Developement of an interpolation routine for gridded EOP data

Transcription

1 Developement of an nterpolaton routne for grdded EOP data Mchael Gerstl DGFI Report No The problem In space geodesy, the transformaton between a terrestal and a celestal reference frame requres, amongst others, fve tme dependent Earth orentaton parameters (EOP) ( xpol (t), y pol (t), UT1(t), X CIP (t), Y CIP (t) ), whch are gven as tme seres on a shared durnal grd (see e.g. the C04 seres of the IERS). Now, a vectoral nterpolaton scheme would be the best choce. Unfortunately, some projects requre dfferent nterpolaton methods for the sngle parameters. Just to top t, UT1(t) s provded and to be delvered together wth ts tme dervatve (lod) whereas the other parameters are handled as smple functon values. That agan suggests dfferent nterpolaton methods for each component of the EOP vector. We seek a mult-method nterpolaton routne whch gves nterpolated values of the fve EOP functons, ther tme dervatves for dynamcal methods, and the partal dervatves of the nterpolatng functons wth respect to the gven parameters f the EOP parameters have to be corrected. 2 Three types of algorthm We begn wth a comparson of three dfferent algorthms for the evaluaton of a polynomal of degree n = 3 nterpolatng the n+1 support ponts (t 0,f 0 ),...,(t n,f n ), the f beng values f(t ) of an unknown functon f. The underlyng problem s to determne an nterpolant ϕ approxmatng the functon f everywhere n [t 0,t n ]. (A) The recursve scheme of Atken-Nevlle: Ths s a sngle-stage algorthm resultng n a sngle functon value of the nterpolant. f 0 = p 00 (t) f 1 = p 11 (t) f 2 = p 22 (t) f 3 = p 33 (t) p 01 (t) p 12 (t) p 23 (t) p 02 (t) p 13 (t) p,k s the polynomal of degree k whch nterpolates the ponts (t,f ),...,(t k,f k ). p 03 (t) Recurson(p,k 1,p +1,k ) p,k for0 < k n p,k (t) = p +1,k (t) + t t k t k t ( p+1,k (t) p,k 1 (t) )

2 2 Algorthms of the Nevlle type do not provde dervatves and they are less effcent f values of p are sought for several arguments t smultaneously, whch s true f hundreds of observatons per day have to be processed. Therefore, ths class of algorthms wll be dscarded n the followng. (B) The recursve scheme of the dvded dfferences When we extract the t-ndependent part from the Nevlle recurson, we gan Newton s representaton of the nterpolant p(t) = [t 0 ]f + (t t 0 )[t 0,t 1 ]f + (t t 0 )(t t 1 )[t 0,t 1,t 2 ]f + (t t 0 )(t t 1 )(t t 2 )[t 0,t 1,t 2,t 3 ]f, whose coeffcents are the so-called dvded dfferences [t ]f = f, [t,t +1 ]f = f +1 f t +1 t, [t 1,t,t +1 ]f = They fulfl a smlar recurson f 0 = [t 0 ]f f 1 = [t 1 ]f f 2 = [t 2 ]f f 3 = [t 3 ]f [t 0,t 1 ]f [t 1,t 2 ]f [t 2,t 3 ]f f +1 f t +1 t f f 1 t t 1 t +1 t 1,... [t,t +1,...,t k 1,t k ]f = [t +1,...,t k 1,t k ]f [t,t +1,...,t k 1 ]f t k t [t 0,t 1,t 2 ]f [t 1,t 2,t 3 ]f [t 0,t 1,t 2,t 3 ]f Ths algorthm conssts of two stages: (1) the constructon of the nterpolant gven by the upper dagonal row of the recurson scheme and (2) the evaluaton of the Newton form. It s the best known algorthm for the evaluaton of the nterpolant and ts tme-dervatves, but cannot provde the partal dervatves wth respect to the parameters. (C) The bass representaton Let {B, = 0,...,n} be any bass of IP n, the lnear space of all polynomals of degree n. An nterpolant p IP n has an unque representaton p(t) = c 0 B 0 (t) + c 1 B 1 (t) + c 2 B 2 (t) c n B n (t). (1) That knd of representaton s well-known from B-splnes and wavelets. We have for a moment assumed that the gven data f = (f 0,...,f n ) are functon values wth dfferent abscssas. Then we can nvert the lnear system of nterpolaton condtons B 0 (t 0 ) B 1 (t 0 ) B n (t 0 ) c 0 f 0 B Bc = 0 (t 1 ) B 1 (t 1 ) B n (t 1 ) c = f 1.. (2) B 0 (t n ) B 1 (t n ) B n (t n ) If the matrx B cannot be nverted analytcally (an example s splne nterpolaton), we end up wth a three-stage algorthm c n f n

3 Three types of algorthm 3 (1) Buld a system of equatons for the unknown coeffcents (c ) n the mathematcal representaton of the nterpolant, and solve these equatons. (2) Evaluate the bass functons and, f necessary, ther tme dervatves. (3) Sum up the mathematcal representaton of the nterpolant. Ths algorthm gves always access to partal dervatves wth respect to the parameters, p(t) f = n k=1 p(t) c k c k f = n B k (t) ( B 1) k. k=1 Frst of all s the truncated Taylor seres a bass representaton, because the monomals {1, (t t ), (t t ) 2,..., (t t ) n } form a bass of the polynomal space IP n. The same holds for the Newton form whose bass functons 1 ω 0 (t) = 1, ω (t) = (t t k ), = 1,...,n turn the matrx of nterpolaton condtons (2) nto a lower trangular matrx. k=0 For two bases {B (1) } and {B (2) } wth coeffcents c (k) = (c (k) 0,...,c(k) n ) exsts a transfer matrx T such that c (1) c 0. (2) B = T 0. (2) 0 (t) B (1) 0 and. = (t) TT.. (3) c (1) n c (2) n B n (2) (t) B n (1) (t) We can now construct a partcular bass wth c = f. For any gven bass {B (1) } wth representaton coeffcents c (1) the soluton of the nterpolaton condtons (2) yelds c (1) = B 1 f. Set T = B 1 and create a new bass ( B (2) (t) ) =0,n = TT( B (1) (t) ). The correspondng =0,n coeffcents are by (3) c (2) = T 1 c (1) = Bc (1) = f. Ths specal bass offers the smple partal dervatves p(t)/ f k = B (2) k (t). It turns out that the specal bass only depends on the grd {T } and on the type of nterpolaton condtons, but not on the functon f to be approxmated. In textbooks always appears the specal bass of plan polynomal nterpolaton whch s buld of the Lagrange polynomals L n (t) = n k=0 k t t k t t k ( = 0,1,...,n). They are the unquely determned polynomals L n IP n wth L n (t k ) = δ k (k = 0,...,n). Wth t the matrx B of nterpolaton condtons (2) turns nto the unt matrx, and t follows c = f. But for numercal puposes the Lagrange representaton s far from beeng effcent. All representatons lnked together by a relaton (3) descrbe the same nterpolant whch s only defned by the nterpolaton condtons. Thus, notaton must dstngush between the mathematcal representaton of the nterpolant and the nterpolaton method descrbed by the nterpolaton condtons.

4 4 Interpolaton methods are (entre/plan) polynomal nterpolaton, Hermte nterpolaton (osculatng polynomal nterpolaton) Bessel nterpolaton, splne nterpolaton etc. Representaton forms are Taylor form (truncated Taylor seres), Newton form, Lagrange representaton, boundary-stable representaton etc. 3 Lmtaton of polynomal approxmaton (A) Condton: The condton of the entre (as opposed to pecewse) polynomal nterpolaton descrbes the mpact of data errors on the nterpolant. Ths condton, known as Lebesgue constant Λ n, depends on knot spacng and polynomal degree: n equdstant knots Čebyšev knots Choosng ponts whch concentrate at the boundares (Čebyšev knots) s the sutable remedy for the bad condton of polynomal nterpolaton. Unfortunately, t cannot be appled for long tme seres. We have to lmt the degree by a pecewse polynomal representaton. (B) Error bounds: Another crteron are bounds for some error norm. Usually gven s the maxmum norm n the space C (n+1) [t 0,t n ]. The maxmal absolute error for the Runge example, the even ratonal functon t 1/(1+t 2 ), shows the typcal two-branch exponental behavour: Maxmal absolute error of Runge example 25 odd degree polynomal even degree polynomal polynomal degree The error bound strongly depends on the space of functons to be approxmated. Snce the EOP functons should be trgonometrc functons wth bounded frequency spectrum and the

5 Boundary-stable representaton 5 Runge example s not part of those functons, the error bounds for pecewse polynomal approxmaton should be less than shown above. On the other hand, the Runge example reflects the case of a local dsturbance n the data and should therefore be taken nto account. (c) Localty : The global dependence on local dsturbances can be avoded when the specal bass functons have a local support. The consequences for polynomal EOP approxmaton: 1. Use pecewse polynomal representaton. 2. Keep the degree of the polynomal peces not greater than fve. 3. Prefer odd-degree polynomals. 4. Use a bass wth local support. 4 Boundary-stable representaton Intally regard lnear nterpolaton over the nterval [t 1,t ] wth mdpont t m = (t 1 +t )/2. Mathematcal representaton 1. Newton representaton p (t) = f 1 +(t t 1 ) f f 1 t t 1 2. Offset and drft representaton p (t) = f(t m )+(t t m )f (t m ) 3. Boundary-stable representaton p (t) = t t f t t 1 + t t 1 f 1 t t 1 Roundng error mnmal error at t = t 1 maxmal error at t = t mnmal error at t = t m maxmal error at t = t 1 and t = t mnmal error at t = t 1 and t = t maxmal error at t = t m The frst two representatons rsk a dscontnuty due to roundng errors across subnterval bounds at the nner knots t. That behavour s corruptve for the numercal ntegraton of dfferental equatons such as the orbt computaton n SLR. GPS and VLBI need not worry about that problem as long as ther sessons contan only one subnterval. The thrd representaton does not suffer from numercal dscontnuty at the nner knots and wll therefore be called boundary-stable. Let s stand for the argument t over the -th nterval [t 1,t ] normalsed to [0,1]. Then t t 1 t t 1 = s, t t t t 1 = 1 s The boundary-stable representaton for the cubc nterpolaton over [t 1,t ] based on the four data {f 1, f 1, f, f } reads [ ] p (t) = f 1 (1 s) + f s + h f 1 (f f 1 ) (1 s) 2 s + [ ] + (f f 1 ) h f s 2 (1 s), p (t) = 1 h {f f [ ] h f 1 (f f 1 ) (1 s) ( (1 s) 2s ) + [ (f f 1 ) h f ] s ( 2(1 s) s )}. (4)

6 6 where h := t t 1 and s = (t t 1 )/h. 5 Pecewse polynomal representaton When nterpolatng long tme seres, one has to depart from an entre polynomal nterpolant and nstead of that to concatenate several peces of polynomals as far as possble wthout jumps ( contnuty ) and wthout vertces ( contnuous dfferentablty ). For the numercal ntegraton of second order dfferental equatons (as satellte movement), theory requres twce contnuously dfferentable nterpolants! The abscssas τ k where the subpolynomals are concatenated are called knots, the ordered set of knots forms a grd = {τ 0,...,τ m }. The knots need not concde wth the support abscssas. The relaton between subpolynomal and nterpolant ϕ s expressed by grd nterval j = 1,...,m : ϕ(t) := p j (t) for τ j 1 t τ j, p j IP k. The noton of the latter formula can be concentrated as restrcton to the nterval [τ j 1,τ j ], ϕ [τj 1,τ j ] = p j. We lmt ths work to the case of concdng knots and support abscssas, whch mples that, because of the homgenety of support ponts, the number of condtons posed n both subnterval endponts s even, hence the degree of the subpolynomals must be odd. In the followng we dscuss some pecewse nterpolaton methods. The order of the approxmaton s gven by the maxmal mesh sze h max = max h. Pecewse lnear nterpolaton Support ponts : {(t 1,f 1 ), (t,f )} Approxmaton order : O(h max ) Global dfferentablty : not dfferentable. Cubc wndow Lagrange nterpolaton Support ponts : {(t 2,f 2 ), (t 1,f 1 ), (t,f ), (t +1,f +1 )} Approxmaton order : O(h 2 max) Global dfferentablty : not dfferentable. Ths s the nterpolaton method recommended by the IERS. Its name Lagrange nterpolaton s msleadng. It s an ordnary polynomal nterpolaton wth sldng wndow whch s not bound to the use of the Lagrange representaton. Ths method s not recommended and only used for the sake of contrast. Cubc Hermte nterpolaton Support ponts : {(t 1,f 1 ), (t 1,f 1 ), (t,f ), (t,f )} Approxmaton order : O(h 4 max) Global dfferentablty : contnuously dfferentable.

7 Contnuous pecewse lnear nterpolaton (polygon) 7 Cubc Bessel nterpolaton Support ponts : {(t 1,f 1 ), (t 1, f 1 ), (t,f ), (t, f )} where f k s an approxmaton to f k whch s ganed by a quadratc nterpolaton of {(t k 1,f k 1 ), (t k,f k ), (t k+1,f k+1 )}. Approxmaton order : O(h 3 max ), but O(h4 max ) n case the grd s unform. Global dfferentablty : contnuously dfferentable. The approxmaton of f only runs for the nner knots. At the boundares we must provde dervatves or addtonal knots or other condtons. Exponental splne nterpolaton to be done 6 Contnuous pecewse lnear nterpolaton (polygon) The nterpolant ϕ s represented over the -th nterval [t 1,t ] by a lnear polynomal ϕ [t 1,t ] (t) = p (t) = f 1 + f f 1 t t 1 (t t 1 ) = f 1 t t t t 1 + f t t 1 t t 1 Taylor and Newton form bass form = f 1 (1 s) + f s wth s = t t 1 t t 1 The concatenated nterpolant s a globally contnuous polygon through {(t,f ) = 0,...,n}. Its dervatve s dscontnuous n the support ponts t k and hence not qualfed for the computaton of e.g. the varaton of the length of day (lod). If we combne the two segments whch are multpled wth the faktor f, we get the followng contnuous, but not contnuously dfferentable functon (whch also s the lnear splne bass functon) B [f ] = t t 1 für t t t 1 t t, 1 B[f ](t) = B +1 [f ] = 1 t t für t t +1 t t t +1, 0 sonst. Ths s a bass functon of the lnear nterpolaton, and t holds ϕ(t) = n f B[f ](t). =1 7 Cubc Hermte nterpolaton Here, the nterpolant s defned by the support ponts{(t 1,f 1 ), (t 1,f 1 ), (t,f ), (t,f )}. That gves a completely localsed procedure.

8 B[f ](t) t 1 t t +1 B[f ](t) Fg. 1: Bass functons B[f ](t) of the contnuous pecewse lnear nterpolaton wth sngle knots (above) and one two-fold knot (below). 0 t 1 t =t +1 t +2 (A) Representaton n Newton s form for double (confluent) knots t 1 and t b 0, + b 1, (t t 1 ) + b 2, (t t 1 ) 2 + b 3, (t t 1 ) 2 (t t ), t [t 1,t ], p (t) = b 0, + h b 1, s + h 2 b 2, s2 + h 3 b 3, s2 (s 1), s = 1 (t t h 1 ) [0,1], (5) the coeffcents of whch are computed by the recurson of dvded dfferences, b 0, = [t 1 ]f = f(t 1 ) = f 1 b 1, = [t 1,t 1 ]f = f (t 1 ) = f 1 b 2, = [t 1,t 1,t ]f = 1 ( ) h 2 (f f 1 ) h f 1 b 3, = [t 1,t 1,t,t ]f = 1 ) (h h 3 f 1 +h f 2(f f 1 ) (6) (B) Boundary-stable polynomal representaton We know the bass functons of both, the Newton form : {1, s, s 2, s 2 (s 1)} and the boundary-stable form : {(1 s), s, (1 s) 2 s, s 2 (1 s)}. From that we deduce the transpose of transfer matrx T, then T = T maps the Newton coeffcents (b 0,, h b 1,, h 2 b 2,, h3 b 3, ) to the coeffcents of the boundarystable form, e 0, = f 1 Ths form was already shown by (4). e 1, = f e 2, = h f 1 (f f 1 ) e 3, = (f f 1 ) h f

9 Cubc Bessel nterpolaton 9 (C) Representaton by the specal bass of Hermte nterpolaton The specal form of cubc Hermte nterpolaton wll be p (t) = f 1 H [f 1 ](s) + f 1 H [f 1 ](s) + f H [f ](s) + f H [f ](s) (7) wth the normalsed argument s = s(t) = (t t 1 )/(t t 1 ) [0,1] Snce the Hermte coeffcents (f 1,f 1,f,f ) are the gven data of nterpolaton as well, the transfer matrx from the Hermte coeffcents to the Newton coeffcents has already been defned by equaton (6) to be the trangular matrx 1 0 h T = 1 h 1 2 h 2 h Followng equaton (3), the transpose oft maps the bass of the Newton form, {1, s, s 2, s 2 (s 1)}, to the sought specal Hermte bass of thrd degree H [f 1 ](s) = (1 s) 2( 1+2s ). H [f ](s) = s 2( 1+2(1 s) ) = H [f 1 ](1 s) H [f 1 ](s) = h (1 s)2 s (8) H [f ](s) = h s 2 (1 s) = H [f 1](1 s) The tme dervatve of the polynomal p s represented by the tme dervatves of the bass functons. Those read: d dt H [f 1 ](s) = 6 (1 s)s h d dt H [f ](s) = + 6 s(1 s) h d dt H [f 1](s) = (1 s)(1 3s) d dt H d [f ](s) = s(1 3(1 s)) = = d dt H [f 1 ](1 s) dt H [f 1 ](1 s) The partal dervatves of the polynomal pece p wth respect to the support parameters f 1, f 1, f, and f are exactly the values of the bass functons : p f 1 (t) = H [f 1 ](s), p f (t) = H [f ](s), p f 1 (t) = H [f 1](s), p f (t) = H [f ](s). 8 Cubc Bessel nterpolaton Gven only the the functon valuesf, we try to generate the contnuously dfferentable Hermte nterpolant by approxmatng the unknown dervatves f. At least at the nner knots t a quadratc nterpolaton of the trple {(t 1,f 1 ), (t,f ), (t +1,f +1 )} can be made. At the boundares t 0 and t n we need ether addtonal support ponts (t 1,f 1 ) and (t n+1,f n+1 ) or dervatves (t 0,f 0 ) and (t n,f n) or other condton equatons.

10 10 If parameters {f } are not corrected, take (t 1,f 1 ) and (t n+1,f n+1 ) from the tme seres f possble. That s not the rght way when parameters are estmated: f 1 and f n+1 have to be fxed for lack of observatons; by the correcton of {f 0,...,f n } a dscontnuty can arse between corrected and uncorrected values whch corrupts the nterpolant. I choosed the way mapped out by the natural splne : the free-end condtons ϕ (t 0 ) = ϕ (t n ) = 0. Some other condtons stll reman to be tested. The Besselan nterpolant over [t 1,t ] depends on (t 2,f 2 ) and (t +1,f +1 ), but does not nterpolate these ponts! The quadratc nterpolaton yelds for the approxmate dervatves h f 1 = f 1 1+p f 1 (1 p) f 2 p 2 1+p h f = f +1 q 2 1+q + f (1 q) f q wth wth p = p = h h 1, q = q = h h +1. (9) The correcton factors p and q account for the devaton from unformty of the grd n the left and rght neghbour ntervals. On an equdstant grd s p = q = 1, and the latter formula reduces to h f 1 = 1 ) (f 2 f 2, h f = 1 ) (f 2 +1 f 1. The Bessel formulas are obtaned when the approxmate dervatves are ntroduced nto the Hermte formulas. Alternatvely we can use the transfer matrces whch are no dfference from the Hermtan ones. specal bass of the Bessel nterpolaton n the nterval [t 1,t ] B [f 2 ](s) = p2 1+p (1 s)2 s B [f 1 ](s) = (1 s) + p (1 s) 2 s q 1+q s2 (1 s) B [f ](s) = s p 1+p (1 s)2 s + q s 2 (1 s) (10) B [f +1 ](s) = q2 1+q s2 (1 s) wth the dervatves d dt B [f 2 ](s) = 1 [ h d dt B [f 1 ](s) = 1 [ h d dt B [f ](s) = 1 [ h d dt B [f +1 ](s) = 1 [ h wth 0 p2 1+p d p (s) + 0 d q (s) ] 1 + p d p (s) q 1+q d q (s) ] 1 p 1+p d p(s) + q d q (s) d p (s) q2 1+q d q(s) d p (s) = d ds d q (s) = d ds [ ] (1 s) 2 s [ ] s 2 (1 s) [ = (1 s) ] ] ] (1 s) 2s, [ ] = s s 2(1 s). (11)

11 Boundary condtons for the Bessel nterpolaton 11 And the second dervatves d 2 dt 2 B [f 2 ](s) = 1 h 2 d 2 dt 2 B [f 1 ](s) = 1 h 2 d 2 dt 2 B [f ](s) = 1 h 2 d 2 dt 2 B [f +1 ](s) = 1 h 2 wth [ [ [ [ p2 ] 1+p d2 p(s) + 0 d 2 q(s) p d 2 p (s) q 1+q d2 q (s) ] p 1+p d2 p(s) + q d 2 q(s) 0 d 2 p(s) q2 1+q d2 q(s) [ ] d 2 p(s) = d2 ds 2 (1 s) 2 s [ d 2 d2 q (s) = ds 2 s 2 (1 s) ] ] ] = 2 [ 3(1 s) 1 ], = 2 [ 3s 1 ]. (12) 9 Boundary condtons for the Bessel nterpolaton (A) Vanshng second dervatve at border ponts Left boundary polynomal: B 1 [f 1 ](s) = 0 B 1 [f 0 ](s) = (1 s) 1 q 21+q (1 s)2 s q 1+q s2 (1 s) B 1 [f 1 ](s) = s q (1 s)2 s + q s 2 (1 s) q 2 B 1 [f 2 ](s) = q (1 s)2 s q2 1+q s2 (1 s) Left boundary polynomal: B n [f n 2 ](s) = 0 p2 1+p (1 s)2 s 1 21+p s2 (1 s) B n [f n 1 ](s) = (1 s) + p (1 s) 2 s p s2 (1 s) B n [ f n ](s) = s p 1+p (1 s)2 s 1 p 21+p s2 (1 s) B n [f n+1 ](s) = 0 p 2

12 12 Bass functons of the Bessel nterpolaton 1 B[t 1 ] B[t ] 0 t 2 t 1 t t +1 t +2 h d/dt B[t 1 ] d/dt B[t ] 0 h h = t +1 t 4h 2 0 4h 2 d 2 /dt 2 B[t 1 ] d 2 /dt 2 B[t ] t 2 t 1 t t +1 t +2 Fg. 2: Bass functons of the Bessel nterpolaton wth dervatves

13 Comparson 13 Bass of Bessel nterpolaton wth left border polynomals 1 B[f 0 ] B[f 1 ] B[f 2 ] B[f 7 ] 0.8 quadratc border functons second dervatve = 0 at the border (t 0 ) nner bass functon t 0 t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8 t 9 t 10 Fg. 3: Bass functonen of the Bessel nterpolaton at the left border 10 Comparson For a gven data vector d, contanng functon values f and f representaton n ϕ(t;d) = d B[t ](t), s a lnear functon of the support values, thus =0 for example, the specal bass ϕ( ;d+δd) = ϕ( ;d) + ϕ( ;δd). Then we can test the robustness of the nterpolaton procedure wth an error δd whch descrbes an outler n a sngle pont, say (t 5,f 5 ), whereas the error n the other ponts s neglgble. It means to set δd = δ,5. The result s shown n fgure 4.

14 14 Comparson of bass functons (mpact of an outler) Hermte nterpolaton Bessel nterpolaton splne nterpolaton wndow Lagrange (IERS nterpolaton) t 0 t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8 t 9 t 10 h frst dervatve d/dt Hermte nterpolaton Bessel nterpolaton 0 h splne nterpolaton wndow Lagrange 4h 2 second dervatve d 2 /dt 2 Hermte nterpolaton Bessel nterpolaton 0 4h 2 splne nterpolaton wndow Lagrange t 0 t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8 t 9 t 10 Fg. 4: Comparson of the bass functons and ther dervatves for the mpact of an outler n the pont (t 5,f 5 ). To compare Hermte nterpolaton wth the other methods, f 5 s supposed to be error-free.

15 Descrpton of the Fortran routne Descrpton of the Fortran routne module MOD_PARAMETR use LOC_NUMERIK, only: r8 mplct none real(r8),allocatable :: PARVAL(:),PARSIGMA(:),... end module MOD_PARAMETR PARVAL contans the grdded EOP parameters n the order x p (t 0 ),y p (t 0 ), UT1(t 0 ),lod(t 0 ), X CIP (t 0 ), Y CIP (t 0 ), x p (t 1 ),y p (t 1 ), UT1(t 1 ),lod(t 1 ), X CIP (t 1 ), Y CIP (t 1 ),... The onedmensonal storage of parameters has been choosen, because PARVAL or a subset of PARVAL s used n parameter estmaton as vector of the unknowns. At the moment, PARVAL contans the a-pror values. module MOD_EOP use LOC_NUMERIK, only: PI,r8 mplct none nteger,parameter :: & NPEOP = 5! number of dfferent EOP parameter functons, here we! have fve: x_pol(t), y_pol(t), Delta UT1(t),! Delta X_CIP(t), and Delta Y_CIP(t) nteger :: & IPAREOP,&! Offset of EOP parameters n PARVAL IPMOD(1:NPEOP),&! nterpolaton methods used for EOP parameter functons,! = 0 : lnear nterpolaton,! = 1 : cubc Bessel nterpolaton,! = 2 : cubc Hermt nterpolaton,! = 3 : reserved for exponental splne ITPOL,&! Offset of the EOP parameters at the nstant t_{} n! the parametervector PARVAL. MEOP,& NTEOP = 0! number of model parameters at a sngle nstant t_{}! NTEOP+1 s the number of nstants of tme whch form! the grd of support abscssas t_0,...,t_nteop. real(r8),prvate :: & P(0:3),Q(0:3)! Coeffcents of Besselan bass functonen from equaton (10) P(0:3) Q(0:3) B [f 2 ](s) = p2 1+p (1 s)2 s + 0 s 2 (1 s) B [f 1 ](s) = (1 s) + p (1 s) 2 s q 1+q s2 (1 s) B [f ](s) = s p 1+p (1 s)2 s + q s 2 (1 s) B [f +1 ](s) = + 0 (1 s) 2 s q2 1+q s2 (1 s)

16 16 real(r8),allocatable :: & EOP_REG(:,:),&! Regularsaton correcton (UT1-UT1R) and ts dervatve! wth respect to tme, TEOP(:),&! support abscssas for EOP values [days],! allocated as TEOP(0:NTEOP) TTmUTC(:)! TTmUTC() = TT - UTC for TT=TEOP() [sec].! TT-UTC s a step functon, contnuous from the rght,! wth dervatve 0 almost everywhere. real(r8) :: & TNPOL,&! = (t-teop())/(teop(+1)-teop())! normalsed argument for TEOP() <= t < TEOP(+1). DTPOL,&! = (TEOP(+1)-TEOP()) grd wdth for current subnterval. The regularsaton correcton of UT1 s not already subtracted when readng n the data, but straght before the nterpolaton, because the uncorrected values located n PARVAL are requred as a-pror values n parameter estmaton. If the program s not used for parameter estmaton, one could proceed n a dfferent way. Smlarly TT UTC s subtracted, whch s a rght-contnuous step functon and gves no contrbuton to the computed dervatves. Thus we remove any dscontnuty, because a functon to nterpolated by polynomals of degree hgher than one must be free from dscontnutes. = pol f (TDYN < TEOP()) = 1! restart do whle (TDYN >= TEOP(+1)) ; = +1 ; enddo f ( /= pol) then! changeover to new subnterval pol = DTPOL = TEOP(pol+1) - TEOP(pol) ITPOL = IPAREOP + pol*meop j = ITPOL + 1 do k = 1,NPEOP select case (IPMOD(k)) case (0)! pecewse lnear nterpolaton! c(0,k) = 0._r8! not necessary c(1,k) = PARVAL(j)! f_{} c(2,k) = PARVAL(j+MEOP)! f_{+1}! c(3,k) = 0._r8! not necessary f (k == 3) then c(1:2,3) = c(1:2,3) - EOP_REG(1,pol:pol+1) - TTmUTC(pol:pol+1) endf j = j + 1 case (1)! cubc Bessel nterpolaton c(1,k) = PARVAL(j)! f_{} c(2,k) = PARVAL(j+MEOP)! f_{+1} f (k == 3) then c(1:2,3) = c(1:2,3) - EOP_REG(1,pol:pol+1) - TTmUTC(pol:pol+1)

17 Descrpton of the Fortran routne 17 endf f ( > 1) then! not left boundary nterval c(0,k) = PARVAL(j-MEOP)! f_{-1} c(0,3) = c(0,3) - EOP_REG(1,pol-1) - TTmUTC(pol-1) P(1) = DTPOL / (TEOP()-TEOP(-1)) P(2) = -P(1)/(P(1)+1._r8) P(0) = P(1)*P(2) P(3) = 0._r8 endf f ( < NTEOP-1) then! not rght boundary nterval c(3,k) = PARVAL(j+2*MEOP)! f_{+2} c(3,3) = c(3,3) - EOP_REG(1,pol+2) - TTmUTC(pol+2) Q(0) = 0._r8 Q(2) = DTPOL / (TEOP(+2)-TEOP(+1)) Q(1) = -Q(2)/(Q(2)+1._r8) Q(3) = Q(1)*Q(2) else! rght boundary nterval c(3,k) = 0._r8! f_{+2} Q(0:3) = P(0:3)*0.5_r8! 2nd dervatve = 0 at border endf f ( == 1) then! left boundary nterval P(0:3) = Q(0:3)*0.5_r8! 2nd dervatve = 0 at border c(0,k) = 0._r8! f_{-1} endf! Compute the coeffcents of the boundary-stable representaton dp0 = c(0,k) c(0,k) = dp0*p(0) + c(1,k)*p(1) + c(2,k)*p(2) + c(3,k)*p(3) c(3,k) = dp0*q(0) + c(1,k)*q(1) + c(2,k)*q(2) + c(3,k)*q(3) j = j + 1 case (2)! cubc Hermte nterpolaton end select enddo endf c(0,k) = PARVAL(j+1)*DTPOL! h*f _{} c(1,k) = PARVAL(j)! f_{} c(2,k) = PARVAL(j+MEOP)! f_{+1} c(3,k) = PARVAL(j+MEOP+1)*DTPOL! h*f _{+1} f (k == 3) then c(1,3) = c(1,3) - EOP_REG(1,pol) - TTmUTC(pol) c(2,3) = c(2,3) - EOP_REG(1,pol+1) - TTmUTC(pol+1) endf! Compute the coeffcents of the boundary-stable representaton c(0,k) = c(0,k) - (c(2,k)-c(1,k))! h*f_{} - (f_{+1}-f_{}) c(3,k) = (c(2,k)-c(1,k)) - c(3,k)! (f_{+1}-f_{}) - h*f _{+1} j = j + 2 Interpolaton n the boundary-stable representaton

18 18! Interpolaton. TNPOL s the normalzed coordnate n the nterval [t_{},t_{+1}] TNPOL = (TDYN-TEOP(pol))/DTPOL ; S1 = 1._r8 - TNPOL ; S2 = TNPOL EOP(1:5) = c(1,1:5)*s1 + c(2,1:5)*s2 + (c(0,1:5)*s1 + c(3,1:5)*s2)*s1*s2 dt_eop(1:5) = (c(2,1:5) - c(1,1:5) & & + c(0,1:5)*s1*(s1-2*s2) + c(3,1:5)*s2*(2*s1-s2))/dtpol Computaton of partal dervatves wth respect to the parameters S1 = 1._r8 - TNPOL ; S2 = TNPOL select case (IPMOD(k)) case (0) PolBas(0,k) = 0._r8 PolBas(1,k) = S1 PolBas(2,k) = S2 PolBas(3,k) = 0._r8! bass functons of pecewse! lnear nterpolaton case (1)! bass functons of pecewse S12 = S1*S2! cubc Bessel nterpolaton PolBas(0,k) = P(0)*S1*S12 + Q(0)*S2*S12 PolBas(1,k) = S1 + P(1)*S1*S12 + Q(1)*S2*S12 PolBas(2,k) = S2 + P(2)*S1*S12 + Q(2)*S2*S12 PolBas(3,k) = P(3)*S1*S12 + Q(3)*S2*S12 case (2)! bass functons of pecewse! cubsche Hermte nterpolaton PolBas(0,k) = S1**2 * (1._r8 + 2*S2) PolBas(1,k) = S1**2 * S2 * DTPOL PolBas(2,k) = S2**2 * (1._r8 + 2*S1) PolBas(3,k) = - S2**2 * S1 * DTPOL end select