Scaling and Analysis of Long Time Series of Soil Moisture Content

ByGabriel Katul

Nicholas School of the Environment and Earth Sciences, Duke University

NSF Workshop: Data-Model Assimilation in Ecology: Techniques and Applications

Norman, Oklahoma, October 22-24, 2007

Background – Soil moisture dynamics and climate

Because of storage effects within the soil pores, the dynamics of soil moisture posses a memory that is often considerably longer than the integral timescale of many atmospheric processes.

Background – Soil moisture dynamics and climate Hence climate anomalies can be

‘‘sustained’’ through land surface feedbacks primarily because they can ‘‘feed off’’ on this long-term memory.

Experimental Results

Canonical findings across experiments are:

1) The amplitude of soil moisture variations decreases with soil depth.

2) Soil moisture ‘memory’ across various geographic regions increases for dryer states when compared to wetter conditions.

3) Soil moisture is generally in-phase with precipitation at long-time scales but can be out-of-phase for short time scales.

Robock et al.,2000

Objective

A simplified analytical theory that predicts the spectrum (and phase) of soil moisture content at time-scales ranging from minutes to inter-annual.

Focus on a case study in which 8 years of 30-minute spatially and depth - averaged soil moisture time series is available along with precipitation, throughfall, and eddy-covariance based evapotranspiration.

Precipitation

Transpiration

Evaporation

Drainage

Through-fall

Soil Porosity Root-DepthRL

Dimensionless

( ) ( ) ( );rL t ET t D t ( ) ( ) / Ls t w t R

( ) ( ) ( )

L L

ds t L t p t

dt R R

( )( ) ( ) ( )r

dw tp t ET t D t

dt

4 rods per ring

1998-2005 – 8 years of 30 min. data

Pi ~ 1280 mm y-1 [Measured]Interception ~ 40% of P ~ 512 mm y-1 [See data below]ET ~ 650 mm y-1 [Measured by EC]

Through-fall ~ Pi-Interception ~ 768 mm y-1 = P(t)ET/Through-fall ~ 85%

L(t) ET( ) ( ) ( )

L L

ds t ET s p t

dt R R

Modeling Soil Moisture Dynamics: ET-s relationship

ET/ETmax

S1.00

1.0

Linear Model

NonlinearModel

Uniform Model

from Porporato et al. (2004)

Models for Soil Moisture Dynamics

1

( ) ( )( )

L

ds t p ts s

dt R

max1( ) ( )

L

ETs f s

R

Linear Model: f(s) = 1

Spectral Analysis of Soil Moisture

Soil moisture spectrum Es(f)

Fourier-Transform:

( ) ( )

1( ) ( )

2

i f t

i f t

H f h t e dt or

h t H f e df

2

2 21

| ( ) |( ) ;s

P fE f

f

max

1( )L

ETs

R

Phase Shifts (from Katul et al., 2007)

(1) By increasing the rooting zone depth (dr), the rainfall and soil moisture variability become increasingly out-of-phase.

(2) for long time scales (e.g., decadal), f0 and soil moisture and rainfall variability become in-phase with each.

(3) Lowering ETmax, rainfall and soil moisture become out-of-

phase.

Consistent with linear phase shift analyses reported by Amenu et al. [2005] (Illinois Climate Network stations).

1max( ) tan ( / )rf f d ET

Precipitation

Evapotranspiration

Soil moisture

Duke Forest Experiment – 8 years of 30-min. Data

2| ( ) |

.

P f

Const

1 max

1 1

45L

ETR d

2

2 21

| ( ) |( )s

P fE f

f

=0.55 and 300LRmm maxET=0.9 mm h-1

Unbounded variance as f0 or time

RandomForce Langevin

e.g. Langevin Equation: dx/dt = v [random]Unbounded trajectories.

2

2

1( ) ( ) ( )s equE f P f ET f

f

( )0.6 ( ) ( )

dw tp t ET t

dt

ET max varies with time only

S

ET/ETm

( ) ( )equ nET t R t

2

2 21

| ( ) |( )s

P fE f

f

2

2

1( ) ( ) ( )s equE f P f ET f

f

Summary and Conclusions

•Simplified hydrologic balance suggests that for white-noise precipitation, soil moisture becomes red (decaying as f-2).

•Analytical model for memory

1 max

1 LR

ET

Summary and Conclusions• If soil moisture memory (here ~ 45 days) is >> 12 hours, then diurnal dynamics of soil moisture do not contribute much to the overall variance.

• 45 day memory is much larger than those of many atmospheric processes. Hence, climate anomalies can be sustained through land-surface feedbacks primarily because they can ‘feed-off’ on this long-memory.

Summary and Conclusions

Simplified analytical model predicts that reduced ETmax results in

(1) ‘longer’ soil moisture memory and

(2) out-of-phase relationship between rainfall and soil moisture variations.

References Amenu, G. G., P. Kumar, and X. Z. Liang (2005), Interannual

variability of deep-layer hydrologic memory and mechanisms of its influence on surface energy fluxes, J. Clim., 18, 5024–5045.

Katul, G. G., A. Porporato, E. Daly, A. C. Oishi, H.-S. Kim, P. C. Stoy, J.-Y. Juang, and M. B. Siqueira (2007), On the spectrum of soil moisture from hourly to interannual scales, Water Resour. Res., 43, W05428, doi:10.1029/2006WR005356

Koster, R. D., and M. J. Suarez (2001), Soil moisture memory in climate models, J. Hydrometeorol., 2, 558– 570.

Koster, R. D., et al. (2004), Regions of strong coupling between soil moisture and precipitation, Science, 305, 1138–1140

Porporato, A., E. Daly, and I. Rodriguez-Iturbe (2004), Soil water balance and ecosystem response to climate change, Am. Nat., 164, 625–632.

Robock et al. (2000), The global soil moisture bank, Bulletin of the American Meteorological Society, 81, 1281-1299.

Variable Interception and ETmax

LAInThrough-fall