ParametrizationTemporal

This class gives a parametrization of time depending parameters (gravity field, positions, ...). It will be used to set up the design matrix in a least squares adjustment. If multiple parametrizations are given the coefficients in the parameter vector are sequently appended.

Useally time intervals are defined half open meaning the last time belongs not to the interval. This behaviour can be changed for the last interval with includeLastTime.

Constant

Represents a parameter being constant in time in each interval.

The parameter names are *:*:*:<interval>.

Name	Type	Annotation
interval	timeSeries
includeLastTime	boolean

Trend

A time variable function is given by a linear trend \[ f(x,t) = \frac{1}{T}(t-t_0) \cdot f_t(x), \]with $t_0$ is timeStart and $T$ is timeStep in days. A constant term is not included and must added separately.

The parameter name is *:*:trend.<timeStep(days)>*(t-<timeStart>):*.

Name	Type	Annotation
timeStart	time	reference time
timeStep	time

Splines

A time variable function is given by \[ f(x,t) = \sum_i f_i(x)\Psi_i(t), \]with the (spatial) coefficients $f_i(x)$ as parameters and the temporal basis functions $\Psi_i(t)$. Basis splines are defined as polynomials of degree $n$ in intervals between nodal points in time $t_i$, for details see basis splines.

The parameters are ordered timewise. First all parameters of $f_{i=1}(x)$ then $f_{i=2}(x)$ and so on. The total parameter count in each interval is $N=N_t+d-1$, where $N_t$ is the count of time points from timeSeries in each interval and $d$ is the degree.

The parameter names are *:*:spline.n<degree>:<interval of each spline>.

Name	Type	Annotation
degree	uint	degree of splines
timeSeries	timeSeries	nodal points in time domain
intervals	timeSeries
includeLastTime	boolean

Polynomial

A time variable function is represented by Legendre polynomials in each interval. The time is normed to $[-1,1)$ in each interval.

The total parameter count is $(N+1)M$, where $N$ is the polynmial degree and $M$ the number of intervals with the parameter names *:*:legendrePolynomial.n<degree>:<interval>.

Name	Type	Annotation
polynomialDegree	uint	polynomial degree
interval	timeSeries	intervals of polynomials
includeLastTime	boolean

Oscillation

A time variable function is given by a oscillation \[ f(x,t) = f^c(\M x)\cos(\omega_i(t)) + f^s(\M x)\sin(\omega_i(t)) \]with $\omega_i=\frac{2\pi}{T_i}(t-t_0)$, $t_0$ is timeStart and $T$ is timePeriod in days.

The parameter names are *:*:oscillation.cos(2*pi/<period(days)>*(t-<timeStart>)):* and *:*:oscillation.sin(2*pi/<period(days)>*(t-<timeStart>)):*.

Name	Type	Annotation
period	time	[day]
time0	time	reference time

Fourier

A time variable function is given by a fourier expansion \[ f(x,t) = \sum_{m=1}^M f_m^c(\M x)\cos(2\pi m \tau) + f_m^s(\M x)\sin(2\pi m \tau) \]with the normalized time \[ \tau = \frac{t-t_A}{t_B-t_A}, \]and $t_A$ is timeStart, $t_B$ is timeEnd in each interval and $M$ is the fourierDegree.

The total parameter count is $2MN$, where $N$ is the number of intervals. The parameters are sorted in following order: $f_1^c, f_1^s, f_2^c, \ldots$ with the parameter names *:*:fourier.cos(<m>*x):<interval> and *:*:fourier.sin(<m>*x):<interval>.

Name	Type	Annotation
fourierDegree	uint
interval	timeSeries	intervals of fourier series
includeLastTime	boolean

DoodsonHarmonic

The time variable function is given by a fourier expansion \[ f(x,t) = \sum_{i} f_i^c(x)\cos(\Theta_i(t)) + f_i^s(x)\sin(\Theta_i(t)), \]where $\Theta_i(t)$ are the arguments of the tide constituents $i$ \[ \Theta_i(t) = \sum_{k=1}^6 n_i^k\beta_k(t), \]where $\beta_k(t)$ are the Doodson's fundamental arguments ($\tau,s,h,p,N',p_s$) and $n_i^k$ are the Doodson multipliers for the term at frequency $i$. The multipliers must be given by doodson coded as Doodson number (e.g. 255.555) or as names intoduced by Darwin (e.g. M2).

The major constituents given by doodson can be used to interpolate minor tidal constituents using the file inputfileAdmittance. This file can be created with DoodsonHarmonicsCalculateAdmittance.

The total parameter count is $2N$ with $N$ the number of doodson frequencies. The parameters are sorted in following order: $f_1^c, f_1^s, f_2^c, \ldots$ with the parameter names *:*:doodson.cos(<doodsonName>):* and *:*:doodson.sin(<doodsonName>):*.

Name	Type	Annotation
doodson	doodson	code number (e.g. 255.555) or darwin name (e.g. M2)
inputfileAdmittance	filename	interpolation of minor constituents