Velocity-weighted Doppler profile correction

A velocity-weighted Doppler profile correction converts an integrated sampled velocity into an integrated area-mean velocity by weighting each instantaneous correction according to the sampled velocity at that time. For a fixed opening of cross-sectional area $A$, suppose its area-mean normal velocity is $\bar{v}(t)=\alpha (t)u(t)$. Thenprovided the denominator is nonzero. Applying the time average of $\alpha$ to a velocity-time integral generally answers a different question.

The error has an exact covariance form. Define $⟨g⟩=T^{-1}{\int }_0^{T}g(t)dt$ and time covariance ${\mathrm{Cov}}_t(\alpha ,u)=⟨\alpha u⟩-⟨\alpha ⟩⟨u⟩$. ThenConsequently an unweighted correction is exact if that covariance vanishes, even when the correction varies in time. Its failure is about how profile changes align with transport, not merely whether either trace varies.

In an original two-second example, equal one-second intervals have sampled velocities 10 and 40 cm/s and corrections 1.0 and 0.5. The unweighted mean correction is 0.75, but ${\alpha }_{\mathrm{eff}}=(10+20)/50=0.60$. With $A=2\mathrm{c}{\mathrm{m}}^2$, actual modeled transport is 60 mL; unweighted correction predicts 75 mL, a 25% overestimate. Here time covariance is $-3.75$ cm/s and mean sampled velocity is 25 cm/s, recovering $0.75-3.75/25=0.60$.

If $u$ is nonnegative, normalized velocity weights are nonnegative and ${\alpha }_{\mathrm{eff}}$ lies between the instantaneous corrections. For reversing fluid flow, the denominator can instead approach zero through cancellation. Forward and reverse transported volumes can then be more informative than a single unstable ratio. If opening area changes, the integral must also retain $A(t)$; this fixed-area correction does not remove that additional dependence.

The primary Doppler report requires appropriate matching of sampled velocity and anatomical area and distinguishes modal spectral velocities from the fastest envelope.^[ref] The covariance calculation is an original integration identity, not a universal clinical profile correction.