# Use small dt to "cope with" ocean sector blow up
# (due to spatially-constant infl)
OneYear = 0.05 * (24 / 6) * 365
t = dpr.Chronology(0.005,
dtObs=0.05,
T=110 * OneYear,
Tplot=Tplot,
BurnIn=10 * OneYear)
land_sites = np.arange(Nx // 2)
ocean_sites = np.arange(Nx // 2, Nx)
jj = land_sites
Obs = dpr.partial_Id_Obs(Nx, jj)
Obs['noise'] = 1
Obs['localizer'] = nd_Id_localization((Nx, ), (1, ), jj)
HMM = dpr.HiddenMarkovModel(Dyn,
Obs,
t,
X0,
LP=LPs(jj),
sectors={
'land': land_sites,
'ocean': ocean_sites
})
####################
# Suggested tuning
####################
def obs_inds(t):
return jj + random_offset(t)
@modelling.ens_compatible
def hmod(E, t):
return E[obs_inds(t)]
# Localization.
batch_shape = [2, 2] # width (in grid points) of each state batch.
# Increasing the width
# => quicker analysis (but less rel. speed-up by parallelzt., depending on NPROC)
# => worse (increased) rmse (but width 4 is only slightly worse than 1);
# if inflation is applied locally, then rmse might actually improve.
localizer = nd_Id_localization(shape[::-1], batch_shape[::-1], obs_inds, periodic=False)
Obs = {
'M': Ny,
'model': hmod,
'noise': modelling.GaussRV(C=4*np.eye(Ny)),
'localizer': localizer,
}
# Moving localization mask for smoothers:
Obs['loc_shift'] = lambda ii, dt: ii # no movement (suboptimal, but easy)
# Jacobian left unspecified coz it's (usually) employed by methods that
# compute full cov, which in this case is too big.
Dyn = {
'M': Nx,
'model': step,
# It's not clear from the paper whether Q=0.5 or 0.25.
# But I'm pretty sure it's applied each dto (not dt).
'noise': 0.25 / tseq.dto,
# 'noise': 0.5 / t.dto,
}
X0 = modelling.GaussRV(mu=x0(Nx), C=0.001)
jj = linspace_int(Nx, Nx // 4, periodic=True)
Obs = modelling.partial_Id_Obs(Nx, jj)
Obs['noise'] = 0.1**2
Obs['localizer'] = nd_Id_localization((Nx, ), (1, ), jj, periodic=True)
HMM = modelling.HiddenMarkovModel(Dyn, Obs, tseq, X0)
HMM.liveplotters = LPs(jj)
# Pinheiro et al. use
# - N = 30, but this is only really stated for their PF.
# - loc-rad = 4, but don't state its definition, nor the exact tapering function.
# - infl-factor 1.05, but I'm not sure if it's squared or not.
#
# They find RMSE (figure 8) hovering around 0.8 for the LETKF,
# and conclude that "the new filter consistency outperforms the LETKF".
# I cannot get the LETKF to work well with so much noise,
# but we note that optimal interpolation (a relatively unsophisticated method)
# scores around 0.56:
Nx = KS.Nx
# nRepeat=10
t = dpr.Chronology(KS.dt,
dkObs=2,
KObs=2 * 10**4,
BurnIn=2 * 10**3,
Tplot=Tplot)
Dyn = {'M': Nx, 'model': KS.step, 'linear': KS.dstep_dx, 'noise': 0}
X0 = dpr.GaussRV(mu=KS.x0, C=0.001)
Obs = Id_Obs(Nx)
Obs['noise'] = 1
Obs['localizer'] = nd_Id_localization((Nx, ), (4, ))
HMM = dpr.HiddenMarkovModel(Dyn, Obs, t, X0)
HMM.liveplotters = LPs(np.arange(Nx))
####################
# Suggested tuning
####################
# Reproduce (top-right panel) of Fig. 4 of bocquet2019consistency # Expected rmse.a:
# --------------------------------------------------------------------------------
# xps += LETKF(N=4 , loc_rad=15/1.82, infl=1.11,rot=True,taper='GC') # 0.18
# xps += LETKF(N=6, loc_rad=25/1.82, infl=1.06,rot=True,taper='GC') # 0.14
# xps += LETKF(N=16, loc_rad=51/1.82, infl=1.02,rot=True,taper='GC') # 0.11
#
import numpy as np
import dapper.mods as modelling
from dapper.mods.Lorenz05 import Model
from dapper.tools.localization import nd_Id_localization
# Sakov uses K=300000, BurnIn=1000*0.05
tseq = modelling.Chronology(0.002, dto=0.05, Ko=400, Tplot=2, BurnIn=5)
model = Model(b=8)
Dyn = {
'M': model.M,
'model': model.step,
'noise': 0,
'object': model,
}
X0 = modelling.GaussRV(mu=model.x0, C=0.001)
jj = np.arange(model.M) # obs_inds
Obs = modelling.partial_Id_Obs(model.M, jj)
Obs['noise'] = 1
Obs['localizer'] = nd_Id_localization((model.M, ), (6, ))
HMM = modelling.HiddenMarkovModel(Dyn, Obs, tseq, X0)
