Every signal received by the instrument is subject to some amount of noise. The velocity measurement obtained from a single ping is typically too noisy to be used alone, but the average of a number of these pings is less noisy and therefore more useful. The Doppler noise characteristics can be summarized as:
- Random and non-biased. That means if the measurements are averaged over a long enough time period, the correct velocity will be obtained.
- The distribution of the velocity is Gaussian, meaning that the velocities measured are symmetric around the true velocity.
- Averaging reduces uncertainty. The more measurements that are averaged, the better is the estimate of the mean velocity.
The Doppler velocity uncertainties comprise two types of errors; the short-term error (random) and the long-term error (bias).
Short-term error
One velocity estimate is commonly the average of many velocity measurements (also called pings). The uncertainty of each ping is dominated by the short-term, or random, error. The short-term error depends partly on internal factors such as the size of the transmit pulse, the measurement volume and the beam geometry (which is collectively called Doppler noise), and external factors such as signal strength of the return echo, turbulence, and instrument motion. The random error is uncorrelated from ping to ping, so by averaging together a number of pings, the measurement uncertainty is reduced to acceptable levels according to the formula:
| \( \sigma_{mean}=\frac{\sigma_{single ping}}{\sqrt{N}} \) | (1) | |||
| \( \text{sigma} \) | - | standard deviation | ||
| \(N\) | - | number of pings averaged together | ||
As seen from equation 1, the standard deviation of a velocity estimate decreases with increasing number of data points included in the average. The number of pings that your velocity estimate is based on depends on how you configure your instrument, and the Nortek Deployment software therefore predicts the instrument error based on the short-term error of a single ping and the number of pings averaged together and reports it under Horizontal and Vertical Precision. The concept of "precision" is related to idea of "repeatability" as it is being used for acoustic Doppler systems. The precision value given is a theoretical estimate of the standard deviation of the velocity measurements based on how the instrument is set up. The velocity precision is always calculated along beam first and is a function of frequency, bandwidth, cell size and velocity range. From there the horizontal and vertical precision are calculated based on number of beams and the geometry of the head.
In many situations, external factors such as the environment itself dominate the short-term error. This is true near an energetic surface and in turbulent flow such as boundary layers and rivers. In situations like these, the data collection strategy should take into account the nature and the time scales of the environmental fluctuations. Here are two examples:
- Waves: When measuring mean velocities in the presence of waves, sample velocity at roughly ¼ the interval of the dominant wave period, and measure through 6-10 wave cycles.
- Turbulent flow: In boundary layers, a rough rule of thumb is that the root mean square (RMS) turbulent velocity is 10% of the mean velocity. If, for example, the mean velocity is 1 m/s, it is possible to estimate turbulent fluctuations to be 10 cm/s. Obtaining 1 cm/s RMS uncertainty would require at least 100 pings.
Long-term error
Random errors can be reduced, but never eliminated. When averaging several pings to reduce the error, there will be a difference between the resulting “mean current” and the actual current. This deviation from the actual current measurement is called bias, and is often also referred to as accuracy. Bias is not random and cannot be reduced by averaging, it has a fixed magnitude and direction that is either proportional or constant to the measured velocity. The bias is often much smaller than the random errors removed by averaging, and it represents the limit to how much it is possible to reduce the short-term error. The long-term bias depends on internal signal processing, especially filters. This bias for each individual instrument can be found it its respective technical documentation.
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