What is the minimum velocity an ADCP can measure?
The minimum measurable velocity will vary with the type of instrument and configuration, and there are a few factors that will dictate the quality of the data (Signal-to-noise ratio / SNR , for example). The below answer is meant as a theoretical approach to the question.
Speed/Velocity
First off a reminder about the difference between speed and velocity, and how these two parameters relate to statistical uncertainty. We distinguish between the term velocity v (which involves direction of motion as well as the rate) and speed s, which only involves the rate and not direction.
| \begin{equation} v(t)= \frac{dx}{dt}= x'(t) \end{equation} |
(1) |
Speed is the absolute value of the velocity:
| \begin{equation} s(t)= |v(t)| = |\frac{dx}{dt}| \end{equation} |
(2) |
If we have a velocity component U, then the mean and its standard deviation σ are calculated using (3) and (4) respectively:
| \begin{equation} \bar{U} = \frac{1}{N} \sum_{i=1}^{N} U_i \end{equation} |
(3) |
| \begin{equation} \sigma(\bar{U}) = \frac{\sigma (U)}{\sqrt{N}} \end{equation} |
(4) |
using (5)
| \begin{equation} \sigma(U) = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (U_i-\bar{U})^2} \end{equation} |
(5) |
The standard deviation gives a measure of how spread out the variance of the mean is. The smaller the standard deviation, the more of the variance is concentrated close to the mean. That is; the mean is an arithmetic mean and the uncertainty in the mean is the uncertainty of each separate measurement divided by number of measurements. That means that the uncertainty in the velocity measurement gets smaller and smaller towards zero as long as one continue to measures. Therefore, there is theoretically no limit to the measurement uncertainty you can achieve.
If we work with speed S, then: (ref. Pythagoras)
|
\begin{equation} |
(6) |
where
|
\begin{equation} |
(7) |
The uncertainty will get very low if we first take the mean of U and V. Doing that we can get good speed estimates. If, instead, one takes the mean of each speed measurement, we will not get rid of the uncertainty and speed will not converge towards zero even though both velocity component does. Depending on instrument setup, the sig(U) and sig(V) can be relatively large so that the error may be larger than one initially would expect.
The uncertainty in speed estimates can be minimized by first averaging the velocity components U and V. This approach leads to more accurate speed calculations. However, if we instead average the individual speed measurements (i.e., calculate the speed for each pair of U and V and then take the mean), the uncertainty remains. In this case, even if both U and V approach zero, the calculated speed will not converge to zero.
Furthermore, depending on the instrument setup, the standard deviations of U and V (denoted as σ(U) and σ(V)) can be relatively large, which means the error in the speed estimate may be greater than initially anticipated.
Regarding the definition of accuracy (ref. instrument specific data sheet):
|
\begin{equation} |
(8) |
Direction
The problem with direction is related to the above. Direction dir is normally defined as:
|
\begin{equation} |
(9) |
When the U-component of the velocity approaches zero, the current direction measurements can become scattered in all directions. To avoid this, it's important to first calculate the mean velocity components and then compute the mean direction as a separate step.
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