craig.cahillane@LIGO.ORG - posted 21:20, Wednesday 15 June 2016 - last comment - 23:59, Friday 17 June 2016(27765)
LHO Calibration Uncertainty - Now With Covariance
C. Cahillane I have revamped the uncertainty budget to include covariances between all stages of actuation and all time-dependent parameters. I computed each parameter's covariances in real and imaginary coordinates to provide a consistent basis. I then compiled an6 x 6Actuation Covariance MatrixC_A, a2 x 2Sensing Covariance MatrixC_S, and an8 x 8Kappa Covariance MatrixC_K. Then I compile them into a giant covariance matrixC:_ _ | C_A 0 0 | C = | 0 C_S 0 | |_ 0 0 C_K _|Then, I multiply by some conspicuous Jacobian vectorsJ(f)to get the final2 x 2uncertainty matrix σ_R^2(f):σ_R^2 = J * C * J'whereJlooks like:_ _ | d Re(R) d Im(R) | | --------- --------- .... | | d Re(p_i) d Re(p_i) | J(f) = | | | d Re(R) d Im(R) | | --------- --------- .... | |_d Im(p_i) d Im(p_i) _|(I was able to use complex differentiation and Cauchy-Riemann here to make the derivatives easier. Recall that R = 1/C + D*A. Now I can compute dR/dA = D and dR/dC = -1/C^2 to formJ(f), thanks to 200 year old mathematics) Finally, to make the uncertainties readable by humans, I divideσ_R^2(f)by|R(f)|^2, rotateσ_R^2(f)byangle(R(f))via a rotation matrix, and read off the square roots of the diagonal of the rotatedσ_R^2(f)to get the magnitude and phase uncertainties plotted below. I have plotted the uncertainty at GPSTime = 1135136350, the time of the Boxing Day Event. The plot shows an overall increase in magnitude uncertainty of about 1% at low frequency. Phase uncertainty increased by about 0.5 degrees at low frequency. The effects are more dramatic at Livingston. Check out LLO aLOG 26542.
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C. Cahillane I have reproduced the uncertainties including covariance for GW150914 for the calibration companion paper. We will have to update the associated uncertainty calculation sections of the paper. I have also attached two .txt files for the R_C01/R_C03 response comparison and the associated uncertainty. Something I failed to emphasize above: Our uncertainties in the response function are now fully covariant... the plots I show of the magnitude and phase are only approximations to the true uncertainty. I have looked at the 3D plots of the covariant ellipses, and it's a fairly good approximation in this case.
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C. Cahillane
I have attached and printed my relative covariance matrix. Please see DCC T1600227 for an explanation of the relative covariance matrix.
Basically, the below is percentage covariances.
Re(A_U) Im(A_U) Re(A_P) Im(A_P) Re(A_T) Im(A_U) Re(C_R) Im(C_R) Re(K_T) Im(K_T) Re(K_P) Im(K_P) Re(K_C) Im(K_C) Re(f_C) Im(f_C)
Re(A_U) 0.0166 0.0083 0.0139 0.0079 0.0146 0.0067 0 0 0 0 0 0 0 0 0 0
Im(A_U) 0.0083 0.0209 0.0091 0.0169 0.0071 0.0178 0 0 0 0 0 0 0 0 0 0
Re(A_P) 0.0139 0.0091 0.0163 0.0052 0.0157 0.0066 0 0 0 0 0 0 0 0 0 0
Im(A_P) 0.0079 0.0169 0.0052 0.0181 0.0057 0.0156 0 0 0 0 0 0 0 0 0 0
Re(A_T) 0.0146 0.0071 0.0157 0.0057 0.0251 0.0047 0 0 0 0 0 0 0 0 0 0
Im(A_T) 0.0067 0.0178 0.0066 0.0156 0.0047 0.0187 0 0 0 0 0 0 0 0 0 0
Re(C_R) 0 0 0 0 0 0 0.0207 0.0079 0 0 0 0 0 0 0 0
Im(C_R) 0 0 0 0 0 0 0.0079 0.0208 0 0 0 0 0 0 0 0
Re(K_T) 0 0 0 0 0 0 0 0 0.0025 -0.0002 0.0019 -0.0018 -0.0004 0 0.0004 0
Im(K_T) 0 0 0 0 0 0 0 0 -0.0002 0.0025 0.0017 0.0019 0.0001 0 0.0001 0
Re(K_P) 0 0 0 0 0 0 0 0 0.0019 0.0017 0.0035 -0.0003 0.0002 0 -0.0003 0
Im(K_P) 0 0 0 0 0 0 0 0 -0.0018 0.0019 -0.0003 0.0035 0.0006 0 -0.0005 0
Re(K_C) 0 0 0 0 0 0 0 0 -0.0004 0.0001 0.0002 0.0006 0.0037 0 -0.0036 0
Im(K_C) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Re(f_C) 0 0 0 0 0 0 0 0 0.0004 0.0001 -0.0003 -0.0005 -0.0036 0 0.0054 0
Im(f_C) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0