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I agree that the original script was wrong and the following modification is correct. The calculation of error of each component should be also fixed to consider suppression by sqrt of (effective) sample number. sotodlib/sotodlib/site_pipeline/calibration/wiregrid.py Lines 496 to 499 in d77a545 |
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The error of
det_angle(γ in following equation) was actually wrong, the dimension is wrong.All the
det_angleare almost uniformly biased so the impact on circular averageddet_angleshould be small but I think it's good to fix it. Also we can consider adding error of θ_wire if it depends on step, but it's not added in this PR.Derivation:
γ = φ / 2 - θ_wire, where φ = arctan(dU/dQ), dU = U - cy and dU = Q - cx
With r = sqrt(dQ2 + dU2), ∂φ/∂dU = dQ/r^2, ∂φ/∂dQ = -dU/r^2
Var(dQ) = Var(Q) + Var(cy), Var(dU) = Var(U) + Var(cx), Cov(dQ, dU) = 0
Assuming uniform error, Var(Q) = Var(U), Var(cy) = Var(cx)
Var(φ) = [dQ^2 Var(dU) + dU^2 Var(dQ)]/r^4 = [ (Var(Q) + Var(U) + Var(cx) + Var(cy))/2 ] / r^2
σ(γ) = sqrt(Var(φ)) / 2 = sqrt([ (Var(Q) + Var(U) + Var(cx) + Var(cy))/2 ]) / 2r