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- etkf_cl
class etkf_cl |
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class for Ensemble Transform Kalman Filter solver
See Feng et al., ACP 2009
Members:
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# control variables
1. nx:<int>: size of control variable
2. ne:<int>: size of perturbation ensemble
3. x:<array, (nx, ne)>: perturbation ensemble
4. mean_x:<array, (nx)>: mean value of control variable
5. xnorm:<float>: normalized factor
# model observation
6. y:<array, (nobs, ne)>: ensemble for model observation perturbation
7. ny:<int>: ny=nobs
8. mean_y:<array, (nobs)>: model observations for self.mean_x
9. r:<array, (nobs, )>: observation err covariance
10. sqr:<array, (nobs,)>: sqr=sqrt(r)
11. scyt:<array, (ne, nobs): scaled y^t scyt=transpose(self.y)/self.sqr
12. uy:<array, (ne,ne)>: U matrix of svd(scyt)
13. vy:<array, (ny, ny)>:V matrix of svd(scyt)
12. wy:<array, (nwy)>: eigenvalues of scyt. newy=min(ne, ny)
13. md:<array, (nwy)>: md=1.0/(1.0+wy*wy)
14 n_wy:<int>: size of non-zeros wy
15. uw: <array, (ne, n_wy)>: reduced U matrix
Functions
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1. __init__: initialization
2. reset: re-do initialization
3. get_posterior: get posterior
4. get_tm: get transform matrix
5. get_k: get gain matrix
6. get_inc_m: get increment matrix
7. get_x_inc: get analysis increment
8. get_y_inc: get analysis increment in observation space
9. do_y_transform: transform maxtrix in y space
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Methods defined here:
- __init__(self, mean_x, mean_y, x, y, r, xref=1, xnorm=1.0, do_debug=False)
- initialize the class
Inputs
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1. mean_x:<array, (nx)>: mean values of control variables x
2. mean_y:<array, (ny)>: mean model observation values.
3. x: <array, (nx, ne)>: ensemble of x perturbations
4. y: <array, (nobs, ne)>: ensemble of model forecast perturbations. ny=nobs
5. r: <array, (nobs)>: observation error covariances given in 1 dimension
6. xref:<float>: value used to scale mean_x
7. xnorm:<float>: value used to scale ensemble purbations
Notes:
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1. self.x=(x-mean(x))/sqrt(xnorm)
2. self.y=(y-mean(y))/sqrt(xnorm)
- do_y_transform(self, y_in, tm, mean_y=None)
- multiply y_in by tm
Inputs:
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1. y_in:<array>, (new_ny, ne)>: 'new' observation perturbations
2. tm :<array, (ne, ne)>: Transformation matrix
Returns:
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1. yt:<array, (new_ny, ne)>: yt=y_in*tm
- get_inc_m(self, yobs)
- get the analysis increment matrix
Inputs:
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1. yobs:<array, (nobs)>: the observations
Returns:
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1. inc_m :<array, (ne,)>:increase matrix
Notes:
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1. inc_m=U*W*(I+WT*W)^-1*VT*R^-0.5*(Yobs-mean_Y)
- get_k(self)
- calculate analysis gain k matrix
Returns:
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1. k matrix:<array, (nx, nobs)>: gain matrix
2. Notes:
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K=X'*U*W*(I+WT*W)^-1*VT*R^-0.5
- get_posterior(self, yobs)
- calculate posterior from assimilation of observations
Inputs:
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1. yobs:<array, nobs>: observations
Returns:
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1. postprior:<array, nx>:
Notes:
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1. xa=xf+k*(yobs-f(x))=xf+dx*inc_m
- get_tm(self)
- the analysis increment on the variables
Returns:
1. tm :<array, (ne, ne)>: matrix for posterior ensemble: x(f)=x(a)*tm
Notes:
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1. tm=U*(1+W*WT)^-0.5)*UT
- get_x_inc(self, inc_m, x_in=None, mean_x_in=None)
- the analysis increment of control variables
Inputs:
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1. inc_m:<array, (ne)>: increment matrix
2. x_in:<array, (nx, ne)>: ensemble perturbations
3. mean_x_in:<array, (nx)>: mean value
Returns:
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1. xinc: <array, nx>: the x increments
- get_y_inc(self, y_in, inc_m, mean_y=None)
- the analysis increment on the variables
Inputs:
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1. y_in:<array, (new_ny, ne)>: the 'new' observation perturbation
2. inc_m :<array, (ne)>: increment matrix
Returns:
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1. yinc: <array, new_ny>: y increments from xinc
- reset(self, mean_x, mean_y, x, y, r, xref=1.0, xnorm=1.0, do_debug=False)
- re set (reinitialize) the class
Input:
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1. mean_x:<array, (nx)>: mean values of control variables x
2. mean_y:<array, (nobs)>: mean model observation values
3. x: <array, (nx, ne)>: ensemble of x perturbations
4. y: <array, (nobs, ne)>: ensemble of model forecast perturbations
5. r: <array, (nobs)>: observation error covariances given in 1 dimension
6. xref:<float>: value used to scale mean_x
7. xnorm:<float>: valuse used to scale en
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