= Velocity-weighted Doppler profile correction
A \b[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 $\bar v(t)=\alpha(t)u(t)$. Then
$$
V=A\int_0^T\alpha(t)u(t)\,dt,\qquad
\alpha_{\rm eff}=\frac{\int_0^T\alpha(t)u(t)\,dt}{\int_0^T u(t)\,dt},
$$
provided 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 $\langle g\rangle=T^{-1}\int_0^Tg(t)\,dt$ and time <covariance> $\operatorname{Cov}_t(\alpha,u)=\langle\alpha u\rangle-\langle\alpha\rangle\langle u\rangle$. Then
$$
\alpha_{\rm eff}=\langle\alpha\rangle+
\frac{\operatorname{Cov}_t(\alpha,u)}{\langle u\rangle}.
$$
Consequently 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 $\alpha_{\rm eff}=(10+20)/50=0.60$. With $A=2\,\mathrm{cm^2}$, 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 $\alpha_{\rm eff}$ 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.<Recommendations for Quantification of Doppler Echocardiography by Miguel A. Quiñones and others>[PW Doppler and CW Doppler, printed pp. 168–169]{ref} The <covariance> calculation is an original integration identity, not a universal clinical profile correction.
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