This article is a more detailed examination of the Sources of Uncertainty referenced in Quantifying and Controlling Error in Salt-Dilution Measurements. It is based on material presented most recently at the CWRA Conference in Lethbridge, Alberta, Canada, April 2017 (Lethbridge_Salt_Dilution_Uncertainty_V0.9).
One of the concerns of regulators tasked with protecting our shared environment, is ensuring that NaCl concentrations do not exceed safe limits for the most sensitive species that may be present in a watercourse. There are several qualifiers that must accompany this statement:
- How long does the acute salt concentration last? The peak concentration occurs at the point of injection and decreases downstream of this point. The higher the concentraion, the shorter the exposure time, and
- It is very difficult to prove that a sensitive species does not live in a particular ecosystem. We must therefore employ conservative estimates of safe concentration levels.
See this article(tbd) for a more complete discussion of Salt Dilution Instream Q (SDIQ) gauging as it relates to water quality guidelines.
For the purpose of this article, we are interested in reducing the peak concentration at the point of injection. This can be done by two means:
- Pre-mix the dry salt in stream water. Water is capable of dissolving 20% NaCl. This also ensures that the salt will be completely dissolved when it enters the stream.
- Inject the salt slowly over 30-60 seconds. This also reduces the peak concentration and appears to have no significant impact on the resulting SDIQ measurement, as demonstrated in Figure 1.
Figure 1 demonstrates an important aspect of the Salt Dilution method: the time over which the salt is injected does not appear to significantly affect the resulting SDIQ. In this figure, Pulse 1 was an instantaneous injection. Pulse 2 was injected over 60 seconds. This is evident in the elongated rising limb of the pulse. If the Uncertainty is considered in both of these cases, there is no significant difference between the two measurements. The Standard Error for Pulse 1 is +/-3.7% and for Pulse 2 is +/-3.5%. The difference between the two SDIQs is only 2.4%, therefore there is no significant difference between the two derived Qs at 95% confidence, that is the two means are not more than 2 sigma away from each other. The sum of Pulse 1 and 2 results in no significant difference in derived SDIQ either. To find the Pulse 1+2 SDIQ, the mass of injected salt is added and the area under both curves is added. Similar results have been found in several other instances. This was first noticed when a second injection was made before the EC.T had returned to the ECBG. It’s also been observed to been true for constant rate injection trials, as discussed in the Mosquito Creek Tracer Method Flotilla workshop.
An important component of obtaining high quality (low uncertainty) salt dilution measurements is to know your sensor resolution and the resulting instrument uncertainty. Figure 2 shows the result of 4 injections of progressively less salt in a steady flow using a Unidata 6536B EC.T probe. The resolution is coarse, ranging from 0.4-0.6 uS/cm, although the granularity is 0.01 μS/cm. The true resolution can be found by examining the difference between consecutive values, indicated by the “step” in Figure 2. Based on the discussion of uncertainty in the top-level article, we know that the signal uncertainty is the larger of the background noise (standard deviation of pre- and post- pulse EC.T) or the Quantization Error (QE). The ratio of the average pulse EC.T to the background noise is the Signal to Noise Ratio (SNR). This is the same ratio as the area under the pulse to the background noise multiplied by the transit time. Note that for the last injection in Figure 2 of 0.17 kg/(m³/s), the EC.T only rises 2 quanta. The resulting error is 104% due to relatively large QE.
Figure 3 shows the same injections, with the probe in the same location, but for an instrument with a higher effective resolution. The Oakton Con110 has a resolution of 0.1 μS/cm in this range (it has a resolution of 0.01 μS/cm below 20 μS/cm). This results in a much larger SNR, and less uncertainty. Note that for the 4th injection, the uncertainty is only +/-13%. Also note that there is no significant difference between the derived SDIQs for the 4 pulses when the uncertainty is taken into account.
It should be clear from this example that the higher the SNR, and in particular the higher the resolution of the sensor, the less salt we can inject OR the lower the uncertainty that can be achieved.
From our initial field trials, it became apparent that we needed some way to know the true value of Q in order to asses the Error. From such a model, we could then derive relationships between SNR and Uncertainty. So we built an instrument model.
The model is a simple Excel-based Monte-Carlo simulation model which uses statistical parameters to generate multiple simulations of a pulse-instrument interaction, and summary statistics are calculated. What we found was a bit counterintuitive at first. The best measurements are noisy ones! The average measured EC.TBG was closer to the true EC.TBG when the noise was equal to or greater than the resolution. This is the opposite of what we found in most commercial sensors. Typically commercial sensors produced results more similar to the plot on the right in Figure 5, rather than the left. The plot on the left, although it seems unstable in the tail of the plot, actually contained valuable information on the true value of the EC.T. Much like Pulse Width Modulation (PWM) or Frequency Modulation (FM) the frequency of the two digital quanta contained information on the true EC.T, if you used a wide enough averaging kernel.
The plot on the left of Figure 6 has a sensor noise greater than the sensor resolution, but a low SNR. The plot on the right has a sensor noise greater than the sensor resolution, and a high SNR.
The best instrument for SD has:
- High Resolution and Quanta (0.1 μS/cm or better for EC.T <200μS/cm)
- Repeatable, Linear, Stable (these are different than accuracy).
- Absolute calibration not necessary if in situ CF.T derivation carried out for each site.
- Benefit of absolute calibration is that it gives expected CF.T values for QA/QC.
Estimate of Uncertainty and Error
We have chosen to define Type A uncertainty using several components:
where EC.TUnc is given by:
and R is the sensor resolution, SEBGEC.T is the standard error in the BG EC.T given by:
Note that the larger the pre- and post-pulse sample (n), the smaller the SEBGEC.T is. This makes sense if you consider the pre- and post- pulse sample as sampling a population whose mean is the BG EC.T. We often use 10 pre- and post- pulse samples, although with 30 samples the sample mean is very close to the population mean.
To estimate the Error, or Type B Uncertainty, we’ve chosen to compare concurrent SDIQ pairs. These are often taken on opposite banks of the water course for the same injection pulse.
As can be seen in Figure 6, generally there is agreement between the two Types of Uncertainty, which is good. Each SDIQ has an associated Type A uncertainty. Adding these in quadrature, and multiplying by 2, gives the 2-sigma (95% confidence) interval. If the DeltaQ is greater than this value, it is significantly different at alpha = 0.05. Often the Type A is overestimating the error, but that is expected as:
- The true error might not be represented by the DeltaQ, it may be larger and
- Even if the DeltaQ is zero, and the true Q is the same as tme measured Q, we still have uncertainty around the measured Q.
We are mostly interested in the cases where Type A uncertainty is less than the DeltaQ, and those have been labeled#30, #57, and #61. Without going into too much detail, the reason for the discrepancy between the two types is:
- #30 incomplete mixing. Indicator is that the shape of pulse was not smooth and DQ is larger than Unc1 + Unc2.
- #57 incomplete mixing. Indicator was the Left Bank pulse was not smooth. and DQ is larger than Unc1 + Unc2.
- #60 temperature influence. These pairs were taken in a northern stream with long mixing length (long transit time) where the water temperature changed by ~1ºC over the course of the measurement, the water temperature was cold (6ºC), and the EC was high (120 μS/cm). This resulted in an EC.T that was very sensitive to temperature since the correction is 2%/ºC. On top of that, salt was limited so the SNR was already low. The change in the water temperature over the course of the measurement caused a change in BG EC.T that was on the same order as the area under the pulse. This example emphasizes the need for accurately calibrated EC and temperature probe, and high resolution in both EC and Temperature, especially when working in low temperature, high EC streams.