Random and . . . Random Errors of . . . Random Errors: . . . Uncertainty in Eddy Sources of Random Error Random Errors: . . . Covariance Measurements: Systematic Error Systematic . . . An Overview Based on How to Use . . . a Recent Book Edited by Systematic Error: . . . Home Page Marc Aubinet, Timo Vesala, Title Page and Dario Papale ◭◭ ◮◮ ◭ ◮ Aline Jaimes, Jaime Nava, and Vladik Kreinovich Cyber-ShARE Center, University of Texas at El Paso Page 1 of 13 El Paso, Texas 79968, USA, contact vladik@utep.edu Go Back Full Screen Close Quit
Random and . . . 1. Gap Filling Random Errors of . . . Random Errors: . . . • Eddy covariance algorithms require that we have data Sources of Random Error from all moments of time. Random Errors: . . . • In practice: 20 to 60% of data points are faulty. Systematic Error Systematic . . . • Examples: wind measurements too high, or tempera- How to Use . . . tures > 20 C degrees away from average. Systematic Error: . . . • Ideal: get missing values from nearby meteostations. Home Page • If not possible: Title Page – use interpolations (linear or nonlinear, e.g., NN) ◭◭ ◮◮ from before and after the gap, ◭ ◮ – from observations on the day before and day after, Page 2 of 13 – use known relations between variables. Go Back • Result: for short gaps, reconstructed values have al- Full Screen most the same accuracy as others. Close Quit
Random and . . . 2. Random and Systematic Error Components Random Errors of . . . Random Errors: . . . • Measurement are never absolutely accurate; the result Sources of Random Error x is, in general, different from the actual value x . � Random Errors: . . . • Moreover, if we repeatedly measure the same quantity Systematic Error x , we get slightly different values � x 1 , . . . , � x n . Systematic . . . def How to Use . . . • Often, the measurement errors ∆ x i = � x i − are purely “random”, with equal prob. of ∆ x i > 0 and ∆ x i < 0. Systematic Error: . . . Home Page • In this case, in the long run, these errors compensate = � x 1 + . . . + � x n Title Page def each other, and we can estimate x as x n . n ◭◭ ◮◮ • The larger n , the more accurate this estimate x n . ◭ ◮ • Sometimes, instruments have bias; in this case, Page 3 of 13 def E [ x ] = lim x n � = x . Go Back def • The difference ∆ s x = E [ x ] − x is called systematic Full Screen def error , and ∆ r x = ∆ x − ∆ s x random error . Close Quit
Random and . . . 3. Random Errors of Eddy Covariance Measure- Random Errors of . . . ments: Empirical Analysis Random Errors: . . . Sources of Random Error • In some places, there are two towers nearby, at similar Random Errors: . . . places with practically the same carbon flux F . Systematic Error • In this case, the difference between estimates � F 1 − � F 2 Systematic . . . is caused only by the random error. How to Use . . . • So, we can get information about the random error by Systematic Error: . . . Home Page observing these differences. Title Page • First observation: differences are normally distributed. ◭◭ ◮◮ • Explanation: ◭ ◮ – In general: the measurement error is caused by Page 4 of 13 many independent factors. – Known: sum of a large number of small indep. ran- Go Back dom variables has an almost normal distribution. Full Screen Close Quit
Random and . . . 4. Empirical Analysis of Random Errors (cont-d) Random Errors of . . . Random Errors: . . . • Reminder: random error is normally distributed. Sources of Random Error • Fact: a normal distribution is uniquely determined by Random Errors: . . . its mean µ and standard deviation σ . Systematic Error Systematic . . . • By definition of a random error, its mean is 0, so it is sufficient to know σ . How to Use . . . Systematic Error: . . . • How σ depends on F ? Home Page • In each measurement: the observation result � F is slightly Title Page different from F . ◭◭ ◮◮ • We can expand the dependence of � F on F in Taylor ◭ ◮ series and ignore quadratic and higher order terms: Page 5 of 13 � F ≈ a 0 + a 1 · F. Go Back • Measurement error can be caused both by the a 0 -term Full Screen and by the a 1 -term. Close Quit
Random and . . . 5. Random Errors: Theoretical Analysis Random Errors of . . . Random Errors: . . . • Reminder: Measurement error can be caused both by Sources of Random Error the a 0 -term and by the a 1 -term in the dependence Random Errors: . . . � F ≈ a 0 + a 1 · F. Systematic Error Systematic . . . – the a 0 -component does not depend on F while How to Use . . . – the a 1 -component is proportional to F . Systematic Error: . . . • So, the random error has two components: Home Page Title Page – a component with σ = a for some a > 0, and – component with σ = b · | F | , for some b . ◭◭ ◮◮ ◭ ◮ • These components are caused by the imperfection of the same measuring instrument. Page 6 of 13 • So, they must be strongly correlated. Go Back • Hence, the overall error is σ ≈ a + b · | F | . Full Screen Close Quit
Random and . . . 6. Random Errors: Comparison with Empirical Random Errors of . . . Data Random Errors: . . . Sources of Random Error • Theoretically: the overall error σ depends on the flux Random Errors: . . . value F as Systematic Error σ ≈ a + b · | F | . Systematic . . . • In the first approximation: this is exactly what we ob- How to Use . . . serve. Systematic Error: . . . • The actual dependence is somewhat more complex. Home Page • Specifically: Title Page ◭◭ ◮◮ – when the flux F is small, – the observed error σ is smaller than the theoretical ◭ ◮ prediction a + b · | F | . Page 7 of 13 • It is not clear what causes this difference. Go Back • Explaining this empirical dependence is an interesting Full Screen challenge . Close Quit
Random and . . . 7. Sources of Random Error Random Errors of . . . Random Errors: . . . • Reminder: the computations of the flux are based on Sources of Random Error the theoretical model. Random Errors: . . . • Assumption: the parameters of the process do not change Systematic Error during the 30 minutes. Systematic . . . How to Use . . . • Resulting theoretical formula: uses integration over the values at all spatial locations. Systematic Error: . . . Home Page • In practice: we only have sensors at some locations. Title Page • As a result: instead of average over all the values, we ◭◭ ◮◮ only average a sample. ◭ ◮ • Similar case: to estimate the average weight, we only Page 8 of 13 use a sample. In both cases, we have a random error. Go Back • Another source: the footprint changes with time. Full Screen • Yet another source: sensors are not perfect. Close Quit
Random and . . . 8. Random Errors: Remaining Challenges Random Errors of . . . Random Errors: . . . • Fact: we do not observe the random error ∆ x i directly, Sources of Random Error we only observe the difference ∆ x 1 − ∆ x 2 . Random Errors: . . . • Good news: if the distribution was symmetric, we can Systematic Error still reconstruct it from observing the difference. Systematic . . . • Fact: in general, the distribution is not symmetric. How to Use . . . Systematic Error: . . . • Corollary: we cannot distinguish the distribution for Home Page ∆ x and for − ∆ x . Title Page • Question: what information can we recover? ◭◭ ◮◮ – for 1-D distribution, there is a huge uncertainty; ◭ ◮ – for 2-D case, e.g., for joint distribution of errors in two fluxes, the reconstruction is a.a. unique modulo Page 9 of 13 ∆ x → − ∆ x. Go Back • Another problem: reconstruction accuracy: skewness Full Screen of ∆ x only affect the 6-th moment of ∆ x 1 − ∆ x 2 . Close Quit
Random and . . . 9. Systematic Error Random Errors of . . . Random Errors: . . . • Two main sources of systematic error: Sources of Random Error – approximate character of the model used in data Random Errors: . . . processing, and Systematic Error – systematic error of sensors. Systematic . . . • Main source of model inaccuracy: non-turbulent fluxes How to Use . . . are ignored. Systematic Error: . . . Home Page • In reality: non-turbulent fluxes are not negligible, es- pecially at night, when turbulence is lower. Title Page ◭◭ ◮◮ • First idea: estimate the non-turbulent flux component. ◭ ◮ • Problem: the Eddy covariance tower is not well suited for such estimation. Page 10 of 13 • Result: the “corrected” flux is less accurate than the Go Back original one. Full Screen • Better solution: ignore periods when turbulence is low. Close Quit
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