“Investigation of Polar Motion with Continuous Mathematical Functions”

Reza Arab Sahebi
I graduated from Tehran university (Iran)
faculty surveying engineering (Geodesy)

Abstract:
Geodesy is the Science that relative to determine shape and size of the earth and positioning on or up the earth’s surface such as satellite. Therefore define a coordinate system for accession to goals of geodesy is necessary. For relative between a point in 3D space and coordinate system, we need to determine 3 parameters:

Origin
Axes orientation direct
The parameters that defines position of point in a coordinate system

One of systems to be used in geodesy, is geocentric coordinate system, that the origin is near the center of the earth and Z axes of coordinate system is in direct axes rotation. Because of rotation of the earth around the Z-axes and variation of pole’s position, Z-axes will change.

The result, always position of point in 3D coordinate system is a variable. In this paper with using observation of X and Y component of pole from 1962 until 1997 we can research with fitting continuous mathematical functions this none-periodic motion.

The results of the paper suggest the following:

With certain continuous mathematical functions that used in this paper on Y component, we can investigate polar’s vector displacement.
With computation of coefficient after fitting, we can predict position of pole at future.
Between used mathematical functions, sum of six-function sinuosity (general model sin6) has detecting power in pointing of variation of velocity and acceleration of polar motion rather than of them.

Figure (1): polar motion in differential year

In table (1) you can see the position of mean pole at ITRF.

Table (1): mean pole coordinate based on result of (EOP ( IERS) 97 C 01)

With application of table (1) information, if we see earth from up and we see project pole position on equator in any year figure (1) will be achieved. It’s obvious that earth rotation main axis position at equator surface, has coordinate (0, 0) that in figure (2) illustrated near pole.

Figure (2): pole position variation

In motions as polar motion we can investigate velocity and acceleration variation based on position variation at time. According to what is stipulated in physics in uniformity motion velocity is constant and it occurs when the moving object for bases of equal time, it will travel equal distances. But when we gain from figure (2) is that this movement isn’t regular but reversely is it is irregular. Hence in order to understand this motion only variable position in X direction is evaluated and computed.

Figure (3): pole X component variation related to time

Then for velocity investigation we can use below simple relation:

Motion acceleration in this direction will be achieved from:

In below figures and tables, fitting mathematical functions to X component and any function’s statistical specifications are appearing.

a) six degree polynomial:
This polynomial has below mathematical equation:

Figure (4): six degree polynomial is fitted to X component

Table (2): estimated coefficient and six degree polynomial statistical parameters

b) nine degree polynomial:
This polynomial has below mathematical equation:

Figure (5): nine degree polynomial is fitted to X component

Table (3): estimated coefficient and nine degree polynomial statistical parameters

c) sum of 3 sinusoidal functions:
This function has below mathematical equation:

Figure (6): sum of 3 sinusoidal functions fitted to X component

Table (2): estimated coefficient and six degree polynomial statistical parameters

d) sum of 4 Gaussian functions:
This function has below mathematical equation:

Figure (7): sum of 4 Gaussian functions fitted to X component

Table (5): estimated coefficient and sum of 3 sinusoidal function statistical parameters

e) Sum of 6 sinusoidal functions:
This function has below mathematical equation:

Figure (8): sum of 6 sinusoidal functions fitted to X component

Table (6): estimated coefficient and sum of 6 sinusoidal function statistical parameters

In above functions mean of is X component and x in right is measuring year. But what is obvious from table (1) that motion function isn’t continuous since it is discontinuous and first and second derivation in discontinuous function only solve with numerical method. One of methods that we use for fitting in discontinuous values is fitting one continuous function on it and derivation from it.

In this paper we use different mathematical function for fitting on data and with statistical parameters optimum function will be determined. From these functions six-degree polynomial and Gaussian function has less RMSE related other and we use them base for research of polar motion acceleration and velocity changes. Then with known coefficient and first and second derivation related to time we gain acceleration and velocity changes.

Figure (9): polar motion velocity graph with derivation from six-degree polynomial

Figure (10): polar motion acceleration graph with derivation from six-degree polynomial

It is understood from figure (9) and (10) since six-degree polynomial hasn’t the power to offer polar motion acceleration and velocity, therefore these changes are presented with linearity but the same data with sum of six sinusoidal functions because of increase power of indication figures (11) and (12) will be appear.

Figure (11): polar motion velocity graph with derivation from six sinusoidal functions

Figure (12): polar motion acceleration graph with derivation from six sinusoidal functions

Conclusion:

With certain continuous mathematical functions that used in this paper on Y component, we can investigate polar vector displacement.
With computation of coefficient after fitting, we can predict position of pole at future.
Between used mathematical functions, sum of six-function sinuosity (general model sin6) has detecting power in pointing of variation of velocity and acceleration of polar motion rather than of them.