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Advances in Water Resources 31 (2008) 418–423

Stability analysis of probabilistic soil moisture dynamics

Jan M. Nordbotten *

Department of Mathematics, University of Bergen, Bergen, Norway

Department of Civil and Environmental Engineering, Princeton University, Princeton, NJ, USA

Received 18 June 2007; received in revised form 16 September 2007; accepted 28 September 2007Available online 13 October 2007

Abstract

We study the dynamics of soil moisture in the presence of precipitation, evapotranspiration and leakage. This study utilizes a prob-abilistic framework, and considers the evolution of the probability density function of soil moisture. Our main objective is to assess whenthe stationary solution is a physically realizable state of the system. Mathematically, this is equivalent to analyzing when the stationarysolutions are stable. We show the following main results: For a general rainfall and evapotranspiration model, in a multiple plant frame-work, the stationary solution is a stable attractor in L1. However, the stationary solution is at best conditionally stable in L1. This hasimportant implications in terms of (1) the validity of moment analysis of the transient problem, (2) the physical importance of transientsolutions, and (3) the numerical solution of the stationary problem as a late time solution of the transient problem.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Ecohydrology; Water balance; Water-controlled ecosystems

1. Introduction

The interaction between precipitation, soil moisture,and evapotranspiration is fundamental for understandingwater-limited ecosystems. One approach to modeling thisinteraction is to consider the problem of water balance ata point [4]. In this setting, a probabilistic viewpoint is nat-ural – where uncertainty arises from the intermittent natureof rainfall [11]. Within this framework, one obtainsexpected values and variances for seasonal evapotranspira-tion. Thus one can obtain a direct understanding of theimpact changes in the rainfall structure and the instanta-neous evapotranspiration functions have on long termbehavior of the ecosystem [1,6,12].

Over the last decade, a series of papers have analyzedstationary probability distributions for soil moisture (seee.g. [2,6,7,9,10,12]). This probabilistic approach to ecohy-drology has given valuable insight, and has proven to com-pare surprisingly well to measured field data [13]. However,

0309-1708/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.advwatres.2007.09.008

* Address: Department of Civil and Environmental Engineering,Princeton University, Princeton, NJ, USA.

E-mail address: jan.nordbotten@math.uib.no

the applicability of the stationary analysis is limited to peri-ods with statistically homogeneous precipitation. This hasusually been taken to be the growing season, however,the initial period must then be discarded, since the proba-bility densities are affected by the initial conditions priorto the start of the growing season.

The transient problem, in which the full evolution ofthe probability density function is studied, has receivedless attention. This is mostly because of the increasedcomplexity; the stationary problem naturally has one lessindependent variable, time. The full time-dependentproblem can be discussed in Laplace space [2], however,the subsequent inversion is only feasible in special cases,such as linear loss functions [14]. Laio et al. [8] circum-vent the problems of Laplace space analysis by studyingthe first moment of the transient problem, allowing formore general loss functions. This analysis is however lim-ited by closure problems which need to be solved numer-ically. A related problem of forest fires can be seen as ananalogy to a precipitation model which always saturatesthe soil. Within the framework of this simplification,Daly and Porporato [3] discuss the dynamics in depth,both analytically and numerically.

J.M. Nordbotten / Advances in Water Resources 31 (2008) 418–423 419

This paper aims at clarifying the transient period, and itsrelationship to the stationary distribution. In particular weare interested in the role the stationary distribution plays asa late time limit of the transient problem. We will base ouranalysis on the model proposed in [10], which generalizesthe usual relationship between soil moisture and evapo-transpiration to a system consisting of an arbitrary numberof plants of different characteristics. The plants are coupledthrough experiencing the same precipitation. When onlyone plant is considered, this is equivalent to the usualmodel relating soil moisture to evapotranspiration.

Our analysis will show that the stationary solution is astable stationary point of the system in an integral (L1)sense. This implies that moment analysis as presented inLaio et al. [8] is appropriate, since it uses integrals on afinite domain of the time-dependent solution. However,our analysis will also show that for evapotranspirationfunctions commonly encountered, the transient solutiondoes not necessarily converge to the stationary solutionin a point-wise (L1) sense. When this is the case, the anal-ysis of the transient solution loses its physical meaning.This has particular implications for numerical methodsfor solving the stationary problem. Solving the stationaryproblem numerically is highly challenging, since probabil-ity density functions typically span many orders of magni-tude, making the solutions prone to effects of numericalround-off error. A common way to avoid this problem isto find an equation which ‘relaxes’ to the stationary solu-tion. The most obvious such equation is the transient equa-tion, however, the loss of strong stability invalidates thisapproach.

The remainder of the paper is structured as follows: Wewill briefly review the governing model equations, and givetheir generic form and interpretation together with theChapman–Kolmogorov forward equations. We then pres-ent the analysis, which exploits the linearity of the problemin probability space. In conclusion, we discuss the implica-tion of the findings in more depth.

2. Governing equations

The probabilistic framework we will analyze is a repre-sentation of the interaction between soil moisture, precipi-tation/infiltration and losses through evapotranspirationand leakage. In this work, we will model the loss functionsas deterministic functions of soil moisture, while the precip-itation will be modeled as stochastic.

We consider the following conservation equation for soilmoisture (see e.g. [11]):

niZidsi

dt¼ RðtÞ � I iðRðtÞÞ � QiðRðtÞ; siÞ � EiðsiÞ

� LiðsiÞ: ð1Þ

Here s = s(t) is vertically averaged soil moisture, R is rain-fall rate, I is canopy interception, Q is run-off rate, E isevapotranspiration, L is leakage, n is porosity, and Z is

root zone depth. The precise interpretation of soil moisturewill be discussed in detail in Section 2.3. We model a num-ber of plants, indexed by i, conceptualized as being closeenough that they experience the same precipitation, butdistant enough that there is no direct interaction betweentheir respective soil moistures [10]. A special case is thatof Ei(si) = 0 for some index i, which corresponds to baresoil. To avoid the possibility of negative saturations, we re-quire all leakage terms to disappear for dry soils, e.g.Ei(0) = Li(0) = 0.

To simplify the notation in the following, we will con-sider the conservation equation of the following genericform:

dsi

dt¼ IiðtÞ � qiðsÞ; ð2Þ

where I represents infiltration due to rainfall and q is thesum of evapotranspiration and leakage. We will for sim-plicity sometimes refer to q simply as evapotranspiration,and it follows from the preceding paragraph thatqi(0) = 0. Both I and q have been scaled by the void vol-ume of the root zone, nZ.

2.1. Precipitation and infiltration model

Within the context of models used in probabilistic soilmoisture dynamics, we will use the standard descriptionof precipitation. Following previous authors (e.g. theaforementioned Refs. [6,7,10,12]), occurrence of rainfall isidealized as a point process in continuous time. The rainfalloccurrence is thus modeled as a Poisson process in time,with rate k. In the general case, with multiple plants, wewill use the usual approximation of exponentially distrib-uted rainfall depths R

fRðrÞ ¼1

mR

e�r=mR for r P 0; ð3Þ

where mR is the mean depth of rainfall events.We will use a simplified model for canopy interception.

In general, there is a complicated relationship betweenmagnitude of rainfall events, time since last event, and can-opy interception. To facilitate the derivation of a forwardChapman–Kolmogorov equation we will approximate thisrelationship as a fixed shift di in the mean rainfall depth.The normalized version of Eq. (3) then becomes

fHiðhÞ ¼1

aie�h=ai for h P 0; ð4Þ

where

Hi ¼R

mR

ai ð5Þ

is the normalized rainfall, and ai = (mR � di)/niZi is thenormalized mean rainfall depth, both corrected for canopyinterception. Note that since mR, di, ni and Zi (and thusalso ai) are constants, the functions Hi will be correlatedthrough their relationship to R.

420 J.M. Nordbotten / Advances in Water Resources 31 (2008) 418–423

For the simpler case of a single plant-type (and thus asingle soil moisture), we will relax the assumptions on thestructure of rainfall depth by considering precipitation(after taking into account for canopy interception) asdrawn from a general distribution f. The choice of appro-priate distributions are discussed in e.g. [5]. Note that eventhis more general precipitation model neglects the fact thatsuccessive precipitation events might have a correlation indepths, allowing us to treat the infiltration depths asMarkovian.

2.2. Equations for the probability density of soil moisture

As outlined in Section 1, the equations for the probabil-ity density of soil moisture have been frequently discussedin literature. We will therefore restrain ourselves from pro-viding the derivation here, referring to Rodriguez-Iturbeet al. [12] for an excellent exposition for the single plantcase, and to Nordbotten et al. [10] for the generalizationto multiple plant equations.

We are concerned with the probability density (in satu-ration space) function p(s,t) of the vector s = [s1, . . ., sm]T ofm saturations occurring in the root zone of m differentplant species at a time t. The forward equation for thisproblem is (in integral–differential form) given by Eq.(3.2) in [10]:

opot¼ rs � ðqpÞ � kp

þ kZ skðsÞ

0

p s� az

akðsÞ; t

� �fHk ðzÞdz: ð6Þ

The subscript on the gradient operator emphasizes that thisis a gradient in saturation space, and we have introducedthe vector notation q = [q1, . . . ,qm]T and a = [a1, . . . ,am]T.Further, the subscript k refers to a saturation dimensionwhere s/a (where vector division is defined component-wise) takes its smallest value. Eq. (6) has a simple physicalinterpretation. It states that the rate of change of probabil-ity density is related to the flow of probability due to soildrying, minus the probability of a rainfall event, plus theprobability of a rainfall event from a different saturationstate leading to the current saturation state.

In the case of a single plant, Eq. (6) simplifies and weobtain the well-known equation (see e.g. [2,12]):

opot¼ o

osðqpÞ � kp þ k

Z s

0

pðz; tÞf ðs� zÞdz: ð7Þ

For the derivatives to be well defined, we will assumehere and in the following that solutions p of Eqs. (6) and(7) lie in H1.

2.3. Boundaries and limits to saturation

Soil moisture is often considered as a fraction, boundedbetween zero and one. This is inconsistent with the massbalance equation and precipitation models we have pre-

sented, unless the leakage term q(s) becomes unboundedfor s > 1. In this work, we adopt the following definitionof soil moisture:

sðtÞ ¼R

sðt; zÞdznZ

; ð8Þ

where the integration in the vertical direction is allowed tocontinue above the soil surface. Thus it is possible, if pool-ing occurs, to have values of saturation exceeding unity.The leakage term q(s) should be appropriately chosen togive the correct order of magnitude of run-off for s > 1.We note that physically, run-off is not instantaneous, andin the analysis we will therefore consider q(s) 6 qmax <1.It follows that the saturation domain is defined as thewhole positive real space, Rn

þ.For the purpose of the soil physicist, who is primarily

interested in what happens in the root-zone, they shouldintegrate the probability density function for values ofs > 1, and assign the value of this integral as an atom ofprobability at s = 1.

We note that a different way of imposing the limit on soilsaturation is by modifying the precipitation model to directlymodel infiltration. Rodriguez-Iturbe et al. [12] includes a nicediscussion on this (including the implication for boundaryconditions for stationary distributions). From a mathemati-cal perspective, this is a more restricted approach, and a lim-iting case of the convention adopted herein. This is seen fromthe fact that the solutions obtained with a modified model forinfiltration are equivalent to the results obtained with a usualprecipitation model in the limit of q(s > 1)!1, however,the modified infiltration model has less flexibility to accountfor pooling. Nevertheless, the results presented for the singlesaturation probability density evolution Eq. (7) herein allowfor general precipitation functions, and thus also apply tothis approach.

3. Stability analysis of stationary probability density

functions

Ultimately, we are interested in the validity and applica-bility of the stationary analysis of the equations for soilmoisture probability density functions. The stationarysolution is obtained by setting the time derivative in Eqs.(6) and (7) equal to zero. In this section we will analyze ifthis stationary solution is a stable stationary solution ofthe system.

We show that stationary solutions of Eqs. (6) and (7) areunconditionally stable in the L1 norm. However, we go onto show that for $s Æ q > k, we do not have stability in theL1 norm.

3.1. Stability in L1

Consider first the multiple soil moisture case, as givenby Eq. (6). Let p0 = p0(s) be a continuous, stationaryand bounded solution. Note that although the solutionwill only be non-zero in a subset of the domain, due to

J.M. Nordbotten / Advances in Water Resources 31 (2008) 418–423 421

restrictions on realizable soil moistures [10], this does notaffect the analysis which follows. We can then write, with-out loss of generality, the transient solution as the sum ofthe stationary part and a transient part, p(s, t) = p0(s) +g(s, t). Substituting this expression into Eq. (6) by linearityyields an equation for the transient part of the probabilitydensity function:

ogot¼ rs � qg

� �� kg þ k

Z sk

0

g s� azak; t

� �fHk ðzÞdz: ð9Þ

It follows from the definition of g that the zeroth momentequals zero. We analyze the stability of the stationary solu-tion p0 by considering whether the transient part g decayswith time.

We proceed by multiplying Eq. (9) by sign(g) and usingthe property that evapotranspiration is strictly positive toobtain

o

otjgj ¼ rs � qjgj

� �� kjgj þ k signðgÞ

�Z sk

0

g s� azak; t

� �fHk ðzÞdz: ð10Þ

Note that the equation is undefined where g = 0 and$sg 5 0. This will be the case in a region of measure zero,and has no consequence for the integral formulation whichfollows. Integrating Eq. (10) over Rn

þ we obtain

d

dtkgk1 ¼

ZRnþ

ru � qjgj� �

du� kkgk1

þ kZ

Rnþ

signðgðuÞÞ

�Z uk

0

g u� azak; t

� �fHk ðzÞdzdu: ð11Þ

Since p(s, t) is strictly positive with a bounded (one) firstmoment by definition, it follows that lims!1p = 0. Fromthe definition g(s, t) = p(s, t) � p0(s), we thus have that also

lims!1

g ¼ 0: ð12Þ

We recall from Section 2 that qi(0) = 0, thus q Æ n = 0,where n is the (outward) normal vector to the finite bound-aries of Rn

þ. Therefore, the first term on the right-hand sideof Eq. (11) vanishes after application of the divergence the-orem and Eq. (12). The last term in Eq. (11) is bounded bythe norm of the function, as shown in Appendix A. Thusthe last term of Eq. (11) is smaller than the preceding term,and the L1 norm of g therefore satisfies

d

dtkgk1 6 0: ð13Þ

The inequality in Eq. (13) will be strict when kgk1 > 0. Eq.(13) is the main result of this section, showing that all per-turbations will decay with time in the L1 norm.

We conclude this section by noting that the analysis forthe single plant system with an arbitrary precipitationmodel as given by Eq. (7) is identical to the one presented

above for multiple plants. Thus the main result, Eq. (13) isalso valid for this model.

3.2. Stability in maximum norm

Consider again Eq. (10). We will proceed to show lackof stability in the maximum norm by demonstrating acounter example. Let g be a continuous differentiable per-turbation with a maximum at sm > 0, such that g P 0 forall 0 6 s 6 sm. In one dimension, such a perturbation mightbe as simple as a n-dimensional sine wave with maximum atsm. Then, for positive normalized mean rainfall depths a,the last term of Eq. (10) is greater or equal to zero. Sinceg is continuous, the gradient of g at sm is zero. The maxi-mum norm of g equals g(sm), thus

d

dtkgk1 P ðrs � qðsmÞ � kÞkgk1: ð14Þ

This equation shows that a perturbation of the stationarysolution may grow if $s Æ q > k. Thus for evapotranspira-tion functions with large derivatives, bounded and contin-uous stationary solutions of Eq. (6) will not be stable.

Again, we note that the argument in the current sectionextends to the case of a single plant with a general precip-itation model, as given in Eq. (7).

In summary, we have shown that the stationary solutionis a stable attractor in L1, even though it is at best condi-tionally stable in L1. This implies that large absolute devi-ations from the stationary solution may occur, and grow,the time-dependent probability density function. However,our analysis shows that the area of any perturbation willdecrease with time, so that the transient solution convergestoward the stationary solution in a weak sense.

4. Implications

While the analysis presented above is technical in nature,it has important implications for the application of probabi-listic soil moisture distributions. We will review these here.

The stability analysis in L1 has an interesting implicationwhen we consider the soil moisture as a bounded functionbetween zero and one, as discussed in Section 2.3. For thiscase, the moments of the equation are taken on a finitedomain, thus the L1 stability implies that all moments ofEqs. (6) and (7) are well defined for all time. This is animportant result which supports the tacit assumption madein Laio et al. [8].

The results presented for L1 are relevant when$s Æ q > k. This situation is frequently encountered. As anexample, considering the four typical water-limited ecosys-tems described in Rodriguez-Iturbe [12]: All four ecosys-tems have leakage terms which satisfy osq > k when thesaturation is above the field capacity. Further, all exam-ples, except for the one identified as ‘C’, also satisfy osq > kfor saturations between the wilting point of the plant andthe saturation value at which the plant transpires at itsmaximum rate (denoted as s*).

422 J.M. Nordbotten / Advances in Water Resources 31 (2008) 418–423

The lack of stability in L1 reveals several points about thenature of the problem. Foremost, it shows that the initialvalue problem, e.g. starting from a known probability distri-bution, may not be well-posed. In other words, the solutionat time 0 < t <1 may not be a continuous function of theinitial condition. Thus, obtaining solutions (in probabilityspace) to the transient problem serves no purpose. However,this does not invalidate the value of solutions in integraltransform spaces (such as Laplace space), which only needthe L1 stability for finite frequencies to be defined.

Finally, we note that numerically obtaining the station-ary solution for a probability density function, defined onan unbounded domain, is notoriously difficult. This followsfrom the observation that the solution must necessarilyspan arbitrary orders of magnitude since it is positive andbounded in an integral sense. Thus numerical methods willbe affected by round-off errors. As noted in Section 1, acommon way of solving stationary problems are by usingrelaxation methods, where one solves a transient problemwhich is known to converge to the stationary problem.The analysis presented herein shows that the naturalchoice, the physical transient problem, cannot be appliedto avoid the numerical difficulties in obtaining a stationarysolution. Such numerical difficulties were experienced whenpreparing Ref. [10].

We will discuss the issue of instabilities in more detailfor the case of a single soil moisture, where the evolutionof the probability density is given by Eq. (7). For this sys-tem, a formal solution can be obtained in terms of inte-grals, as derived in Appendix B and given in Eq. (18).The structure of this solution serves as an example whichgives more insight into the general analysis presentedabove. In particular, the saturation s0 of a characteristicis a monotonically increasing function of t � t0 for any sat-uration s. In contrast, the integrating factor q(s) is only amonotonically decreasing function subject to the derivedcriterion osq 6 k. Therefore, we observe that the influenceof the probability at an earlier time t0, given byexp(q(s0) � q(s)), is increasing with time when the criterionis violated. This is a clear indication of ill-posedness. Nat-urally, any sufficiently accurate method based on solvingalong the characteristics of the system will resolve this, asdiscussed above. In contrast, if we were to discretize Eq.(7) directly, we might be lead astray. Indeed, a first-orderdiscretization of the hyperbolic transport part leads to anartificial numerical diffusion term. By decreasing the ratioof temporal to saturation discretization, Dt/Ds, we canintroduce sufficient numerical diffusion to eliminate theinstabilities. However, while we might still be able to obtaina converged solution by refining Dt and Ds in such a fash-ion that the artificial diffusion term remains constant, wewould now converge to the solution of a wrong equation.

Acknowledgement

The author thanks Ignacio Rodriguez-Iturbe for manyinteresting discussions on the subject of ecohydrology.

Appendix A

In this appendix, we bound the last term in Eq. (11):

kZ

Rnþ

signðgðuÞÞZ ukðuÞ

0

g u� azak; t

� �fHk ðzÞdzdu

¼ kZ

Rnþ

Z ukðuÞ

0

signðgðuÞÞg u� azak; t

� �fHk ðzÞdzdu

¼ kZ

Rnþ

Z 1

0

signðgðuÞÞg u� azak; t

� �fHk ðzÞdzdu

¼ kZ 1

0

fHk ðzÞZ

Rnþ

signðgðuÞÞg u� azak; t

� �dudz:

Here we have used that g(s) = 0 for saturations outsideRnþ.The inner integral above is bounded by the L1 norm,

therefore

kZ 1

0

fHk ðzÞZ

Rnþ

signðgðuÞÞg u� azak; t

� �du dz

6 kZ 1

0

fHk ðzÞkgk1dz ¼ kkgk1:

The last equality is a consequence of fHk being a proba-bility density function with unity zeroth moment.

Appendix B

In this appendix, we construct the formal solution to theone-plant dynamic probability density function describedby Eq. (7), subject to the conditions that p(z, t) and f(0)are bounded. The simplification in the special case wheref is a weighted sum of delta functions is also considered.

Eq. (7) is a first-order integral partial differential equa-tion. We first transform this equation to a set of two decou-pled ordinary differential equations by the method ofcharacteristics. The characteristic curve tðsÞ of the solutionis given by the solution of

t � t0 ¼ �Z s

s0

1

qðuÞ du: ð15Þ

Since q P 0, the integral in Eq. (15) will be monotonicallyincreasing, and can thus be inverted to give the character-istic curves s ¼ sðtÞ, and the initial saturation of the curve,s0 = s0(s, t, t0). We will therefore take these expressions asknown. By substitution of s into Eq. (7) we obtain theequation for evolution along a characteristic

dpdsþ q0 � k

qpðsÞ ¼ � k

q

Z s

0

pðs� z; tÞf ðzÞdz; ð16Þ

where t is defined as a function of s through Eq. (15), and aprime denotes the derivative. Multiplying Eq. (16) with theexponent of the integrating factor

q ¼Z

q0 � kq

ds;

J.M. Nordbotten / Advances in Water Resources 31 (2008) 418–423 423

we obtain

d

dsðexpðqÞpÞ ¼ � k expðqÞ

q

Z s

0

pðs� z; tÞf ðzÞdz: ð17Þ

We now use the condition that p(z, t) and f(0) are bounded,along with the property that the characteristics do notcross. The former implies that the point ðs; tÞ does not con-tribute to the integral, as it has measure zero. Further, theproperty that the characteristics do not cross implies that ifthe solution is known for all points z < s at time t, the inte-gral can be evaluated. Thus the solution can be constructedfrom s = 0 where Eq. (16) reduces to a trivial ODE. Wetherefore give the formal solution of Eq. (7):

pðs; tÞ ¼ expð�qðsÞÞ�

expðqðs0ÞÞpðs0; t0Þ �Z s

s0

k expðqðsÞÞqðsÞ

�Z s

0

pðs� z; tðsÞÞf ðzÞdzds�; ð18Þ

where as discussed after Eq. (15), s0 = s0(s, t, t0).We note in conclusion a special case. Consider f as a

weighted sum of m so-called Dirac delta functions:

f1ðsÞ ¼Xm

k¼1

xkdðsk � sÞ;

which corresponds to the case where the infiltration depthcan only take discrete values sk with probability xk. This isan unrealistic physical description, but may in some in-stances be a consequence of limited availability of data.For this case, the integral in Eq. (7) can be replaced by asummation; which simplifies the double integral in Eq. (18):

p1ðs; tÞ ¼ expð�qðsÞÞ�

expðqðs0ÞÞpðs0; t0Þ

�Z s

s0

k expðqðsÞÞqðsÞ

Xm

k¼1

xkpðs� sk; tðsÞÞds

#: ð19Þ

Here we apply the convention that p(s, t) = 0 for all s < 0,which follows trivially from extending the initial conditionto the negative axis.

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