Why gap filling isn’t always easy

Andrew RichardsonUniversity of New Hampshire

Jena Gap Filling WorkshopSeptember 2006

Three reasons

1. Choice of model or technique

2. Random errors in the data

3. Long gaps and non-stationarity

Choice of model

• Some models will fit better than others– better representation of process?– evaluate with error statistics, cost function, AIC

• Model predictions may differ depending on model (so gap-filled values will differ)

• Separation of NEE into respiration and photosynthesis depends on model chosen (potential for compensating errors even if NEE is unchanged)

A simple gap-filling method

• Nocturnal data (PPFD < 5 mol m-2 s-1):NEEnight = Reco

• Model Reco as a second-order Fourier:Reco = f0 + s1 sin(D) + c1 cos(D)

+ s2 sin(2 D) + c2 cos(2 D)

• Requires no ancillary data or driving variables (air/soil temperature, soil moisture, etc.)

• Works well in temperate systems: – Assumes seasonal cycle is dominant – Fits better than a first order Fourier (but not

excessively over-parameterized)

Examples

Temperate: good representation of seasonal patterns

FI1_2001 IT3_2002

Mediterranean: Poor representation of seasonal patterns (a higher order Fourier is needed)

• Daytime (PPFD ≥ 5 mol m-2 s-1):

NEEday = Reco + Pgross Pgross = Amax (PPFD/PPFD+Km)

• Michaelis-Menten model: Amax is light saturated rate of canopy photosynthesis, Km is the half-saturation constant

• Fit Amax and Km parameters at the monthly time step

daynet

- 30

- 20

- 10

0

10

PPFD

0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600

amax

0

- 10

- 20

- 30

mo

1 2 3 4 5 6 7 8 9 10 11 12

FI1_2001: July FI1_2001: seasonal variation in Amax

Why model selection matters

Multi-site comparison and ranking of simple respiration models (Richardson et al., Ag. & Forest Met., in press): models that fit the best also predicted the most Reco

Grassland example shown

Same pattern seen in deciduous (Harvard) and coniferous (Howland) forests as well.

Will affect partitioning of NEE to Reco and Pgross.

Why?

Soil temperatures around 0º C exert a large influence on model fit, but make only a small contribution to the modeled annual sum of respiration. Warm soil temperatures exert only a small influence on model fit, but account for most of the annual respiration, since respiration is an increasing function of temperature.

Random errors

• Uncertainty in model parameterization

• Place an upper bound on agreement between measurements and model

• Error propagation: zero mean but non-zero variance over the course of the year (approximately ± 25 g C m-2 y-1 at 95% confidence)

• Non-Gaussian and non-constant variance: Affects choice of cost function (maximum likelihood vs. least squares), which may result in a different parameterization

Non-stationarity

• Ecosystem properties change over time

• If there is a long gap, we don’t really know what those changes might have been

• The longer the gap is, the greater the potential changes—especially when the ecosystem is changing rapidly

• Question: How much uncertainty do long gaps add?

Long Gap Experiment

• Used synthetic data set • Systematic insertion of gaps (i between 1

and 28 days in length) beginning on each day of year (j between 1 and 365)

• Random insertion of additional small (half-hourly) gaps

• Total missing observations ~ 30%• Compared annual sum of NEE for gap-

filled data with and without the large gap

Results: HowlandDepending on the particular start date, gap-filled NEE

could vary by up to ± 75 g C m-2 y-1 for the 28-day gaps.

The standard deviation of this variation, (NEE), differed among months, and in relation to gap length.

Generalizing the results: DeciduousUncertainty CO2 Flux

Generalizing the results: MediterraneanUncertainty CO2 Flux

Application• Generate a table of the slope of the relationship between

(NEE) and gap length

• Process real data sets to find out the length and start date of each gap

• Estimate total additional uncertainty that can be attributed to long gaps

Month IT3_02 FR4_02 BE1_00 BE1_01 FR1_01 DE3_00 Howland1 0.19 0.12 0.20 0.24 0.09 0.02 0.112 0.20 0.25 0.30 0.40 0.07 0.03 0.043 0.59 0.35 0.59 0.34 0.09 0.32 0.574 0.57 0.33 0.30 0.48 1.15 1.00 1.185 0.68 0.64 0.65 0.82 2.11 1.65 0.896 1.25 1.08 0.65 0.71 0.90 0.98 0.987 0.85 0.70 0.41 0.64 0.95 0.65 0.688 0.23 0.33 0.33 0.83 2.09 1.23 0.379 0.53 0.16 0.53 0.33 0.75 1.10 0.33

10 0.48 0.19 0.36 0.38 0.59 1.15 0.3211 0.59 0.21 0.26 0.27 0.35 0.12 0.2612 0.24 0.12 0.11 0.23 0.05 0.11 0.26

Results

95% CILongest

Gap 95% CILongest

GapHowland 96 ±28.0 10 IT3_01 ±39.0 101Howland 97 ±19.3 15 IT3_02 ±13.0 5Howland 98 ±24.0 8 FR4_02 ±9.7 4Howland 99 ±28.7 20 BE1_01 ±13.4 8Howland 00 ±11.1 7 BE1_00 ±12.3 8Howland 01 ±7.8 5 FR1_01 ±10.0 6Howland 02 ±10.9 7 FR1_02 ±8.4 10Howland 03 ±10.5 6 DE3_00 ±19.1 8Howland 04 ±9.9 6 DE3_01 ±35.2 14

• Over the nine year record at Howland, long gaps add ±10 ~ ±30 g C m-2 y-1 additional uncertainty to the annual gap-filled NEE. Results comparable at other sites.

• Additional uncertainty due to long gaps can be comparable to that due to random measurement errors in half-hourly fluxes… but long gaps are avoidable!

Take-home message

• Method provides a mechanism for quantifying uncertainty due to long gaps.

• Patterns are robust across similar sites (i.e., within vegetation types):– Uncertainty increases most quickly with gap length

when system is changing rapidly (e.g., spring in temperate forests)

– Uncertainties are smaller when fluxes are smaller (e.g. mediterranean systems vs. deciduous forests).

Summary

• Choice of model or gap-filling method is important

• Random errors in measurements cause uncertainty in gap-filled NEE

• Long gaps (especially at the wrong time of year) are to be avoided at all costs because of non-stationarity in ecosystem function.