Error is the difference between reality and our representation of it (Unwin 1995).

Transcription

1 9. Analysis a. Analysis tools for dam removal vi. Estimating Measrement Error 1.0 Rationale For most dam removal or restoration monitoring projects, the emphasis is on changes in attribtes, not the vale of the attribtes themselves. Inclsion of estimates of ncertainty is essential to certifying change. Hoever, the vales of attribtes may have inherent meaning, sch as spaning gravel sies. In that case, qantifying the degree of ncertainty in measrements is valable to establish hether or not the vale trly falls ithin defined favorable verss nfavorable ranges. Withot inclsion of ncertainty, any comparison of data lacks contet. There is no sch thing as a perfect measrement (Coleman and Steele 1999). There are to types of error to consider: random and systematic. Random error is likely to be different for every measrement and can be attribted to the hman component: interpolation, reaction time, etc. (Taylor 1997). Systematic error is consistent for a grop of measrements taken the same ay and it is sally a fnction of eqipment: offsets de to temperatre sensitivity, miscalibration, etc. Different approaches need to be tilied to calclate the to different kinds of error..0 Definitions Error is the difference beteen reality and or representation of it (Unin 1995). Accracy is the closeness of reslts to vales that are accepted to be tre (Coleman and Steele 1999; Unin 1995). Precision/Random error is the difference beteen an observation and the average of a nmber of observations (Coleman and Steele 1999; Taylor 1997). Random errors lead to a normal or Gassian freqency distribtion of vales arond the average of a nmber of observations (Coleman and Steele 1999; Taylor 1997). For eample, hen reading a thermometer or tape measre, random error is introdced by the need to interpolate vales beteen markings and from the angle at hich the thermometer or tape measre is read. Or, if a stopatch is sed to time a velocity measrement, random error is introdced by the reaction time of the person stopping and starting the stopatch reslting in slightly longer or shorter times. Bias/Fied/Systematic error is the difference beteen the average of a nmber of observations and the tre vale (Coleman and Steele 1999; Taylor 1997). Systematic error is consistent for all measrements of the same qantity taken the same ay, and, therefore, cannot be detected by repeated measres (Coleman and Steele 1999; Taylor 1997). For eample, if a thermometer sed to measre ater temperatre as miscalibrated, it read 10 C in ice ater, then all measrements made ith the thermometer old have a systematic error of +10 C. Similarly, if a net sed for macroinvertebrate samples had a small tear, there old be an nknon systematic error in the nmber of individals, and types and nmber of species sampled de to loss of some individals smaller than the tear.

2 Independence: Measrement errors are independent if the error associated ith one measrement does not inflence the error for another measrement i.e. ftre errors cannot be predicted by prior errors (Taylor 1997). Random errors, by definition, are independent and systematic errors are not. Error propagation is the transference of errors from a measred qantity to calclated reslt (Rabinovich 000). For eample, area is typically calclated as the prodct of idth and depth. Therefore, the error associated ith the calclated area is a combination of the error in the measrements of idth and depth. 3.0 Methodology 3.1 Error estimation There are three common methods sed to estimate measrement error: 1) Repetition of measrements; ) Comparison to reslts from measrements made sing a different method; 3) Pblished or previosly established vales Repetition One of the most poplar ays to measre error for individal measrements is throgh repetition. To make this methodology sccessfl, one mst ensre that the same qantity is being measred each time, and systematic errors mst be small (Taylor (1997) defines this as smaller than reqired precision ). This can range in effort from a handfl of repetitions to statistically rigoros qantities: sample sies smaller than 31 reslt in nderestimates of ncertainty sing the sample standard deviation for a normal distribtion nless the Stdent s t statistic is sed as the mltiplier (Dieck 006). There are three main ays to calclate and report error from repeated measrements: 1) Maimm difference; ) Standard deviation; or 3) Probable error (PE) (Hbbard and Glasser 005). If there are fe repetitions, typically the error estimate (±) sed is half the maimm difference beteen measrements, e.g. largest mins smallest divided by (Donard 1995; Hbbard and Glasser 005). For nmeros repeated measrements ith random error and minimal systematic error, a normal or Gassian distribtion can typically be sed as a good approimation (Taylor 1997). To check if the assmption of a normal distribtion is acceptable, a Chi-sqared test can be performed (see tets sch as Taylor 1997 for details). By assming a knon freqency distribtion, e can find the range (the mean ± the ncertainty) ithin hich measred vales ill fall ith a desired probability as a mltiple of the standard deviation (Taylor 1997). For eample, for a normal distribtion there is a probability of approimately 95 % that an individal measrement is ithin standard deviations of the tre vale (Taylor 1997). 1 Sample standard deviation: s ( ) N 1 i An alternative ay to report ncertainty is as the probable error (PE): there is a 50 % probability of a measrement being ithin ± PE of the tre vale (Taylor, 1997). The probable error is approimately eqal to 0.67 times the standard deviation (Taylor, 1997).

3 Eamples from the literatre: Donard (1995) delineated channel bondaries from aerial photographs and maps in GIS. In order to establish the ncertainty in digitiing those bondaries, Donard (1995) chose a bondary location from one map to repeatedly digitie. Slithers ere created beteen a control bondary line and each digitied bondary line; error as calclated as the standard deviation for the assmed normal freqency distribtion of mean individal slither displacement (slither area/arc length) (Donard 1995) Comparison: Another method sed to estimate error is comparison to reslts from a more accrate or better established techniqe. Systematic and random errors are measred together sing this techniqe. This is a common method in remote sensing here the coordinates for locations on images or photos are checked against maps or images of the same landscape ith knon errors. In the case of sediment sampling, volmetric samples are typically the standard, so calclated vales from other measrement techniqes can be compared to reslts from volmetric sampling, sch as blk samples (Fripp and Diplas 1993). Eamples from the literatre: Simonson and Lyons (1995) compared vales for species abndance, richness, and assemblage strctre for 9 small (< 8 m ide) streams in Wisconsin from a single catch per effort (CPE) to-barge electrofishing pass verss mltiple electrofishing to-barge passes ith block-nets and removal (REM). Both types of sampling ere performed on each stream, ith CPE pstream of REM sampling for 4 of the streams, and REM pstream of CPE for 5 for the streams (Simonson and Lyons 1995). The Wilcoon s signed-rank test, Spearman s rank correlation, and a similarity inde ere sed to compare vales calclated from the CPE sampling and the removal sampling (Simonson and Lyons 1995) Established vales: Finally, there may already be error estimates for standard methods or eqipment. In that case, as long as the same protocols are observed and conditions apply, finding and sing pblished error estimates may be the best se of time and resorces. Eample Sorces of Error Estimates Category Methodology Citation Sediment Pebble Cont (Fripp and Diplas 1993; Green 003; Wohl et al. 1996; Wolman 1954) Blk and Freee Core Sampling (Fergson and Paola 1997; Zimmermann et al. 005) Geomorphology GPS and DEM (Brasington et al. 000; Cheng and Granata 007)

4 Total station (Donard 1995) Macroinvertebrates Srber (Li et al. 001) Physical Habitat USFS: AREMP, PIBO; (Whitacre et al. 007) EPA: EMAP Fish Electrofishing (Reid et al. 009; Rosenberger and Dnham 005; Simonson and Lyons 1995) Hydrology Stage, Discharge, Rating crves (Carter and Anderson 1963; Rant and others 198; Saer and Meyer 199) 3. Error propagation Once individal errors for measrements are estimated, there is the isse of error propagation. Error propagation is the combination of mltiple types of error for a given end vale. The maimm error for a qantity is alays the sm of the individal errors in the orst case scenario here all errors lead to either overestimation or nderestimation of the qantity of interest (Taylor 1997). If errors are independent and random, there is a 50 % probability that the individal errors ill be opposite (some nderestimating and some overestimating) and at least partially cancel each other (Taylor 1997). Therefore, the combined error is the qadratic sm of the individal errors (Taylor 1997). Modified from Taylor (1997): If a qantity, q, is the sm and/or difference of mltiple terms: q ( ), then the ncertainty in the vale of q, δq, is: q ( ) ( )... ( )... ( ) If a qantity, q, is the prodct and/or qotient of mltiple terms: q, then: q q Eample: Error in channel idth from coordinates If channel idth is measred from coordinates, taken for eample ith a total station, instead of ith a tape measre, then the error in the channel idth is a fnction of the error in the coordinates for each end point. To establish the random error in each end point (assming systematic error is either smaller than precision or consistent for both end points), the coordinates for each point ere taken 3 times. Assming a normal freqency distribtion, the

5 error in and y coordinates for each end point can be calclated as the standard deviations for each. End point Error/standard deviation in (δ) Error/standard deviation in y (δy) River left (RL) 0.15 m 0.09 River right (RR) 0.1 m 0.1 Since the random errors hich create a freqency distribtion of and y coordinates for the end points are de largely to decision-making and rod placement by the srveyor, they can be assmed to be random and independent for each direction and each end point. Based on that assmption, the qadratic sm of the individal errors for each direction and both endpoints can be calclated as the total error for the channel idth. RL RR Error in (δ) = (verss 0.4 for the simple sm) RL y RR Error in y (δy) = y (verss 0. for the simple sm) Error in channel idth (δ) = y m (verss 0.4 for simple sm) 4.0 Additional Information Taylor (1997) has a very coherent and applicable tetbook on error analysis for physical measrements: An introdction to error analysis: the stdy of ncertainties in physical measrements. Sasalito, Calif., University Science Books. The International Organiation for Standardiation (1995) has created the Gide to the Epression of Uncertainty in Measrement hich has the official definitions of terms, orldide standard for methods to determine and epress ncertainty, and detailed eplanations of the methods themselves, the reasons for the methods, and eamples of the methods being pt to se. 5.0 References Brasington, J., Rmsby, B. T., and McVey, R. A. (000). "Monitoring and modelling morphological change in a braided gravel-bed river sing high resoltion GPS-based srvey." Earth Srface Processes and Landforms, 5(9), Carter, R. W., and Anderson, I. E. (1963). "Accracy of crrent-meter measrements." American Society of Civil Engineers Jornal, Cheng, F., and Granata, T. (007). "Sediment transport and channel adjstments associated ith dam removal: Field observations." Water Resorces Research, 43(W03444), 14. Coleman, H. W., and Steele, W. G. (1999). Eperimentation and ncertainty analysis for engineers, Wiley, Ne York. Dieck, R. H. (006). Measrement ncertainty: methods and applications, Instrment Society of America, Research Triangle Park, N.C.

6 Donard, S. R. (1995). "Information from topographic srvey." Changing river channels, A. M. Grnell and G. E. Petts, eds., Wiley, Ne York, Fergson, R. I., and Paola, C. (1997). "Bias and precision of percentiles of blk grain sie distribtions." Earth Srface Processes and Landforms, (11), Fripp, J. B., and Diplas, P. (1993). "Srface sampling in gravel streams." Jornal of Hydralic Engineering, 119(4), Green, J. C. (003). "The precision of sampling grain-sie percentiles sing the Wolman method." Earth Srface Processes and Landforms, 8(9), Hbbard, B., and Glasser, N. F. (005). Field techniqes in glaciology and glacial geomorphology, John Wiley & Sons, Chichester, West Ssse, England ; Hoboken, NJ. International Organiation for Standardiation. (1995). Gide to the epression of ncertainty in measrement, International Organiation for Standardiation, Siterland. Li, J., Herlihy, A., Gerth, W., Kafmann, P., Gregory, S., Urqhart, S., and Larsen, D. P. (001). "Variability in stream macroinvertebrates at mltiple spatial scales." Freshater Biology, 46(1), Rabinovich, S. G. (000). Measrement errors and ncertainties: theory and practice, AIP Press, Ne York. Rant, S. E., and others. (198). "Measrement and comptation of streamflo: volme 1. Measrement of stage and discharge." 175, United States Geological Srvey, Washington, D. C. Reid, S. M., Ynker, G., and Jones, N. E. (009). "Evalation of single-pass backpack electric fishing for stream fish commnity monitoring." Fisheries Management and Ecology, 16(1), 1-9. Rosenberger, A. E., and Dnham, J. B. (005). "Validation of abndance estimates from mark recaptre and removal techniqes for rainbo trot captred by electrofishing in small streams." North American Jornal of Fisheries Management, 5(4), Saer, V. B., and Meyer, R. W. (199). "Determination of error in individal discharge measrements." U. S. Geological Srvey, Norcross, Georgia. Simonson, T. D., and Lyons, J. (1995). "Comparison of catch per effort and removal procedres for sampling stream fish assemblages." North American Jornal of Fisheries Management, 15(), Taylor, J. R. (1997). An introdction to error analysis: the stdy of ncertainties in physical measrements, University Science Books, Sasalito, Calif. Unin, D. J. (1995). "Geographical information systems and the problem of 'error and ncertainty'." Progress in Hman Geography, 19(4), Whitacre, H. W., Roper, B. B., and Kershner, J. L. (007). "A comparison of protocols and observer precision for measring physical stream attribtes." Jornal of the American Water Resorces Association, 43(4), Wohl, E. E., Anthony, D. J., Madsen, S. W., and Thompson, D. M. (1996). "A comparison of srface sampling methods for coarse flvial sediments." Water Resorces Research, 3(10), Wolman, M. G. (1954). "A method of sampling coarse river-bed material." Transactions, American Geophysical Union, 35(6), Zimmermann, A., Colombe-Pontbriand, M., and Lapointe, M. (005). "Biases of sbmerged blk and freee-core samples." Earth Srface Processes and Landforms, 30(11),