Spherical harmonics coefficients format (extended GRACE format)

This is the description of the extended GRACE format, which is an extension of the original GRACE gravity models format (document GR-GFZ-FD-001 GRACE 327-732(v1.1) November 27,2003).

New acronyms for the gravity coefficients have been introduced in order to enable a modelling of the coefficients
more complex than a single bias or bias+drift.

Existing acronyms:
——————
GRCOEF: bias, the date indicates the mid-point of data used for this coefficient
GRCOF2: bias, the dates indicate the time span of data used for this coefficient
GRDOTA: rate, the date indicates the epoch to which coefficients with rates are mapped
        (GRDOTA needs to be associated with either a GRCOEF or GRCOF2 coefficient)

New acronyms:
————-
G_BIAS, GDRIFT, GCOS…, GSIN… (see below)

G_BIAS and GDRIFT:
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They replace GRCOEF and GRDOTA; they have the same format as GRCOF2 (with 2 dates, t1 and t2),
except that the dates have a different meaning.

G_BIAS: is the bias at a reference epoch T_refep.
        The first date t1 has a double meaning:
        It is this reference epoch (t1 = T_refep), it is also the start of the validity range for this bias coefficient.
        The second date t2 is the end of the validity range for this bias coefficient.
GDRIFT: is the rate /year. The dates t1 and t2 indicate the range of validity of this rate.

If G_BIAS is not associated with a GDRIFT coefficient having the same range of validity, then the potential at any given
date T is given by:
  coef(T) = G_BIAS(t1,t2) with T belonging to [t1,t2[

If G_BIAS is associated with a GDRIFT coefficient having the same range of validity, then the potential at any given
date T is given by:
  coef(T) = G_BIAS(t1,t2) + GDRIFT(t1,t2) * (T – t1) with T belonging to [t1,t2[

It is therefore possible with G_BIAS and GDRIFT to create a modelling of the coefficients
with successive bias and drift values, similar to the one used by IGS for the Earth orientation parameters.
The resulting modelling of the coefficients can take the shape of a piece-wise-linear function, if the function at the
end of an interval coincides with the function at the beginning of the following interval, but it can also be discontinuous
in order to accomodate for earthquakes, for instance.

GCOS1A, GSIN1A, GCOS2A, GSIN2A,…:
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Periodic coefficients at the annual (1A), semi annual (2A), etc, periods.

The periodic signal is computed at any given date T using the formula:
coef_perio(T) = GCOS1A * cos( 2 PI * (T – 1st January) / year ) + GSIN1A * sin( 2 PI * (T – 1st January) / year ) + etc.

GCOS1A: Amplitude of cosine term at the annual period
GSIN1A: Amplitude of sine term at the annual period
GCOS2A: Amplitude of cosine term at the semi annual period
GSIN2A: Amplitude of sine term at the semi annual period
etc…
In all cases, the two dates indicate the range of validity of the periodic coefficient. It is possible
therefore to have, for instance, a different set for each year.

These periodic functions must be combined with a Bias (+ optionally Drift) function, in order to obtain the complete
modelling of the time variations of the gravity field.

In summary:
***********
coefficients with one date: GRCOEF, GRDOTA
coefficients with two dates: GRCOF2, G_BIAS, GDRIFT, GCOS… and GSIN…
For these coefficients, the first date is inclusive (T >= first date), the second date is exlusive (T < second date).
 

Computation of the periodic coefficients

For the determination of the periodic coefficients the following equation is to be used:

coef_perio(T) = GCOS1A * cos( 2 PI * (T – T0′) / EYD ) + GSIN1A * sin( 2 PI * (T – T0′) / EYD ) + etc.

With:

T – T0′	Time of computation, in days, from T0′; with T0′ = the first of January of the current year, at 0. h
EYD	Exact duration, in days, of the current year (= 365 or 366 days, depending on the year)
GCOS1A	Amplitude of cosine term at the annual period
GSIN1A	Amplitude of sine term at the annual period
GCOS2A	Amplitude of cosine term at the semi annual period
GSIN2A	Amplitude of sine term at the semi annual period