Error Analysis and Data Quality B-1 Error Analysis The overall accuracy of water budget calculations depends on the relative accuracy of the input data. A simplified example is discussed as follows to illustrate the importance of using accurate data. The water balance in one of its most basic forms can be expressed as; dw = P r + M ET R dt dw/dt = rate of change of soil water content over time; P r = precipitation rate; M = rate of snowmelt; ET = rate of evapotranspiration; and R = rate of runoff. For experimental watersheds, the simplified annual water balance equation can be written as: ET = P R ± S ET = evapotranspiration; P = precipitation; R = runoff; and ± S = change in groundwater storage. For long-term experiments, it has been found that the groundwater storage change term is generally insignificant compared to the other terms, and hence: ET = P R The precipitation and runoff are sample values with a corresponding standard error of estimate. B-1
The variations in parameters (P r, M, ET or R) may affect the behaviour or results of the water budget analysis. Sensitivity analysis could therefore be used to determine which parameter is most critical for the error analysis. If the standard errors of estimate of the rainfall and runoff (P SE, R SE ) are assumed to be independent; then the corresponding standard error of the estimated evapotranspiration is given by: ET SE = P SE 2 + R SE 2 2 ρ PR P SE R SE ρ PR is the coefficient of correlation between the variables (P and R in this case; see NOTE 1 ). The standard error of ET can be large, and may even approach the estimated mean evapotranspiration in situations where both precipitation and runoff are large in comparison with ET and where both have significant variations due to measurement errors and natural random variations. When undertaking water budget analyses, the errors associated with precipitation measurements must be recognized. Measurement errors include the following factors (see Note 2 ): Inadequate spatial sampling; 1 NOTE: Where the parameter being estimated (Y) is a function of multiple parameters with known errors, then the estimation error can be expressed by the following general equation as: Y SE = n n n Σ 2 i X 2 ise ± 2Σ Σ i j ij X ise X ise i=1 i=1 i=i+1 Y SE = combined standard error of estimate = weighting factor (normally = 1) X ise = standard error of estimate of variable X i ij = coefficient of correlation between variables i and j n = the number of parameters 2 NOTE: Legates, D. R., and C. J. Willmott, Evaluating the Terrestrial Water Balance from the Historical Climate Record in Oliver, H.R., 1998. B-2
Bias in gauge measurement; effect of wind, wetting losses, evaporation from the gauge, splashing effects, blowing and drifting snow, trace precipitation events, impact of automatic recorders, and random errors. The average annual (areal-averaged) bias in annual precipitation can be up to 10-12% (i.e. the gauge measurement of precipitation can be up to 10-12% different than the bias adjusted precipitation). Systematic errors in streamflow estimation result from uncertainties in the stage-discharge relationship for flow measuring stations. The precision of the measurement and short-term fluctuation in the stage-discharge relationship are sources of random error. Average Precipitation Gauge Error (%) Winter 16.6 19.9 Spring 10.0 12.0 Summer 5.9 7.1 Autumn 8.1 9.7 Annual 10.0 12.0 In our opinion, the relative standard error in streamflow measurements is approximately ± 5%. For example, for a watershed with an average annual precipitation of 900 mm (SE = 100 mm) and runoff of 300 mm (SE = 15 mm), and assuming a correlation coefficient of 0.9, the average annual evapotranspiration (using the above equation) is estimated to be 600 mm with a corresponding standard error of approximately 87 mm (or 15% of the estimated mean annual evapotranspiration). When applying hydrologic models to calculate runoff, the most basic form of the water budget equation can be simplified to the following form: R = P ET In this case, P and ET are available for input to the model, and the standard error of the estimated runoff becomes dependent on the measurement precision of the P and ET inputs. Assuming P = 900 mm (SE = 100 mm) and ET = 600 mm (SE = 87 mm), then R = 300 mm with a model standard error of approximately ± 37 mm (or 12% of the estimated mean annual runoff), assuming a correlation coefficient between P and ET of 0.93. It is B-3
clear that the input accuracy of ET and P estimates is significant when undertaking water budget calculations. The overall usefulness of a given water balance analysis should therefore be evaluated with reference to the natural random variations inherent in the input data. Some knowledge of the quality control (QC) and quality assurance (QA) practices inherent in the underlying data collection programs is therefore useful when evaluating the overall accuracy of a water budget. B-4
B-2 Quality Assurance/Quality Control The quality assurance function (QA) includes organization of the monitoring system, planning, data collection, quality control, documentation, evaluation and reporting activities. The quality control function (QC) covers the routine technical activities such as error checks, and accuracy in the field, laboratory and office. Likewise, the combined QA/QC program ensures that the data collected is consistent, which enhances the credibility of the watershed management activities. Precision: refers to the degree of agreement among repeated measurements of a parameter. It defines consistency and reproducibility. From replicate samples, precision can be quantified by standard deviation, relative standard deviation or relative percent difference. Accuracy: refers to a measure of confidence in the measurements. In principle, the more precise or reproducible the data, the more accurate the results. Accuracy equates to average value minus true value. Representativeness: refers to measurements that reflect the true local environmental conditions. A number of factors or parameters may be used as indicators of conditions. Completeness: means the number of samples needed to complement the samples originally planned. Percent completeness is the number of planned samples judged to be valid to the total number of measurements. Comparability: is the evaluation of similar data, e.g. seasonal data collected at different periods. Measurement range: is the measurement range of an instrument or a measuring device. Detection Limit: is the lowest concentration of a given pollutant that equipment can detect and report as greater than zero. It should be noted that readings falling below the detection limit are not reliable. Spiked Samples: used to measure accuracy. These are samples containing a known concentration of the analyte of interest. When done in the field, it reflects the effects of preservation, shipping of samples to the laboratory, QA/QC Defines standards of quality and accuracy in data, and establishes a stated level of confidence. Essentials are precision, accuracy, representation, sensitivity, and confidence in the data collected. Standard Deviation S = [ (x x^) 2 /(n-1) ] 1/2 S = standard deviation x = sample x^ = average of samples n = number of samples Relative Standard Deviation (%): S R = (S/x^) 100 S R = relative standard deviation S = standard deviation x^ = average of samples Relative Percent Difference (%): R = (x 1 -x 2 )/(1/2(x 1 +x 2 ))100 R = relative percent difference x 1, x 2 are samples B-5
and laboratory preparation and analysis. When done in the laboratory, it reflects the effects of laboratory analysis. Replicate Samples: are used to detect the natural variability in the watershed and that caused by the field sampling. If two or more samples are taken from the same site at the same time using the same method, these represent replicate samples. Split Sample: used to measure precision. A sample that is divided into two or more samples, then analyzed by different analysts or laboratories. The field split measures both the field and analytical sampling precision, while the lab split measures the analytical precision. Field Blank: a clean sample produced in the field. It is used to detect analytical problems during transport and laboratory analysis. Equipment Blank: a clean sample used to check the cleanliness of the sampling equipment. B-6