Characterization of ecosystem responses to climatic controls using artificial neural networks

Characterization of ecosystem responses to climaticcontrols using artificial neural networksA N T J E M . M O F FA T *, C L E M E N S B E C K S T E I N w , G A L I N A C H U R K I N A z, M A R T I N A M U N D *

and M A R T I N H E I M A N N *

*Max Planck Institute for Biogeochemistry, Hans-Knoll-Str. 10, 07745 Jena, Germany, wDepartment of Mathematics and Computer

Science, Friedrich Schiller University, Ernst-Abbe-Platz 1-4, 07743 Jena, Germany, zLeibniz-Centre for Agricultural Landscape

Research (ZALF), Eberswalder Strasse 84, 15374 Muncheberg, Germany

Abstract

Understanding and modeling ecosystem responses to their climatic controls is one of the major challenges for

predicting the effects of global change. Usually, the responses are implemented in models as parameterized functional

relationships of a fixed type. In contrast, the inductive approach presented here based on artificial neural networks

(ANNs) allows the relationships to be extracted directly from the data. It has been developed to explore large,

fragmentary, noisy, and multidimensional datasets, such as the carbon fluxes measured at the ecosystem level with the

eddy covariance technique. To illustrate this, our approach has been systematically applied to the daytime carbon flux

dataset of the deciduous broadleaf forest Hainich in Germany. The total explainable variability of the half-hourly

carbon fluxes from the driving climatic variables was 93.1%, showing the excellent data mining capability of the

ANNs. Total photosynthetic photon flux density was identified as the dominant control of the daytime response,

followed by the diffuse radiation. The vapor pressure deficit was the most important nonradiative control. From the

ANNs, we were also able to deduce and visualize the dependencies and sensitivities of the response to its climatic

controls. With respect to diffuse radiation, the daytime carbon response showed no saturation and the light use

efficiency was three times greater for diffuse compared with direct radiation. However, with less potential radiation

reaching the forest, the overall effect of diffuse radiation was slightly negative. The optimum uptake of carbon

occurred at diffuse fractions between 30% and 40%. By identifying the hierarchy of the climatic controls of the

ecosystem response as well as their multidimensional functional relationships, our inductive approach offers a direct

interface to the data. This provides instant insight in the underlying ecosystem physiology and links the observational

relationships to their representation in the modeling world.

Keywords: artificial neural networks (ANNs), climatic controls, ecological data mining, ecosystem physiology, eddy covariance

carbon flux, FLUXNET, Hainich forest, inductive modeling

Received 20 August 2009 and accepted 8 November 2009

Introduction

The change of the earth’s climate strongly affects terres-

trial biological ecosystems (IPCC, 2007a), but the response

of the ecosystems to the changing environmental condi-

tions is largely unknown. Even basic phenomena are still

under debate: The observed net uptake of CO2 by the land

biosphere implies an unexplained large, increasing land

sink, also called missing sink or residual land sink

(Burgermeister, 2007; IPCC, 2007b). For the Northern

Hemisphere, the average estimate of the land carbon sink

from atmospheric inversions is almost a factor of two

larger than the bottom-up estimate, and the longitudinal

partitioning of the northern sink is subject to large un-

certainties (IPCC, 2007b). Furthermore, it is now recog-

nized that biological processes influence the climate of the

earth system significantly (Heimann & Reichstein, 2008).

Therefore, understanding the climatic controls of the

ecosystem response is fundamental and essential in the

context of global change.

To tackle this question, towers equipped with the

eddy covariance technique have been established, and

these are measuring the carbon flux in a wide range of

vegetation types and climate zones all over the world

(Baldocchi, 2008). The flux measurements have a high

temporal resolution of half-hourly to hourly, but, due to

the limitations of the eddy covariance technique, they

are fragmentary and noisy (Papale et al., 2006). Main

limitations are the theoretical requirement of stationar-

ity of the flow, turbulent atmospheric conditions, and

no residual vertical wind speed or horizontal advection,Correspondence: Antje M. Moffat, tel. 1 49 3641 576220, fax 1 49

3641 577200, e-mail: [email protected]

Global Change Biology (2010) 16, 2737–2749, doi: 10.1111/j.1365-2486.2010.02171.x

r 2010 Blackwell Publishing Ltd 2737

mailto:[email protected]

but also varying source areas of the fluxes in hetero-

geneous environments (Goeckede et al., 2004).

In contrast to controlled lab experiments, the ecosys-

tem response is driven by external weather conditions.

To capture the climatic controls, there are concurrent

measurements of a wide range of meteorological vari-

ables, such as radiation, temperature, and humidity.

This results in large, complex, and multidimensional

datasets from which the causalities cannot be obtained

just by visual evaluation of the measurements. There-

fore, additional modeling is required.

Two basic modeling approaches can be distinguished:

the hypothetic-deductive and the inductive (Hempel &

Oppenheim, 1948; Young & Jarvis, 2002). The hypothetic-

deductive approach (Fig. 1, top) begins with hypotheses

about how the controls in the ecosystem work. The

controlling processes are then implemented in an ecosys-

tem model as parameterized equations (deduction). The

carbon flux datasets are used to constrain the parameters

and to test the validity of the model. A good agreement of

the model’s predictions with the measurements is

assumed to corroborate the hypotheses.

This paper presents a fully inductive approach (Fig. 1,

bottom), where a priori assumptions are avoided as

much as possible. It is based on a purely empirical

model with a very general function class, here artificial

neural networks (ANNs). The functional relationships

of the carbon fluxes to the climatic controls are inferred

solely and directly from the observations. These purely

empirical relationships are then used to characterize the

ecosystem response to its climatic drivers, e.g., the

hierarchy of the controls, the multivariate dependen-

cies, and the sensitivities of the response. Only at the

last step are the results put in the context of current

hypotheses. Hence, the inductive approach described

below can be used to answer the question of what

controls the carbon flux in terrestrial ecosystems di-

rectly from the observations.

Method

ANN modeling framework

In the past, purely empirical models have been used broadly,

e.g., for the spatial or temporal interpolation of the carbon

fluxes (Papale & Valentini, 2003; Gove & Hollinger, 2006;

Stauch & Jarvis, 2006). These models are generally used as a

black box. Only a few of them are used in an inductive manner

also aiming to provide a physiological interpretation, such as

data-based mechanistic modeling (Young & Jarvis, 2002).

Our inductive approach is based on statistical multivariate

modeling with ANNs (Bishop, 1995; Rojas, 1996). It exploits

their ability to recognize the underlying patterns even in large

sets of (noisy) observational datasets. Owing to their out-

standing data-mining ability, the ANNs often outperform

classical semi-empirical methods (e.g., Abramowitz, 2005;

Moffat et al., 2007) and can thus be used as a benchmark for

process-based model descriptions (Abramowitz, 2005).

The modeling framework used in this paper is based on feed-

forward ANNs with a sigmoid activation function trained with

the backpropagation algorithm (Bishop, 1995; Rojas, 1996). A

feed-forward ANN consists of nodes, interconnected by

weighted links. Information moves only in a forward direction,

from the input node layer through the hidden node layer(s) to

Inductive modeling approach

?

?

XY

Characterization

HypothesesPurely

empiricalmodel

Extraction of thefunctional

relationshipsand evaluation

∂f∂x

∂f∂y

Data

Hypothetic-deductive modeling approach

XY

Hypotheses Parameterizationand evaluation

Ecosystemmodel with

fixedequations

Constrained set of

parameters

Data

Fig. 1 Conceptual flow of the two different modeling approaches. The shaded areas depict the special features of the inductive

approach presented in this paper to characterize the underlying functional relationships of the ecosystem response.

2738 A . M . M O F F AT et al.

r 2010 Blackwell Publishing Ltd, Global Change Biology, 16, 2737–2749

the output node layer. This type of ANN provides the features

required by our inductive approach: a very general function

class with a closed-form expression representing the network

response, trained by a supervised learning algorithm that is

suited for nonlinear regression tasks. Cybenko (1989) proved

that a single hidden layer, feed-forward ANN is capable of

approximating any continuous, multivariate function to any

desired degree of precision. This means that a complex enough

feed-forward ANN has the built-in flexibility to map the

individual conditions without needing prior assumptions about

the shape of the response.

The training procedure is used to constrain the weights of

this purely empirical model to the functional relationships

present in the data. The weights can be viewed as nonlinear

regression parameters. Since the dataset is presented as snap-

shots, one tuple at a time, the ANN models picks up correla-

tions of the responding output variable to the controlling input

variables (drivers) at the presented time scale, here half-

hourly. An optimally trained ANN model is able to map the

temporal correlations in the data while maintaining the ability

to generalize beyond the training dataset.

To yield a good generalization of the data, the ANN model

is systematically varied in each training run. The training

begins with a large, usually single layer network with many

nodes and randomly chosen initial values of the weights. Then

the network is trained in online or batch mode and pruned to

reach an optimum size. The final structure and weight para-

meters of the network depend on the initial number of nodes

at each layer, the initial values of the weights, and the training

progression (pattern shuffling, online or batch mode, pruning,

early stopping). Further details on the technical implementa-

tion can be found in A. M. Moffat, 2010. The parameterized

ANN model with fixed weights after training will be referred

to as the ANN model.

For the mapping to be robust, two ANNs trained on the

same dataset should result in the same functional relation-

ships, even though their final node structure and weight

parameters differ due to the systematically varied training

procedure. To get a measure of the robustness, each ANN

training scenario in this paper has been repeated 10 times.

Analysis tools

Mapping performance. The quality, or performance, of the ANN

model can be used to estimate how much of the response can

be mapped (explained) with the input drivers provided.

During the ANN training, the sum of squared errors

(SSEerr) is optimized:

SSEerr ¼1

N

Xpi � oið Þ2 ¼ RMSE2; ð1Þ

where oi are the individual observed data, pi are the values

predicted by the ANN, and N is number of data points. The

SSEerr is equal to the squared root mean square error (RMSE2).

The coefficient of determination (R2) is directly related to

SSEerr normalized by the total variance SSEtot of the dataset:

R2 ¼ 1� SSEerr

SSEtot¼ 1�

Pðpi � oiÞ2Pðoi � oÞ2

; ð2Þ

where �o is the mean of the observed values. For the model

residuals, the standard deviation (SD) will be used according

to Richardson et al. (2006):

SD ¼ffiffiffi2p

MAE ¼ffiffiffi2p 1

N

Xpi � oij j; ð3Þ

where MAE is the mean absolute error. If the ANN model will

be used for predictions, the mean bias error should also be

considered (Moffat et al., 2007).

Driver relevance. If a climatic variable d1 has more correlation

with the responding output variable than another climatic

variable d2, the mapping performance P1 of the ANN with d1

as the single input will be higher than the performance P2 of

the ANN with single d2:

P1 > P2: ð4Þ

The ANN mapping performance with single inputs can

thus be used to quantify their importance as primary input

drivers.

In the same manner, the improvement in ANN per-

formance with a new driver added to an existing network

can be used as a measure of importance as an additional

driver. The more new information the additional driver dA

adds, the greater is the improvement in the network

performance and the more relevant is this climatic variable

dA for the response (van de Laar et al., 1999). The performance

improvement DPA can be calculated as:

DPA ¼ PþA � P; ð5Þ

where P is the performance without dA and P1A the

performance with dA added.

When using the performance improvement as a measure of

relevance, attention has to be paid to correlations between the

input drivers. If a new driver adds little information to the system,

it might mean that it is irrelevant or that the information is already

present in the existing inputs. The latter fact can be used to detect

correlations by first training the networks separately on two

drivers of interest, and then together. Assuming that the two

drivers showed high relevance when trained separately but only

little added performance when both were used for the training, it

means that the two are closely correlated.

Network function. The ANN model maps the response of the

dependent output variable to the input driver(s) as present in

the data. In a feed-forward network, the input drivers d1 to dn

are mapped unidirectionally, layer by layer, onto the predicted

output. This yields a unique, continuous analytical network

function f describing the response:

fðd1; :::; dnÞ; where f: D! R and D � Rn: ð6Þ

If the input drivers are mapped on multiple outputs m,

then f is a vector of Rm. Each element of this vector is a closed-

form expression of the input space, describing one aspect of

the response.

Numerical partial derivatives. The numerical partial derivative

PaD of f with respect to each input driver di characterizes the

C H A R A C T E R I Z I N G E C O S Y S T E M R E S P O N S E S 2739


change in the ecosystem response for each individual dataset

tuple tj:

PaDi;j ¼@f

@di

� ��tj

: ð7Þ

To calculate the numerical partial derivatives, the standard

ANN backpropagation algorithm was extended to save the

composition of the derivatives at each node. Now the ANN

can be used not only for function approximation but

additionally for the calculation of the numerical partial

derivative (after Rojas, 1996).

The numerical partial derivative PaDi,j describes the

change of the ecosystem response per measured physical

unit. Since the interest of this study is in the overall response

of the ecosystem, the numerical partial derivatives of each

input driver di are transformed from the dynamic range,

estimated from its yearly absolute minimum di,min and

maximum di,max, to unit range:

½di; min; di; max�7!½0; 1�: ð8Þ

The normalized numerical partial derivative is calculated

as:

nor: PaDi;j ¼ di; max � di; min

� �PaDi;j: ð9Þ

The normalized PaD has the same scale for each of the

climatic drivers, namely in units of ecosystem response per

unit-normalized dynamic range.

To get an estimate of the mean absolute change of the

response, the absolute numerical partial derivatives for an

input variable di are averaged over all N tuples tj:

abs: PaDi ¼1

N

XN

j¼1

nor: PaDi;j

�� : ð10Þ

The positive and negative fractions of this sum provide

information on negative and positive changes in the response:

neg: PaDi ¼1

N

XPaDi;j < 0

ðnor: PaDi;jÞ; ð11Þ

and

pos: PaDi ¼1

N

XPaDi;j > 0

nor: PaDi;j

� �: ð12Þ

Methodological stages

Our inductive approach encompasses six stages to character-

ize the ecosystem response directly from large and multi-

dimensional, even fragmented observational datasets with as

few prior assumptions as possible. The stages are based on the

highly flexible, purely empirical modeling framework and on

the analysis tools described above. An extension of this

inductive approach to a full methodology – a body of methods

generally applicable for the exploration of ecological datasets –

is presented in A. M. Moffat, 2010.

1. Preconsiderations. The observational dataset consists of a

responding variable to be induced, such as the net carbon

flux, and driving variables, such as the meteorological data. To

ensure that the dataset is representative for the char-

acterization of the ecosystem response, the following pre-

considerations should be taken into account:

� Quality: Since an empirical model will map the functional

relationships as present in the datasets, it is important to

use accurate and consistent measurements. The quality of

the dataset does not depend on the quantitative amount of

data but on the enclosed information.

� Reducing complexity: The more explicit the information in

the dataset, the more concise will be the mapped relation-

ships. To reduce the complexity, only data relevant for the

queried ecosystem response should be considered. For

example, if the photosynthetic response of the ecosystem

is of interest, the dataset should be restricted to the day-

time data of the active period.

� Candidates for input drivers: The dataset should contain all

climatic variables that are assumed to have an effect on the

ecosystem response.

� Data coverage: A purely empirical model can only map

functional relationships properly within the scope of the

training dataset. Interpolation between underrepresented

regions or extrapolation might lead to physiologically im-

plausible mapping. Therefore, the measurements used for

training should have good data coverage over the full

range of interest.

2. Benchmarking. The ANN training is first performed with all

available climatic variables in the dataset as input drivers.

Assuming an optimally trained ANN, this gives a benchmark

measure of the maximum mapping between the responding

variable and all the provided meteorological observations. If

the determination coefficient R2 is used as the performance

measure, then the benchmark describes the total explainable

variability in the dataset.

If all relevant climatic controls, and, if necessary, all

information about the state of the ecosystem such as the

phenology are included in the benchmark dataset, the

remaining unexplained variability can be attributed to the

noise in the measurement. The model residuals can then be

used to give an estimate of the uncertainty (random error) in the

flux measurements (Richardson et al., 2008).

3. Hierarchy of the climatic controls. After determining the total

benchmark, the ANNs are trained with single input drivers at

a time. This ANN mapping exercise determines the relevance

of a climatic variable as a primary driver. The hierarchy of the

climatic controls of the response can be obtained by ranking

the driver’s relevances.

If the response is highly modulated by a certain driver (e.g.,

the response of photosynthesis by light), it acts like a carrier

signal for the response to the minor driver (e.g., temperature).

In this case, the ANN might not be able to pick up the

underlying minor correlation directly. To overcome this

problem, the dominating input driver is identified using the

primary driver performance from above. The networks are

then trained with the dominant primary driver, plus each of

the other climatic variables in turn as secondary drivers. The



performance improvement is used to identify the relevance of

the climatic variables as secondary drivers.

4. Function analysis of single to multivariate ANN models. The

hierarchy of the climatic controls allows to confine the function

analysis to the relevant climatic drivers. First, the single ANN

model trained on the primary climatic control is examined,

and, then, the ANN models of the primary control plus

secondary driver(s) are analyzed for their multivariate

dependencies. In order to get plausible functional

relationships, the choice of input drivers is subject to the

following conditions:

� The drivers should be physiologically meaningful.

� The drivers often have obvious or hidden correlations that

may distort the dependencies. Therefore, it is important to

be aware of cross-dependencies and to keep the drivers as

independent as possible.

� Confounding drivers should be included in the driver set

in order to obtain robust relationships.

� The degrees of freedom of the empirical model increase

with each added driving variable and may lead to a phy-

siologically implausible mapping of the response. Conse-

quently, the number of input drivers should be kept as low

as possible.

During the training, the general ANN network function is

constrained by the measurements. Afterwards the ANN net-

work function represents the ecosystem response to its climatic

controls as present in the data. This network function can be

used to characterize the physiological properties of the eco-

system as present in the data:

� Its form (e.g., the basic shape, the offsets at the origin, or

the saturation) shows the functional dependency and can

be used to derive the physiologically relevant parameters.

� Plotting of the network function helps to visualize the

functional dependencies on the climatic controls.

� Its partial derivatives give information about the changes in

the response with respect to the input driver(s): positive and

negative derivatives or potential turning points of the re-

sponse may provide insight into the underlying processes.

� The absolute sums of the partial derivatives and their

positive and negative fractions reveal the sensitivities of

the ecosystem response to the climatic drivers.

The derived physiological properties are then compared with

the existing hypotheses. This comparison may corroborate the

hypotheses or indicate new or different features present in the

data. The function analysis of the purely empirical ANN

models can thus serve as a link between the observations

and their semi-empirical representations in the modeling

world.

5. Data stratification. Our inductive approach works on

fragmented data as long as there are enough representative

samples in the dataset for training the ANN models. With

binning and grouping, different phases of the response can be

examined:

� Data binning: To analyze certain aspects of the overall

response query, the representative dataset can be binned

to certain variable ranges (e.g., flux magnitudes). The ANN

models are trained on the complete dataset, but the analy-

sis is performed on the individual bins.

� Data grouping: To investigate differences in the response,

the representative dataset can be grouped into subsets

(e.g., each month). The ANN models are trained separately

for each subset and the differences between them give

insight into the variability of the response. The setup of the

grouping can be varied to test for diurnal to seasonal to

interannual variability.

6. Theoretical driver variables. The dataset can be extended from

observable to theoretical driver variables. For example, the

phenological state of the ecosystem can be described with a

fuzzy variable for the course of the season (Papale & Valentini,

2003) or with the latent variable of the weekly mean

temperature. Other latent variables, such as the fraction of

diffuse light, might expose a different aspect of the response.

Time lag effects can be included by providing information

about preceding events, e.g., previous productivity rates. The

relevance of the theoretical variable as an additional input

driver gives a measure of its importance to modeling the

ecosystem response.

The six stages described are independent of a specific

domain and can be generally applied to characterize

ecosystem responses to climatic controls hidden in complex

observational datasets. Their capability will be demonstrated

for the daytime carbon flux measurements of the Hainich

forest in the following.

Domain of the study

The domain of this study is the characterization of the daytime

carbon fluxes of the deciduous broadleaf forest Hainich in

Germany obtained with eddy covariance measurements

(Knohl et al., 2003). Hainich is a mature beech forest in a

temperate, continental climate with the flux tower located at

51.071N and 10.451E. The turbulent exchange of CO2 is mea-

sured above the canopy with the eddy covariance technique.

Detailed stand characteristics of the Hainich forest within the

main footprint of the measurements can be found in Table 1 of

Kutsch et al. (2008).

The carbon flux measured is the net ecosystem exchange

(NEE) between the atmosphere and the terrestrial ecosystem.

Throughout this paper, the term net ecosystem productivity

(NEP) will be used to describe the negative of NEE

(NEP 5�NEE). NEP of the forest is the carbon uptake by

photosynthesis minus the release by autotrophic and hetero-

trophic respiration. Since the focus of this study is on the

daytime response during the active period, the response of the



forest is dominated by photosynthesis, while respiration plays

only a minor role.

The quality-checked (level 3) observational datasets of the

three nondrought years 2000, 2001, and 2002 were obtained

from the standardized Carboeurope IP database (Papale et al.,

2006). The climatic variables used for the characterization are

listed in Table 1. In addition to the provided variables, the

diffuse fraction fdif was calculated from the ratio of diffuse

global Rdif to total global radiation Rg : fdif 5 Rdif/Rg. The vari-

able fdif ranges from only direct, 0%, to only diffuse light, 100%.

Since PPFDdif and PPFDdir were not measured directly, they

were calculated from photosynthetic photon flux density

(PPFD) and fdif. Only best quality data (flag 5 0) with complete

input data during daytime (PPFD410 mmol photon m�2 s�1) of

the summer period (June–September, to avoid phenology

effects) were selected. Additionally, five outlier data points

of an exceptionally dry day [vapor pressure deficit

(VPD)418 hPa] and 27 unclean diffuse fractions (fdif 5 0%

and fdif4100%) were removed from the dataset. The total

number of half-hourly data analyzed was 3015.

Results

How much information about the NEP response is presentin the dataset?

The ANNs trained with all 14 climatic drivers yielded

an R2 of 93.1(� 0.1)%, where the value in brackets is the

SD over 10 ANN training scenarios. This benchmark

means that 93.1% of the total variability of NEP in the

half-hourly dataset can be explained with these 14

climatic drivers (Fig. 2). The high R2 also attests to the

excellent data mining capability of the ANNs.

The residuals of the benchmark ANN models can be

used to obtain an estimate of the remaining error. The

SD of the model residuals, binned by the NEP flux

magnitude in steps of 5mmol CO2 m�2 s�1, varied be-

tween 1.1 and 3.4mmol CO2 m�2 s�1 (Fig. 3).

Table 1 List of variables used for the characterization

Net carbon flux

NEP Net ecosystem productivity (mmol CO2 m�2 s�1)

Radiative variables

PPFD (Total) photosynthetic photon flux density

(mmol photon m�2 s�1)

PPFDdir Direct PPFD (mmol photon m�2 s�1)

PPFDdif Diffuse PPFD (mmol photon m�2 s�1)

Meteorological variables

VPD Vapor pressure deficit (hPa)

Rh Relative humidity (%)

SWC Soil water content (%)

Ta Air temperature ( 1C)

Ts1, Ts2 Soil temperature at 5 and 30 cm depth ( 1C)

WD Wind direction (1)

u* Friction velocity (m s�1)

Theoretical variables

Rpot Potential radiation at the top of atmosphere (W m�2)

fdif Diffuse fraction (%)

NEPhh NEP measurement of the previous half-hour

PPFD (μmol photons m–2 s–1)

0 200 400 600 800 1000 1200 1400 1600

NE

P (

μmol

CO

2 m

–2 s

–1)

–5

0

5

10

15

20

25

30

35

MeasuredModeled

Fig. 2 Daytime NEP response of the Hainich forest plotted

vs. PPFD. The response modeled with all 14 climatic dri-

vers (black circles) captures 93.1% of the variability of the half-

hourly measurements (gray circles). For variable descriptions see

Table 1.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

–5 0 5 10 15 20 25S

D (

µmol

CO

2 m

–2 s

–1)

NEP (µmol CO2 m–2 s–1)

Fig. 3 The standard deviation SD of the model residuals binned

by the NEP flux magnitude in steps of 5mmol CO2 m�2 s�1.

The SD ranging from 1.1 to 3.4mmol CO2 m�2 s�1 indicates low

noise in the measurements as well as a good ANN model

performance. (The dotted line is the average SD of the whole

dataset. The error bars show the standard deviation of 10 ANN

training scenarios.) For variable descriptions see Table 1.



To compare our results to previous work using paired

observations or model residuals from Richardson et al.

(2008), the linear relationship between SD and the

magnitude of NEP was calculated for the positive bins:

SD ¼1:32ð�0:05Þ þ 0:079ð�0:003ÞNEP: ð13Þ

The relationship obtained has an offset similar to the

one previously reported for Hainich, but with only half

the slope. This means that the random error estimated

from the ANN benchmark models increases only half as

fast with increasing flux magnitude. ANN training

setups with different sets of input variables showed

that the smaller increase can be attributed to including

the diffuse radiation – this was not included in the

analysis of Richardson et al. (2008); a minimal config-

uration with only PPFD, PPFDdif, VPD, and Ta as input

drivers and only three to five nodes in the hidden layer

of the ANN resulted in almost half the slope.

The fact that the standard deviation of the ANN

residuals is even below the paired observation esti-

mates from Richardson et al. (2008), corroborates the

assumption that the remaining error and thus the un-

explained variability can be mostly attributed to noise

in the measurements. Moreover, it shows that the

relevant climatic drivers were included in the training

dataset, and that the ANNs were able to pick up the

underlying correlations and fully capture the ecosystem

response. The mapped correlations permit the recon-

struction of missing NEP measurements from the asso-

ciated meteorological data. Hence, the benchmark

ANNs can also be used as a so called gap-filling

technique (A. M. Moffat, 2010).

What are the climatic controls of the measured NEP flux?

ANNs trained with only one climatic variable at a time

showed the best performance with total PPFD as the input

variable (Fig. 4, top). The coefficient of determination for

modeling the half-hourly daytime NEP at the Hainich

forest with the main driver total PPFD yielded 82.2%. The

diffuse radiation PPFDdif exhibited a higher relevance

than direct radiation PPFDdir as the single input driver.

It is interesting to note that the NEP measurement for the

previous half-hour explained 76.6% of the total variability.

This indicates the persistency of the meteorological con-

ditions between successive half-hours and can be taken as

a measure of the lower performance limit of models used

for predictions. The SD of R2 over 10 ANN training scena-

rios was generally small, demonstrating the robustness of

this inductive approach; for these two primary drivers,

the formal numerical SD was o0.01%.

Since the response of NEP is dominated by light, the

relevance of the other climatic controls was determined

by training the ANNs with PPFD plus one secondary

climatic driver at a time. The highest improvement in

performance, thus the most relevant secondary control

for the daytime NEP response, was the proportion of

diffuse radiation (Fig. 4, bottom). Each combination,

total PPFD plus PPFDdir, total PPFD plus PPFDdif, total

PPFD plus fdif, or PPFDdir plus PPFDdif (shown in the

Secondary driver

R2

0.70

0.75

0.80

0.85

0.90

0.95

1.00 Dominant driver onlyPerformance improvementwith secondary driver added

Primary driverPPFDdir

PPFDdifPPFD

VPD RhSWC Ta Ts1 Ts2 WD

ustarR_pot

f_dif

NEP_hh

PPFDdir

PPFDdifPPFD

VPD RhSWC Ta Ts1 Ts2 WD

ustarR_pot

f_dif

NEP_hh

R2

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Radiative variableMeteorological variableTheoretical variableBenchmark

2 2

4 6

33

(2)

4 577

1

Fig. 4 Primary R2 performance of the ANN models trained

with a single climatic driver at a time (top). Total PPFD is the

dominating climatic control of the half-hourly daytime NEP

response at Hainich. Then the ANNs were trained with PPFD

plus a secondary climatic driver (bottom) and the improvement

in the performance indicates the relevance. The proportion of

diffuse to direct radiation (provided as PPFDdif, PPFDdir, or

diffuse fraction fdif) is the most important secondary climatic

driver. (The dotted line is the total explainable variability bench-

marked with all 14 drivers. The error bars indicate the standard

deviation of 10 ANN training scenarios; for most drivers this

error bar is so small that it is not visible on the graph. Please note

the different scale of the y-axis for the bottom graph.) For

variable descriptions see Table 1.



next section), yielded the same R2 network performance.

This means that each of these input driver combinations

carries the same amount of information. In addition, it

demonstrates the outstanding ability of the ANNs to

extract this information even when the input drivers

change units, change magnitude, or are (non)linearly

transformed. The independence from the representation

of the input drivers also shows the reliability of the

network performance as a measure of the relevance of

the climatic drivers. The sensitivity of these results to

input uncertainty was tested by introducing an artificial

uncertainty of 5%, the maximum relative error accord-

ing to the specifications of the instrument devices. None

of the tested scenarios (positive or negative offset and

uncorrelated or correlated random noise) had an impact

on the network performances or on the functional

relationships derived below.

The R2 of the ANN models with PPFD plus one of the

three diffuse proportion drivers was 89.6( � 0.1)%, and

the average SD was reduced by over 20%, from 3.7 to

2.9mmol CO2 m�2 s�1. In other words, the diffuse pro-

portion explains an extra 7% of the variability and both

drivers together almost all the explainable variability

(93.1%) in the dataset.

The next most relevant secondary control was the

amount of air moisture represented either as VPD, or as

relative humidity (Rh); this was followed by air tempera-

ture (Ta), and wind direction (WD). As a single driver (Fig.

4, top), the information of WD was concealed, since the

mean NEP per degree was fairly constant. However, with

the modulation of NEP by PPFD included, the effect of WD

as a secondary driver revealed a difference in the NEP

uptake. The uptake was higher for winds from the SW than

from the NE (not shown). This is probably mostly related

to regional weather patterns (prevailing dry and cold air

masses from NE vs. humid and warm air from SW), but

can also be associated with changes in the footprint.

The friction velocity u*, which is a measure of the

turbulent mixing in the atmosphere, should be of little

relevance, since a correlation of NEP with u* indicates a

systematic bias error in the eddy covariance measure-

ments. The soil temperature Ts2 at the greater depth of

30 cm added slightly more new information than Ts1 to

the daytime NEP response modeled with PPFD as the

primary driver. As a slower changing variable, Ts2

might provide some information on the ecosystem state.

The soil water content, SWC, had little effect during

these nondrought summers.

What are the characteristics of the NEP response to light?

PPFD was identified as the dominant primary control of

the half-hourly daytime NEP response. The ANN models

trained on the Hainich dataset with PPFD as the only

climatic control showed the expected functional form: a

steep, almost linear initial increase that levels off to

saturation for high PPFD (Fig. 5). This form is the result

of the projection of the input PPFD via the nodes in the

hidden layer onto the output f(PPFD). One of the ANNs

with four hidden nodes had the following analytical

network function:

Although the regression parameters of Eqn (14) (the

weights and offsets of the logistic sigmoid functions)



0 200 400 600 800 1000 1200 1400 1600

NE

P (

μmol

CO

2 m

–2 s

–1)

dNE

P/d

PP

FD

–5

0

5

10

15

20

25

30

35

MeasuredModeled

0 200 400 600 800 1000 1200 1400 1600–0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

Quantum Yield α

Fig. 5 Daytime NEP response (top) and its numerical deriva-

tive (bottom) modeled with PPFD as a single climatic driver. The

light response curve shows the expected behavior: a steep,

almost linear initial increase leveling off to saturation for high

PPFD. For variable descriptions see Table 1.

NEPANN ¼ fðPPFDÞ ¼ � 38:4þ 105:4

1þ 0:934 e0:440

1þ14:6e�0:00173PPFD þ 0:289�1�5:53e�0:00166PPFD þ 0:380

�1�5:08e�0:00152PPFD þ 2:5991þ 2:15e0:00360PPFD

� ! : ð14Þ



have no direct physiological meaning, the curve pro-

gression of this function and of its derivative can be

used to derive the physiological characteristics: The

derivative starts off almost constant at the onset of light,

corresponding to a linear initial slope. This initial

slope of 0.050 mmol CO2/mmol photons is the initial

quantum yield a, the maximum light use efficiency

of the ecosystem. The offset of NEP at zero light

is the daytime respiration and has a value of

�2.9mmol CO2 m�2 s�1. Towards high PPFD values,

the derivative approaches zero, denoting the saturation

of the NEP response. The optimum (saturated) NEP at

the highest irradiance of 1750 mmol photons m�2 s�1 is

22.5mmol CO2 m�2 s�1.

The obtained properties, the initial linear increase, the

leveling off to saturation, and the magnitude of physio-

logical parameters, meet the behavior expected for the

light response of a deciduous broadleaf forest (e.g.,

Larcher, 2003). This agreement demonstrates that our

inductive approach is able to extract the underlying

functional relationship directly from the data.

Since the relationships were derived solely from the

observations without a priori assumptions, the agree-

ment also provides an independent corroboration of the

light response hypotheses at the ecosystem level.

Although the need for such corroboration might not

be obvious, an assessment of all commonly used semi-

empirical light response curves showed that some

curves (e.g., the rectangular hyperbola) do not reflect

the required physiological characteristics (A. M. Moffat,

2010). Their incorrect behavior at the edges, right where

the physiological parameters are derived, leads to large

differences in the estimates of the physiological para-

meters, despite a comparable overall performance.

What is the effect of diffuse radiation?

The dependency of the daytime NEP response on the

diffuse light was extracted from the dataset by training

the ANN models with the diffuse and direct PPFD as

inputs (Fig. 6). The simplicity of these ANN models (see

Fig. 6), their high R2 of 89.6% and their low SD of

2.9mmol CO2 m�2 s�1 (see section ‘What are the climatic

controls of the measured NEP flux?’) demonstrate, that

the extracted functional relationship NEPANN(PPFDdif,

PPFDdir) is well suited to display and quantitatively

characterize the response. The numerical partial deri-

vatives reveal a significant difference in the functional

relationship to diffuse radiation compared with direct

radiation (Fig. 7, bottom): The initial quantum yield of

PPFDdif is almost three times higher, its light use

efficiency (magnitude of the derivative) is enhanced

throughout the response, and the response shows no

saturation even for high PPFDdif. These results are in

full agreement with Gu et al. (2002), who found simi-

larly enhanced light use efficiencies and weakened

tendencies to cause canopy saturation for the diffuse

radiation. Since the response to PPFDdif does not satu-

rate, this effect is even more pronounced for high values

of total PPFD.

The high input relevance and enhanced light use

efficiency and sensitivity of PPFDdif compared with

PPFDdir stresses the importance of the diffuse radiation

for the ecosystem response. As the dominant secondary

control of the half-hourly daytime NEP response, it

should be included in ecosystem models trying to

predict the carbon flux at half-hourly or hourly time-

scales (see also Roderick et al., 2001). The hypotheses

needed for the implementation can be based on the

functional relationships derived by the ANNs. The

presented study shows the dependencies of the NEP

response to diffuse radiation at the Hainich forest

(Figs 6 and 7). Herein lies the strength of our inductive

approach: in addition to the detection and quantifica-

tion of the impact of diffuse radiation, it provides an

explicit characterization of the functional relationship.

The enhanced light use efficiency of diffuse light

leads to an increase in the NEP response of the Hainich

forest. However, less of the potential radiation Rpot is

received at the surface for high diffuse fractions due to

the absorption and reflection by clouds and aerosols,

and less light leads to a decrease in the NEP response.

Therefore, the question arises whether the overall effect

is positive or negative?

To provide insight into this aspect, the ANN model

was trained with the following three climatic input

PPFDdir

(μmol photons m–2 s–1)

200 400 600 800 100012001400PPFDdif

(μmol photons m –2 s –1)

100200

300400

500600

700800

NE

P (

μmol

CO

2 m

–2 s

–1)

05

101520253035

Fig. 6 Closed symbolic representation of the ANN modeling

the half-hourly daytime NEP response to the climatic controls

diffuse PPFDdif and direct PPFDdir. The simplicity of the ANN

model is well suited to display and characterize the functional

relationship NEPANN(PPFDdif, PPFDdir). For variable descriptions

see Table 1.



drivers: the potential radiation Rpot and the diffuse

fraction fdif, plus the vapor pressure deficit VPD to

include confounding effects with diffuse radiation.

Since the daytime NEP response is now modeled with

three inputs, the analytical function has too many

dimensions to be directly visualized. For these multi-

dimensional relationships, the partial derivatives are of

great value to examine their behavior. Figure 8 shows

the numerical partial derivatives of the modeled re-

sponse with respect to fdif: At first, the NEP response is

enhanced (positive derivative) until it reaches an opti-

mum (zero derivative) and then the NEP response is

reduced (negative derivative). This means that the net

effect of the diffuse radiation is at an optimum for

diffuse fractions from 28% to 44% at the Hainich site.

This range is close to the optimum of 45% found in a

recent study by Knohl & Baldocchi (2008) for NEP fluxes

at Hainich using a biophysical multilayer model of the

canopy. Both approaches thus depict optima where there

is less diffuse than direct light. But, as one can see in Fig. 8,

the majority of the half-hourly measurements are beyond

the optimum range counteracting the increase in the NEP

response. This leads to a slightly negative average numer-

0 200 400 600 800 1000 1200 1400 1600

–5

0

5

10

15

20

25

30

35

PPFDdir (μmol photons m–2 s–1)

0 200 400 600 800 1000 1200 1400 1600–0.01

0

0.01

0.02

0.03

0.04

0.05

0.06 Numerical partial derivatives

0 100 200 300 400 500 600 700 800

–5

0

5

10

15

20

25

30

35(a) (b)

(c) (d)

PPFDdif (μmol photons m–2 s–1)

0 100 200 300 400 500 600 700 800

∂NE

P/∂

PP

FD

dif

∂NE

P/∂

PP

FD

dif

NE

P (

μmol

CO

2 m

–2 s

–1)

NE

P (

μmol

CO

2 m

–2 s

–1)

–0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

MeasuredModeled

MeasuredModeled

Numerical partial derivatives

Fig. 7 ANN model predictions (black circles) and half-hourly measurements (gray circles) of the daytime NEP response plotted vs. the

two climatic drivers: (a) diffuse PPFDdif and (b) direct PPFDdir. The ANN model captures 89.5% of the variability of the half-hourly

measurements. The numerical partial derivatives correspond to the light use efficiency, which is about three times higher for diffuse

compared with direct radiation. For variable descriptions see Table 1.

fdif

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

∂NE

P/∂

f dif

–30

–20

–10

0

10

20Numerical partial derivatives

fdif

NE

P

Fig. 8 Numerical partial derivatives of the daytime NEP re-

sponse to the diffuse fraction fdif for each half-hourly data point.

The small sketch depicts the functional relationship of NEP to fdif.

The net effect of fdif reaches its optimum between 28% and 44%.

For variable descriptions see Table 1.



ical derivative of �0.5mmol CO2 m�2 s�1 per half-hourly

data point, thus a negative but small overall effect from

the diffuse radiation.

How does the VPD affect the daytime NEP response?

After the diffuse proportion, the next most important

secondary driver is a measure of the air humidity, Rh, or

dryness, VPD, respectively. To investigate the effect, the

half-hourly daytime NEP response was modeled with

PPFDdif, PPFDdir, and VPD as the climatic input drivers.

Adding VPD improved the R2 by 1.1 to 90.7%. The

numerical partial derivatives show the characteristics of

the NEP response with respect to VPD (Fig. 9): first, a

slight increase in NEP (positive derivative), an opti-

mum (zero derivative) around 4 hPa, and, then, a strong

down-regulating effect (negative derivative) with in-

creasing dryness of the air.

ANN models trained on individual months can be

used to investigate whether the sensitivity of the NEP

response to VPD varies over the summer period. To

detect primarily the response to VPD, only early after-

noon hours (11:30am–2pm hours) with stable light

conditions but high changes in VPD were extracted

for the analysis. Figure 10 shows that the negative

sensitivity to VPD peaks in August, the hottest and

driest month. In a study by Schulze (1970) on the carbon

gas exchange of single beech trees in Sollingen, 100 km

north-east of Hainich, the strongest effect due to dry

atmospheric conditions occurred also in August. Thus,

the response to air moisture found at the tree level can

be observed in the carbon flux measurements at the

stand level – analogously to the light response hypoth-

eses in section ‘What are the characteristics of the NEP

response to light?’ above.

Discussion

The strength of a fully inductive approach to rely only

on the information present in the data has its own

specific challenges. The following points need to be

taken into consideration to avoid pitfalls in the inter-

pretation of the ANN models:

Adequate response space: To reach the goal of modeling

the overall response, an annual dataset is appropriate. If

the interest is in the light response curve, the dataset

should span time periods where the ecosystem stays in

the same phenological and ecological states with re-

spect to the photosynthesis response. For example,

including months with leaves off would smear out the

photosynthesis response. Taking summer months but

including months with drought conditions might result

in a light response curve where the saturation has a

drop for the highest irradiances. This will look like

photoinhibition, but will actually be caused by the

superposition of the light response curve with a re-

duced optimum NEP under water stress. As an alter-

native to limiting the dataset to the same state, the

entire dataset can also be used but with an additional

input variable describing the changing condition, for

example a proxy for the water stress. With this, the

ANN is able to distinguish between drought and

VPD (hPa)0 2 4 6 8 10 12 14 16

∂NE

P/∂

VP

D

–0.8

–0.6

–0.4

–0.2

0

0.2Numerical partial derivatives

VPD

NE

P

4

Fig. 9 Numerical partial derivatives of the daytime NEP re-

sponse to the VPD for each half-hourly data point. The small

sketch depicts the functional relationship of NEP to VPD. There

is a negative, down-regulating effect on NEP for high values of

VPD. For variable descriptions see Table 1.

–20

–10

0

10

20

30

40

MonthJun Jul Aug Sep

Sen

sitiv

ities

(P

aD)

PPFDdirPPFDdifVPD

Fig. 10 The positive and negative sensitivities of the daytime

NEP response to PPFDdir, PPFDdif, and VPD during early after-

noon hours, modeled separately for each month. The sensitivity

of NEP to VPD is most negative in the hottest and driest month

of August. PPFD, photosynthetic photon flux density. For vari-

able descriptions see Table 1.



nondrought conditions and will map the responses

accordingly.

Artifacts: To avoid modeling artifacts present in a

specific dataset or nonobvious changes in the phenolo-

gical or ecological states, the identified relationships

should prove to be robust for different time periods,

e.g., individual months vs. the whole summer period or

summer months of different years.

Missing relevant driver: The ANN can show a good

model performance though a physiologically relevant

driver was missing. It means that the effect of the

missing driver was mapped onto the included drivers

through cross-correlations. Usually, the found relation-

ships are then not independent, and, therefore, not

robust. The mapped functional relationships will

change as soon as another driver with some cross-

correlation or the actual missing driver is added. If

adding drivers does not change the main properties of

the numerical partial derivatives, this is a good sign for

robustness.

Confounding factors: The hidden biases or indirect

effects caused by confounding phenological, ecological,

or climatic factors are much harder to detect. To rule out

known confounding factors, these can be added to the

data used for training as observed or theoretical drivers.

This way, their impact is included in the modeled

response, provided that the confounding factors are

not correlated to any of the other input drivers, that

they are well defined over the whole range, and that

they do not add too many degrees of freedom to the

network. An alternative solution is to perform marginal

sampling, where the dataset is grouped into subsets for

certain ranges of the confounding factor. The ANN

models are then trained on each of the subgroups.

Robust relationships will hold true for all of the sub-

groups.

Ecophysiological plausibility: Since the ANN models are

constrained solely by the data, some prior knowledge of

ecosystem physiology is required to ensure a proper

choice of the representative dataset and to judge the

plausibility of the results under anticipation of con-

founding factors. Only then does this inductive ap-

proach produce meaningful results.

The systematic approach presented in this paper has

been implemented as a toolbox. Once the dataset is

configured, the setup of different ANN routines is

simple and highly flexible. The ANN training proce-

dure is fully automated and takes only a few minutes

on a typical desktop computer. Although the analysis

tools have been tailored to extract information from

large datasets, the ANNs also appear to work with

small amounts of data. We have tested their ability to

model the light response curve for single days with as

few as 10 data points; the physiological quantities

estimated from the ANNs were consistent with the

estimates of a prescribed semi-empirical equation.

Conclusions and outlook

As demonstrated for the daytime carbon fluxes of the

Hainich forest, the inductive approach presented here

can be used to characterize the functional dependencies

solely from the half-hourly eddy covariance measure-

ments, without prior assumptions about the shape of

the response. The extracted purely empirical light re-

sponse curve provides an independent corroboration of

current plant physiological hypotheses. Estimates of the

random measurement error from the ANN model re-

siduals were lower than previous estimates. This could

be attributed to the inclusion of the proportion

of diffuse radiation, which was the second most im-

portant input variable to explain the daytime carbon

fluxes after total radiation. This key finding stresses the

importance of the diffuse radiation for the short-term

light response.

The functional dependency of the daytime response

to diffuse radiation showed no saturation, and it would

be of great interest to investigate the generality of this

relationship for other types of ecosystems. The net effect

of the diffuse radiation was determined by modeling

with two theoretical drivers – the potential radiation at

the top of the atmosphere and the diffuse fraction. Since

the light conditions at the Hainich forest were mainly

beyond the optimum diffuse fraction of 30–40%, the

overall effect of the diffuse light was on average slightly

negative for the 3 years 2000–2002.

The most important nonradiative drivers in the hier-

archy of the climatic controls were the vapor pressure

deficit, followed by air temperature and wind direction.

Multidimensional relationships in the data were further

characterized using numerical partial derivatives. For

example, the vapor pressure deficit showed a strong

down-regulating effect with increasing dryness of the

air.

Our inductive approach offers the potential to serve as

a new key instrument for the explanation of observa-

tions, for instant testing and independent validation of

hypotheses, and for the detection of new findings. The

worldwide network of eddy flux towers in FLUXNET

offers the opportunity to investigate ecosystems span-

ning from the arctic to the savannah. For managed

ecosystems, the ability to include theoretical variables,

such as a fuzzy variable to describe the harvesting event,

will be of benefit. The theoretical variables also offer the

possibility to include time lag effects and determine their

relevance for the ecosystem response. The approach is

not limited to the net carbon flux, but can be extended to



the partitioned GPP/RE carbon flux, the energy and

momentum flux, or other greenhouse gases.

By supplying the link between the observations and

their representation in the modeling world, the pre-

sented inductive approach is complementary to

the classic hypothetic-deductive approach. This will

further the understanding of the underlying processes

as well as promote their implementation in models,

which, in turn, will help the prediction of the effects

of changing environmental conditions on the terrestrial

biosphere.

Since purely empirical models adapt to the particular

conditions of the ecosystem as present in the training

dataset, they can also be used to identify differences in

the response over time. If changes in the climate lead to

changes in the ecosystem response to its climatic con-

trols, the presented methodology would be able to

detect these directly in the measurements.

Acknowledgements

We would like to thank the following people for their contribu-tions to this paper: Olaf Kolle for introducing Antje Moffat to theHainich flux site, Corinna Rebmann for sharing her broadknowledge about the measurements, Andrew Richardson, JohnGrace, Alessandro Cescatti, and Detlef Schulze for in-depthdiscussions of the results, Petra Werner and Andrew Jarvis fortheir comments on the general scope, Gill McLean for proof-reading of this manuscript, Bryce Moffat for proofreading of thevarious drafts, and the ROOT team at CERN for providing theirextensive C 11 programming framework. Furthermore, wewould like to thank the editor Ivan Janssens and the threeanonymous reviewers for their thorough comments and con-structive criticism, which greatly helped to improve this paper.

The datasets used in this paper were obtained from theCarboEurope-IP database (EU project GOCE-CT-2003-505572).The site PIs Alexander Knohl, Corinna Rebmann, and WernerKutsch are thanked for making the Hainich data available to thedatabase.

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