3. Setting up the functional model

class kf.timefunction.TimeFct(time, model, origintime=0.0, verbose=False)

Initialization of the class, which build parametrised function of time. This creates an object that can build functional representations and spit out names, etc.

• time : vector of the time (decimal years )

• originTime : origin of the time used here

• model : contain string of function and associated parameters with syntax described in table below

model =	description
[('POLY'   ,deg),	polynomial of degree deg
('COS'    ,freq),	cosine of frequency freq
('SIN'    ,freq),	sine of frequency freq
('STEP'   ,t1,t2,...),	earthquake at time ti
('HTAN'   ,t1,w1,t2,w2,...),	slowslip(s) centred on ti
('EXP'    ,t1,w1),	starting and characteristic time
('LOG'    ,t1,w1),	starting and characteristic time
('BSPLINE',order,t1,w1,t2,w2,...),	peak(s) of deformation centred on ti
('ISPLINE',order,t1,w1,t2,w2,...),	slowslip(s) centred on ti
('LISEG'  ,t1,t2,...)]	line segment(s) beween ti and t(i+1)

check_model()

Verify number of parameters are consistent and write down summary of model description

comp_phase_shift(m, P=None)

Compute oscillation amplitude and phase shift with its variances from amplitudes of cosine and sine terms with their variances.

• m : state vector containing at least all the model parameters (length > self.L)

• P : if error is known =P the state covariance (default None)

create_A(k, M)

Create A, the transition n × n matrix which converts m into mf Matrix used during prediction in the Kalman Filter

• k : the timestep

• M : the state vector length

cut_model(N)

Function that removes some of the component of the functional model based on the length of parameters (N). This allows to include informations when needed and not before.

draw_model(coeff)

Gives f(t) as defined in model

• coeff : contain multiplying coefficients of each function in the same order as in model (correspond to first elements of state vector m)

expend_model(k, dt, verbose=True)

Add events in reference model not in self.mod, if we are getting temporarily close to the occurence of the event

• dt : anticipation time

WORKS ONLY IF CHRONOLOGICAL ORDER IN MODEL

find_coeff_lsq(evolv, err)

Basic linear least-squares which finds of defined model

• evolv : time series of phase change (shape of time)

• err : standard deviation of shage (shape of time)

get_label(L, unit, tunit='day', phase=False)

Get an array of labels for each model parameters (name and units) which can be used for plotting

• L : number of parameters

• unit : string of length unit

• tunit : string of time unit

• phase : True if sine and cosine amplitudes converted into sine amplitude and phase shift

identify_outdated(dtmax)

Function optimizing model as a function of starting time of time series, localize modification that will be berformed later on every state vectors/cov

• dtmaxtime after event allowed to optimize localized function in time

  (apply only on ‘STEP’,’EARTHQUAKE’,’HEAVISIDE’ and cst of ‘POLY’)

remove_oldstuff(m, P)

Require the outputs of identify_outdated()

• m, P : specify local state vector and covariance

shift_t0(t0, coeff)

• t0 : shift in t0 (t0_new -t0_old)

• coeff : parameter (assuming: offset, slope, sin, cos along last axis)

transition_vect(t)

Build linear transition vector that gives the model

• t : time can be one date or an array of dates

Note

This trajectory model is used to make the forecast step of Kalman filtering. It helps fill holes in the interferometric network on a given pixel. If so, the error on the model knownledge will propagate on the estimated phase change uncertainty. It is NOT used to filter the time series of phase change.