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THE PROBABILISTIC APPROACH TO THE DESCRIPTION OF THE CHANDLER WOBBLE
I.J.. Tsurkis, M.S. Kuchai, E.A. Spiridonov, S.V. Sinyuhina
Schmidt's Institute for Physics of the Earth RAS, B. Gruzinskaya st., 10, Moscow, Russia, sp287@mail.ru, tsurkis@ifz.ru
INTRODUCTION. The probabilistic approach to the description of the Chandler wobble (CW) was proposed by Arato, Kolmogorov and Sinai in [Arato et al., 1962]. They assumed that the moment of the forces causing the CW is a stationary random process with correlation time cor which is small in comparison with length of the row of observations. Then, the CW itself can be regarded as a diffusion Markovian process with discrete time; wherein the sampling step must satisfy the condition: cor . There was shown in [Tsurkis et al. 2009], that the probabilistic model does not contradict with observations. Besides, evaluations for cor and the diffusion coefficient d were obtained: cor 100 days, d 1.11016.....1.8 1016 rad 2 / day. (1) An equally important task is the studying of processes causing CW. Polar motion is due to several reasons, the main of which, apparently, is the impact of ocean and atmosphere to the solid Earth [Gross et al, 2003], [Barnes et al., 1983]. The article [Tsurkis et al., 2012b] is devoted to analysis of the data on ocean angular momentum. This report is based on the results obtained there.
1. Data The series k (t Service for 1982-2003 where axes x1 and x2 plane. Components M 1

), k 1, 2 of the ocean excitation functions of the International Earth Rotation and Reference System (IERS, http://www.iers.org) are analyzed. These data refer to a right rectangular coordinate system, are lying in the equatorial plane, and x1 is directed along projection of the Greenwich Meridian to this and M 2 of the torque exerted by the ocean, are M 1 C 2 1 , M 2 C 1 2 , 37 2 where 2 / day is the average frequency of the Earth's rotation; 7.04 10 kg m is the equatorial moment of inertia of the Earth; 0, 0145 day 1 is the frequency of free nutation (Chandler frequency). The procedure for calculating the excitation functions 1 (t ) and 2 (t ) is described in [Gross et al, 2003].
2. Statement of the Problem The CW is described by the linearized Liouville equation expressing the law of angular momentum conservation [Munk, MacDonald, 1964]. d 1d d 1d x1 x2 x2 f1 , x2 x1 x1 f 2 , (2) dt 2Q dt dt 2Q dt

In the interval 0...50 days; if 50 days it remains close to value

F12 0.1110

17

rad 2 / day,

(11)

and also performs oscillations whose amplitude increases with increasing of sampling parameter, Fig. 3c. So, results presented in Figure 3 are consistent with the basic hypothesis; for the correlation time we find:


rad 2 / day

cor

0 100 days.

a The coefficient of diffusion and the anisotropy constant one can roughly estimate, using (10), (11) and formulas (5) and (6):

a 1.6 10

17

rad 2 / day, 0.47

k here xk , , 1, 2 are dimensionless coordinates of the pole; Q is mantle quality factor (at frequencies of the order of ); f k M k / (C ). Within the framework of the probabilistic approach f k are random functions of time. The pair f1 , f 2 will be referred to as a random load. If nothing but the ocean impacts the pole motion, then 2 1 , f 2 1 (3) f1 2 where k (t ) are the ocean excitation functions. Obviously, the right-hand side of (2) contains law-frequency, year and half-year modes. However, we will mean below f k as random parts of functions (3), so f1 f 2 0. The hypothesis to be checked is as follows: the load f1 (t ), f 2 (t ) is a normal stationary stochastic process with small time of correlation. A mathematical model of such process is a two-dimensional white noise: we assume that
f1 (t1 ) f1 (t2 ) F11 (t2 t1 ), f 2 (t1 ) f 2 (t2 ) F22 (t2 t1 ) f1 (t1 ) f 2 (t2 ) F12 (t2 t1 ), (4)
where F11 , F 22 , F
12

Fig 3. ).
rad 2 / day

To find confidence intervals for a and we must use the fact that estimations (9) for N 1 are normally distributed values (it is the sequence of (4)). We have to find a statistical relationship between these estimates, to build a threedimensional "confidence region" P for matrix F ( F11 , F12 , F22 ) and probability P; and then to find the marginal values of functions a(F) and (F) on the set P , using (5) and (6): marginal values will be the required boundaries of the confidence intervals. Thus, we must consider the matrix F on the whole as three-dimensional object.

rad 2 / day

are components of a non-negative symmetric matrix, which we call the diffusion matrix and denote by F : F12 F F 11 . F12 F22
cor

The aim of this work is to test the statistical hypothesis (4). Along the way, we shall evaluate the correlation time parameters characterizing the matrix F, namely the diffusion coefficient a a( F) Tr F F11 F22 (5) and anisotropy constant

and

2 ( F11 F22 ) 2 4 F12 F2 2 (F ) 1 , 2 2 F1 F11 F22 ( F11 F22 ) 4 F12

(6)
Fig 3. b). Fig 3. c).

here F1 and F2 F1 eigenvalues of the matrix F. We disclaim the assumption that the ocean load is isotropic, i.e. that F2 F1. 3. Course of the solution. Main results Let us consider the equation (2) without dissipation (i.e. in case Q ). It can be written in the form:

FIG. 3. Estimates (8) as functions of sampling step: ) - F11 (), b) - F22 (), c) - F12 () The plan invented gives the following result: parameters a and belong to intervals:

x i x f (t ),
where x x1 ix2 , f f1 if 2 . If x(0) 0, then In terms of the excitation functions,



a 1.3 1017.....2.2 10

17

rad 2 / day, 0.06...0.65.

(12)

x(t ) f ( ) exp i (t ) d
0

t

with probability P 0.92. The confidential interval for the anisotropy constant lies entirely in the positive area; therefore we must consider the ocean load acting to the solid Earth as anisotropic.

x
where 1 i 2 , see Fig. 1.






(t ) (0) exp it (1




) i ( ) exp i (t )d ,
0

t

(7 )

rad

rad

4. The contribution of the ocean to the CW excitation The diffusion coefficient a Tr F is the main characteristic of the random load. Indeed, if sin() / 1 and Q 1, then regardless of the relationship between the eigenvalues of the matrix F, the sequence x(n), n 1, 2, ... can be interpreted as a discrete-time process with isotropic diffusion matrix, the trace of which equals to a [Tsurkis et al., 2014]. But in the case of an 2 isotropic load the CW amplitude A(t ) x12 x2 is also a Markovian process. If Q this process can be characterized by stationary amplitude, i.e. the expectation Ast A(t ), which is proportional a [Tsurkis et al., 2011]. a / d , where d is "general" diffusion coefficient Therefore, the ocean share of CW can be estimated by value characterizing the pole motion as a whole. Comparing (1) and (12), we see that a ~ 0.1 d . This means that in absence of other sources of excitation, the average amplitude of CW would be approximately one-third of the real one. On the other hand, if we subtract the ocean torque from general angular moment acting on CW, the expectation of CW amplitude decreases slightly, by approximately 5%. It is explained by non-linear dependence of average amplitude from a Tr F. 5. Conclusions It is shown that the data on the torque exerted by the ocean on the solid Earth do not contradict the statistical hypothesis. Namely, it can be interpreted as an anisotropic stationary random process with a time of autocorrelation cor 100days. In context of the probabilistic approach, an estimate of the contribution of the ocean to the CW was obtained: the one-third part of the CW amplitude can be explained by ocean.

rad 01.01.1980

rad

REFERENCES Fig.1. The function x(t ) calculated by formula (7) for 1000 days. The moment t 0 corresponds to 1.01.1980 . Fig.2. Ocean component of CW for 1000 days (from 01.01.1980)

If the hypothesis (4) is true, then the function x(t ) is a realization of a Markovian process in virtue of Doob's theorem (see, for example, [Tikhonov et al., 1977]). Next, we must eliminate the deterministic part of function (7) consisting of low-frequency component, the mode with a period of 1 year and its first overtone. This is the problem of independent interest; a variant of spectral analysis proposed in [Tsurkis et al., 2012a] allows one to do it correctly. Below, we denote as x(t) the random part of the function (7), see. Fig. 2; it will be referred to as the ocean component of CW. Let us consider the function xu ,v (t ) : t _____ __ xu ,v u x(t ) v x (t ) 2i x( ) exp i (t ) d , 0 which is the solution of the differential equation

xu ,v i x



u ,v

uf (t ) v f (t )

______

(8)

If conditions (4) are satisfied, xu ,v (t ) is a Markovian process too. It turns out that the elements of the matrix can be estimated by formulas 1 F11 x1/ 2,1/ 2 , x1/ 2,1/ 2 F22 x1/ 2,1/ 2 , x1/ 2,1/ 2 F12 x1,i , x1,i xi ,1 , xi ,1 , (9) 8 where *, * is Wiener-Liouville scalar product, which was taken into account in [Tsurkis et al., 2012a]:

1 z, w T


n 0

N 1

( z ((n 1)) e

i

z (n))( w((n 1)) e

_______________

i

w(n))
and the 0 , if

________

Arato, M., Kolmogorov, A.N., Sinai, Va. G (1962) On estimation of parameters of complex stationary Gaussian processes. Dokl. Akad. Nauk, 146(4), 747-750. Barnes, R., Hide, R., White, A., Wilson, C. Atmospheric angular momentum fluctuations, length-of-day changes and polar motion. //Proc. R. Soc. London, Ser.A, 387, P.31-73 Gross, R., Fukumori, I., Menemenlis, D. // Atmospheric and oceanic excitation of the Earth's wobbles during 1980-2000.// J. Geophys. Res., 2003, V. 108, No.B8. Munk, W.H. and G.J.F. MacDonald, 1960. The Rotation of the Earth. Cambridge University Press. 323pp. Tsurkis, I. Ya., Spiridonov E. A. On the applicability of the mathematical apparatus of Markovian processes to the description of the Chandler wobble. Izvestiya Physics of the solid Earth (2009) 45, 273-286, April 01, 2009 Tsurkis, I. Ya., Spiridonov E. A., Kuchay, M. S. Probabilistic model of the polar motion and the Chandler anomaly of the early XX century// Geofizicheskie issledovaniya, V.11, 4, 2010 Tsurkis, I. Ya., Kuchay, M. S., Spiridonov, E. A., Probabilistic analysis of the data on atmospheric angular momentum for January 1, 1980 to March 27, 2003. //Izvestiya Physics of the solid Earth April 2012a, Volume 48, Issue 4, pp 339-353. Tsurkis, I. Ya., Kuchay M. S., Spiridonov, E. A., Probabilistic analysis of the data on the oceanic angular momentum for January 1, 1980 to March 27, 2003. // Geofizicheskie issledovaniya, 2012b. Tikhonov, V.I., Mironov, M.A. Markovian processes. Moscow, Soviet Radio, 1977, 488 p. Tsurkis, I. Ya., Kuchay, M. S., Sinvukhina S. V., Polar motion under anisotropic random load // Izvestiya Physics of the solid Earth, January 2014, Volume 50, Issue 1, pp 137-149

here T N is length of the series of observations. If the main hypothesis (4) is true, the estimates (9) are consistent xu ,v ((n 1)) ei xu ,v (n), nonshifted; proof of this fact uses statistical independence of increments which presents sequence of absence of dissipative term in the left-hand side of (8). Starting with some value of the sampling parameter estimations (9) should not depend on . And, indeed, functions F11 () and F22 () increase with 0 100 days; then, 0 , they range near the values:

F11 (0 ) 1.00 10

17

rad 2 / day, F22 (0 ) 0.57 10

17

rad 2 / day,

(10)
17

Fig. 3, b. For the function F12 () the "threshold" value is 50 days : this function decreases from 0 to 0.15 10

rad 2 / day