Estimation of the precision by the tidal analysis programs ETERNA and VAV

Transcription

1 11189 Estimation o the precision by the tidal analysis programs and B. Ducarme*, L. Vandercoilden*, A.P. Venedikov** * Royal Observatory o Belgium, ducarme@oma.be ** Geophysical Institute & Central Laboratory on Geodesy, Soia, venedikov@yahoo.com Abstract A comparison between the algorithms, used by the programs and or estimation o the precision is made. It is shown that the algorithm o does not use the principles o the least squares method, because it does not use sum o squares o residuals. uses sum o squares o residuals o iltered numbers in a time/requency domain. However, the variant o the algorithm used till now does not take into account the deviation o the noise rom the normal distribution. In a new version o a test o normality is included and a urther development o the analysis method by applying weight o the iltered numbers. 1. Introduction. The initial intention o this paper was to present results o comparative analyses o Superconducting Gravimeters (SG) data by the programs (Wenzel, 199) and (Venedikov et al., 1, 3,, 5, Ducarme et al.,, 5, a, b). The aim was, by using the estimates o the precision, to make conclusions about the advantages o one or another o the programs. In this connection we have careully studied, actually or the irst time, the algorithm used by or the computation o the root mean square errors (RMS). It was surprising to establish that the RMS o are not Least Squares (LS) estimates, i.e. they are not root mean square errors. In these circumstances the comparison between the RMS o and has lost its sense. Thus the paper had been reoriented to compare the algorithms o the programs used or the estimation o the precision. In this connection we would like to recall a undamental principle o the LS estimation o the precision. The precision is estimated through the root mean square errors (RMS) o the data and the estimated unknowns. In all cases the RMS are to be determined through the sum o squares o residuals (SSQR) o the data. The theoretical reason is that SSQR is expected to have the Pearson χ distribution. Then an expression like x = 1.18 ±, where is the RMS o x has the meaning o a conidence interval with conidence probability %. On the basis o the expression x = 1.18 ± we can conclude that the true value o x, actually its mathematical expectation E((x), will be ( ) E( x) to the conidence interval 1.18 t, t (1) where t is the coeicient o Student, depending on a chosen conidence probability. E.g. or probability 95% and a great number o data t = 1.9. I the RMS is not computed according to the LS rules through the SSQR, the expression x = 1.18 ± has not any concrete statistical meaning. SSQR can be directly used in this way provided the data are charged by a white noise (WN). This is not the case o the tidal data. Both and solve the problem by using requency dependent RMS o the data, but this is made in completely dierent ways

2 1119. Estimation o the precision by. We shall ollow how determines the RMS σ ET (δ g ) o the amplitude actor δ g o a tidal group g in one o the main tidal amilies at = 1,, 3 or (cycles/day) cpd. For revealing the algorithm used we had to decipher it in the FORTRAN code o the program. nm/s LP? B1 A1 B A B3 A3 B A.. _spect_wen_bruss Frequency deg/hr Figure 1. SG data Brussels, spectrum o the residuals, obtained with application o the Pertsev ilter. The average noise levels L( ) at the requencies = 1,, 3 & cpd (cycles/day) are determined as arithmetic means o the amplitudes in the intervals, given in Table 1 and indicated in Figure 1 by A1, A, A3 & A respectively. Table 1. Frequency ranges or the determination o the average noise levels L( ). Name Tidal amily (cpd) Angular speed (deg/hr) From To L(1) = L() = L(3) = L() = The colored character o the noise is taken into account through the spectrum o the residuals o the hourly iltered numbers. It uses the so called average noise levels L( ) at requency, which are arithmetic means o the amplitudes, as shown by Figure 1 and Table 1. It also uses the RMS or unit weight σ, called in the output Standard deviation and computed according to LS under the condition o WN. Then determines the so called Average noise level at white noise L(wn) through L(wn) = σ π/ n, where n is the number o the hourly data. () It does not correspond to the description o the program ANALYZE in Manual 3.3.hlp, where it is stated that L(wn) is the estimated Fourier amplitude o the white noise in the requency band cpd. It seems that L(wn) is intuitively computed as the square root o the energy in the basic requency interval π/n, as σ is the total spectral energy.. Further determines the RMS σ WN (δ g ) o δ g, again or the case o WN, as σ (δ ) = C.σ (3) WN g g where C g is a coeicient, obtained through the matrix o the LS normal equations. Finally, determines the colored RMS o δ g as 1119

3 11191 ( ) σ ET (δ ) = σ (δ ) L = C L(wn) g WN g g With a good enough approximation Cg 1/ ( hg n/) amplitude in the group g. By using this we get ET ( hg ) σ L( ) Cg. L( ) π/ n σ π/ n = (), where h g is the min theoretical σ (δ ) L( )/ π/ (5) g In a similar way we get, or the RMS o the phase lag α g o ET ( hg g ) σ (α ) L( )/ π/.δ in radians () g It is strange, that σ, which is obtained through the SSQR, inally disappears in these expressions. Also strange is the presence o the π in the deinition o L(wn), without any theoretical explanation. The expressions look as intuitive ones, which had been adapted empirically, e.g. by including the term π = 1.77, until gets some reasonably looking values. Further remarks to this way o estimation o the precision are the ollowing. (i) The RMS σ ET are not LS estimates because they use arithmetic means L( ) o amplitudes instead o SSQR, i.e. the RMS are actually not root mean square errors. (ii) The levels L( ) depend on how the intervals A are chosen. I we do not obey to the dogmatic deinition by o the intervals (Table 1), this can be done in many dierent ways. E.g., i A in Figure 1 are replaced by B, we may get dierent values o L( ). (iii) As shown by the example in Section 3 hereater, in particular cases o the noise can provide completely disorientating estimates o the precision. (iv) The spectrum is computed with a ixed requency step o deg/hr, which is too coarse or long series o data. E.g. or Brussels the requency step should be deg/hr, i.e. the spectrum looses 9% o the inormation in the residuals. Possibly, the large step deg/hr has been chosen in order to save computer time, which was important earlier, when has been developed. (v) The spectrum is computed till requencies at cpd, may be again in order to save computer time. Thus it does not allow to obtain L( ) values at higher requencies, e.g. or the shallow water tides. is not prepared or analysis o the ocean tides. (vi) The precision o the multi-channel analysis is estimated under WN assumption only. Unlike in, these coeicients are not requency dependent. 3. Estimation o the precision by the program The main algorithm o, is based on the partition o the data into N intervals I(T) o equal length T and central epochs T = T1, T,... TN. As shown by Figure, the intervals are generally without overlapping. I there is a gap, it should remain between the corresponding neighboring I(T). In the case o a gap allows an overlapping o the I(T) in order to avoid loosing data. When a moving iltration is used, as in the method o Chojnicki/, every gap brings a loss o data o one length o the ilter used

4 1119 T 1 T T 3 T T 5... gap Figure. Scheme o the organizing o the intervals IT ( ) used by. Figure 3 shows details o an I(T). nm/s 3 1 Central epoch T t = T + τ Internal time τ y(t) Time t = T + τ in hours Figure 3. Example o an interval IT ( ) with T = h. The drit is represented independently in all IT ( ) by polynomials o power K d, i.e. by polynomials having arbitrarily dierent coeicients in the dierent I(T). Our recent experiments have shown that the case o K d = is a very eicient one, especially or T = h, provided the analysis includes the determination o the LP tides in parallel with the shorter D, SD, tides (Ducarme et al., ). The variant K d = means that the drit is represented by a stepwise unction, remaining a constant in every given IT ( ), which is allowed to reely change between the neighboring IT ( ). h h In this section we shall consider only the case o Kd = and T = & 8, because the general case o these parameters needs sophisticated explanations. In a irst stage o the hourly data y(t) in every IT ( ) are transormed through iltration into even and odd iltered numbers (u,v), as shown by (): θ ( u T v T ) = F τ y T + τ = ( T ) ( ), ( ) ( ) ( ), where θ 1 /, i.e or 3.5 (7) τ = θ The iltration is made by using a set o orthogonal ilters F ( τ ), which are complex unctions o the internal time τ. For the case considered here π F ( τ) = / T.Exp τ which ampliies the requency cpd h (8) The ilters F (τ) eliminate an arbitrary constant, i.e. a polynomial o power K d =. We have a ull set o (u,v) quantities, when the (u,v) used are ( ) ( ) For T = : u ( T), v ( T) at requencies = 1,,... µ = 1 cpd For T = 8 : u ( T), v ( T) at requencies =.5,1.,1.5,, , µ = 1. cpd (9) 1119

5 11193 Due to the orthogonality o F (τ) the ull set o (u,v) in (9) is a legal transormation o the hourly data y(t) rom the time domain in a time/requency domain in the ollowing sense. Let LS is applied on y(t) with corresponding observation equations. We shall get identical results through the application o LS on (u,v) by using the same observation equations, modulated by the ilters. The transormation remains practically legal, i some o the requencies, at which useul signals do not exist, are omitted. E.g., i some high requencies are omitted, i.e. i the highest requency is µ < 1 cpd. Following this possibility, applies LS on (u,v) as i (u,v) are the observations. As a result it provides the estimates o the unknowns, in which we are interested, the adjusted ( uv %,%) o the observed ( uv, ) and the residuals u ( T) = u ( T) u% ( T) & v ( T) = v ( T) v% ( T) or all values o T and. (1) I we had data with WN, all u & v, = 1,... µ would have one and the same standard deviation. Then it would be estimated by the RMS or unit weight µ σ = ( ) + ( ) = 1 T T u T v T ( Nµ m) (11) where m is the number o the unknowns. In act, the data are charged by Colored Noise (ColN), i.e. the standard deviation o u & v, = 1,... µ is dependent on the requency, usually increasing with decreasing (red noise). In the same time we have established, that the sequence ( 1 1 ) ( ) ( N N ) u ( T), v ( T), u ( T ), v ( T ),... u ( T ), v ( T ) or a ixed requency (1) can be reasonably accepted as non-correlated. On the basis o this solves the problem o the ColN by using separately the residuals ( u, v ) or getting the RMS σ ( uv, ), as an estimate o the standard deviation o ( u, v ). Namely, we use the RMS o the data at requency computed through ν where = + (13) σ ( uv, ) u ( T) v ( T) ( N m) T T m is the number o unknowns, whose determination is mostly depending on (, ) u v.. An extreme case o a parallel application o and programs. As said above, the quantities (u,v) are a legal transormation o the data y(t). I this is neglected and (u,v) are examined at only one ixed requency, it seems that uses a h decimation o the data with a time step T. Thus or T = 8, earlier used by MV (Venedikov, 19), the Nyquist requency becomes.5 cpd, which is an extremely large requency. Then, i e.g. the tide O1 with requency ν(o1) is considered, the waves A1 with requency (O1).5 cpd & A with requency (O1) +.5 cpd are aliases o O1 which cannot be separated rom O1 and which may aect the results or O1. (1) 11193

6 1119 Under such assumptions, Schüller (1978) has shown that i the waves A1 and A are added to a series o data the MV results or O1 are strongly aected. It has been shown (Venedikov, 1979) that A1 and A can be separated rom O1 and thus they are not aliases o O1. Nevertheless, on the basis o the result o Schüller, Wenzel (1997) has declared that MV (and thus now) violates the sampling theorem. It is strange that this severe critic has been made on the basis o the eect o A1 & A on MV only, but it has not been shown, what is the eect o the same A1 & A on other programs, in particular on. In order to avoid conusions we have made a similar experiment, somewhat extended (Table ), but this time testing both and. Table. Variants o testing and. The amplitudes o all waves are 38 µgal. Variant : Original SG data Brussels without added waves Variant 1: SG data Brussels 1987 with added A waves: A1 with requency (O1).5 cpd A with requency (O1) +.5 cpd The aliases used by Schüller A3 with requency (O1).5 cpd A with requency (O1) +.5 cpd Two more waves, related with the supposed Nyquist requency Variant : SG data Brussels 198 with added A waves: A1 with requency (O1).5 cpd A with requency (O1) +.5 cpd The aliases used by o Schüller A3 with requency (O1) + 1. cpd A with requency (O1) cpd Two more aliases A5 with period 7 hours A wave, suggested by W. Zürn has been used with highest requency µ Kd h = & T = 8 and nearly ull set o (u,v) in (9) with = 11.5 cpd. Let δ is the amplitude actor o a given tide, obtained by the analysis o the data in variant (without A waves) and δ A is the amplitude actor obtained by the analysis o the data, ater the waves A are added. Then the eect o the waves A on the amplitude is naturally measured by the dierence = δa δ. The result o the comparison is shown by Figure Variant Variant Q1 O1 K1 N M S Q1 O1 K1 N M S Figure. The eect = δa δ o the added waves A, variants 1 &. 1119

7 11195 Surprisingly or not, the eect o the A waves is, irst, considerably higher or and, second, it is expanded on all D and SD waves. There is a third item, also not in avor o, concerning the eect o the A waves on the estimates o the precision. Figure 5 shows that the eect o the A waves, variant 1, is the addition o a noise, completely deorming the tidal curve. Same is the situation with the data, obtained in variant. Obviously, a reasonable result should show a considerably lower precision, i.e. considerable higher RMS, when the data in variant 1 and are analyzed. nm/s //87 31/1/87 3/1/87 9/1/87 8/1/87 7/1/87 /1/87 5/1/87 /1/87 3/1/87 /1/87 1/1/87 /1/87 19/1/87 18/1/87 17/1/87 1/1/87 15/1/87 1/1/87 13/1/87 1/1/87 11/1/87 1/1/87 9/1/87 8/1/87 7/1/87 /1/87 5/1/87 /1/87 3/1/87 /1/87 1/1/87 Figure 5. Sample o the data ater adding the A waves, variant 1. Indeed, as shown by Tables 3 and, the analyses show considerably lower precision ater adding the A waves. E.g. the RMS o δ(m) in Table 3 is increased rom ±.5 to ±.19, i.e. more than 1 times. Table 3. RMS σ o the δ actor o the main D & SD tides o the original data (variant ) and the data with added A waves (variant 1 in Table ); SG data Brussels Q1 O1 K1 N M S, RMS σ Variant, No added waves ±.3 ±.8 ±. ±.5 ±.5 ±.11 Variant 1 o added waves ±.377 ±.713 ±.55 ±.31 ±.19 ±.137, RMS Variant, no added waves ±.37 ±.7 ±.5 ±. ±. ±.9 Variant 1 o added waves ±.17 ±.9 ±.1 ±. ±. ±.9 Table. RMS σ o the δ actor o the main D & SD tides o the original data (variant ) and the data with added A waves (variant in Table ); SG data Brussels 198. Q1 O1 K1 N M S, RMS σ Variant, No added waves ±. ±.8 ±.5 ±.3 ±. ±.1 Variant o added waves ±.18 ±.7 ±.18 ±.819 ±.157 ±.3, RMS Variant, No added waves ±.35 ±.7 ±.5 ±. ±. ±.9 Variant o added waves ±.1 ±.8 ±. ±.118 ±. ±.93 in Table 3 shows only a weak increase o the RMS or the D tides, but absolutely NO changes in the RMS or the SD tides. In Table shows more important increase o the RMS or the SD tides, but almost no changes in the RMS o the D tides. σ ET σ ET 11195

8 1119 Thus generally, with the exception o the SD case in Table, ailed to notice the huge noise, added to the data. The explanation o the anomalous estimates o the precision by is the dependence o these estimates on the intervals in Figure 1 and Table 1. In the case their requencies all out o the intervals, their eect remains inappreciable. This is an exaggerated case but, anyway, it shows the dependence o the estimates o on the choice o the intervals. An attempt to escape rom the situation has been made by returning to the option o (Table 5) based on the assumption o WN, although this is a non-standard and nonrecommended procedure or the D and SD tides. In such a way we have got indeed a considerable increase o the RMS or all tides, i.e. the strong noise became visible. However, the new RMS o are still considerably dierent rom and they are strongly dependent on the ilters used, which is somewhat strange. Table 5. An attempt to ind more reasonable estimates o the precision by applying with the non-standard procedure under the assumption or WN, or variant in Table. Q1 O1 K1 N M S, Assumption o white noise, ilter NMM, 17 coeicients Variant o added waves ±.3 ±.33 ±.8 ±.339 ±.33 ±.135, Assumption o white noise, ilter PERTSEV59, 51 coeicients Variant o added waves ±.3 ±.58 ±.358 ±. ±.5 ±.111 There are very ew comparative analyses by and. Many o them (but not all o them) show a numerical agreement between σ and σ ET, σ ET always being lower than σ by some 1 to %. Such is the case o the RMS or the unperturbed data in Tables 3 &. The closeness, which appeared in most cases, should not be considered as universal and thus as a justiication o the estimates o. The results with the added waves are cases when the similitude completely disappears. Something more! Let us accept that the similitude o σ (Q1) = 3 and σ ET (Q1) = 37 means that σ ET (Q1) = 37 is a reasonable RMS. Then, since σ ET (Q1) < σ (Q1), we have also to accept that Q1 is better determined by. Actually, since σ ET (Q1) is not an LS estimate, rom the comparison o σ ET and σ cannot be made anyone o the conclusions: neither Q1 is better determined by, nor Q1 is better determined by. Same is the situation i we had σ (Q1) < σ ET (Q1). The reality is that due to the uncertainty on σ ET (Q1) we do not know how well Q1 is estimated. 5. A new approach by the variant o the program The algorithm, described in Section 3 is used till now and is implemented in (Venedikov et al., ). Now we have introduced a urther development o this algorithm, described hereater. In a discussion Dr. W. Zürn made the reasonable critical remark that implies the assumption that the iltered numbers have a normal Gaussian distribution (ND) and this is actually not proven. Indeed, the technique considered above is based on the assumption that the set o data u ( T), v( T), T = T1, T,... TN at given requency has a ND. In this connection has included a test o normality, based on the χ criterion o Pearson. 1119

9 11197 builds the observed or empirical distribution o the residuals u( T), v ( T), T = T1, T,... TN or everyone o the requencies and the corresponding theoretical N(, σ), i.e. central ND with standard deviation σ. The dierence between the distributions is measured by the observed χ quantity. In the examples here the distributions Obs are determined over 31 intervals with equal expected probability, so that χ Obs has 3 degrees o reedom. The value o σ in N(, σ) is ound by making χ Obs = Min. The critical χ Crit = 3.5 o Pearson or testing the hypothesis or ND is corresponding to conidence probability 95%. The examples are obtained by the processing o the SG data Vienna Figure A shows the distribution o the (u,v) at = 1 cpd. In practice, such pictures χ, we are oten accepted as a satisactory approximation to a ND. However, according to have NOT an ND, because χ Obs exceeds considerably χ Crit. Crit. A χ Obs = B χ Obs = > χ Crit = > χ Crit = Residual intervals nm/s Residual intervals nm/s Figure. Observed density distribution o the residuals at = 1 cpd (iled grey curve) and the corresponding ND (thick line). A: all residuals; B: eliminated large residuals (7.5% data). Such is the case o most o the SG data we have tested. It appeared that one o the reasons or the deviations rom the ND is that there is proportionally too many large residuals. Indeed, as shown by Figure B, when the option (Venedikov et al., ) o automatic elimination o data with big residual is applied, the dierence between χ Obs and χ Crit is considerably reduced, although strictly considered, the inerence is that the distribution remains not ND. We came considerably closer to ND in the ollowing new way o applying. is run in two general loops A & B. In A ollows Section 3. In addition, it provides a separate determination o the RMS o u & v, instead o the common RMS (13). This is made through ( ) ( ). (15) σ ( u) = u ( T) N m & σ () v = v ( T) N m T T The idea is to use these RMS in order to apply weights to the iltered (u,v) and thus make them all with (i) nearly normal distribution and (ii) nearly equal variances. By accepting that a quantity with unit weight should have an RMS equal to the RMS or the WN σ, obtained by (11), the weights to be applied become or u : (σ σ ( u)) and or v : (σ σ ()) v (1) 11197

10 11198 Practically, these weights are introduced through the replacement o the iltered numbers u ( T) & v ( T ) by what we call weighed iltered numbers u ( T) = σ. u ( T)/σ ( u) & v ( T) = σ. v ( T)/σ () v (17) Theoretically, or perectly well determined RMS, the weighted quantities (17) become observations with equal precision and RMS approximately equal to σ D residuals = 1 cpd χ Obs =.1 < χ Crit = SD residuals = cpd χ Obs = 85. < χ Crit = Residual intervals nm/s Residual intervals nm/s TD residuals = 3 cpd χ Obs = < χ Crit = QD residuals = cpd χ Obs = 77.5 < χ Crit = Residual intervals nm/s Residual intervals nm/s 5 Figure 7 Observed density distribution o the residuals WITH APPLIED WEIGHTS and eliminated large residuals (.% o all data) with the corresponding ND χ Obs = 77.5 < χ Crit = Residual intervals nm/s 3 5 Figure 8 Distribution o the residuals ( uν, vν) or all ν = 1,, 3 & cpd WITH APPLIED WEIGHTS and eliminated large residuals (.% o all data). Figure 7 shows the eect o the weights on the distribution o the residuals o the weighed quantities, separately or the main requencies. In all cases we get a rather acceptable approximation to ND with χ clearly under the χ. Obs Very important is that the approximation o the ND o the residuals o all main requencies together, as shown by Figure 8 is also satisactory. Crit 11198

11 11199 Unortunately, the positive results in Figures 7 & 8 are obtained by applying an elimination o a considerable quantity o data (.%) with too high residuals. One o the sources o these big residuals may be due to the preprocessing o the data. is based on a moving iltration o the data, which does not support a great number or gaps. Due to this the authors o the data are obliged to keep in the data obviously anomalous data, as well as to apply massive data reparations and interpolations, which operations introduce noise o very bad properties. Conclusions. We hope that this paper can attract the attention o the tidal community to the importance o the estimation o the precision. Actually, the result about a parameter x has to be represented by two numbers: the value o x and its RMS. The value without the RMS is as useless, as the RMS without the value. No need to say, that the RMS has to be determined according to the rules o the mathematical statistics, i.e. according to the LS rules. Otherwise, an incorrect RMS is equivalent to or worse than a missing RMS. The analysis o the tidal data has to solve a diicult problem. In principle, now is irmly accepted, that the LS is perectly suitable general method. The problem is that the data are correlated, i.e. they are charged by a colored noise. In the same time we do not dispose a priori by the covariance matrix, in order to correctly apply the MLS. The compromise, accepted by both and is to estimate the unknowns by creating the observation equations under the condition o WN. This does not aect seriously the values o the unknowns, because we always deal with very great number o data. In the same time, since the colored noise has a requency dependent eect, the colored character o the noise is taken into account through the estimation o the precision by requency dependent RMS. Nevertheless, the RMS should observe the general LS principle to be determined through sums o squares o residuals, in order to be able to apply undamental statistical tools, like the conidence intervals o Student. It has been demonstrated that the RMS o are, unortunately, not LS estimates and they depend on some intuitive assumptions. Something more, there are cases in which these RMS can be completely disorientating. In addition to the extreme case, considered in Section 3, such is the doubtul case o the RMS o the parameters o the auxiliary channels in the multi-channel analysis. The case o LP waves is even more complicated and is treated in a separate paper (Ducarme et al. c). The paper has discussed the estimates o the precision, proposed by. They are certainly not the ideal solution but, at least, they ollow the LS principle to use the squares o the residuals. In the same time, as shown in Section 5, we have ound a weak element in the RMS estimation, applied until now. Namely, it appeared that the distribution o the residuals, rigorously checked, deviates rom the ND o Gauss. We have not hesitated to introduce corresponding improvement in, which has brought the residuals nearly to the ND. In addition, it seems that the solution o the problem will generally improve the precision o the analysis results. We believe that the decision to immediately modiy when a weak element is ound is a good example o a scientiic approach. In our opinion the researchers, tightly devoted to the use o, should ollow this example and critically reconsider the method o preprocessing and analysis they are currently using, especially in its options or the estimation o the precision

12 11 Acknowledgements The authors are grateul to Walter Zürn or his suggestions and to the members o the Global Geodynamics Project (Crossley et al., 1999), who provided the SG data. Reerences. 1. Chojnicki, T., 197. Determination des parameters de marées par la compensation de observations au moyen de la méthode des moindres carrées. Publications o the Institute o Geophysics, Polish Academy o Sciences, Marees Terrestres, 55, Crossley, D., Hinderer, J., Casula G., Francis O., Hsu, H.T., Imanishi, Y., Jentzsch G., Kääriäinen J., Merriam, J., Meurers B., Neumeyer J., Richter B., Shibuya K., Sato T., Van Dam T., Network o superconducting gravimeters beneits a number o disciplines. EOS, 8, 11, 11/ Ducarme B., Venedikov A.P., Arnoso J., Vieira R.,. Determination o the long period tidal waves in the GGP superconducting gravity data. Journal o Geodynamics, 38, Ducarme B., Van Ruymbeke M., Venedikov A.P., Arnoso J., Vieira R., 5. Polar motion and non-tidal signals in the superconducting gravimeter observations in Brussels. Bulletin d Inormations des Marées Terrestres, 1, Ducarme B., Venedikov A.P., Arnoso J., Vieira R., a. Analysis and prediction o the ocean tides by the computer program. Journal o Geodynamics, 1, Ducarme B., Venedikov A.P., Arnoso J., Chen X.D., Sun H.P., Vieira R., b. Global analysis o the GGP superconducting gravimeters network or the estimation o the polar motion eect on the gravity variations. Journal o Geodynamics, 1, Ducarme B., Vandercoilden L., Venedikov A.P., c. The analysis o LP waves and polar motion eects by and methods. Bulletin d Inormations des Marées Terrestres, 11, Schüller K, About the sensitivity o the Venedikov tidal parameter estimates to leakage eects. Bulletin d Inormations des Marées Terrestres, 78, Venedikov A.P., 19. Une méthode d'analyse des marées terrestres à partir d'enregistrements de longueurs arbitraires. Acad. Royal Belg., Bull. Cl. Sci., LII, 3, also in Commun. Observ. Roy. Belg., S. Géoph., 71, Venedikov A.P., The Nyquist requency and the substitution o the hourly ordinates by iltered values in the Earth tidal analysis. Bulletin d Inormations des Marées Terrestres, 81, Venedikov A.P., Arnoso J., Vieira R., 1. Program / or Tidal Analysis o Unevenly Spaced Data with Irregular Drit and Colored Noise. J. Geodetic Society o Japan, 7, 1, Venedikov A.P., Arnoso J.,Vieira R., 3. : A program or tidal data processing. Computer & Geosciences, 9, Venedikov A.P., Vieira R.,. Guidebook or the practical use o the computer program version 3. Bulletin d Inormations des Marées Terrestres, 139, Venedikov A.P., Arnoso J., Vieira R., 5. New version o program or tidal data processing. Computer & Geosciences, 31, Wenzel H.-G., 199. Earth tide analysis package 3.. Bulletin d Inormations des Marées Terrestres, 118, Wenzel H.-G., Analysis o Earth tide observations. Lecture Notes in Earth Sciences,, Springer,