Advances in Water Resources 24 (2001) 1119±1132
www.elsevier.com/locate/advwatres
Multiscale analysis of vegetation surface ¯uxes: from seconds to years
Gabriel Katul
a,*
, Chun-Ta Lai a, Karina Sch
afer a, Brani Vidakovic b, John Albertson c,
David Ellsworth a, Ram Oren a
a
b
c
School of the Environment, Duke University, Box 90328, Durham, NC 27708-0328, USA
Institute of Statistics and Decision Sciences, Duke University, Durham, NC 27708-0251, USA
Department of Environmental Sciences, University of Virginia, Charlottesville, VA 22903, USA
Received 5 June 2000; received in revised form 7 December 2000; accepted 19 March 2001
Abstract
The variability in land surface heat (H), water vapor (LE), and CO2 (or net ecosystem exchange, NEE) ¯uxes was investigated at
scales ranging from fractions of seconds to years using eddy-covariance ¯ux measurements above a pine forest. Because these ¯uxes
signi®cantly vary at all these time scales and because large gaps in the record are unavoidable in such experiments, standard Fourier
expansion methods for computing the spectral and cospectral statistical properties were not possible. Instead, orthonormal wavelet
transformations …OWT† are proposed and used. The OWT are ideal at resolving process variability with respect to both scale and
time and are able to isolate and remove the e€ects of missing data (or gaps) from spectral and cospectral calculations. Using the
OWT spectra, we demonstrated unique aspects in three appropriate ranges of time scales: turbulent time scales (fractions of seconds
to minutes), meteorological time scales (hour to weeks), and seasonal to interannual time scales corresponding to climate and
vegetation dynamics. We have shown that: (1) existing turbulence theories describe the short time scales well, (2) coupled physiological and transport models (e.g. CANVEG) reproduce the wavelet spectral characteristics of all three land surface ¯uxes for
meteorological time scales, and (3) seasonal dynamics in vegetation physiology and structure inject strong correlations between land
surface ¯uxes and forcing variables at monthly to seasonal time scales. The broad implications of this study center on the possibility
of developing low-dimensional models of land surface water, energy, and carbon exchange. If the bulk of the ¯ux variability is
dominated by a narrow band or bands of modes, and these modes ``resonate'' with key state and forcing variables, then low-dimensional models may relate these forcing and state variables to NEE and LE. Ó 2001 Published by Elsevier Science Ltd.
1. Introduction
Temporal variability in land surface sensible (H) and
latent (LE) heat ¯uxes and net ecosystem exchange
(NEE) occur over a wide range of scales ranging from
fractions of seconds to decades. These ¯uxes are interwoven at the most fundamental level, the leaf stomata.
Describing the dynamics of H, LE, and NEE requires
resolving an extensive high-dimensional spectrum of
variability [2]. To date, no one model is able to reproduce the variability in these ¯uxes across such spectrum
of scales. Over seconds, biosphere±atmosphere exchange
of water vapor …q†, heat …Ta †, and carbon dioxide (C) are
governed by complex turbulent eddy motion, rich in
spectral properties. At hours to monthly scales, much of
the variability is induced by interactions between
*
Corresponding author. Tel.: +1-919-613-8033; fax: +1-919-6848741.
E-mail address: [email protected] (G. Katul).
weather patterns, bulk turbulent ¯ow characteristics,
eco-physiological and biochemical characteristics of the
canopy, and the hydrologic state of the ecosystem. The
latter variable is strongly subject to biological feedbacks
by means of stomatal closure and xylem cavitation [40].
At monthly to annual time scales, synoptic weather
patterns superimposed on seasonal variations are responsible for changes in ecosystem properties and,
consequently, dominate the variability in these land
surface ¯uxes. This multiscale nature of land surface
¯uxes is now motivating the development of a new
generation of canopy±atmosphere models accompanied
by long-term measurements [11]. The objective of this
study is to explore structural aspects and controls on the
temporal spectra of H, LE, and NEE from fractions of
seconds to years. A case study is made using measurements of H, LE, and NEE collected over three years at
10 Hz above a 17 year old loblolly pine stand at Duke
Forest, near Durham, North Carolina. The Duke Forest
site is part of a long-term NEE monitoring initiative
0309-1708/01/$ - see front matter Ó 2001 Published by Elsevier Science Ltd.
PII: S 0 3 0 9 - 1 7 0 8 ( 0 1 ) 0 0 0 2 9 - X
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G. Katul et al. / Advances in Water Resources 24 (2001) 1119±1132
aimed at representing the ecological behaviour of managed pine plantation in the Southeastern US. About
70% of the total forested area in the Southeastern US
…35:6 106 Ha† is pines, and 41% of this area is pine
plantations younger than 25 years [33]. The site index
( ˆ 16 m at 25 years) of the Duke Forest pine plantation
also represents the mode of managed pine plantations in
the Southeast.
2. Methods of analysis
We propose multiscale methods that can analyze the
spectral and cospectral properties of H, LE and NEE.
Traditional spectral and cospectral analysis in the
Fourier domain is primarily complicated by random
gaps in the ¯ux time series measurements and the
convolution between these ¯uxes and the suite of
hydrologic and environmental variables at multiple
scales. Wavelet transformations can localize the gaps in
the ¯ux record and eliminate them from the spectral
and cospectral calculations. Additionally, the multiscale
interactions between the ¯uxes and forcing variables
are naturally resolved in the standard multiresolution
formulation of OWT [13,14,30,31,42,43,47]. Towards
this end, all spectral and cospectral properties will be
analyzed using OWT, as described in the following
section.
2.1. Fast wavelet transform (FWT)
The Haar wavelet basis is selected for: (i) its temporal
di€erencing characteristics, (ii) its locality in the time
domain, (iii) its short support which eliminates any edge
e€ects in the transformed series, and (iv) its wide use in
atmospheric turbulence research (e.g. see review in [18]),
and (v) its computational eciency. The Haar wavelet
coecients …d† and coarse grained signal (or scaling
function coecients) …S …m† † can be calculated using the
usual multiresolution decomposition on a measurement
vector …f …xj †† originally stored in S …0† for m ˆ 0 to
…0†
M 1, where S ˆ 0 and M ˆ log2 N is the total
number of scales. Throughout, n is time averaging any
variable n over intervals ranging from 20 to 30 min. The
decomposition produces N 1 wavelet coecients de®ning the Haar OWT of …f …xj †† [26,27].
2.2. Wavelet spectra and cospectra
The N 1 discrete Haar wavelet coecients satisfy
the energy conservation (analogous to Parseval's identity)
N 1
X
jˆ0
2
f …j† ˆ
M
X
m† 1
2…MX
mˆ1
iˆ0
2
…d …m† ‰iŠ† ;
…1†
where m is the scale index and i the time (or space) index. The total energy TE contained in a scale Rm is
computed from the sum of the squared wavelet coef®cients in Eq. (1) (recall S…0† ˆ 0) at a scale index …m†
using
TE …Rm † ˆ
1
N
m† 1
2…MX
2
…d …m† ‰iŠ† ;
…2†
iˆ0
where Rm ˆ 2m dy (with dy ˆ fs 1 ) is the time scale, and fs
is the sampling frequency. Hence, the power spectral
density function P …Rm † is computed by dividing TE in
Eq. (2) by the change in frequency DRm …ˆ 2 m dy 1
ln…2†† so that
P …Rm † ˆ
hh…d …m† ‰iŠ†2 iidy
;
ln…2†
…3†
where hh ii in Eq. (3) is averaging overall values of the
position index ‰iŠ at scale index …m†. Notice that the
wavelet power spectrum at Rm is directly proportional to
the average of the squared wavelet coecients at that
scale. Because the power at Rm is determined by averaging many squared wavelet coecients as evidenced in
Eq. (3), the wavelet power spectrum is generally
smoother than its Fourier counterpart and does not
require windowing and tapering. Such averaging in the
wavelet domain is statistically stable because of the
whitening properties of wavelets (later discussed in
greater details). The Haar wavelet cospectrum of two
variables (e.g. w0 and c0 , vertical velocity and CO2 ¯uctuations, respectively) can be calculated from their respective wavelet coecient vectors …dw…m† and dc…m† † at
scale Rm using
Pwc …Rm † ˆ
hhdw…m† ‰iŠ…dc…m† ‰iŠ†iidy
;
ln…2†
…4†
where
w0 c0
M
1 X
ˆ
N mˆ1
m† 1
2…MX
iˆ0
…dw…m† ‰iŠdc…m† ‰iŠ†:
…5†
While w0 and c0 are turbulent ¯uctuations of vertical
velocity and CO2 concentration, with w0 ˆ c0 ˆ 0 (see
[18,19,21]), the above formulations in Eqs. (4) and (5)
are valid for all three scalars.
2.3. Robustness to frequency shifts and gaps in time series
To illustrate the robustness of OWT to time±frequency shifts and to the presence of gaps in the time
series, we consider two synthetic experiments. The ®rst
experiment considers a frequency change within the time
series and contrast its detection by Fourier spectra and
wavelet half-plane imaging. The second experiment
considers a fractional Brownian motion (fBm) time
series, randomly infected by large gaps (typical in longterm ¯ux monitoring), and contrasts the wavelet spectra
G. Katul et al. / Advances in Water Resources 24 (2001) 1119±1132
1121
Notice in Fig. 1 that the Fourier power spectra for
these two signals are nearly identical suggesting that
Fourier spectra alone are not well suited to discern these
frequency shifts. The squared wavelet amplitudes are
shown in the wavelet time±frequency plane in Fig. 1. In
contrast to the Fourier power spectrum, the energy is
resolved with respect to both scale (or frequency) and
temporal extent in the wavelet half-plane. This is an
important quality for studying geophysical phenomenon
that possess temporal variability in the scale-wise distribution of energy (for example, studies of interdecadal
drought variability).
An additional challenge in analyzing long-term ¯ux
measurements is the frequent occurrence of random gaps
in the measurement record. In fact, many long-term ¯ux
monitoring sites within the AmeriFlux and EuroFlux
programs have a data recovery rate below 70% and gap
events whose duration exceeds 10% of the record size [6].
The occurrence of large gaps precludes the use of traditional Fourier type power spectra. Wavelets, and OWT
in particular, o€er distinct advantages here. The wavelet
coecients are localized in space, and orthogonal in their
relationship to each other, thus limiting the e€ects of a
record gap to the coecients local to the gap. In computing the global spectra the a€ected coecients are
simply omitted from the averaging operator. To illustrate, consider a class of non-stationary stochastic processes, known as fractional Brownian motion (fBm)
whose statistical properties in the Fourier and wavelet
domains are well known (see Appendix A for a brief
review). Fig. 2(a) shows an fBm time series with a Hurst
exponent He ˆ 1=3. At random, three unevenly spaced,
large gaps are injected into this time series (Fig. 2(b)).
The gaps collectively represent 30% of the sampling duration for this experiment (a representative value for the
Duke Forest AmeriFlux site). The wavelet image of these
two time series is also shown in Fig. 2(c) and (d). Notice
that the gaps are well localized and can be readily detected in the wavelet half-plane. The wavelet power
spectra of these two fBm series are compared in Fig. 3
with excellent agreement between the original and gapinfected time series. In the gap-infected time series, the
hh ii in the power spectrum de®nition is applied only to
the non-gap-infected coecients at scale index …m†.
Fig. 1. (a) Synthetic time series with and without time±frequency
shifts; (b) standard Fourier power spectra for the two series; (c)
squared wavelet coecients in the time±frequency half-plane.
Fig. 2. The e€ects of gaps on wavelet spectral properties: (a) a standard fBm(1/3) time series; (b) a standard fBm(1/3) time series with 30%
gaps in the record; (c) and (d) the wavelet coecients in the time±
frequency half-plane.
for the original and gap-perturbed synthetic signals. For
the ®rst experiment, consider the two signals …y1 …t† and
y2 …t†† shown in Fig. 1. These two signals are constructed
as follows:
y1 …t† ˆ 0:5 sin…4t† ‡ 0:5 sin…32t†
and
y2 …t† ˆ
sin…4t†;
sin…32t†;
t 6 p;
t > p:
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G. Katul et al. / Advances in Water Resources 24 (2001) 1119±1132
Fig. 3. Comparison between wavelet spectra for an fBm(1/3) signal
with gaps and no gaps. For reference, the standard Fourier power
spectrum for the original fBm(1/3) time series and the theoretical
power-law ( ˆ )5/3) associated with a Hurst exponent He ˆ 1=3 are
also shown.
Fig. 4. Two standard test-signals of Donoho±Johnstone (Doppler and
Blocks). The autocorrelation q…s† as a function of lag …s† is shown for
the time series (dotted) and their wavelet coecients (solid) at the ®nest
level.
3. Experiment
2.4. Whitening property of wavelet transformations
An additional desirable feature of wavelets in dealing
with time series measurements is their decorrelation
property [8,43,47]. Informally speaking, when correlated
measurements in the time domain are transformed to
wavelet domain, the wavelet coecients exhibit little or
no autocorrelation. Because of this property, wavelets
are often called approximate Karhunen±Loeve transformations. In terms of a random P
process X …t†, the Karhunen±Loeve transformation is i Zi /i …t†, where Zi are
uncorrelated random variables and /i …t† are functions
from an orthonormal system, see [43]. To illustrate the
decorrelation property of wavelets, we consider two
standard test functions: blocks and doppler, for a sample
size N ˆ 16384. The two time series are shown in Fig. 4
(top panels). These two test-signals represent frequency
shifts (doppler) and rapid transients (blocks) in time. The
lower panels show the autocorrelation function for both
time series (dotted) as well as the autocorrelation function for their Haar wavelet coecients at the ®nest scale
(or level). Notice that the autocorrelation function of
the Haar wavelet coecients decays very rapidly with
lag for both signals when compared to the autocorrelation function in the time domain. This rapid decorrelation amongst coecients is attributed to the so-called
whitening property of wavelet transformations. This
property is desirable when computing Haar spectral and
cospectral functions in Eqs. (3) and (4) because uncorrelated wavelet coecients provide more stable averages (or statistics) when contrasted to correlated and
long-range dependent coecients. Having discussed
the methods of analysis, the experimental setup for
measuring H, LE, and NEE as well as the physiological
variables and forcing functions is brie¯y described.
3.1. Study site
Hydrometeorological
and
eco-physiological
measurements were conducted at the Blackwood Division of the Duke Forest near Durham, North Carolina
…36°20 N; 79°80 W, elevation ˆ 163 m). These measurements, initiated in August of 1997 and continuing to the
present date, are part of the (AmeriFlux) long-term ¯ux
monitoring initiative. Here, we focus on a long-term
period from mid-August 1997 to August 2000. The site
is a loblolly pine (Pinus taeda L.) forest, planted in 1983.
It extends 300±600 m in the east±west direction and 1000
m in the north±south direction. The mean canopy height
…hc † was 13.5 (0:5 m) in 1998. The topographic variations are small (terrain slope changes < 5%) so that the
in¯uence of topography on the turbulent transport can
be neglected [12].
3.2. Eddy-covariance measurements
The turbulent ¯uxes (of CO2 , water vapor, and heat)
between the land and the atmosphere were measured by
an eddy-covariance system comprised of a CO2 =H2 O
infrared gas analyzer (LI C O R -6262, LI-COR, LI N C O L N ,
NE, USA), a triaxial sonic anemometer (CSAT3,
CA M P B E L L SC I E N T I F I C , LO G A N , UT, USA), and a
krypton hygrometer (KH2 O, Campbell Scienti®c). The
anemometer and hygrometer were positioned 15 m
above the ground surface and anchored on a horizontal
bar extending 1.5 m away from the walkup tower. The
hygrometer was used to assess (and support corrections
of) the tube attenuation e€ects in the infrared gas analyzer. The open path hygrometer also provided a reference against which the lag time of the closed path
G. Katul et al. / Advances in Water Resources 24 (2001) 1119±1132
Fig. 5. Eddy-covariance measured time series of ¯uxes H, LE, and
NEE as well as the spatially averaged soil moisture content h in the top
30 cm and net radiation Rn from August 1997 to August 2000. The
horizontal line in the moisture content time series is the critical at
which further reduction in h induces bulk canopy stomatal closure
(after [36]).
infrared gas analyzer could be measured, as discussed in
[16,17]. Analog signals from these instruments were
sampled at 10 Hz using a Campbell Scienti®c 21X data
logger. All the digitized signals were transferred to a
portable computer via an optically isolated RS232 interface for future processing. Raw 10 Hz measurements
were processed to de®ne 30 min average turbulent ¯uxes
using the procedures described in [16,17,23]. The approximate data recovery rate of the eddy-covariance
measurements, shown in Fig. 5, was 65% with downtime due to storms, hurricanes, pump failure, non-optimal wind directions, loss of electrical power, on-site
calibration of gas analyzer, and periodic maintenance
and factory calibration of instruments. The annual NEE
and LE for this stand vary from 580 to 670 g C m2 yr 1
and from 520 to 580 mm yr 1 (using a 12 month moving
average).
3.3. Other meteorological variables
In addition to the ¯ux measurements, a Ta =RH probe
(HMP35C, CA M P B E L L SC I E N T I F I C ) was positioned at
15.5 m to measure the mean air temperature …T † and
relative humidity. A Fritchen-type net radiometer and a
quantum sensor (Q7 and LI-190SA, respectively, LICOR) were installed to measure net radiation …Rn † and
PAR, respectively. All meteorological variables were
sampled at 1 s and averaged every 30 min using a 21X
Campbell Scienti®c datalogger.
3.4. Hydrologic measurements
Precipitation was measured by a tipping bucket gage
(TE X A S EL E C T R O N I C S , MO D E L 525, DA L L A S , TX)
1123
above the canopy every 1 s and integrated every 30 min.
The gage has a 476 cm2 circular opening catchment
area. Soil moisture content h was measured at 24 locations by an array of Campbell Scienti®c CS 615 soil
moisture content vertical rods, each 30 cm in length. The
mean of all 24 rods is also shown in Fig. 5. The calibration of these rods is described in [35]. We note that
when soil moisture drops below 0.2, cavitation in the
xylem-root system is likely to occur thereby reducing the
stomatal conductance [29,35]. Such low soil moisture
content regimes in the root zone coincide with some of
the extended summer droughts recently experienced in
the Southeast. In fact, based on our three year record at
the site, the time fraction in which the root-zone experienced such low h (or drought) is about 16% as
evidenced from Fig. 5.
3.5. CO2 =H2 O pro®les within the canopy
A multi-port system was installed to measure the
mean CO2 …C† and H2 O…q† concentration inside the
canopy at 10 levels (0.1, 0.5, 1.5, 3.5, 5.5, 7.5, 9.5, 11.5,
13.5 and 15.5 m). Each level was sampled for 1 min (45 s
sampling and 15 s purging) at the beginning, the middle,
and the end of a 30 min sampling duration. The ¯ow
rate within the tube (internal diameter ˆ 1=600 ) was
0:9 l min 1 to dampen all turbulent ¯uctuations. The
modeled concentration pro®les within the canopy volume were compared to these measurements every 30 min
to assess the accuracy of the modeled source distribution
and dispersion calculations. These measurements were
also used to investigate the storage ¯ux terms within the
canopy volume.
3.6. Leaf-level physiological parameters
The leaf-level physiological parameters were determined from gas exchange measurements by a portable
infra-red gas analyzer system for CO2 and H2 O
(CIRAS-1, PP-SY S T E M S , HE R T S ., U.K.) operated in
open ¯ow mode with a 5.5 cm long leaf chamber and an
integrated CO2 gas supply system. The chamber was
modi®ed with an attached Peltier cooling system to
maintain chamber temperature near ambient atmospheric temperature. The data were collected for
upper canopy foliage at heights 95% of the total tree
height, accessed with a system of towers and mobile,
vertically telescoping lifts [5]. These measurements were
collected over a broad range of environmental and
hydrologic conditions spanning a period of two and a
half years (May 1997±October 1999) and were used to
de®ne values of all the canopy physiological parameters
included in a one-dimensional vegetation±atmosphere
coupled photosynthesis±turbulence model (hereafter
referred to as CANVEG).
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G. Katul et al. / Advances in Water Resources 24 (2001) 1119±1132
3.7. Plant area density
The vertical distribution of the plant area density
(PAI) was measured by gap fraction techniques following the theory presented in [34]. A pair of optical sensors
with hemispherical lenses (LAI-2000, LI-COR) was used
for canopy light transmittance measurements from
which gap fraction and plant area densities were calculated. The measurements were made at 1 m intervals
from the top of the canopy to 1 m above the ground to
produce the vertical pro®le in plant area index. PAI
measurements were made within two weeks of each of
the measurement periods at peak PAI, from the same
tower used for measuring the ¯ow statistics, CO2 concentration pro®les and turbulent ¯uxes.
4. Results and discussion
The H, LE, NEE, h, and Rn time series, shown in Fig.
5, clearly exhibit rich variability at multiple scales. When
discussing the spectral properties of land surface ¯uxes it
is helpful to consider three fundamental time scales:
fractions of seconds to minutes (turbulent time scales),
hours to tens of days (meteorological time scales), and
months to years (seasonal time scales). At the very ®ne
time scales, much of the ¯ux variability is induced by
turbulent motion whose production is partly governed
by the existing mean meteorological conditions. At the
hour to days time scale, the interaction between the
physiological and biophysical characteristics of the
canopy and its micro-climate become the dominant
modes of ¯ux variability. Over these ®ne- and mid-range
time scales, the vegetation and its physiological characteristics are assumed, in a ®rst-order analysis, to be
static. At longer time scales, seasonal patterns and
vegetation dynamics (phenology) share importance with
interannual climate variability. The interaction between
available energy, hydrologic status, and the vegetation
function control the critical modes of temporal variability in heat, water vapor and CO2 ¯uxes. Below, we
present a framework for representing the processes at
their respective time scale.
4.1. Turbulent time scales
At time scales shorter than an hour, the processes
governing the land surface ¯uxes are complex eddy
motion. Spectrally, the eddy motion is commonly decomposed into three regimes: production, inertial cascade, and dissipation. The production time scales are
the scales in which turbulence extracts energy from the
meteorological forcing, with in¯uences by canopy
morphology and mean meteorological conditions, particularly the mean longitudinal velocity …U †. Scale-wise
energy distribution in this range can be well described
1=2
by hc , the friction velocity …u ˆ … u0 w0 † †, and the
characteristic scalar concentration ¯uctuation scale
c ˆ w0 u0 =u [38]. Near the canopy±atmosphere interface (i.e. z=hc ˆ 1), the friction velocity (a variable related to the mechanical production of turbulence) is
strongly correlated with U (a variable related to the
mean meteorological forcing) with U =u 3:3 [19,38]
for near-neutral conditions. In the vicinity of z=hc ˆ 1
for rough canopies, the ¯ow is predominantly nearneutral (e.g., >70% of the time for this study). Recent
advances in hydrodynamic stability theory suggested
that much of the organized motion near the canopy±
atmosphere interface, through which the turbulence is
produced, is attributed to Kelvin±Helmholtz type instability which scales with hc and u . In fact, such
theory explained well the spectral peaks in scalar ¯uxes
(e.g. [19]) via a mixing layer analogy. The latter analogy challenged the common paradigm that the structure of turbulence near the canopy±atmosphere
interface resembles a rough-wall boundary layer. The
role of this organized eddy motion on mass and heat
transfer cannot be overemphasized, with more than
90% of the net transport occurring in 10% of the time
fraction [21,23]. In the mid-range of turbulent eddy
scales we encounter the onset of an inertial cascade,
described well by Kolmogorov [9,15,25] type scaling (or
K41). This scaling is evident in many scalar spectra and
cospectra, with Pwc proportional to Rm 7=3 as in Fig. 6,
with the exception of w0 T 0 , for which the power-laws
reported near the canopy±atmosphere interface vary
from Rm 5=3 to Rm 7=3 . The departure from Rm 7=3 has been
attributed to the active role of T 0 in buoyant production or destruction of turbulent kinetic energy, active at
these ®ne scales. At yet smaller scales, the action of
molecular viscosity becomes dominant and the turbulent ¯uctuations are dissipated away as heat. These
small scales are in fact ®ner than the ®nest scales (10
Hz) measured in this study.
4.2. Meteorological time scales
In this range of time scales, the variations in land
surface ¯uxes are due to the strong and often non-linear
interaction between the physiological, radiative, biochemical, and drag properties of the canopy and its
microclimate.
To represent such interactions, we developed a onedimensional multilevel canopy-vegetation model that
resolves the key processes in such interaction. The
model, now known as CANVEG after [1], is described in
[28] but is brie¯y reviewed below. In its primitive form,
the model considers three coupled one-dimensional
scalar conservation equations at a level within the canopy volume of thickness dz, in the absence of advective
transport:
G. Katul et al. / Advances in Water Resources 24 (2001) 1119±1132
(a)
(b)
(c)
Fig. 6. Measured Haar wavelet ¯ux cospectra for w0 T 0 (a), w0 q0 (b), w0 c0
(c), as a function of frequency (Hz) for 41 half-hour runs. For comparisons, the w0 , T 0 , q0 , and c0 time series are normalized to zero-mean
and unit variance. The 7=3 power-law, as derived from K41's locally
homogeneous and isotropic turbulence, is also shown.
oT
ˆ
ot
oq
ˆ
ot
oC
ˆ
ot
oFT
‡ ST ;
oz
oFq
‡ Sq ;
oz
oFc
‡ Sc ;
oz
where P …z; t j z0 ; t0 † is the transition probability density
function that can be estimated from the velocity statistics within the canopy volume via a dispersion matrix as
discussed in [37]. Higher-order Eulerian closure models
for estimating these velocity statistics within the canopy
from the drag and foliage distribution are described
elsewhere (e.g. [20,24]) and are used to compute
P …z; t j z0 ; t0 † or the dispersion matrix. The boundary
conditions used in modeling the ¯ow statistics inside the
canopy needed to compute P …z; t j z0 ; t0 † are presented in
Table 1. In short, Lagrangian ¯uid mechanics principles
provide three additional equations to the set of three
equations in (6). The Lagrangian frame of reference (or
Eq. 7) does not uniquely de®ne such relationships and
an equivalent formulation in the Eulerian frame of reference is also possible (e.g. [22]). The later formulation is
based on higher-order turbulence closure principles for
scalar transport.
To mathematically ``close'' this problem de®ned by
Eqs. (6) and (7), three additional equations are needed
and are derived by assuming the transfer of scalars from
the leaf to the atmosphere is governed by Fickian diffusion through the stomatal cavities and within a thin
boundary layer at the leaf surface. This approximation
leads to:
…T i …z; t† T …z; t††
;
rb …z; t†
…q …z; t† q…z; t††
Sq …z; t† ˆ a…z† i
;
rb …z; t† ‡ rs …z; t†
ST …z; t† ˆ a…z† Sc …z; t† ˆ a…z† …6†
where, FT ˆ w0 T 0 , Fq ˆ w0 q0 , and Fc ˆ w0 c0 are the turbulent ¯uxes at a level z and time t; ST ; Sq , and Sc are the
heat, water vapor, and CO2 sources (‡S ˆ source,
S ˆ sink) within a canopy layer of thickness dz. At
z=hc > 1, q Cp FT ˆ H , Lv Fq ˆ LE, and Fc ˆ
NEE, where Lv is the latent heat of vaporization, q is the
mean air density, and Cp is the speci®c heat capacity of
dry air under constant pressure. Eq. (6) contains nine
unknowns. Using Lagrangian ¯uid mechanics [41], it is
possible to establish a relationship between the sources
and mean concentrations via
Z Z
T …z; t† ˆ
P …z; t j z0 ; t0 †ST …z0 ; t0 † dz0 dt0 ;
Z Z
q…z; t† ˆ
P …z; t j z0 ; t0 †Sq …z0 ; t0 † dz0 dt0 ;
…7†
Z Z
C…z; t† ˆ
P …z; t j z0 ; t0 †Sc …z0 ; t0 † dz0 dt0 ;
1125
…8†
…C i …z; t† C…z; t††
;
rb …z; t† ‡ rs …z; t†
where rb and rs are the boundary layer and stomatal
resistances, respectively, and T i ; qi ; C i are the stomatalcavity temperature, water vapor and carbon dioxide
concentrations, respectively. Hence, while the Fickian
di€usion approximation in Eq. (8) provided the
necessary equations to close the primitive equations, it
introduces ®ve additional unknowns: rb ; rs ; T i ; qi ; C i . In
the CANVEG approach in [28], these unknowns are
determined from the leaf-level energy balance (at each
layer within the canopy) and radiation attenuation
model of Norman [3], the photosynthesis model of
Farquhar [7], the boundary layer resistance …rb † model
of [39], and the stomatal conductance …1=rs † model of
Collatz [4], given by:
1
RH A
ˆm
‡ b;
rs
Cs
…9†
where, m and b are species-speci®c parameters, A is the
carbon assimilation rate which depends on SRc and leaf
h
area density a…z† (related to LAI via LAI ˆ 0 a…z† dz),
and RH …z; t† and Cs are the mean leaf relative humidity
and CO2 concentration at time t and level z. The coupling between the radiative transfer, energy balance, and
1126
G. Katul et al. / Advances in Water Resources 24 (2001) 1119±1132
Table 1
Key eco-physiological, radiative, and drag properties of the pine forest used in the CANVEG model
Variable
Value
Leaf area index
August, 1999
July, 2000
Canopy height …hc †
Characteristic length for calculating rb
Clumping factor
4:52 m2 m
4:31 m2 m
14 m
0.001 m
0.8
Maximum Rubisco capacity (Vcmax †
Ball±Berry slope parameter …m†
Ball±Berry intercept parameter …b†
Maximum quantum eciency
Leaf absorptivity for PAR
64 lmol m 2 leaf s
5.9
0.017
0:08 mol mol 1
0.83
Drag coecient
At z ˆ 1:15hc
ru =u
rv =u
rw =u
dU =dz
0.2
Estimated from [20]
2.4
1.9
1.25
u =k…z
See
See
See
See
[28]
[28]
[28]
[28]
See
See
See
See
[28]
[28]
[28]
[28]
At z ˆ 0
dru =dz
drv =dz
drw =dz
U
0
0
0
0
Comments
2
Measured using LAI-2000
Measured using LAI-2000
Measured in 1999
Measured
Estimated in [28]
2
d†
1
Estimated from porometry
Estimated from porometry
Estimated from porometry
Assumed as in [28]
Assumed as in [28]
The boundary conditions for longitudinal, lateral, and vertical velocity standard deviations (ru , rv , rw , respectively) used in the second-order closure
model are also shown. The mean wind speed U boundary conditions are also presented, with d representing the zero-plane displacement, iteratively
calculated by the closure model, and k ˆ 0:4 is von Karman's constant.
the Farquhar photosynthesis model requires all three
scalars to be simultaneously considered in Eqs. (6)±(8)
and at each canopy layer. These equations are solved for
FT …z; t†, Fq …z; t†, Fc …z; t†, T …z; t†, q…z; t†, C…z; t†, ST …z; t†,
Sq …z; t†, Sc …z; t†, rs …z; t†, rb …z; t†, T i , qi , and C i for speci®ed
mean wind speed …U †, mean air relative humidity, mean
air temperature, mean CO2 concentration, and PAR in
the free space above the canopy every 30 min. The
parameters for the Farquhar photosynthesis model,
Eq. (9), and the radiation/energy balance are shown in
Table 1. The details of implementing these radiative,
energy, and eco-physiological schemes in Eqs. (6)±(9)
are discussed in [28,29]. While several multilayer radiatiative-photosynthesis models have been proposed (e.g.
[32,44,45]), the present CANVEG model has the added
advantage of fully integrating the turbulent transport
dynamics and the microclimate to the sources and sinks
via Eqs. (6) and (7). Hence, canopy morphology (via
a…z†) a€ects the radiation attenuation, the ¯ow statistics
within the canopy, the energy balance, and the scalar
source strength at each level in a consistent manner.
To illustrate the CANVEG model performance, a
comparison between modeled and measured mean CO2
concentration inside the canopy volume is shown in
Fig. 7. The over-all features of CO2 buildup during
night-time and the well-mixed features during day-time
are well reproduced by the CANVEG model. Using a
similar approach but with prescribed within-canopy
velocity statistics, other studies (e.g. [10]) also report
good agreement between measured and modeled CO2
concentration pro®les for a boreal forest. We restate
that many of the coupled photosynthesis±radiative
transfer schemes (e.g. [32,44,45]) cannot predict the
space±time evolution of the mean CO2 concentration,
temperature, and water vapor concentration, but rather
require such concentration measurements as input. A
comparison between eddy-covariance measured and
modeled values of H, LE, and NEE, on a 30 min time
step, are shown in Fig. 8 for the same periods as in Fig. 7
with good agreement between measured and modeled
¯uxes. To assess the spectral recovery of measured H,
LE, and NEE by CANVEG, the measured and modeled
spectra in Fig. 9 are also compared, with reasonable
agreement between these spectra for a wide range of
time scales (1 h to 21 days). We restate that in the above
calculations, the vegetation was assumed static (i.e. a…z†
and all the physiological parameters of the Collatz [4]
model, drag properties, and radiative characteristics
including foliage clumping are not permitted to vary in
time). If the temporal evolution of these canopy physical
and physiological properties are known or speci®ed,
then the CANVEG model can be used to reproduce the
full spectrum of H, LE, and NEE from hours to years.
4.3. Seasonal to annual time scales
While the CANVEG approach can be integrated to
seasonal time scales, the temporal dynamics of the
G. Katul et al. / Advances in Water Resources 24 (2001) 1119±1132
1127
Fig. 7. Comparison between measured and CANVEG modeled mean CO2 concentration for two periods: August 1999 (a) and July 2000 (b).
physiological parameters required by such an approach
are rarely measured at such temporal resolution [5]. For
example, one of the key physiological parameters
needed in the Collatz [4] model adopted in our CANVEG is the maximum Rubisco capacity per unit leaf
area …Vcmax †. For pines, Vcmax depends on leaf nitrogen
concentration, which varies at monthly to annual time
scales. Furthermore, the leaf area density pro®le (as well
as LAI) also varies at seasonal time scales thereby
altering the radiation attenuation, clumping, and
the fraction of sunlit foliage throughout the canopy. Additionally, a…z† dynamics commonly result in phenological and physiological changes with over-wintering
foliage having di€erent m and Vcmax when compared to
newly grown foliage. In short, the land surface ¯ux
variability at monthly and longer time scales is attributed to changes in forcing and meteorological variables
(e.g. Rn ), hydrologic state (e.g. h), eco-physiological
properties (e.g. Vcmax ; m), and ecological properties (e.g.
a…z†† all convolved at multiple time scales. While such
1128
G. Katul et al. / Advances in Water Resources 24 (2001) 1119±1132
Fig. 8. Comparison in the time domain between eddy-covariance
measured and CANVEG modeled H, LE, and NEE time series for the
two periods: August 1999 (a) and July 2000 (b).
Fig. 9. Wavelet spectral comparison between eddy-covariance
measured and CANVEG modeled H, LE, and NEE for the time
series shown in Fig. 8.
physiological changes are likely to pose major challenges
to modeling water and carbon ¯uxes at longer time
scales, a recent study on six European coniferous forests
suggests that long-term water and carbon ¯uxes can be
modeled without using detailed physiological and site
speci®c information [46]. In [46], arti®cial neural networks were used to select a minimal set of input variables to model LE and NEE without resorting to
detailed physiological information. The appeal to a
neural network type analysis is to empirically derive
highly non-linear relationships between forcing variables and ¯uxes. Here, rather than producing such empirical non-linear relationships between forcing
variables and ¯uxes, we explore the characteristic time
scales which exhibit the largest variability in ¯uxes and
forcing variables as well as the scales which exhibit
strongest wavelet spectral correlations.
For this reason, we investigate both measured
wavelet spectral and cospectral characteristics of the
land surface ¯uxes with respect to Rn (energy forcing)
and soil moisture …h† (hydrologic state) in Figs. 10 and
11. In Fig. 10, it is clear that at hourly time scales, the
three scalar ¯uxes exhibit larger variability than Rn and
h (particularly NEE). More important from Fig. 10 is
that the seasonal variability for all the variables is at
least one order of magnitude more energetic (or important) than the diurnal or daily variability. That is,
when annual carbon or water budgets are required, it is
the seasonal variability that is critical to resolve. We
restate that this pine forest is evergreen and retains
foliage (and water ¯ux as well as carbon uptake) yearround. In all the cospectral calculations, the interactions amongst the variables are strongest for seasonal
(200 d) time scales.
For all three land surface ¯uxes, the seasonal time
scales are at least one order of magnitude more im-
Fig. 10. Measured Haar wavelet ¯ux spectra for H, LE, NEE, Rn , and
h time series in Fig. 5. For spectral intercomparisons, all the time series
are normalized to zero mean and unit variance.
G. Katul et al. / Advances in Water Resources 24 (2001) 1119±1132
Fig. 11. Measured Haar wavelet ¯ux cospectra for the time series in
Fig. 5. For comparisons, the cospectra are normalized by covariances
between signals in the Fig. 5 caption.
portant than the diurnal or daily time scales for the
correlations considered. Interestingly, time scales at
which the variability in h ``resonates'' most with variability in LE are quite localized to seasonal time scales
despite several drought events (i.e. h < 0:2† that induced-stomatal closure at shorter time scales (e.g.,
Fig. 5). The correlation at the annual scale is a hydrologic interaction induced mainly by the combined
drought and high LE in the summer and the high soil
moisture and low LE in the winter (despite an even
precipitation distribution at monthly time scales). Note
that the sample size of the soil moisture record is different from the LE record (e.g. Fig. 5). Such a discrepancy in sampling sizes prevent the use of Fourier-type
cospectral calculations, and further lends support to the
use of OWT for such long-term ¯ux analysis. To further explore the relationship between Rn and LE, we
consider their correlation at 30 min and 30 days averaging intervals in Fig. 12. While the relationship between LE and Rn is poor on 30 min time scale, the
opposite is true at monthly (or longer) time scales. This
increased correlation between Rn and LE, may appear
obvious at ®rst glance, as summer months have higher
maximum daily net radiation, longer day-lengths, and
higher LAI when compared to winter months. Less
apparent in this strong correlation between Rn and LE is
that the temporal dynamics of LAI and Rn are not
synchronous as shown in Fig. 12. Note that the rapid
change in LAI occurs over one to two months with little
change thereafter. On the other hand, monthly Rn exhibits variability amongst all months. In other words,
while monthly LE is strongly related to monthly Rn ,
monthly latent heat ¯ux per unit leaf area can be large or
small for a given Rn as evidenced from Fig. 12. The time
scale of foliage dynamics may be further ampli®ed for
deciduous forests in highly seasonal latitudes. Another
1129
Fig. 12. Relationship between Rn and LE at multiple time scales: (a)
the relationship for the 30 min time scale; (b) variations of LE per unit
leaf area at monthly time scales with Rn at the canopy-top; (c) monthly
time scales with the solid line representing equilibrium evaporation; (d)
monthly variation of LAI with Rn .
interesting point is that for Rn > 120 W m 2 the
oLE=oRn is well described by equilibrium evaporation
…D=…D ‡ c††, where D is the slope of the saturation vapor
pressure±temperature curve, and c is the psychrometric
constant. However, for monthly Rn < 80 W m 2 (i.e.
winter months), we note that oLE=oRn < D=…D ‡ c†
coinciding with low foliage and temperature regulation
on stomata. For pines, stomatal closure occurs when the
air temperature drops below 10 °C [5]. Over the entire
year, we found that LE < …D=…D ‡ c††Rn suggesting that
the decrease in LE ¯ux by stomata exceeds the net increase in LE because of ®nite atmospheric drying power.
Finally, we note that the increased correlation between
LE and Rn with increasing time scales is not restricted to
water vapor exchange. We also found similar patterns
for NEE and H as evidenced in Fig. 13.
Fig. 13. Variations of sensible heat H (a) and net ecosystem exchange
NEE (b) on Rn for monthly time scales.
1130
G. Katul et al. / Advances in Water Resources 24 (2001) 1119±1132
5. Conclusions
This study, the ®rst to explore the measured wavelet
spectra and cospectra of land surface ¯uxes from fractions of seconds to three years, demonstrated the following:
· Orthonormal wavelet transformation provides a robust framework for analyzing the spectral and cospectral properties of long-term ¯ux records that
exhibit (1) frequency shifts in time, and (2) multiple
gaps or missing data.
· The wavelet spectra of the three land surface ¯uxes
(carbon, water, and heat) demonstrated that the
largest variability occurs at seasonal time scales. In
fact, the energy or activity at seasonal time scales is
at least one order of magnitude larger than variability
at diurnal (12 h) time scales. As expected, the measured soil moisture spectrum exhibits large activities
at seasonal time scales with little activities at sub-daily scales.
· Three broad categories of time scale models must be
considered when describing temporal dynamics of
land surface ¯uxes: (1) turbulent time scales (seconds to an hour), (2) meteorological time scales
(hours to days), in which physiological properties,
radiative transfer, and diurnal boundary layer dynamics interact, and (3) seasonal time scales (weeks
to year). The wavelet cospectral comparison suggests that for short turbulent time scales, the Kolmogorov theory K41 describes well observed
spectral power-laws. For longer time scales characteristic of turbulent production (100±1000 s), the
mixing layer analogy of Raupach [38] reproduces
well the observed cospectral peaks in the turbulent
¯ux of water and carbon. The CANVEG model reproduced well the H, LE, and NEE spectra over a
broad intermediate range of time scales (hours to
days), as long as the vegetation characteristics can
be assumed static.
· We showed that on seasonal time scales, the interaction between radiative forcing and land surface ¯uxes
is substantially enhanced when compared to the 30
min time scales consistent with a recent study on six
European conifer sites. This enhancement is, in part,
attributed to the intricate balance between the ¯ux
per unit leaf area, leaf area dynamics, and the
seasonal dynamics of Rn .
The broad implications of this study center on the
possibility of developing low-dimensional models of
land surface water, energy, and carbon exchange. If the
bulk of the ¯ux variability is dominated by a narrow
band or bands of modes, and these modes ``resonate''
with key state and forcing variables, then low-dimensional models may relate these forcing and state variables to NEE and LE. In fact, current use of remotely
sensed data to infer the land surface ¯uxes in combi-
nation with low-dimensional models bene®t from such
an analysis in two ways: (1) it identi®es the time scales at
which the remotely sensed observations are likely to
``resonate'' with the ¯uxes and hence the time scales at
which remotely sensed observations must be resolved,
and (2) it identi®es the level of predictability that these
models can o€er given the observed degree of spectral
correlation between these observations and the scalar
¯uxes.
Acknowledgements
The authors would like to thank Yavor Parashkevov
and Cheng-I Hsieh for their assistance in the data collection and Judd Edburn for his overall help at the Duke
Forest. This research was supported by the Department
of Energy (DOE) Na- tional Institute for Global Environmental Change (NIGEC) through the NIGEC
Southeast Regional Center at the University of Alabama in Tuscaloosa (DE-FC03-90ER61010) and the
FACE-FACTS project, the National Science Foundation (NSF) Grants DMS-96-26159, EAR-99-03471, and
BIR-95-12333 at Duke University. The MA T L A B programs can be obtained from the authors on request. The
Duke Forest data sets can be accessed from http://
152.16.58.129/facts-1/facts.html.
Appendix A. Properties of fractional Brownian motion
The fBm is a Gaussian, zero mean, continuous, nonstationary process, indexed by the parameter He (Hurst
exponent, 0 < He < 1) such that
BH …0† ˆ 0; and BH …t ‡ r†
BH …t† N…0; r2H j r j2He †;
where
r2H ˆ C…1
2He †
cos pHe
pHe
R1
and C…x† ˆ 0 tx 1 e t dt is the C function. From its
de®nition, the sample paths of fBm satisfy the scaling
d
equation BH …at†ˆaH BH …t†, and
BH …t†jn i ˆ CfBm jrjnHe ;
hjBH …t ‡ r†
where the CfBm is
r2H
2n=2 C…n ‡ 1=2†
C…1=2†
d
and where ˆ is equal in distribution [8]. The Fourier
transformation of the scaling equation is proportional
n…H ‡1=2†
to 1=jxj e
with an average power spectrum pro2H ‡1
portional to 1=jxj e .
G. Katul et al. / Advances in Water Resources 24 (2001) 1119±1132
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