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DACost3.py
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DACost3.py
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import numpy as np
from numpy import matrix
from numpy import linalg
import DAmodel4 as DAm
import Data2 as D
import math
import matplotlib.pyplot as plt
from scipy.optimize import fmin_bfgs
from scipy.optimize import fmin_ncg
d=D.dalecData()
B=matrix([[d.sigB_cf,0,0,0,0],[0,d.sigB_cw,0,0,0],[0,0,d.sigB_cr,0,0],[0,0,0,d.sigB_cl,0],[0,0,0,0,d.sigB_cs]]) #Background error covariance matrix
XB=np.matrix([58.,102.,770.,40.,9897.]).T #Background state
lenrun=365 #assimilation window length
#Matrix factoral function
def Mfac(Mlist,a): #Matrix factoral to find product of M matrices
if a==1:
return Mlist[0]
else:
Mat=Mlist[0]
for x in xrange(0,a-1):
Mat=Mlist[x+1]*Mat
return Mat
#Adjoint Matrix factoral function
def MfacAdj(Mlist,a): #Matrix factoral to find product of M matrices
if a==1:
return Mlist[0].T
else:
Mat=Mlist[0].T
for x in xrange(0,a-1):
Mat=Mat*Mlist[x+1].T
return Mat
#Creates size x size R matrix
def Rmat(y1_err):
if len(y1_err)==1:
return y1_err[0]
matrix = [[0]*len(y1_err) for i in range(len(y1_err))]
for i in range(len(y1_err)):
matrix[i][i] = float(y1_err[i]) #uses elements of y_err as stnd devs
return np.matrix(matrix)
#Cost function that takes initial state as argument
def J(X):
nlhx=DAm.nldalec(X,lenrun)[0] #Run nonlinear model for length run outputing models obs guess
X1=[float(X[0]),float(X[1]),float(X[2]),float(X[3]),float(X[4])]
X0=np.matrix(X1).T
#Creates H_i's, y_i's and y_err_i's
hx=[] #h(x)
y=[] #obs
y_err=[] #obs error
for x in range(lenrun):
temp=[]
yi=[]
yi_err=[]
obs=False
if d.gpp[x]!=None:
temp.append(nlhx[x,0])
yi=np.append(yi,[d.gpp[x]])
yi_err=np.append(yi_err,[d.gpp_err[x]**2])
obs=True
if d.lf[x]!=None:
temp.append(nlhx[x,1])
yi=np.append(yi,[d.lf[x]])
yi_err=np.append(yi_err,[d.lf_err[x]**2])
obs=True
if d.lw[x]!=None:
temp.append(nlhx[x,2])
yi=np.append(yi,[d.lw[x]])
yi_err=np.append(yi_err,[d.lw_err[x]**2])
obs=True
if d.rt[x]!=None:
temp.append(nlhx[x,3])
yi=np.append(yi,[d.rt[x]])
yi_err=np.append(yi_err,[d.rt_err[x]**2])
obs=True
if d.nee[x]!=None:
temp.append(nlhx[x,4])
yi=np.append(yi,[d.nee[x]])
yi_err=np.append(yi_err,[d.nee_err[x]**2])
obs=True
if d.cf[x]!=None:
temp.append(nlhx[x,5])
yi=np.append(yi,[d.cf[x]])
yi_err=np.append(yi_err,[d.cf_err[x]**2])
obs=True
if obs==False:
temp.append(0)
yi=np.append(yi,0)
yi_err=np.append(yi_err,0)
hx.append(np.matrix(temp))
y.append(np.matrix(yi))
y_err.append((yi_err))
#Create y-hx increments
yhx=[y[i].T-hx[i].T for i in range(len(y))]
Incr=[0]*lenrun
for i in range(lenrun):
if np.linalg.norm(y_err[i])!=0:
Incr[i]=yhx[i].T*np.matrix(Rmat(y_err[i])).I*yhx[i]
#Obs elements to be summed
#Incr=[yhx[i].T*np.matrix(Rmat(y_err[i])).I*yhx[i] for i in range(len(y))]
#Cost fn
J=0.5*(X0-XB).T*B.I*(X0-XB) +0.5*np.sum(Incr)
return float(J)
#Derivative of costfunction takes initial state as argument
def dJ(X):
Mlist,GPP_diff,M=DAm.ldalec(X,lenrun) #run linear model out list of linear model matrices and value for GPP derivative at each time step
nlhx=DAm.nldalec(X,lenrun)[0] #run nonlinear model for obs guess output
X1=[X[0],X[1],X[2],X[3],X[4]]
X0=np.matrix(X1).T
#Creates H_i's, y_i's and y_err_i's
H=[] #linear obs operator
hx=[] #h(x)
y=[] #obs
y_err=[] #obs error
for x in range(lenrun):
temp=[]
temp2=[]
yi=[]
yi_err=[]
obs=False
if d.gpp[x]!=None:
temp.append([GPP_diff[x],0,0,0,0])
temp2.append(nlhx[x,0])
yi=np.append(yi,[d.gpp[x]])
yi_err=np.append(yi_err,[d.gpp_err[x]**2])
obs=True
if d.lf[x]!=None:
temp.append([d.p_5,0,0,0,0])
temp2.append(nlhx[x,1])
yi=np.append(yi,[d.lf[x]])
yi_err=np.append(yi_err,[d.lf_err[x]**2])
obs=True
if d.lw[x]!=None:
temp.append([0,0,d.p_6,0,0])
temp2.append(nlhx[x,2])
yi=np.append(yi,[d.lw[x]])
yi_err=np.append(yi_err,[d.lw_err[x]**2])
obs=True
if d.rt[x]!=None:
temp.append([d.p_2*GPP_diff[x],0,0,d.p_8*d.T[x],d.p_9*d.T[x]])
temp2.append(nlhx[x,3])
yi=np.append(yi,[d.rt[x]])
yi_err=np.append(yi_err,[d.rt_err[x]**2])
obs=True
if d.nee[x]!=None:
temp.append([-(1-d.p_2)*GPP_diff[x],0,0,-d.p_8*d.T[x],-d.p_9*d.T[x]])
temp2.append(nlhx[x,4])
yi=np.append(yi,[d.nee[x]])
yi_err=np.append(yi_err,[d.nee_err[x]**2])
obs=True
if d.cf[x]!=None:
temp.append([1,0,0,0,0])
temp2.append(nlhx[x,5])
yi=np.append(yi,[d.cf[x]])
yi_err=np.append(yi_err,[d.cf_err[x]**2])
obs=True
if obs==False:
temp.append([0,0,0,0,0])
temp2.append(0)
yi=np.append(yi,0)
yi_err=np.append(yi_err,0)
H.append(np.matrix(np.vstack(temp)))
hx.append(np.matrix(temp2))
y.append(np.matrix(yi))
y_err.append((yi_err))
yhx=[y[i].T-hx[i].T for i in range(len(y))] #Create y-h(x) incriment
#stacklist1=[H[0]] #Creates H hat matrix
#for x in xrange(1,len(y)):
# stacklist1.append(H[x]*Mfac(Mlist,x))
stacklist2=[H[0].T] #creats M^T*H^T for each time step
for x in xrange(1,len(y)):
stacklist2.append(MfacAdj(Mlist,x)*H[x].T)
#Incr=[stacklist2[i]*np.matrix(Rmat(y_err[i])).I*yhx[i] for i in range(len(y))] #Creates obs incriment
Incr=[]
for i in range(lenrun):
if np.linalg.norm(y_err[i])!=0:
Incr.append(stacklist2[i]*np.matrix(Rmat(y_err[i])).I*yhx[i])
dJ=B.I*(X0-XB) -np.sum(Incr, axis=0) #Cost function first derivative
dJlist=[float(dJ[0]),float(dJ[1]),float(dJ[2]),float(dJ[3]),float(dJ[4])]
return dJ.T #np.array(dJlist)
#2nd derivative of costfunction takes initial state as argument
def d2J(X):
Mlist,GPP_diff=DAm.ldalec(X,lenrun)
nlhx=DAm.nldalec(X,lenrun)
X1=[X[0],X[1],X[2],X[3],X[4]]
X0=np.matrix(X1).T
#Figure out Hi's, y and y_err
H=[]
hx=[]
y=[]
y_err=[]
for x in range(lenrun):
temp=[]
temp2=[]
yi=[]
yi_err=[]
obs=False
if d.gpp[x]!=None:
temp.append([GPP_diff[x],0,0,0,0])
temp2.append(nlhx[x,0])
yi=np.append(yi,[d.gpp[x]])
yi_err=np.append(yi_err,[d.gpp_err[x]**2])
obs=True
if d.lf[x]!=None:
temp.append([d.p_5,0,0,0,0])
temp2.append(nlhx[x,1])
yi=np.append(yi,[d.lf[x]])
yi_err=np.append(yi_err,[d.lf_err[x]**2])
obs=True
if d.lw[x]!=None:
temp.append([0,0,d.p_6,0,0])
temp2.append(nlhx[x,2])
yi=np.append(yi,[d.lw[x]])
yi_err=np.append(yi_err,[d.lw_err[x]**2])
obs=True
if d.rt[x]!=None:
temp.append([d.p_2*GPP_diff[x],0,0,d.p_8*d.T[x],d.p_9*d.T[x]])
temp2.append(nlhx[x,3])
yi=np.append(yi,[d.rt[x]])
yi_err=np.append(yi_err,[d.rt_err[x]**2])
obs=True
if d.nee[x]!=None:
temp.append([-(1-d.p_2)*GPP_diff[x],0,0,-d.p_8*d.T[x],-d.p_9*d.T[x]])
temp2.append(nlhx[x,4])
yi=np.append(yi,[d.nee[x]])
yi_err=np.append(yi_err,[d.nee_err[x]**2])
obs=True
if obs==False:
temp.append([0,0,0,0,0])
temp2.append(0)
yi=np.append(yi,0)
yi_err=np.append(yi_err,0)
H.append(np.matrix(np.vstack(temp)))
hx.append(temp2)
y.append(yi)
y_err.append(yi_err)
#yhx=np.matrix((np.hstack(y)-np.hstack(hx))).T
#R=Rmat(np.hstack(y_err))
stacklist=[H[0]] #Creates H hat matrix
for x in xrange(1,lenrun):
stacklist.append(H[x]*Mfac(Mlist,x))
#Hhat=np.matrix(np.vstack(stacklist))
#d2J=B.I+Hhat.T*R.I*Hhat
#Incr=[np.matrix(Rmat(y_err[i])).I*yhx[i] for i in range(0,lenrun,1)]
d2jlist=[0]*lenrun
for i in range(lenrun):
if np.linalg.norm(y_err[i])!=0:
d2jlist[i]=stacklist[i].T*np.matrix(Rmat(y_err[i])).I*stacklist[i]
d2J=B.I+sum(d2jlist)
return d2J
def Plot(X):
nlhx=DAm.nldalec(X,lenrun)
gppB=nlhx[:,0]
gppO=d.gpp[0:lenrun]
xlist=np.arange(0,lenrun,1)
#xa=fmin_bfgs(J, X, dJ2, maxiter=4000)
xa=fmin_ncg(J, X, dJ2, fhess=d2J2)
nlhx2=DAm.nldalec(xa, lenrun)
gppA=nlhx2[:,0]
print xa
plt.plot(xlist,gppO,label='GPP_O')
plt.plot(xlist,gppB,label='GPP_B')
plt.plot(xlist,gppA,label='GPP_A')
plt.legend()
plt.show()
def Plot2(X):
nlhx=DAm.nldalec(X,lenrun)
neeB=nlhx[:,4]
neeO=d.nee[0:lenrun]
#neeO2=[None]*lenrun
#for i in range(lenrun):
# if neeO[i]!=None:
# neeO2[i]=-neeO[i]
xlist=np.arange(0,lenrun,1)
#xa=fmin_bfgs(J, X, dJ, maxiter=2000)
xa=fmin_ncg(J, X, dJ2, fhess=d2J2)
nlhx2=DAm.nldalec(xa, lenrun)
neeA=nlhx2[:,4]
print xa
plt.plot(xlist,neeO,label='NEE_O')
plt.plot(xlist,neeB,label='NEE_B')
plt.plot(xlist,neeA,label='NEE_A')
plt.legend()
plt.show()
def Plot3(X):
nlhx=DAm.nldalec(X,lenrun)
neeB=nlhx[:,1]
neeO=d.lf[0:lenrun]
xlist=np.arange(0,lenrun,1)
xa=fmin_bfgs(J, X)
nlhx2=DAm.nldalec(xa, lenrun)
neeA=nlhx2[:,1]
print xa
plt.plot(xlist,neeO,label='lf_O')
plt.plot(xlist,neeB,label='lf_B')
plt.plot(xlist,neeA,label='lf_A')
plt.legend()
plt.show()
def Plot4(X):
nlhx=DAm.nldalec(X,lenrun)
neeB=nlhx[:,3]
neeO=d.rt[0:lenrun]
xlist=np.arange(0,lenrun,1)
xa=fmin_ncg(J, X, dJ2, fhess=d2J2)
nlhx2=DAm.nldalec(xa, lenrun)
neeA=nlhx2[:,3]
print xa
plt.plot(xlist,neeO,label='rt_O')
plt.plot(xlist,neeB,label='rt_B')
plt.plot(xlist,neeA,label='rt_A')
plt.legend()
plt.show()
def Plot5(X):
nlhx=DAm.nldalec(X,lenrun)[0]
gppB=nlhx[:,0]
neeB=nlhx[:,4]
lfB=nlhx[:,1]
rtB=nlhx[:,3]
#rtO=d.rt[0:lenrun]
#gppO=d.gpp[0:lenrun]
#neeO=d.nee[0:lenrun]
#lfO=d.lf[0:lenrun]
xlist=np.arange(0,lenrun,1)
xa=fmin_ncg(J, X, dJ) #, fhess=d2J)
#xa=fmin_bfgs(J, X) #fprime=dJ)
nlhx2=DAm.nldalec(xa, lenrun)[0]
rtA=nlhx2[:,3]
gppA=nlhx2[:,0]
neeA=nlhx2[:,4]
lfA=nlhx2[:,1]
gpp_err=[]
gppO=[]
gppday=[]
for i in range(lenrun):
if d.gpp_err[i]!=None:
gpp_err.append(d.gpp_err[i])
gppO.append(d.gpp[i])
gppday.append(float(d.D[i]))
nee_err=[]
neeO=[]
needay=[]
for i in range(lenrun):
if d.nee_err[i]!=None:
nee_err.append(d.nee_err[i])
neeO.append(d.nee[i])
needay.append(float(d.D[i]))
rt_err=[]
rtO=[]
rtday=[]
for i in range(lenrun):
if d.rt_err[i]!=None:
rt_err.append(d.rt_err[i])
rtO.append(d.rt[i])
rtday.append(float(d.D[i]))
lf_err=[]
lfO=[]
lfday=[]
for i in range(lenrun):
if d.lf_err[i]!=None:
lf_err.append(d.lf_err[i])
lfO.append(d.lf[i])
lfday.append(float(d.D[i]))
print xa
plt.subplot(2,2,1)
plt.plot(xlist,gppB,label='GPP_B')
plt.plot(xlist,gppA,label='GPP_A')
plt.errorbar(gppday,gppO,yerr=gpp_err, fmt='o', label='GPP_O')
plt.legend()
plt.xlabel('Day')
plt.ylabel('GPPvalue (gCm$^{-2}$)')
plt.title('DA GPP')
plt.subplot(2,2,2)
plt.plot(xlist,neeB,label='NEE_B')
plt.plot(xlist,neeA,label='NEE_A')
plt.errorbar(needay,neeO,yerr=nee_err, fmt='o',label='NEE_O')
plt.legend()
plt.xlabel('Day')
plt.ylabel('NEEvalue (gCm$^{-2}$)')
plt.title('DA NEE')
plt.subplot(2,2,3)
plt.plot(xlist,rtB,label='rt_B')
plt.plot(xlist,rtA,label='rt_A')
plt.errorbar(rtday,rtO,yerr=rt_err, fmt='o',label='RT_O')
plt.legend()
plt.xlabel('Day')
plt.ylabel('RTvalue (gCm$^{-2}$)')
plt.title('DA RT')
plt.subplot(2,2,4)
plt.plot(xlist,lfB,label='lf_B')
plt.plot(xlist,lfA,label='lf_A')
plt.errorbar(lfday,lfO,yerr=lf_err, fmt='o',label='LF_O')
plt.legend()
plt.xlabel('Day')
plt.ylabel('LFvalue (gCm$^{-2}$)')
plt.title('DA LF')
plt.suptitle('DA Run with initial guess x0=%s'%(X))
plt.show()