June 4-15, 2007 Boulder, CO National Center for Atmospheric Research Advanced Study Program Carbon and water flux measurements and applications to regional biogeochemistry - Eddy covariance revisited David Hollinger USDA Forest Service Northern Research Station Durham, NH USA NCAR ASP 2007 Colloquium Regional Biogeochemistry: Needs and Methodologies

1.There’s always a need for top quality data ! 2.To use the data effectively, must understand the assumptions Why ?

Organization: Turbulence and Eddies Eddy Covariance Methodology –History (Baldocchi 2003, GCB 9:479) –Theory –Practice –Using other people’s data – how do you know the data are any good? Uncertainty Crucial assumptions Methods Key problems

Philosophy: All models are wrong –Useful simplifications All data are corrupt –Noise (random component) –Bias (fixed component, compared to an independent measure) –Variable bias But, we need to move forward…

Advantages of eddy flux data for evaluating and parameterizing models Same or finer timestep as models Simultaneous measurements of CO 2, water, & energy fluxes Environmental data also recorded –PFD, temperature, soil moisture, etc. Often abundant site data –LAI, leaf %N, C pool data, etc. Area averaging, whole ecosystem net flux

Eddy flux data useful for the study of biogeochemical processes are collected worldwide and readily available 294 active FLUXNET sites worldwide 57% with woody vegetation Available data on-line: –AmeriFlux380 site-years –EUROFLUX51 site-years a –Fluxnet-Canada, Asiaflux, Ozflux, etc. In a few years we’ve gone from a data poor to a data rich environment. How do we get to a knowledge rich environment? a. Additional data available to network participants or by special arrangement

“…an irregular condition of flow in which the various quantities show a random variation with time and space coordinates, so that statistically distinct average values can be discerned” (Hinze, 1975) Turbulence can be visualized as eddies (local swirling motions). Over forests, eddies may form as coherent “transverse rollers” (Raupach et al. 1996) Turbulence and Eddies Raupach, M. et al Boundary-Layer Met. 78:

Eddies are carried by the mean flow (turbulence is not local) Eddies overlap in space, with large eddies carrying smaller eddies. Turbulent kinetic energy transfers from large eddies to smaller eddies in a cascading process and is dissipated as heat energy once molecular viscosity becomes important. Turbulence and Eddies Finnigan, J Turbulence in plant canopies. Annual Review of Fluid Mechanics 32:

“Big whorls have little whorls that feed on their velocity, And little whorls have lesser whorls, and so on to viscosity.” -- Lewis F. Richardson The supply of energy from and to Atmospheric Eddies' (1920) Proceeding of the Royal Society A Great fleas have little fleas upon their backs to bite 'em, And little fleas have lesser fleas, and so ad infinitum. And the great fleas themselves, in turn, have greater fleas to go on; While these again have greater still, and greater still, and so on. -- Augustus De Morgan

The size of an eddy (  ) is its “turbulent length scale”, above a forest canopy this may be 10’s-1000’s of meters For Monin-Obukhov (surface layer) scaling,   z/u z is measurement height, u is mean windspeed (Bussinger et al. 1971, J. Atmos. Sci 28:181; Wyngaard et al. 1971, J. Atmos. Sci 28:1171) For Raupach-Finnigan (mixing layer) scaling,   h/u h is vegetation height Turbulence and Eddies Finnigan, J Turbulence in plant canopies. Annual Review of Fluid Mechanics 32:

Population >> ’s – 100’s of meters Turbulence integrates sources/sinks and connects them to the measurement system: Transport uncertainty Measurement uncertainty + C

Eddy Method: Simplification of the Continuity Equation Fluxes consist of mean and fluctuating components (Reynold’s decomposition) The mean components (advection) = 0 The only relevant fluctuating component is with the vertical wind Baldocchi et al. (1988) Ecology 69: Aubinet et al. (2000) Adv. Ecol. Res., 30: Massman and Lee (2002) Ag and For Met 113:121-44

x z u is horizontal wind, w is vertical wind Overbar signifies time average, primes deviations from mean

x z u is horizontal wind, w is vertical wind Overbar signifies time average, primes deviations from mean Crucial assumption: F = eddy flux +  storage

The covariance F = w’c’ Cov(w,c) = E{[ w - E(w) ][ c - E(c) ]} –E() expected value (mean) –Insensitive to uncorrelated noise

Cospectra show the distribution of the flux in the frequency domain: Tall tower, low wind Short tower, high wind

Methods (Massman & Lee 2002, Ag. For. Met. 113:121. Papale et al. 2006, Biogeosciences 3:571) Measure vertical wind & scalar (lag scalar) No net w; rotate coordinates as necessary Correct for spectral deficiencies in measurement Correct for density fluctuations (WPL, self- heating) Reject bad data (variances too high/low, bad calibrations,u* threshold, etc.) Fill in gaps for annual sums Software: –MacMillen (ATDD), EddySol, EddyRe, TK2, etc.

ATI & Gill

Crucial Assumption – Site: No horizontal or vertical advection Homogeneous surface  roughness  exchange characteristics Flat Extensive  old rule: 100h No local sources of CO 2  avoid buildings and roads

z T z T z T Unstable - good Stable - bad Neutral - ok Meteorological conditions affect turbulence: a u* threshold ensures there will be sufficient mixing

Crucial Assumptions - Meteorology Turbulence is present  no measurement without transport - u* (mixing) threshold (Goulden, 1996) Conditions are “stationary” (statistical properties not changing or “not changing too quickly”) –Stationarity Tests (Foken & Wichura 1996, Ag For Met 78:83) C

Measurement Diagnostics Does the energy balance? R n = H + LE + G +  S Is there flux divergence? Are fluxes the same at different heights? Do the power spectra and cospectra look o.k.? (raw data) Does the flux plateau with u*? Is the intercept of the light response similar to the mean nocturnal respiration.

NEE = P max I / (K m +I) + R d Compare nocturnal data with parameters estimated from daytime data:

Another solution: A re the data any good? Use data from long established sites. See what sites keep appearing in synthesis papers. Avoid sites with “mountain”, “ridge”, etc. in the name.

All measurements are corrupted by random error: We want the flux, F, but we measure F +  +  where  is the random error and  is a bias The random error is characterized by its PDF; the first moment (  ) referred to as “uncertainty” What is uncertainty and why do we care? Uncertainty information is required to properly fit models to data; e.g. for estimating the parameters of canopy processes models.  Maximum Likelihood  Data assimilation (Kalman filter)  Bayesian methods Uncertainty useful for assessing model fits, annual uncertainties, risk analysis, etc.

Maximum likelihood – “given the data, what are the most likely model coefficients (parameters)?” Determined by minimizing the difference between data and model via a likelihood (cost) function: The likelihood function requires uncertainty information (  ) from the data For Gaussian data For double exponential data Where do we get uncertainty info? What is uncertainty info?

"To put the point provocatively, providing data and allowing another researcher to provide the uncertainty is indistinguishable from allowing the second researcher to make up the data in the first place." –Raupach et al. (2005). Model data synthesis in terrestrial carbon observation: methods, data requirements and data uncertainty specifications. Global Change Biology 11:

a double exponential PDF better represents the random error distribution of eddy fluxes Flux Uncertainties are non- Gaussian with non- constant variance Simultaneous measurements at 2 towers (Howland) 1 Single tower next day comparisons (Howland, Harvard, Duke, Lethbridge, WLEF, Nebraska) 2 Data-model residuals 3 1.Hollinger & Richardson (2005) Tree Physiology 25: Richardson et al. (2006) Ag. Forest Met. 136: Hagen et al. (2006), Journal of Geophysical Research 111, D08S03.

Richardson et al. (2006) Ag. Forest Met. 136: Flux uncertainty increases linearly with flux magnitude ~proportional to flux (for least squares uncertainty is constant) Forests F C > 0  = *F C F C < 0  = *F C

Maximum likelihood – “given the data, what are the most likely model coefficients (parameters)?” Probability Determined by minimizing the difference between data and model via a likelihood function: The likelihood function depends on the uncertainty characteristics (PDF) of the data For Gaussian data For double exponential data

Choices affect the outcome: Weighting of data model deviations R d = R ref e (  /(T-T0)) Richardson & Hollinger 2005 Ag. For Met 131:191

Inverse modeling - choosing the “best” respiration model (Richardson et al. 2006) Linear Q 10 temperature varying Q 10 time varying Q 10 Arrhenius Lloyd and Taylor Lloyd and Taylor (restr) Fourier Neural Networks

Harvard Howland Howland below canopy Howland soil Ranking respiration models based on eddy flux or automated chamber data Results: Q 10 and restricted Lloyd & Taylor performed poorly

The better the model fit, the higher the estimated annual respiration – bias from using different models?

Cost functions? There can be many different cost functions, each yielding different parameters and integral estimates Be humble

Words to remember with Eddy Covariance? Be aware of the assumptions –Turbulence present (stationary) –No advection Reject bad data - Use good data

A Problem with openpath sensors: Large corrections are necessary due to density effects of sensible & latent heat fluxes a FCO 2 is thus affected by measurement error and uncertainty in H and LE Corrections for self heating also necessary a. Webb, E.K., G.I. Pearman, and R. Leuning Correction of flux measurements for density effects due to heat and water vapor transfer. Quart. J. Met. Soc. 106:

WPL (density) Corrections: Self heating Corrections: (Burba et al. AMS 2006)

X X NEE = + Inverse modeling: A simple canopy model example driven by PFD, (Tair +Tsoil)/2, &  W A max –maximum PS K m – half saturation T opt – temperature optimum  – temp shape factor  –  W shape factor R a – base respiration E 0 – activation energy All 2004 flux data from Howland & Bartlett

data uncertainty By guided trial and error, find the parameters that minimize the sum of the weighted deviations between the model and measurements (    The time to do this is linear in N and factorial in parameter number

Example results: The 95% confidence intervals of the parameter sets for Howland and Bartlett are far apart

Functional responses for Howland and Bartlett based on most likely parameter estimates are distinct

Determining parameter covariances help to understand model behavior: Howland Bartlett

Determining measurement uncertainty by multiple measurements of the same thing Compare independent measurements of the same thing (x and y) - The surface must be homogenous Calculate mean and standard deviation of the difference between the towers: q i = x i - y i (~ 0)  q  0 If x and y are independent and have the same random measurement uncertainty: Howland x1x1 x2x2 y2y2 y1y1

Openpath gas analyzers have superior high frequency response and thus smaller corrections:

Corrections have Implications for energy balance closure & u* corrections!

How do we minimize the likelihood function? Traditional non-linear approaches (LM) –Available in SAS and other statistical packages –Often fail if parameters poorly constrained or model contains local minima If functions have local minima, we need a global method –Available in MATLAB or program form –Simulated annealing (SIMANN, Goffe et al ) –Genetic algorithms (SRES, Runarrson & Yao 2000) (See Press et al. “Numerical recipes”, chapters 10 & 15)

Independent data sets from separate towers yielded similar parameter variation:

Simple Respiration models: Fourier series  Unbiased, good fit  No need for other data  Not mechanistic Lloyd and Taylor  Possibly biased, good fit  Mechanistic  Need complete temper- ature data Lloyd and Taylor (1994) Functional Ecology 8:315-23

History of Model Inversions with flux data First to use eddy flux data to determine parameters of canopy models –Wang et al. (2001) 10 parameters (Leuning & Wang 1998), 3 weeks data (pasture) X 6 sites 3-4 parameters constrained –Schulz et al. (2001) 21 parameters, 2 weeks flux data (wheat), 3 parameters Bayesian inversions – prior distribution specified –Braswell et al. (2005) SIPnET model, 23 parameters, 10 years day/night data Harvard Constrained 12 parameters, 7 edge hitting –Knorr and Kattge (2005) Bethy model, 24 parameters, 7 days of forest flux data Constrained 5 parameters, evaluated parameter covariances