ELEMENTS 8.1. ESULTS OF A 6-MONTH COMPASON At the Centre de Meteorologe Spatale (CMS) of Meteo-France, three modes of AAPP navgaton have been run under operatonal condtons, usng respectvely: TBUS bulletns, whch s the standard on of AAPP verson 1 to 4, Two-Lne element sets, whch s the new on proposed n AAPP-5, AOS bulletns, whch s the CMS operatonal mode. Here are presented the results obtaned over a 6-month perod for NOAA-16 and NOAA-17. The consstency of each orbtal element data set s gven by the orbt extrapolaton error, whch s the dstance between the poston predcted wth the current bulletn and the poston of the next bulletn, dvded by the tme nterval between these two bulletns. Statstcs of ths parameter are presented n Table 8-1 and examples of ts temporal varaton are shown n the upper plot of Fgure 8-1 and Fgure 8-. Satellte Method bas sgma r.m.s noaa16 tbus lne argos noaa17 tbus lne argos Table 8-1 : extrapolaton error n km per day, from 3/9/ to 4/3/15 The results obtaned wth the Two-Lne element sets are obvously better than those obtaned wth the TBUS bulletns, for both satelltes. The CMS operatonal sute for AVH magery ncludes an Automatc Navgaton Adjustment (ANA), whch s a correlaton technque based on coastal landmarks. Ths allows calculaton, for each landmark successfully processed by ANA, of the navgaton error as the dstance between the AAPP calculated poston and the ANA measured poston. The AAPP poston s calculated wth a default atttude error, whch s a parameter ntroduced by the user (n the satd fle) and usually derved from ANA atttude earler results. The default atttude values used n the 6-month experment are gven n Table 8-. The default atttude error s calculated separately for the three types of orbtal elements, snce the so-called ptch bas corresponds actually to a ptch error and to an orbt extrapolaton error. Satellte Method yaw roll ptch noaa16 tbus lne..7.6 argos..7.7 noaa17 tbus lne.8..9 argos Table 8- : NOAA-16 and NOOA 17 defaut atttude values (n mrad) Page 6 / 9

27 5_Navgaton.doc Date : October 11 Statstcs of the AVH navgaton error have been calculated over all landmarks of a pass, then over all passes of the 6-month perod and the fnal results are presented n table 3. Examples of the navgaton error temporal varaton are shown n the plot enttled dstance rms error (km) of Fgure 8-1 and Fgure 8-. Satellte Method bas sgma r.m.s noaa16 tbus lne argos.4.8. noaa17 tbus lne argos Table 8-3 : AVH navgaton error n km, from 3/9/ to 4/3/15, usng AAPP default atttude. The statstcs are derved from only the passes for whch the yaw, roll and ptch have been estmated. The navgaton accuracy wth the Two-Lne element sets s better than the one wth the TBUS bulletns. For NOAA-16, there s only a slght dfference, probably because the default ptch error has partly compensated the orbt extrapolaton error of the TBUS. For NOAA-17, there s a sgnfcant dfference. 8.. CONCLUSON A new feature, orbt calculaton usng the Two-Lne element sets, has been added to AAPP navgaton. The AVH magery s navgated more accurately wth these data than wth TBUS bulletn. The AAPP user can easly swtch from TBUS to Two-Lne data and t s recommended to do so. Page 7 / 9

28 5_Navgaton.doc Date : October 11 Fgure 8-1 : ANA NOAA-17 wth TBUS Page 8 / 9

29 5_Navgaton.doc Date : October 11 Fgure 8- : ANA NOAA-17 wth TLE Page 9 / 9