Principles of Operation:
Velocimeter Accuracy
Contents
- Accuracy
- Bias
- Short-term uncertainty
- Spikes
- How individual measurements are collected
- Uncertainty, variance and standard deviation
- Averaging
- Variance and noise level
- Noise level and real data
- How do we specify the noise level?
Accuracy
The word accuracy is often used loosely to cover some combinations of bias and short-term uncertainty. Bias and short-term uncertainty are different, and they have different ramifications for the results of your data collection. Here is the difference:
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Bias is error that remains after taking long-term averages.
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Short-term uncertainty is the random error of individual measurements, which can be removed by averaging.
These errors have different sources, different consequences for your results, and are specified separately. Velocity spikes are distinctive and obvious as a source of error. The following will address each of these errors separately.
Bias
Velocimeters long-term accuracy depends on the geometry of its probes and the stability of its internal oscillators. Over most of the velocity range, bias can be measured in terms of an offset and a scale-factor error. Offsets would appear as a non-zero velocity in still water. If probe geometry were entirely fixed and repeatable, then we could determine the velocity scale factor based entirely on dimensions, frequencies and our physical model. In practice, small variations in probes require that probes be individually calibrated in the factory.
A factory-calibrated velocimeter should have a scale-factor bias that is less than 1% of the measured velocity. A velocimeter will retain this accuracy as long as its probe remains unbent. Thus far we have found the offset to be too small to measure (which is what we expect based on theory).
Bias increases near the extremes of a velocimeter's velocity range because velocities that would be measured beyond the extremes wrap around to the opposite end of the range. You can avoid this bias by keeping the maximum observed speed toward the middle of the velocimeter's maximum range.
Short-term uncertainty
Most of the rest of this page is devoted to short-term uncertainty, expressed in terms of noise level. Velocimeter's short-term uncertainty is small compared with most other acoustic sensors, largely as a result of the algorithm they use. Small short-term uncertainty comes from collecting acoustic echoes with exceedingly high correlation. However, because sources of decorrelation can be both internal and external, a probes noise level is not entirely intrinsic to the probe itself, but it depends also on the environment in which it is measuring. Primary exrternal sources of decorrelation include mean speed, turbulence and nonuniformity in the scatterers.
Spikes
In many velocimeter measurements, velocity spikes are the dominant source of error. Spikes can be disproportionate sources of error because they can have such large values compared with the mean velocity. Large spikes are easy to see, so they can be removed from the data, but it appears that even despiked data have higher noise levels than comparable data without spikes (see Laboratory Velocimeter Tests in a Flume).
The source for spikes is not well understood. Spikes may occur as a result of the environment itself, but it is clear that the velocimeter hardware and algorithms are a primary source of spikes. This conclusion is based on our experiences with the Vector Velocimeter. The Vector incorporates substantial improvements, both in its hardware and its algorithm, and as a consequence, it sees far fewer spikes in its data relative to our older velocimeters. The new algorithm, retrofitted into our olde velocimeters reduces spikes in them as well.
How individual measurements are collected
A velocimeter transmits acoustic pings into the water at a high rate, typically many hundreds per second. The ping rate is fixed within each velocimeter according to its acoustic frequency and velocity range. The sample rate, or the rate at which the velocimeter reports data, is much smaller than the ping rate to allow each measurement to average multiple pings. Every time a velocimeter reports a result, it computes the result using all the available pings, then clears its buffers and starts a new average. If it sends data out at, say, 25 Hz, each output value will include twice as many pings as it would it it were sending data out at 50 Hz.
Uncertainty, variance and standard deviation
Uncertainty is often measured in terms of measurement variance or standard deviation. These two are related:
variance = (standard deviation)2
(1)
The standard deviation is the size of the typical variation you would expect in a measurement, around the true value you are trying to measure. For example, imagine you were to make many measurements of an unvarying velocity. The best estimate of the true velocity is the mean of your measurements, and the uncertainty of each of your measurements is the standard deviation of the collection of measurements.
Averaging
When you average a collection of velocity measurements, your average is a better estimate of the true velocity than each individual estimate. The uncertainty of the average is reduced according to:
variance(mean) = variance(individual measurements)/N
std dev(mean) = std dev(individual measurements)/N1/2
(2)
In other words, if you reduce the variance of individual measurements by a factor of 2, you can collect data twice as fast.
Variance and Noise Level
The velocimeter's noise level, also called the noise floor, is its most fundamental measure of measurement uncertainty. If you know the noise level, you can easily compute the expected variance or standard deviation of your measurements for any sample rate. Velocimeter uncertainty is a random white noise. This means that each velocity estimate is independent of the next, and it means that the noise has no preferred frequency.
The noise level varies widely according to the characteristics of the velocimeter, how it is set up, the characteristics of the flow, and the acoustic scattering environment. Here is how the noise level and measurement variance are related:
variance = (noise level) * (sample rate) /2
(3)
You can use (2) to turn (3) into an equation for standard deviation. Table 1, below, shows how the variance and standard deviation depend on the sample rate. Note that Table 1 is mathematically identical to (2). For example, remember that each 1 Hz measurement has roughly 100 times as many individual measurements in its average as a single 100 Hz measurement. Then (2) tells us that the 100 Hz measurement should have 100 times the variance of the 1 Hz measurement, which is what you see in Table 1.
Sample
Rate |
Variance
(mm2/s2) |
Standard Deviation
(mm/s) |
| 100 Hz |
5000 |
71 |
| 25 Hz |
1250 |
35 |
| 5 Hz |
250 |
16 |
| 1 Hz |
50 |
7 |
Table 1. Measurement variance and standard deviation for different sample rates. This table assumes a noise level of 10-4 m2s-2Hz-1. Remember that noise levels vary widely in practice.
Noise level and real data
The velocity noise level limits the frequency range over which you can obtain useful data. Figure 1 sketches the spectrum of turbulence observations made by a typical velocimeter. The shape of the turbulent spectrum shown is characteristic of the "inertial subrange", the frequency range in which large eddies break into smaller eddies, which break into even smaller eddies, and so on. In the case shown in Figure 1, the noise level limits the maximum useful frequency to around 2 Hz.
Figure 1. Spectra of velocity noise level, turbulence and the resulting measurement. The noise level plotted is typical for velocimeters (actual noise levels can vary several orders of magnitude from this value). The turbulent spectrum shown falls as f-5/3, where f is frequency. The spectrum of the resulting measurement is the sum of the turbulent spectrum plus the noise level.
How do we specify the noise level?
It turns out that noise levels are difficult to specify because they depend on many factors, including factors that are outside the instrument (i.e. flow velocity and turbulence). This dependence is apparent, for example, in Laboratory Velocimeter Tests in a Flume. We specify velocimeter noise level in terms of single-measurement, single-component random noise (i.e. short-term uncertainty) that is equal to 1% of the velocity range at 25 Hz. Results in New Velocimeter Processing Algorithms Reduce Variance are consistent with this specification. For example, for 1 m/s velocity range, this specification gives a 2-component noise level of 1.6 x 10-5 m2s-2Hz-1 (the noise level of a two-component velocity is double double thie level for a single component). The observed 2-component noise level for 1 m/s velocity range in New Velocimeter Processing Algorithms Reduce Variance was 1.8 x 10-5 m2s-2Hz-1 when using our old algorithm, and 7 x 10-6 m2s-2Hz-1 when we used our new algorithm. In Laboratory Velocimeter Tests in a Flume, variance in smooth flow was around 1 x 10-5 m2s-2Hz-1, and about a factor of 10 higher in turbulent flow. This order-of-magnitude difference between smooth and turbulent flow illustrates the general difficulty in specifying the noise level.
This means that the best way to determine a velocimeter's noise level in your application is to collect data in the same conditions. You can also estimate noise level with existing data collected in similar conditions. When you consider existing data, look carefully at how the data was collected and the nature of the flow, keeping in mind that noise level for a velocimeter can vary by orders of magnitudes as these factors change.
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