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Agricultural Water Management 130 (2013) 79– 89

Contents lists available at ScienceDirect

Agricultural Water Management

j ourna l h omepage: www.elsev ier .com/ locate /agwat

ultiscale spectral analysis of temporal variability invapotranspiration over irrigated cropland in an arid region

isheng Dinga, Shaozhong Kanga,∗, Rodrigo Vargasb, Yanqun Zhangc, Xinmei Haoa

Center for Agricultural Water Research in China, China Agricultural University, Beijing 100083, ChinaDepartment of Plant and Soil Sciences, Delaware Environmental Institute, University of Delaware, Newark, DE 19717, USAState Key Laboratory of Simulation and Regulation of Water Cycle in River Basin, National Center of Efficient Irrigation Engineering and Technologyesearch—Beijing, China Institute of Water Resources and Hydropower Research, Beijing 100048, China

r t i c l e i n f o

rticle history:eceived 30 March 2013ccepted 21 August 2013vailable online 13 September 2013

eywords:vapotranspirationpectral analysisavelet transform

oil water contentrrigation

a b s t r a c t

The temporal patterns of evapotranspiration (ET) and its biophysical and physiological controls (e.g., soilwater content, solar radiation, and canopy conductance) occur over a wide range of time-scales rangingfrom seconds to decades. Thus, there is increasing interest in understanding at which scales the maintemporal correlations between ET and its controlling factors occur across different ecosystems. For thisstudy, we used eddy covariance measurements of ET over 2 years at an irrigated maize field in an aridinland region of northwest China. We applied the wavelet transform as a novel technique to examinespectral characteristics of ET and its controlling factors. The ET power spectra displayed a −1 powerlaw in turbulent inertial subrange at <1-h time-scale, and showed substantial power at daily, seasonaland annual time-scales. The cospectra of ET and soil water content (SWC) showed significant temporalcorrelation at 5-days, which has implications for calculation of ET using the soil water balance methodin this region. We found that ET synchronized with the change of net radiation, and led vapor pressuredeficit and air temperature for ∼2 h at the 1-day time-scale (i.e., positive lags), but the phase relationship

between ET and SWC was influenced by irrigation patterns. Canopy conductance influenced ET variabilityat the 1-day time-scale, but the effect was not consistent across the growing seasons. Our results showthe importance of irrigation practices and its influence on the multi-temporal correlations of ET andits controlling factors. Irrigation can sharply change the phase angle relationship of ET and SWC from−10 h to 10 h. These results are important for understanding water cycle processes, improving water

s foo

management, and addres

. Introduction

Evapotranspiration (ET, i.e., latent heat flux) is an important pro-ess for the water cycle, and a major component of the energynd water balance in agricultural ecosystems (Burba and Verma,005). Water used in cropland is mostly lost as evaporation fromoil and transpiration from plants (Rana and Katerji, 2000). Fur-hermore, crops are one of the main users of water consumption inrid inland regions of northwest China, and are highly dependentn irrigation because of low precipitation (Ding et al., 2010; Zhaot al., 2010). Thus, quantification of ET magnitudes and dynamicss necessary to develop efficient irrigation schemes and to improve

ater resources management (Ding et al., 2010; Kang et al., 2003;

hang et al., 1996). Several studies have highlighted that tem-oral variability of ET in natural vegetation occurs over a wideange of time-scales ranging from seconds to decades (Baldocchi

∗ Corresponding author. Fax: +86 10 62737611.E-mail address: kangsz@cau.edu.cn (S. Kang).

378-3774/$ – see front matter © 2013 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.agwat.2013.08.019

d security issues across irrigated croplands in arid regions.© 2013 Elsevier B.V. All rights reserved.

et al., 2001; Jarvis, 1995; Kaimal et al., 1972). Despite the impor-tance of the temporal variability of ET for crop irrigation, it is stillunclear how physical, physiological controls (e.g. soil water con-tent, solar radiation and canopy conductance), and management(i.e., irrigation practices) influence ET in arid inland agriculturalsystems. This understanding is important for improving watermanagement practices and addressing food security issues acrosscroplands in arid regions.

The ET dynamics comprehensively reflects interactions andfeedbacks between micrometeorological and crop physiologicalecology (Allen et al., 1998; Kisi, 2011). For instance, crop ET fluxescould change local and regional weather and climate by influenc-ing vapor pressure (Brutsaert, 1982). The change of climate (e.g.,changes in air temperature and humidity) could influence atmo-spheric evaporation capacity, and this may affect the temporalvariations of crop ET (Baldocchi et al., 2001). Previous studies have

suggested that the controlling factors over the temporal variation ofET change over time-scales (Kang et al., 2003; Lei and Yang, 2010;Suyker and Verma, 2008). At the scale of seconds, ET variabilityis mainly affected by the complex turbulent eddy motion (Katul

8 ater M

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0 R. Ding et al. / Agricultural W

t al., 2001). At daily to monthly scales, much of ET variation isnfluenced by weather patterns, crop ecophysiological character-stics, and soil water state (Allen et al., 1998; Steduto and Hsiao,998). For monthly to seasonal time-scales, ET is determined by theeasonal pattern of solar radiation, and crop phenology (i.e, tim-ng of leaf expansion and growth, and leaf-out) (Baldocchi et al.,001; Steduto and Hsiao, 1998; Suyker and Verma, 2008). There-ore, it is important to determine the main factors affecting ET inrder to understand the temporal variation of ET and to providenformation for better parameterization of ET models over differ-nt time scales (Burba and Verma, 2005; Katul et al., 2001; Stedutond Hsiao, 1998).

For this study, we used spectral analysis techniques (i.e., waveletransform) to better understand the temporal patterns of ET andhe temporal correlations with its controlling factors over multipleime-scales (Kaimal et al., 1972; Torrence and Compo, 1998; Vargast al., 2010). Spectral analysis is an alternative technique to evaluateomplex temporal patterns in the frequency domain by identifyinghe main periodic components (Kaimal et al., 1972). Furthermore,he analysis of the temporal correlations between two time seriesould draw insights about relationships of control factors at mul-iple time-scales and can provide information on the nature andrigin of coupling between the processes, and causality under thessumption that the effect must follow the cause (Baldocchi et al.,001; Torrence and Compo, 1998; Vargas et al., 2010).

Previous studies have used Fourier transform to examine peri-dic features of atmospheric turbulence (Kaimal et al., 1972) andatural land surface fluxes (Baldocchi et al., 2001). However, thisethod fails in presence of non-stationary and non-continuous

ime series (Katul et al., 2001; Vargas et al., 2010). Due to irrigationulses and crop phenology, maize ET fluxes were typically non-tationary time series and had large data gaps between growingeasons. Wavelet transform could localize gaps and eliminate themrom the spectral values, and resolve the non-stationary timeeries in both time and scale (frequency) domains (Kumar andoutoula-Georgiou, 1997). We refer the reader to previous stud-es that have described the use of wavelet transform (Grinstedt al., 2004; Kumar and Foutoula-Georgiou, 1997; Li et al., 2012;artal, 2009; Sen, 2009; Torrence and Compo, 1998; Yoshida et al.,010). The scientific questions addressed in this study are: (1)hich time-scales are the most dominant for the temporal vari-

bility of ET in a maize plantation for an arid inland region inorthwest China? (2) Which are the temporal correlations betweenhe biophysical and physiological factors (i.e., solar radiation, soilater content, and canopy conductance) that control maize ET

n an arid region? We hypothesize that: (a) ET will show strongaily and seasonal fluctuations that will reflect biophysical pro-esses controlled by patterns of solar radiation (i.e., day–night,lant phenology); and (b) irrigation practices may have a substan-ial influence on the temporal variability of ET and the temporalorrelations between ET and its biophysical controls. This study fills

gap in the understanding of the temporal variability of ET to betternderstand the underlying water processes and to improve wateresources management across croplands in arid inland regionsSteduto and Hsiao, 1998; Suyker and Verma, 2008). Finally, thiss a pioneer work for maize plantation in an arid region in northernhina.

. Materials and methods

.1. Study site and measurements

The experiment was conducted at Shiyanghe Experimentaltation for Water-saving in Agriculture and Ecology of China Agri-ultural University, located in Gansu Province of northwest China

anagement 130 (2013) 79– 89

(N 37◦52′, E 102◦50′, Altitude 1581 m) during two full growing sea-sons between 2008 and 2009. The sunlight hours have a meanannual sunshine duration >3000 h, the mean annual temperatureis 8 ◦C, and the site has a mean of 150 frost-free days. The regionis limited in water resources with a mean annual precipitationof 164 mm and a mean annual pan evaporation of 2000 mm. Theaverage groundwater table is below 30 m depth.

Spring maize was sown in the experimental field with anorth–south length of 700 m and a west–east width of 300 m onMay 3 2008 and April 21, 2009. It was harvested on September 25,2008 and September 28, 2009. Soil surfaces were partly mulchedwith plastic film, which is a well-established management strat-egy and widely used in the region (Hou et al., 2010; Zhou et al.,2009). The width of plastic film was 100 cm, and there was a 65 cmbare soil between two plastic films rows. Maize was sown in holesof 5.0 cm diameter under the plastic film, with a row spacing of50 cm and a plant spacing of 23.8 cm, i.e., maize was not sown overthe bare soil. Therefore, this planting pattern had an actual densityof 76,300 plants ha−1, and actual mulching fraction of ∼0.5, whichwas defined as the bare soil surface and holes-sowing areas per unitsoil area. For the 0–1.0 m soil depth, soil type is silt loam, with bulkdensity of 1.45 g cm−3, a field capacity of 0.32 m3 m−3, and a wiltingpoint of 0.10 m3 m−3. The irrigation regime and crop managementwere listed in Table 1.

Evapotranspiration was measured using an eddy covariance sys-tem, which was installed in the center of maize field during two fullgrowing seasons between 2008 and 2009. The eddy covariance (EC)system consists of a fast response 3D sonic anemometer (CSAT3,Campbell Scientific Inc., Logan, UT, USA), a Krypton hygrometer(KH20, Campbell Scientific Inc., Logan, UT, USA), a temperature andhumidity sensor (HMP45C, Vaisala Inc., Helsinki, Finland) all con-nected to data loggers (CR5000, Campbell Scientific, Inc., Logan,UT, USA). The sonic anemometer and Krypton hygrometer wereinstalled at a 3.5 m height above ground level. Net radiation (Rn)was measured by a net radiometer (NR-LITE, Kipp & Zonen, Delft,Netherlands), which was installed also at a height of 3.5 m. Twosoil heat flux plates (HFP01, Hukseflux, Delft, Netherlands) wereinstalled at 8.0 cm soil depth under the plastic film and the bare soil,respectively. Soil temperature above the soil heat flux plates wasmeasured with thermocouples at depths of 2.0 cm and 6.0 cm in linewith each soil heat flux plate, and soil moisture at 0–10.0 cm wasmeasured using an EnviroSMART soil moisture reflectometer (Sen-tek Sensor Technologies, SA, Australia), which was calibrated byoven drying of soil samples. Ground heat fluxes (G) were estimatedby using the heat fluxes at 8.0 cm along with soil heat storage abovethe transducers. The soil heat storage above 8.0 cm was determinedfrom changes in soil temperature and moisture above the trans-ducers. The instrumentation and fluxes correction were describedin detail in Ding et al. (2010).

Volumetric soil water content (SWC) in the root zone (0–1.0 m)was measured with eight PVC access tubes using the portabledevice Diviner 2000 (Sentek Sensor Technologies, SA, Australia).Measurements were made at an interval of 0.1 m with maximalsoil depth of 1.0 m at intervals of 3–5 days. Additional samplingwas conducted before and after irrigation events, and after rain-fall events. The SWC measurements were calibrated by oven dryingof in situ soil samples (gravimetric method). Linear interpolationswere applied between measured days to determine the SWC for thegrowing season.

Ten maize plants were randomly selected to measure leaf lengthand width at intervals of approximately 10 days during the grow-ing season. Leaf area was calculated by summing rectangular area

of each leaf (i.e., [leaf length × maximum width]) multiplied by afactor of 0.74, which was obtained by analyzing the ratio of rectan-gular area to real area, measured by an AM300 (ADC BioScientificLtd., UK). Continuous Leaf Area Index (LAI) was obtained by fitting

R. Ding et al. / Agricultural Water Management 130 (2013) 79– 89 81

Table 1Crop management and irrigation scheduling over the whole growing seasons in 2008 and 2009.

Year Planting date Emergence date Harvest date Irrigation scheduling

Irrigation date Irrigationamount (mm)

2008 May 2 May 11 September 25 June 12 110July 7 100July 27 105August 23 95

2009 April 21 May 1 September 28 June 15 105July 6 105July 29 105

N

ota

2

w

R

wfiabt

cMaae

G

wa�((a

G

wf

2

apcqs2ttbti

ote: Irrigation water amount was measured using pump meter in each event.

bservations with the Days After Sowing (DAS) using a single equa-ion (LAI = a × tb × exp(−r × t), where t is DAS, r is a rate of LAI change,nd a, b are fitted coefficients) (Hashimoto, 1990).

.2. Parameters calculation

The relative extractable water (REW) of soil in the crop root layeras defined as:

EW = SWC − SWCwSWCF − SWCw

(1)

here SWC (cm3 cm−3) is soil water content, SWCF (cm3 cm−3) iseld capacity, and SWCw (cm3 cm−3) is wilting point. The total avail-ble soil water (TAW) in the root zone was defined as the differenceetween SWCF and SWCw. The 50% of TAW is referred to as thehreshold of maize water stress (Suyker and Verma, 2008).

The canopy conductance (Gc) is a key variable reflecting arop’s physiological response to changing environment (Jarvis andcNaughton, 1986). Gc represents a strong response to net radi-

tion (Rn), vapor pressure deficit (VPD), air temperature (Ta), LAI,nd SWC, and was calculated by inverting the Penman–Monteithquation (Allen et al., 1998):

c = ��ETGa�(Rn − G) + �aCpGaVPD − �ET(� + �)

(2)

here �a (kg m−3) is air density, Cp (J kg−1 K−1) is specific heat of dryir at constant pressure, � (J kg−1) is the heat of water vaporization,

(kPa ◦C−1) is the slope of the saturation vapor pressure curve, �kPa ◦C−1) is psychrometric constant, Rn − G is the available energyW m−2). Ga (mm s−1) is aerodynamic conductance and estimateds follows (Monteith and Unsworth, 2008):

a =(u

u2∗+ 6.2u−2/3

∗

)−1

(3)

here u (m s−1) is the mean wind velocity and u* (m s−1) is theriction velocity measured by the EC system.

.3. Wavelet spectral analysis

We start by describing terms commonly used for time seriesnalysis, which are relevant for our study. The power spectrumrovides information on how much variance is associated with spe-ific frequencies (Kaimal et al., 1972). In addition, the cospectrumuantifies the amount of covariance that occurs between two timeeries, x and y, across a spectrum of frequencies (Baldocchi et al.,001). Coherence spectrum gives information on the local correla-ion between two time series (Torrence and Compo, 1998). Finally,

he phase angle spectrum reflects synchronization of oscillationetween the two time series (Grinsted et al., 2004). Importantly,he phase leads or lags between two time series and could providenformation on the nature and origin of coupling between two

August 20 105

processes, or causality under the assumption that the effect mustfollow the cause (Grinsted et al., 2004).

We used the orthonormal wavelet transforms (OWT) to cal-culate the power spectrum and cospectrum between ET and itscontrolling factors (Yoshida et al., 2010). We used this techniquedue to fast processing speed, low computational cost, and the abilityto remove the effects of data gaps (nearly 55% of our measurements)due to the defined growing seasons of maize (i.e., about half of theyear there are no plants in the field). We used continuous wavelettransforms (CWT) to calculate coherence and phase angle spectrabetween ET and its controlling factors because CWT can capturelocal information and causality between both time series (Grinstedet al., 2004; Li et al., 2012; Torrence and Compo, 1998). For spectralinter-comparisons, all time-series are normalized to zero mean andunit variance.

The OWT of time series x(t) with square-shaped Haar functionis described as follows:

Wx(j, k) =N−1∑t=0

x(t)2− j2 (2−jt − k) (4)

where Wx(j,k) is the Haar wavelet transform coefficient of timeseries, x(t); (t) is mother wavelet function; N is the total num-ber of time series (power of two); j is the scale index (0 to M − 1,M = log2 N); k is the time or space index (0 to 2M−j). The global powerspectrum was defined as the averaged variance contained in allwavelet coefficients at the scale range:

Px(j) = 12M−j

2M−j−1∑k=0

(Wx(j, k))2 = (Wx(j, k))2 (5)

where the overbar represents the averaging overall values ofwavelet coefficients square at scale index (j). The cospectrabetween two time series, x(t) and y(t), were given as:

Coxy(j) = (Wx(j, k)) · (Wy(j, k)) (6)

where Wx(j,k) and Wy(j,k) represent wavelet transform coefficientsof two time series, respectively.

Eddy covariance (EC) technique directly measured vertical windspeed (w) and water vapor density (�v) at a frequency of 10 Hz.Therefore, instant ET was calculated by covariance of fluctuationbetween w and �v (w’ and �v’) in the sampling period (such as30 min), i.e., ET = w’�v’. The numerical scheme applied for OWT wasa power of two in this study, where the number of 16,384 (214) isthe closest power of two for 10 Hz ET. The total number of 30 min ETtime series was 32,768 (215), equivalent to 682 days. As defined bythe Nyquist frequency, the highest frequency being resolved was

half of the sampling frequency. So, the lowest and highest frequen-cies we could resolve with OWT were 0.00122 Hz and 5 Hz for 10 Hzsampling frequency, and 0.000122 h−1 and one cycle per hour (1 hto 341 days duration) for 30 min ET.

82 R. Ding et al. / Agricultural Water Management 130 (2013) 79– 89

F over t( as thL

M

W

w(ada

G

wC

C

wwad

˚

wpbv

urac

ig. 1. Seasonal variation of (a) soil water content (SWC), and (b) leaf area index (LAI)P) are included. The total available soil water (TAW) of 50% in the root zone is referredAIo is observed value and LAIe is fitted value.

The CWT of a time series x(t) with length N, with respect to aorlet wavelet function (t), was defined as (Li et al., 2012)

x(s, �) = 1√s

N−1∫0

x(t) ∗(t − �

s

)dt, s > 0 (7)

here * is the complex conjugate of Morlet wavelet function Sen, 2009); s is the scale dilation; and � is the time index. The snd � are variables within length of time series according to theifferent wavelet function. The cross spectrum (Gxy) was defineds:

xy(s, �) = Wx(s, �)W∗y (s, �) (8)

here Wy* is the complex conjugate of wavelet coefficient Wy.

oherence spectrum was defined as:

oh2xy(s) =

∣∣⟨s−1Gxy(s, �)⟩∣∣2⟨

s−1∣∣Wx(s, �)

∣∣2⟩ ⟨s−1∣∣Wy(s, �)

∣∣2⟩ (9)

here 〈〉 indicates smoothing in both scale and time; the factor s−1

as used to convert energy to power spectral density. The phasengle spectrum, i.e., complex argument of cross spectrum (Gxy), wasefined as:

xy(s) = tan−1

(Im{⟨s−1Gxy(s, �)

⟩}Re{⟨s−1Gxy(s, �)

⟩})

(10)

here Im indicates the imaginary part, and Re indicates the realart of cross spectrum. The phase shift describes the delay or leadetween two time series at a specific scale and time, with possiblealues ranging from −180◦ to 180◦.

The statistical significance of coherence spectra was assessed

sing Monte Carlo methods (Torrence and Compo, 1998). The sur-ogate datasets of 10,000 were generated with the same first orderutoregressive coefficients for each time series of two, and theorresponding coherence calculated. The 5% significance level was

he whole growing seasons of maize in 2008 and 2009. Irrigation (I) and precipitatione threshold of crop water stress, below which crop development will be constrained.

estimated for each scale using only values outside the cone of influ-ence (COI), in which wavelet transform might as well suffer fromedge effects, so the results in COI are unreliable (Torrence andCompo, 1998). Wavelet transforms were performed using MAT-LAB R2010a (The MathWorks Inc.), the WAVELAB 850 package(http://www-stat.stanford.edu/∼wavelab/), and wavelet analysissoftware written by Grinsted et al. (2004), Torrence and Compo(1998) and Torrence and Webster (1999).

3. Results

3.1. Wavelet power spectra of hydrometeorological variables andleaf area index

Mean seasonal Ta was 19.1 and 18.1 ◦C, VPD was 1.36 and1.20 kPa, Rn was 115 and 106 W m−2, and total precipitation was78.0 and 118.8 mm for 2008 and 2009, respectively. These resultsindicate that drought was stronger during the growing season in2008 than 2009. The annual precipitation also shows that 2008was dryer (124 mm) than 2009 (158 mm), which was closer to thelong-term mean precipitation record. SWC in the root zone var-ied greatly over the whole growing season (Fig. 1a). The variabilityof SWC was attributed to irrigation scheduling that depended onirrigation amount and timing (Table 1). SWC had a peak value afterirrigation and reduced gradually until the next irrigation (Fig. 1a).LAI showed a clear “single peak” pattern over the whole growingseason for both years (Fig. 1b), with maximum values of 4.5 and5.4 m2 m−2, and mean values of 2.7 and 3.1 m2 m−2 for 2008 and2009, respectively.

ET totals were 503.1 and 562.4 mm (see Ding et al., 2013), withsupply water amount of net irrigation plus net precipitation (I + P)of 488.0 and 538.8 mm for the two years, respectively. The ratios ofET to I + P were close, with 1.03 and 1.04, for the two years, respec-

tively, indicating that water budgets were nearly in balance. Theexcess ET originated from soil water storage before crop sowing.These results suggest that ET measurements using eddy covariancewere accurate for the two years.

R. Ding et al. / Agricultural Water Management 130 (2013) 79– 89 83

10-2 10-1 100 101 102 103

Nor

mal

ized

pow

er s

pect

ra

10-6

10-5

10-4

10-3

10-2

10-1

100

101

Rn

Ta

VPDu

100 101 102 103

10-6

10-5

10-4

10-3

10-2

10-1

100

101

LAISWC

(a) (b)

Time-scales (d) Time-scales (d)

F it (VPD( the v

afRodtdiwa

aihttsd

3c

b(d5ot

Fts

ig. 2. Haar wavelet global power spectra of net radiation (Rn), vapor pressure deficSWC). The spectral densities are multiplied by nature frequency and normalized by

The Haar wavelet global power spectra of meteorological vari-bles (Rn, VPD, Ta, and u) presented both common and distincteatures involving the position of spectral peaks and gaps (Fig. 2a).egarding common features, the power spectra for all four mete-rological variables showed pronounced spectral peaks at daily (1ay at x axis) and annual (341 days) time-scales. In addition, most ofhe meteorological variables had spectral gaps at monthly (21–43ays) and semi-annual time-scales (171 days). There were signif-

cant differences in the power magnitude at the daily time-scales,here Rn spectrum was an order higher than VPD and u spectra,

nd two orders higher than Ta spectrum (i.e., Rn > VPD > u > Ta).The power spectra of LAI exhibited a spectral cascade from

nnual to daily time-scales, with a slope of about 2.0 (Fig. 2b). Thenter-annual variation for LAI was at least five-orders of magnitudeigher than daily variation. The power spectrum of SWC displayedhe pronounced spectral cascade with the slope of 2.0 from seasonalo daily time-scales (Fig. 2b). The gap of SWC spectrum occurred atemi-annual scale, corresponding to the end of irrigation eventsuring the growing seasons of maize.

.2. Wavelet power spectrum of ET and its cospectra withontrolling factors

The dynamic of 10 Hz ET presented rich variability at unsta-le atmospheric condition for a typical half-hourly measurementFig. 3a). The wavelet power spectrum of ET time series was

escribed by a −1 power law with an energetic cascade from 0.04 to.0 Hz (Fig. 3b), indicating that the spectral energy of ET dependednly on the turbulence at less than 5.0 Hz, beyond which the spec-rum is affected by the viscosity (Katul and Parlange, 1995).

Time (s)

0 30 0 60 0 90 0 120 0 150 0 180 0

ET (m

m s

-1)

-3e-3

-2e-3

-1e-3

0

1e-3

2e-3

3e-3

4e-3(a)

ig. 3. Wavelet power spectrum for 10 Hz ET time series: (a) raw data at unstable atmosransform. The −1 power law (thick solid line) is shown. The spectral densities are multieries.

), air temperature (Ta), wind speed (u), leaf area index (LAI), and soil water contentariances of the associated time series.

The data recovery rate of eddy covariance measurements wasnearly 55% due to a large gap between the two years, but this didnot affect our measurements during two full growing seasons ofmaize (Fig. 4a). ET time series clearly presented large variability atmultiple scales, and the ET global power spectrum exhibited dis-tinct spectral peaks at daily, seasonal and annual scales (Fig. 4b).At week time-scale, the global power spectrum of ET in 2009 hadlarger variability than that in 2008 (Fig. 4c). In contrast, at monthlytime-scales, the spectral energy of ET in 2008 was larger than thatin 2009.

The wavelet cospectra between 10 Hz wind speed (w′) and vapordensity (�v

′) showed a −4/3 power law line with energetic cas-cade from 0.04 to 5.0 Hz (Fig. 5a). The cospectra between ET and allcontrolling factors demonstrated the strongest interactions at theannual time-scale (Fig. 5b). The inter-annual magnitude of cospec-tra was at least twice more energetic than that at daily time-scale.Furthermore, the cospectra were similar, indicating that long-termET variability was commonly influenced by all controlling variables.At daily time-scale, the cospectra between ET and meteorologicalvariables were greater than those of ET and both SWC and LAI, andthe cospectrum of ET and Rn was the greatest, suggesting that Rn

was the main influential factor on ET variation at the scale.

3.3. Wavelet coherence and phase spectra between ET andcontrolling factors

The higher coherence spectra between ET and Rn indicated thatthere was higher correlation at daily time-scale during the wholegrowing seasons of 2008 and 2009 (Fig. 6a and b). At daily time-scale, ET and Rn were in phase for the two measured years (Table 2).

Frequency (Hz)

10-3 10-2 10-1 100 101

Nor

mal

ized

pow

er s

pect

ra

10-3

10-2

10-1

100

101

Haar wavelet-1

(b)

pheric condition for half-hour run; (b) global power spectrum using Haar waveletplied by nature frequency and normalized by the variances of the associated time

84 R. Ding et al. / Agricultural Water Management 130 (2013) 79– 89

0 50 100 150-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

350 400 45 0 500 680

Date (d)

ET (

mm

h-1

)

(a)

Time-scales (d)

10-2 10-1 100 101 102 103

Nor

mal

ized

pow

er s

pect

ra

10-5

10-4

10-3

10-2

10-1

100

101

20082009Two years

(b)

Fig. 4. Wavelet power spectrum for half-hourly ET in 2008 and 2009: (a) raw time series during the whole growing seasons of maize; (b) global power spectrum using Haarwavelet transform. The spectral densities are multiplied by nature frequency and normalized by the variances of the associated time series.

Fig. 5. Haar wavelet global cospectra: (a) between 10 Hz vertical wind speed fluctuations (w′) and vapor density (�v′) for 36 half-hour runs. The −4/3 power law line is also

s eratua spectro

SldIt2s2dTc(

crsisw

TAa

N

hown; (b) between half-hourly ET measurements, and net radiation (Rn), air temprea index (LAI), canopy conductance (Gc) and aerodynamic conductance (Ga). The

f the associated time series.

imilar to the coherence spectra of ET and Rn, a strong significantlyocal correlation between ET and VPD was observed at daily scaleuring the whole growing seasons of 2008 and 2009 (Fig. 6c and d).

n contrast, the phase angles of ET and VPD were out of phase at dailyime scale (Table 2), where variation of ET preceded VPD by about

h. Furthermore, the coherence spectra between ET and VPD weretronger and in phase (i.e., no lags) at intermediate time-scales (i.e.,-days to monthly scales). The phase angles between ET and Ta alsoisplayed similar observations as between ET and VPD (Table 2).he correlation between ET and u was not so strong compared tooherence spectra between ET and other meteorological variablesTable 2).

The coherence spectra between ET and SWC showed that higherorrelation was associated with discrete irrigation pulses as rep-esented by the dark red areas inside the contour lines at daily

cale (Fig. 7a and b). Irrigation pulses appeared to have a strongernfluence on ET during the early growing season (before days afterowing; DAS of 50). The phase angle relationship of ET and SWCas sharply changed by the irrigation pulses (Fig. 7c and d). For

able 2verage phase angles (plus and minus one standard deviation in hours) at the daily timengles greater than 0 represented ET leading meteorological variables; otherwise ET lagg

Year ET − Rn ET − VPD

2008 −0.5 ± 0.41 2.1 ± 1.12

2009 −0.4 ± 0.30 2.2 ± 0.72

ote: Rn is net radiation, VPD is vapor pressure deficit, Ta is air temperature, and u is horiz

re (Ta), vapor pressure deficit (VPD), wind speed (u), soil water content (SWC), leafal densities are multiplied by nature frequency and normalized by the covariances

instance, the phase shift varied from -10 h to 10 h at the first eventof irrigation (DAS 40 and 42 in Fig. 7c and d), indicating that phaserelationship was changed from SWC limiting ET before irrigation toET determining SWC variation after irrigation.

The coherence spectra between ET and Gc presented that highercorrelation occurred at daily time-scale (Fig. 8a and b). The meanphase angles between ET and Gc were −1.0 and −0.45 h in 2008 and2009 (Fig. 8c and d), indicating that mean ET lagged Gc for the twoyears. In addition, the amplitude of daily phase angle had greaterranges from less than −2 h to close to 1 h. Our findings presentedthat the phase angle between ET and Gc was dynamic during thewhole growing period, whereas at the early and later growing sea-sons, ET lagged Gc more, suggesting that Gc had the greater controlof ET during the period.

4. Discussion

The motivation for this study was to understand at which scalesthe main temporal correlations between ET and its controlling

scale for the relationship between ET and meteorological variables. Average phaseing meteorological variables.

ET − Ta ET − u

2.1 ± 1.07 2.2 ± 3.172.3 ± 0.73 1.7 ± 1.95

ontal wind speed.

R. Ding et al. / Agricultural Water Management 130 (2013) 79– 89 85

Fig. 6. Morlet wavelet coherence and phase spectra between ET and meteorological variables during the whole growing season of maize in 2008 and 2009: (a) and (b) netradiation (Rn), and (c) and (d) vapor pressure deficit (VPD). DAS represents days after sowing. The black thick contour is 95% confidence level. The darker red areas inside thecontour lines represent the significant local correlation at 5% level. The phase relationship between two time series, X and Y, is shown as arrows: in-phase pointing right (nolagging between X and Y), anti-phase pointing left (lagging by 180◦ between X and Y), X leading Y by 90◦ pointing down, and X lagging Y by 90◦ pointing up. The white dashl

ftwrlpat21idwfo

stpsoTwapi

tbqF

ine represents cone of influence outside which edge effects become significant.

actors occur in an irrigated cropland of an arid region. Informa-ion to optimize crop water use with respect to yield can provideater resource managers with the tools necessary to improve water

esources efficiency and address food security issues in water-imited regions. Our findings showed that the Haar wavelet globalower spectra of meteorological variables had the spectral peakst 1-day and 341-days, representing meteorological day-night fluc-uations and inter-annual variability (Fig. 2a). The spectral gaps at1–43 days represent synoptic events of weather fronts and the71-days represents the fallow cycle of the maize (i.e., the grow-

ng season for this crop). The highest power magnitude of Rn at theaily time-scales supports that Rn was the meteorological variableith the strongest fluctuations at the daily time-scale, and there-

ore is the driving variable for the daily patterns observed for thether variables (Baldocchi et al., 2001; Steduto and Hsiao, 1998).

The spectral cascades of LAI and SWC at annual to daily time-cales, and seasonal to daily time-scales show the relevance ofhe growing seasons to LAI and SWC and represent crop long-termhenology (Fig. 2b, Allen et al., 1998). The slope in the region of thepectral cascade was 2.0, indicating that the variation in amplitudef LAI and SWC was a function of power with respect to time-scales.his result suggests that the temporal variability of LAI and SWCas well related to time-scales. The peak of SWC spectrum at inter-

nnual scale demonstrated the strong influence of human irrigationractices on the annual cycle of SWC as seen previously in other

rrigated crops (Kang et al., 2000).From Fig. 3b, the spectral slope of −1 of 10 Hz ET suggests that

he spectral energy of ET in the range of cascade was not depletedut transported by turbulence from higher frequency to lower fre-uency according to the −1 power law (Foken, 2008; Stull, 1988).urthermore, the results of the wavelet cospectra between 10 Hz

wind speed (w′) and vapor density (�v′) also suggest that that the

processes governing vapor transport (i.e., maize ET) were complexturbulent eddy motion at time-scales less than an hour (Katul et al.,2001).

From Fig. 4b, we can see that the annual variability of ET hadabout two orders of magnitude more spectral power than the dailyvariability, but the magnitude varied between years. This resultsuggests that the annual variation of ET could be larger for differ-ent hydrological years and therefore provides insights about theamount of water needed for crop irrigation (Kang et al., 2000; Ranaand Katerji, 2000; Zhang et al., 2011). For example, lower spec-tral power may represent less irrigation to compensate for the soilwater deficit. However, it is necessary to determine the annualamount of ET across multiple hydrological years, which is criticalfor proper managing regional water allocation and improving wateruse efficiency (Howell et al., 1998; Zhang et al., 2011).

The ET spectral peak at daily time-scale was expected, andsupports previous observations that ET variation experienced a pro-nounced daily fluctuation (Allen et al., 1998; Howell et al., 1998). Inaddition, the fact that the greater variability of power spectrum ofET in 2009 than that in 2008 at weekly time-scale was attributed toET primarily affected by meteorological variables in 2009 and notsuffering from soil water deficit (Fig. 1a). Conversely, the greatervariation of the global ET power spectrum in 2008 at monthly time-scale was due to ET controlled by soil water supply that was mainlydetermined by irrigation in the arid region because of low precipi-tation (Ding et al., 2013; Zhang et al., 2012). These results support

our hypothesis that ET will show strong daily and seasonal fluc-tuations, but suggest that irrigation and climate variability couldinfluence especially the seasonal fluctuations of ET. The fact thatET had pronounced spectral peaks at different time-scales is a key

86 R. Ding et al. / Agricultural Water Management 130 (2013) 79– 89

Fig. 7. Morlet wavelet coherence and phase spectra between ET and soil water content (SWC) during the whole growing season of maize in 2008 (a) and 2009 (b). DASrepresents days after sowing. Phase angle dynamics between ET and SWC are shown in (c) and (d) at the daily time scale. In (a) and (b), the black thick contour is 95%confidence level. The darker red areas inside the contour lines represent the higher local correlation at 5% significance level. The phase relationship between two time series,X and Y, is shown as arrows: in-phase pointing right (no lagging between X and Y), anti-phase pointing left (lagging by 180◦ between X and Y), X leading Y by 90◦ pointingd nfluenr

aa(

wpqdTikattlf1

dssbtse1caDr

own, and X lagging Y by 90◦ pointing up. The white dash line represents cone of iepresent irrigation events in the growing season.

ttribute that process-based ET models should test and take intoccount in order to reproduce ET dynamics at multiple time-scalesKatul et al., 2001; Zhang et al., 1996).

Spectrally, the eddy motion at turbulent scale less than an houras commonly decomposed into three characteristic regimes asrovided by Kolmogorov scaling theory: production in lower fre-uency region (spectral peak), inertial cascade (constant slope), andissipation in higher frequency region (Foken, 2008; Stull, 1988).he three processes were linked through the relationship as shownn Fig. 3b and 5a. First, turbulence extracted energy from meaninetics in the production region (lower frequency left in Figs. 3bnd 5a). Second, energy was transported to smaller eddies throughhe inertial cascade region (straight line in Figs. 3b and 5a). Third,he turbulent fluctuations were dissipated as heat due to molecu-ar viscosity at smaller scales, which were finer than the samplingrequency (10 Hz) in this study (Baldocchi et al., 2001; Kaimal et al.,972).

At scales of 5 to 10-days, the cospectra of ET and SWC showed aistinct platform, indicating that ET variation was independent ofoil moisture dynamics (Fig. 5b). This result suggests that the timetep of 5-days might be used as the time-period when ET coulde calculated by a soil water balance method, which is shorterhan previously reported (Allen et al., 1998). Allen et al. (1998)uggested that the soil water balance method could only give ETstimates usually over long time-scales of the order of week or0-days. This was originally proposed because the component of

apillary rise from shallow water table for the soil water bal-nce is difficult to assess at short time-scales (Allen et al., 1998;ing et al., 2010; Zhang et al., 2011). Thus, longer time-scales are

equired to estimate ET using the soil water balance approach.

ce outside which edge effects become significant. Arrow lines at the top of panels

At the study site, contribution from water table might be negli-gible because the groundwater table was below 30 m depth (Zhanget al., 2012). Thus, we propose that a shorter time step (i.e., 5-days) could be used to estimate ET using the soil water balancemethod. Developing real-time ET rates in this region and otherareas with deep water table is critical to develop precision irrigationschedules (Allen et al., 1998; Kang et al., 2003). In this study, wecompared ET rates estimated by the soil water balance methodusing 5-days and 10-days time-steps of mean ET measured by EC,and found that there were not significant differences between 5-days and 10-days ET with respect to measurements (P < 0.05, datanot shown). Therefore, this study supports the fact that shortertime steps could be used to estimate real-time ET and should bebeneficial for developing more accurate irrigation practices. Werecognize that this application needs to be widely tested and weinvite the scientific community to address issues that link the multi-temporal variation of ET with water balance methods for calculationof ET.

As shown in Fig. 5b, the interactions among meteorological forc-ing (Rn, VPD, Ta, and u), hydrologic status (SWC), physiology (Gc,and Ga) and crop phenology (LAI) determined the critical patternsof temporal variability of ET at multiple time-scales (Katul et al.,2001; Steduto and Hsiao, 1998; Stull, 1988). The power spectralgap of ET, which occurred as the large valley in a monthly scalein Fig. 4b, separated daily from seasonal and inter-annual spectralpeaks. Variation of ET to the left of the gap was mostly associated

with the meteorological forcing variables and physiology (Fig. 5b).The temporal variability of ET to the right was mainly influenced byseasonal climate dynamics, soil water content, and crop ecophysi-ology (Fig. 5b).

R. Ding et al. / Agricultural Water Management 130 (2013) 79– 89 87

Fig. 8. Morlet wavelet coherence and phase spectra between ET and canopy conductance (Gc) during the whole growing season of maize in 2008 (a) and 2009 (b). DASrepresents days after sowing. Phase angle dynamics between ET and Gc are shown in (c) and (d) at the daily time scale. In (a) and (b), the black thick contour is 95% confidencel ation

s pointl de wh

egsc1sPat0mom

evel. The darker red areas inside the contour lines represent the higher local correlhown as arrows: in-phase pointing right (no lagging between X and Y), anti-phaseagging Y by 90◦ pointing up. The white dash line represents cone of influence outsi

The high coherence spectra between ET and Rn provided directvidence that daily ET was primarily driven by Rn during the wholerowing season for the two years (Fig. 6a and b). These resultsupport previous findings from other crops in less water-stressedlimates (Baldocchi, 1994; Lei and Yang, 2010; Steduto and Hsiao,998; Suyker and Verma, 2008). Our results imply that ET at ourite can be estimated using the radiation method, such as theriestley–Taylor model (Priestley and Taylor, 1972) when net radi-tion is available. A previous study at the study site demonstratedhat the slope of linear regression between daily ET and Rn was

.86 (R2 = 0.69), and measured ET can be well reproduced by aodified Priestley–Taylor model (Ding et al., 2013). The variation

f ET leading VPD at daily time-scale suggests that the surface ofaize canopy could act as a source of water vapor, and that the

Fig. 9. Seasonal variation of 1 − ̋ against days after sowing (DAS) over the whole gro

at 5% significance level. The phase relationship between two time series, X and Y, ising left (lagging by 180◦ between X and Y), X leading Y by 90◦ pointing down, and Xich edge effects become significant.

atmosphere could be a sink for water vapor during the measuredyears (Brutsaert, 1982).

Our findings presented that irrigation pulses had a strongerinfluence on ET during the early growing season through the coher-ence spectra between ET and SWC. The phenomena were explainedby the fact that soil evaporation was likely to be the main compo-nent of ET, and could have been remarkably affected by irrigationas LAI was low and the plants were likely not very active at the earlygrowing stage (Allen et al., 1998). The phase angle relationship ofET and SWC was sharply changed by the irrigation pulses (Fig. 7c

and d). These results support the hypothesis that irrigation prac-tices could have a substantial control on the temporal variability ofET, which provides the evidence for using artificial deficit irrigationor limited irrigation to reduce ET and improve water use efficiency

wing season of maize in 2008 (a) and 2009 (b). The ̋ is the decoupling factor.

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8 R. Ding et al. / Agricultural W

Kang et al., 2000; Kang and Zhang, 2004; Steduto and Hsiao, 1998).n other words, when estimating the ET rate, special considerationhould be given to management practices (e.g., irrigation sched-ling) that influence abiotic factors affecting ET processes.

The differences of mean phase angle between 2008 and 2009ndicated that Gc had stronger control on ET during 2008 than in009 (Fig. 8c and d). Furthermore, the phase angles between ETnd Gc were dynamic during the whole growing period. So, tourther explore the dynamic control of Gc on ET, we calculatedensitivity of ET to variation of Gc. The sensitivity of ET to Gc wasystematically developed by Jarvis and McNaughton (1986), whouantified the degree to which canopy conductance controlled ETy calculating a decoupling factor (˝). The relative change in ET for

prescribed change in Gc was given by (dET/ET)/(dGc/Gc) = 1 − ˝,here ̋ = (ε + 1)/(ε + 1 + Ga/Gc), ε = �/� . The sensitivity of daily ET

o Gc (1 − ˝) indicated that Gc variation exerted stronger controln ET processes at the early and later growing seasons than at theiddle growing season (Fig. 9a and b), being consistent with results

n Fig. 8c and d. The averaged values of 1 − ̋ were 0.56 and 0.50or 2008 and 2009, respectively, indicating the effect of Gc on ETas larger in 2008 than in 2009. These results are consistent with

hose obtained by the coherence spectra. Furthermore, they sug-est that Gc can be used: a) as an indicator of ET variability; and b)n estimating ET rates when the coupling of canopy and atmospheres strong (i.e., larger 1 − ˝) (Jarvis and McNaughton, 1986; Zhangt al., 2011).

. Conclusions

Crop evapotranspiration processes are complex because theombined interactions between meteorological forcing and eco-hysiological components act over multiple time scales. In order tonderstand these non-linear interactions, we analyzed the spectralharacteristics of ET and its controlling factors using wavelet trans-orm across two full growing seasons of maize in an arid region.his study demonstrated the following:

1) Wavelet transform provided a robust approach for exploringthe spectral properties of ET records that present non-stationary and gapping data. Wavelet coherence and phasespectra drew insights to local correlation and information ofpotential causality between ET and its controlling factors.

2) The global power spectra of ET showed an energetic cascadeof −1 power law driven by turbulence at less than 1-h scale,and exhibited significant peaks at daily, seasonal, and inter-annual scales. The multiscale characteristics of ET suggest thatprocess-based or empirical models should pay attention to thisvariability for proper simulation of ET dynamics.

3) The cospectra of ET and controlling factors presented that thecombined interactions of meteorological forcings, hydrologi-cal status, and crop ecophysiology determined the temporalvariations of ET at multiple time-scales. The time step of 5-days might be used as the scale where real-time ET could becalculated using the soil water balance method.

4) The coherence and phase spectral relationship between ETand meteorological variables provided the basis for estimatingET by selecting appropriate methods at different time-scales.Irrigation scheduling could have a substantial control on thetemporal variability of ET by influencing soil water content andcanopy conductance.

Our results provided a new dimension for the analysis of timeeries of ET in croplands of arid regions. Studying multiscale tem-oral patterns of crop ET could assist in understanding cropland

anagement 130 (2013) 79– 89

water cycle processes, and improve water resources managementto preserve water and increase crop yield.

Acknowledgments

We are grateful to the research grants from the Chinese NationalNatural Science Fund (91225301 and 51222905), the NationalHigh-Tech Research and Development Program (2011AA100502,2013AA102904, and 2013AA103004), the Chinese Universities Sci-entific Fund (2013XJ018 and 2013QJ042), and the Ministry of WaterResources of China (201201003). We thank the valuable com-ments from two reviewers that helped to improve substantiallythis manuscript.

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