paper number 96jd01448. i--ira [1- (1- a) l/hi] (1)blyon/references/p25.pdf(7), (8), and (9) are all...

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 101, NO. D16, PAGES 21,403-21,422, SEPTEMBER 27, 1996

One-dimensional statistical dynamic representation of subgrid spatial variability of precipitation in the two-layer variable infiltration capacity model

Xu Liang Department of Civil Engineering and Operations Research, Princeton University, Princeton, New Jersey

Dennis P. Lettenmaier

Department of Civil Engineering, University of Washington, Seattle

Eric F. Wood

Department of Civil Engineering and Operations Research, Princeton University, Princeton, New Jersey

Abstract. The two-layer variable infiltration capacity (VIC-2L) model is extended to incorporate a representation of subgrid variability in precipitation, using an analytical one- dimensional statistical dynamic representation for partial area coverage of precipitation. The analytical approach allows the effects of subgrid-scale spatial variability of precipitation on surface fluxes, runoff production, and soil moisture to be represented explicitly. With this method, spatially integrated representations of surface fluxes, runoff, and soil moisture due to subgrid-scale spatial variability in precipitation, infiltration, and vegetation cover are obtained. The results are compared with those obtained using an exhaustive pixel-based approach, and the results obtained by applying uniform precipitation over the precipitation- covered area. The precipitation coverage over a grid cell is shown to play a primary role in estimating the surface fluxes, runoff, and soil moisture. In general, the spatial distribution of precipitation within the precipitation-covered area plays a secondary role, in part because VIC-2L represents the subgrid spatial variability of soil properties. While the analytical approach gives good approximations to the pixel-based approach, and is superior to the uniform precipitation approach in general, the differences are not large.

1. Introduction

Feedbacks from the land surface to the atmosphere can be important determinants of regional and global climate. The representation of the feedbacks in land surface parameterizations used in atmospheric general circulation models (GCMs) poses challenging tradeoffs between space-time resolution and model complexity. For a large area (e.g., a GCM grid cell), the effects of subgrid-scale spatial variations of precipitation on surface energy fluxes, soil moisture, and runoff production may be significant [Wood, 1991]. In practice, subgrid land surface variations have been largely ignored in GCM land surface schemes. Most GCMs, for instance, assume uniform soil characteristics within a grid cell, and ignore spatial variability in precipitation. Recently, however, some schemes have been developed which ad- dress this problem at least in part [e.g., Dumenil and

Copyright 1996 by the American Geophysical Union.

Paper number 96JD01448. 0148-0227/96/96JD-01448509.00

Todini, 1992; Rowntree and Lean, 1994]. Two major types of heterogeneities are of potential concern. One is related to the subgrid-scale hydrologic and topographic heterogeneity of the land surface itself; the other is related to the subgrid-scale variability of meteorological inputs, such as precipitation, downward shortwave and longwave radiation, wind speed, surface air temperature, and humidity. Among the meteorological inputs, the subgrid-scale variability in precipitation is particularly important [Raupach, 1993].

The motivation of this study is to explore the effects of subgrid spatial variabilities in soil properties and precipitation on surface moisture and energy fluxes, runoff, and soil moisture. The VIC-2L model [see Liang et al., 1994 for details] parameteri•.es the subgrid variability of soil properties through a spatial probability distribution. Specifically, it uses the infiltration function suggested by Zhao et al. [1980] (also known as the Xinan- jiang model), to partition precipitation into infiltration and direct runoff. The infiltration function is expressed

i--ira [1- (1- A) l/hi] (1) 21,403

21,404 LIANG ET AL.: SUBGRID SPATIAL VARIABILITY IN VIC-2L MODEL

where i and/m are the infiltration capacity and maximum infiltration capacity, respectively, A is the fraction of an area for which the infiltration capacity is less than i, and bi is the infiltration shape parameter, which is a measure of the spatial variability of the infiltration capacity, defined as the maximum amount of water that can be stored in the soil column. As such, it is a sur- rogate for spatial heterogeneities in soil properties and topography. This study concentrates on incorporating the spatial subgrid variability of precipitation into the model. The issue addressed in this paper is the relative importance of the fractional precipitation coverage over a large area (like a GCM grid cell) and the representation of the spatial distribution of precipitation within the precipitation-covered area.

The spatial variability of precipitation has been widely recognized to have a major effect on evaporation, soil moisture variability, and runoff production [see for example, Warrilow et al., 1986; Shuttleworth, 1988; En- tekhabi and Eagleson, 1989; Famiglietti and Wood, 1991; Pitman et al., 1990; Henderson-Sellers and Pitman, 1992]. Two approaches can be taken to model the spa- tim variability of precipitation. One is the pixel-based approach which discretizes precipitation over a spatial domain. The work of Fami#lietti et al. [1992], and Wi#mosta et al. [1994], among others, falls into the pixel-based category. Although such pixel-based representations are able to account for the spatial variability in precipitation throughout a grid cell (or a catchment), the computation time and data demands using this method are inconsistent with the macroscale land surface representations used in GCMs.

Another option is the statistical dynamic approach to representing the spatial variability in precipitation, which makes use of derived distribution methods to

evaluate the effects of subgrid variability in precipitation. The advantage of this approach is that, if an appropriate statistical distribution is assumed, a closed form solution will result which is computationally much less demanding than pixel-based approaches. Also, this approach requires less data, since only the statistical distribution of precipitation needs to be specified, and not the intensities associated with specific spatial locations. This is the approach we pursue.

A key assumption in this paper is that within a grid cell (or an area), precipitation, infiltration capacity, and/or other features (or attributes) only vary in one direction, which is arbitrarily taken as the z axis, and they are kept constant along the orthogonal (y) axis (one- dimensional concept), where z and y are scaled to give ß y = unit area. Under this assumption one-dimensional expressions that account for both the spatial subgrid variabilities of soil properties and precipitation can be obtained. The general form of the one-dimensional statistical dynamic approach for runoff rate Q•a over an area with a precipitation coverage/z during a time step can be expressed as,

-f* (z)] f(P=)dP=g(z= )dz=

P(z)f(P=)dP=g(z= )dz= } ds

(2)

where P(•) and f* (z) are precipitation and infiltration capacity within the fraction of area • in which rain occurs, f(P=) and g(z•), both of which vary with z, are probability density functions (pdf) of P• - P(z) and z=, respectively, and z• is a variable which varies along z axis. This variable could be an effective relative soil

saturation, a •opography soils index, or some o•her variable, depending on •he specific formulation of interest. The parameters P•,•, z•,•, P•,4, and z•,4 are •he upper limits of •he integrals in equation (2); •he parameters P•,l, z•,l, P•,s, and z•,s are •he lower limits of •he integrals in •he same equation. In equation (2), •he firs• •erm represent s •he infiltration excess runoff, and •he second •erm represents •he saturation excess runoff. In general, it is difficult to determine f(P•) and g(z•), and also to evaluate the triple integrals, therefore, this equation needs to be simplified.

A few previously reported studies can be character- ized as the simplified versions of equation (2). Warrilow el al. [1986] combined a subgrid precipitation distribution with a constant maximum surface infiltration rate

to estimate runoff from a grid cell. They assumed that over a fraction, •, of a grid cell (or an area), the precipitation intensity, Pt, is exponentially distributed and can be expressed as,

o (3) - where Pm is the grid cell average precipitation generated in the GCM. The runoff from the grid cell is then given by

/; q- • (P• - F*)f(P•)dP• - P.•ezp(- •F* ) * -•m (4) where q is the runoff rate, and F* is the maximum surface infiltration rate.

Shuttleworth [1988] derived an expression for canopy throughfall based on the assumption that the precipitation rate over a fraction of a grid cell was expressed by equation (3). Assuming that C',,• is the difference between the canopy storage capacity and the water stored on the canopy divided by the model time step, the throughfall over the grid cell is given by

(Pi - C',•) f(P• )dP• - Pmezp( - !•C"• )

LIANG ET AL.: SUBGRID SPATIAL VARIABILITY IN VIC-2L MODEL 21,405

and the runoff over the grid cell is

q-P,•ezp[ /•(F* + C,•)] (6)

Both equations (4) and (6) are simplifications of equation (2) which assume that •he infiltration rate is not a random variable, and also that the integrands in the brackets in equation (2) do not vary with •.

Pitraan et M. [1990, 1993b] incorporated •he •hrouõh- fall and runoff equations suggested by Shuffleworth [1988] into the Biosphere-Atmosphere Transfer Scheme (BATS) of Dickinson e• M. [1986] to study the sensitivity of evaporation and runoff due to the spatial distribution of precipitation defined by equation (3) within a grid cell. They assumed that the soil moisture and intercepted water were distributed uniformly within a grid

where f[ is the infiltration rate, z - ln(bTs/To•an/5) is the local value of the topography soils index of Top- model, z* represents the critical value of the Topmode! index at which saturation occurs, and f• (z) is the pdf of z, which is assumed to follow a three-parameter gamma distribution according to $ivapalan et al. [1987] and Wolock et al. [1989]. The expected value of the depth of saturation excess runoff, E[Q•], is expressed as,

f•oo •oOO Pif(Pi)dPif•(z)dz + .

i

where Si is the storage deficit. The first integral in equation (9) represents runoff generated on those areas

cell at the end of each time step. They performed a that are saturated at the start of a time step, the second sensitivity study that showed that the monthly evapo- integral represents the runoff generated on those areas ration and runoff resulting for/• values of 0.1, 0.5, 1.0, as compared with BATS standard formulation without •- parameterization (i.e., assuming that both precipitation and infiltration rates are uniform over a grid cell) were

that become saturated during a time step. Equations (7), (8), and (9) are all simplifications of equation (2), which assume that the integrands inside the brackets in equation (2) are independent of z.

quite different. Their results were sensitive to both the Among the parameterizations discussed above, only spatial precipitation distribution and the fractional cov- those of Entekhabi and Eagleson [1989], and Famiglietti erage of precipitation/•. Also, the sensitivity is shown and Wood [1991] consider the effects of interactions of to be higher in their studies of 1990 than in 1993b. subgrid spatial variability in precipitation and subgrid

Entekhabi and Eagleson [1989] represented subgrid variability in land surface characteristics. However, the hydrologic process in their GCM land surface scheme by runoff computed by both of these models (equation (7), combining the precipitation distribution given by equa- or equations (8) and (9)) is a point average over the tion (3) with a two-parameter gamma pdf of the surface fraction of the grid cell on which precipitation falls. In layer point effective relative soil saturation to describe other words, the runoff given by these models is equiva- the spatial heterogeneity in surface soil moisture. By as- lent to the runoff that would be generated by assuming suming independence of the precipitation intensity, Pi, that each point within the fraction is independent of and the surface layer effective relative soil saturation, all other points within the area covered by rainfall, and s, they derived a general relationship for runoff rate for that all of the points have the same statistical distribu- the entire GCM grid during a time step as, tion of precipitation, soil moisture, and/or topography

q ____ • (P• - f*)f(P•)dPif•(s)ds .

(7)

where f* is the infiltrability (infiltration rate) of the first soil layer, f(Pi) is defined by equation (3), and f,(s) is a two-parameter gamma distribution.

soils index. Note that equation (2) would be reduced to the formulations of Entekhabi and Eagleson [1989] and Famiglietti and Wood [1991], if the assumptions of independence and identical distribution are imposed. Such approaches ignore the absolute spatial location within the grid cell where precipitation occurs [Thomas and Henderson-Sellers, 1991].

In this paper, an analytical approach which combines the spatial subgrid variability of precipitation with the

Famiglietti and Wood [1991] considered the subgrid- subgrid variability of other land surface features like in- scale variability in topography, soils, soil moisture, and filtration is described. The major differences between precipitation by combining the precipitation distribu- our approach and those by Entekhabi and Eagleson tion given by equation (3) with the distribution of [1989], and Famiglietti and Wood [1991] are that ours the topography soils index from Topmodel [Beven and considers the effects of precipitation correlations at dif- Kirkby, 1979]. They obtained an expression for the ex- ferent locations within a GCM grid cell. As a result, pected depth of infiltration excess runoff for a large area,

as,

(Pi- f•)f(P•)dP•f•(z)dz (8)

effects of spatial correlations in derived variables such as infiltration are also considered. In other words, it is not necessary in our approach to require that the probability density functions of precipitation f(P•) in equation (2), for example, be the same at different loca-


tions. The parameters of the pdfs of precipitation can be different at different locations within a GCM grid cell, and even the pdfs of precipitation can be different at different locations in our approach.

To distinguish the statistical dynamic models developed by Entekhabi and Eagleson [1989], and by Famigli- etti and Wood [1991] from the one described here, we will refer to their models as point statistical dynamic models. The model developed here will be referred to as a one-dimensional statistical dynamic model.

2. Two-Layer VIC Model

The one-dimensional statistical dynamic model is an extension of the VIC-2L model which is briefly described in this section. For more details, the reader is referred to Liang et al. [1994]. The VIC-2L model char- acterizes the subsurface as consisting of two soil layers. Infiltration into the upper layer is defined by the spatial probability distribution given by equation (1). Down- ward movement of soil moisture from the upper to the lower layer is assumed to be gravitational, as parameterized using the Brooks-Corey formulation. Drainage and subsurface flow from the lower layer (baseflow) is parameterized using a nonlinear recession. The area of interest is partitioned into n land surface cover types, where n - 1, 2, ..., N represents N different types of vegetation, and n - N + 1 represents bare soil. The upper soil layer (soil layer 1) is designed to represent the dynamic behavior of the soil that responds to rainfall. The lower layer (soil layer 2), which is used to charac- terize the seasonal soil moisture behavior, is spatially lumped. Plant roots can extend to layer 1 or layers 1 and 2, depending on the vegetation and soil type. There are three types of evaporation considered in the VIC- 2L model. They are evaporation from the canopy layer of each vegetation class, transpiration from each of the vegetation classes, and evaporation from bare soil.

3. One-Dimensional Statistical

Dynamic Model

The one-dimensional statistical dynamic model is described in this section. Equation (2) is too complicated for direct application, and needs to be simplified. It can be also written as,

-f* f(e )ee g

d•

(10)

This form is amenable to simplification if dQ•d(z) can be expressed in an efficient way which takes into account of the spatial variabilifies of precipitation and soil properties. In the derivation of the simplified version of equation (2) for VIC-2L, four assumptions are made:

1. The precipitation P is a one-dimensional function of z (i.e., P(z)) within the fractional coverage. This assumption is, in a sense, equivalent to assuming that storms are distributed as circles around the storm cen-

ters.

2. At the end of each time step the soil moisture content of each strip yd• (see Figure 1) within the fractional coverage of the same vegetation cover becomes the same. This assumption avoids the necessity for tracking the storm center movement within a storm.

P(x 1 )

P(x) la

0 •

lb

dA1 = ydx 1

dx I

• ß i

i

i

i

i

A2 = ydx 2 l

dx2[ x [1 1C

P(x 2)

io+P(x 1)

/////////////////

i0+P(x 2)

i0

for ydx 1 for ydx 2

Figure 1. Schematic representation of the analytical approach to represent spatial variability of rainfall for VIC-2L model: (a) exponential precipitation distribution in z direction; (b) plan view of a grid cell (or area) with strips of ydzx and yd•2; (c) VIC-2L representation from the strips yd•x and yd•2.


3. Prior to the beginning of the next storm, the soil moisture over the fractional area • is assumed to be the same as the moisture over the unwetted fractional

area 1 - •, which is accomplished by spatial averaging. This assumption is reasonable in practice, if the interarrival time between two storms is long enough that the recently wetted soil drains to a comparable moisture level to that which is not covered by the storm. Here we define the interarrival time as the time between two

storms whose magnitudes are above a specified threshold (taken to be 1.0 mm/hr in this study). If a storm with magnitude below a specified threshold occurs, the soil moisture over • and 1-• is not averaged. It should be noted that if storms smaller than the threshold oc-

cur, they are not ignored, but from the standpoint of the soil moisture distribution, they are treated as a con- tinuation of the previous storm. The effect of this assumption is that the track of the storm center from one storm to another does not need to be specified.

4. Each strip of area ydz is assumed to have an identical infiltration capacity distribution defined by equation (1).

3.1. One-Dimensional Representation for Bare Soil

We begin with the simplest case by assuming that only bare soil is present in a grid cell (or an area) with a fraction coverage of precipitation p at time step t. In the following, the direct runoff concept of the VIC- 2L model is used to obtain an expression for dQ•d in equation (10). If we discretize the area within the fractional coverage p into infinitesimal strips of area ydz (see Figure 1), then the precipitation rate within each such strip is a constant. From the fourth assumption and the VIC-2L runoff calculation [œiang et al., 1994], the direct runoff dQ•d[N-•- 1] of bare soil from strip ydz due to precipitation P(z) can be expressed as

dQ•a[N-+ 1] - P(z)- • 4- At ' i0 + •(•)•X• >_ i.• (XX)

dQ•d[N -+ 1] _ wr w•[•v + •] w• [ •0+

io + P(•)zxt < i.• } dz,

(•2)

where W• is the maximum soil moisture content of layer 1, At is the time step which is taken as 1 hour in the model calculation, W•i[N + 1] is the soil moisture content in layer 1 within p, i0 represents the point infiltration capacity corresponding to W•i[N -•- 1]. The argument IN 4- 1] refers to the bare soil cover class index.

For the vegetation surface c9ver class, the index is In]. Throughout the remainder of this paper the dependence of many of the surface and subsurface characteristics on surface cover class is implied by this argument even if not noted specially. Therefore, the total direct runoff at time step t from p is,

i.• j

where the first term represents the direct runoff generated when io-•-P(z)At _• i,•, is satisfied, and the second term represents the runoff when io-•-P(z)At < i,•, is satisfied. The integral limit, a, represents the location,

•0 + •(•)•x• _> •.•, •n • = • in •qu•ion (•). In equation (13), W•, hi, i,•, and At are constant. On

the basis of assumption 2, the soil moisture W•[N + 1] in equation (13) does not vary with z. In addition, from equation (1) we can obtain the following relationship between i0 and W•[N-•- 1],

(14)

Thus, i0 is independent of z. Therefore, only P(z) varies with z in equation (13). If the expression for r(•) i• •nown, tn•n tn• aicpa •uno• Q•[•V + •] with the effects of spatial precipitation and infiltration can be calculated. For a constant precipitation rate within p, equation (13) reduces to the original runoff expressions given by Liang et al. [1994] for a values of 1 and 0, respectively.

For evaporation, it is reasonable to ignore the effects of spatial variations of precipitation within the fractional coverage of precipitation for large storms, which is equivalent to assuming that evaporation is small during storm periods compared with dry periods. For small storms, the range of magnitudes of precipitation between the maximum and minimum over the fractional

coverage • would not be as large as for large storms, and hence the effects of spatial variations of precipitation might not be significant. Therefore, the evaporation from bare soil within the precipitation coverage • can be determined by

fo A' pEpIN -•- 1] dA -•-

/: , i,,r,[1 - ({ '- A)•/•'] dA (•5)


where E•[N + 1] is the potential evaporation rate, and A, is the fraction of the bare soil that is saturated.

Similarly, the drainage term Q•12[N + 1], and the baseflow Q•b[N + 1] can be calculated by the following equations,

Q•i:•[N+i]-p,K. (W•i[N + 1]- 8•) •-• +3 w•-½ (•)

q•[• + •] - •w,w•w;=[• + •], 0 • w2=[• + 1] • w,w• (lZ)

and soil moisture in layer i and layer 2, W(i_/z)i[N-[- 1] and W(i_•)•[N + 1], are calculated in the same way as described above, except W•x[N + 1], W•[N + 1], and /• are replaced by W(i_•)x[N + 1], W(i_•)•[N + 1], and (1 -/•) in the related equations. The direct runoff in this case, Q(1-•)d[N + 1], is zero.

According to assumption 3, the soil moisture within areas /• and 1 -/• are averaged immediately prior to the beginning of the next storm that is larger than the specified threshold so that the entire area has the same soil moisture content when next storm arrives. Thus, the soil moisture of layer j (j = 1, 2) immediately prior to a storm exceeding the threshold is

p Wsw•W•-•[N + 1]

+ (D,• Ds D,• w•-w.w• '

wf•[• + •] >_ w.w• (18)

wf•. [• + •] - •(w•. [• + •] + w•_•)• [• + •]) (•)

W(•_•)j [N + 1] (•- •,)(w•. [•v + •] +w•_•)• [•v + •]) (22)

3.2. One-Dimensional Representation for Vegetation Cover

For vegetated areas, the approach described above applies, except that the rainfall rate P(z) is replaced by the throughfall rate. It is assumed that the throughfall

where Ks is the saturated hydraulic conductivity, O, rate P•(z) is equal to P(z) minus interception. The is the residual moisture content, Dr• is the maximum approach for incorporating different vegetation covers baseflow rate, Ds is a fraction of Din, W• is the maximum soil moisture content of layer 2, Ws is a fraction of W•, with Ds •_ Ws, and W•2[N + 1] is the soil moisture content at the beginning of the time step in layer 2 within/•.

The water balance in layer 1 over the precipitation coverage/• is then,

w•+•[•v + •] -

where Wfi[N + 1] and Wfx[N + 1] are the soil moisture content at the end and the beginning of each time step in layer 1 within/•, respectively. The water balance for the lower layer within/• can be expressed as,

W•+•. [N + 1]

and bare soil within • is the same as described by Lierig

Like the bare soil case, the soil moisture for different cover classes over the wet and dry fractions/• and I -/• are averaged (or "smeared") prior to the beginning of the next storm that is larger than the specified threshold. That is, the soil moisture of bare soil (W•[N + 1], j - 1,2) and each vegetation type [n]

fO (Wl• j [hi n -- 1 2 ... N)over the fractional coverage of Wfl[N + 1] + {p P(z)dz - E•i - ' ' ' ' precipitation/•, and the soil moisture of each such cover

q•[N + 1] - q•=[N + 1]} ZX• (1•) •p. o•.= •. a=• •=•aion•l •o•.=•. 1-. •=. •.=•.a, so that the entire area (or grid cell) has the same soil moisture content at the onset of the next storm. For the

fractional area/• covered by precipitation, the average soil moisture W•i of layer j (j - 1, 2) can be expressed as,

(20) Wf•.[N + 1] + (Q•i•[N + 1] -Q•[N + 1]- E•)At

where Wf•[N + 1] and Wf•[N + 1] are the soil moisture content at the end and the beginning of each time step in layer 2 within /•, respectively, and E•2 is the bare soil evaporation from layer 2 within/• (it is taken to be zero).

For the area within the dry fraction 1 - p, the evap-

N+i

w• - • c• [•]w• [•] (•3)

where C% [n] is the fraction of the nth surface cover area, •-•N+i and /-n=l Cv [•,] -- 1. Similarly, the average soil mois-

ture W(x_•)j of layer j (j - 1, 2) over the dry fractional area 1 - p can be expressed as,

N+i

- (24)

oration from bare soil E(1-/z)l, drainage from layer 1 The average soil moisture of layer j (j - 1,2) for the to layer 2, Q(•_•)•a[N + 1], baseflow Q(•_•)•[N + 1], entire area, at the beginning of the next storm is,

LIANG ET AL.' SUBGRID SPATIAL VARIABILITY IN VIC-2L MODEL 21,409

+ n-l,2,.'-,N+l (26)

To summarize, there are two points in the scheme at which the soil moisture is averaged. The first is over strips of area yd• within p at each time step for each cover class using equation (19). This avoids tracking the movement of the storm center during a storm. This averaging results in equal soil moisture within the same surface cover type within p. However, the soil moisture is different among the different surface cover classes, and is different also from the soil moisture in the dry fraction 1-p for the same surface cover class. The second averaging, described by equations (21) and (22)or equations (25) and (26), is carried out at the beginning of the next storm exceeding the threshold, and effec- tively "smears" the soil moisture over the entire area, subject to the subgrid variability inferred by equation (1).

The two advantages of this analytical approach are that it avoids the need to identify the specific area that receives a given precipitation rate within the fraction p of an area (or a grid cell), and it considers the spatial variability of precipitation within the fractional area covered by the storm. This is because for P(•) (or P•(•)) within a strip •dz, P(•) (or P•(•))is considered to be a constant, and the spatial variability of the infiltration function (equation (1))is considered over a strip area/4d• with the same initial soil moisture. The dry fraction 1 - p of the area (or grid cell) is taken to have no precipitation throughout the storm. The fraction p is assumed to be a constant, but it can vary with time as p(t). It should be noted that the number of model prognostic and diagnostic variables, for example, soil moisture and water fluxes, is doubled by this approach, since such variables are now predicted for the dry and wet areas.

3.3. Derivation of Precipitation as a Function of ß

The one-dimensional statistical dynamic model requires that the precipitation function P(•) be known. In this section, an appropriate form of P(•) is derived.

Eagleson [1984], Eagleson and Wang [1985], and Ea- gleson et al. [1987] have reported studies on the fraction p of a grid area that is affected by precipitation reach- ing the surface. Warrilow et al. [1986] assumed that over a fraction of a grid cell (or an area), the precipitation intensity is exponentially distributed (equation (3)). This exponential distribution assumption for precipitation seems appropriate in some cases as shown by an analysis of hourly observed precipitation data by A b- dulla [1987]. F.A.M. Abdulla (personal communication,

1993) also found that precipitation over a large area in Oklahoma appears to follow the exponential distribution. Others [for example, Collier, 1993; Gao and Sorooshian, 1994; Sauvageot, 1994] found that for some conditions, the log-normal relationship might be more appropriate. In this paper, though, only the exponential distribution will be used.

Following Warrilow et al. [1986], the precipitation intensity within p is assumed to be described by equation (3). The percentage ß that receives a precipitation rate greater than or equal to precipitation rate P over the fraction p of a grid cell (or an area) can then be expressed as (see Figure la),

area that receives precipitation rate • P total area of p

•o P -- 1 - F(P) -- 1 - f(Pi)dPi

•o P -- 1- •P e•p( -pPi

= The inverse of equation (27) is then,

(27)

P(•)- P'• In(•), 0 < ß _< I (28)

In equation (28), P(1) - 0, that is, the probability of the area within fraction p that can receive precipitation greater than or equal to zero is one. This result is due to the form of equation (3). By substituting equation (28) into equation (13), we obtain,

Q•,•[N + 1] P'• - P At W•i[N + 1]) l+b4

d•

(29)

3.4. Estimation of Fractional Coverage of Precipitation p

Warrilow et al. [1986] note that p may be on the order of 0.95 and 0.60 for large-scale and convective rainfall, respectively. Currently the U.K. Meteorological Of- fice (UKMO) sets p to 1.0 for large-scale rainfall and 0.3 for convective rainfall in their GCM. The observed spatial variability of total storm depth for air mass thunderstorm rainfall in Arizona [Eagleson et al., 1987] sup- ports a wetted fraction of 0.5 to 0.66. While Warrilow et al. [1986], Entekhabi and Eagleson [1989], Pitman et al. [1990], and Famiglietti and Wood [1991] all used a prescribed fraction p of a grid Cell area, Eftahit and


Bras [1993a] found significant temporal variability in the fractional coverage of rainfall. They presented a procedure for estimating the fraction/• as,

Pm

•- E(P•) (30) where E(P,) is the areal mean precipitation over the rain-covered fraction/• of the grid cell. They suggested that P,• can be estimated by

V

P• - AX. AX. AT (31)

2a i i

i

e ß e

where v is the volume of rainfall simulated by the climate model within a grid cell area, AX, and AT are the spatial and temporal resolutions of the model, respectively. By invoking the ergodicity assumption, they estimated E(P•) by the mean of the rainfall rate at a point using rainfall records from a single location. Al- though the volume of rainfall, V, could be taken from a numerical weather prediction model, this can result in large biases due to its dependence on the model- generated precipitation, but not on the characteristics of the regional climate; it is probably more realistic to use an average of observed station data instead of equation (31) to estimate P,•.

On the basis of the observed rainfall data, the fractional area covered by rainfall exceeding a fixed threshold is highly correlated with the mean areal rainfall rate [for example, Chiu, 1988; Atlas et al., 1990; Braud et al., 1993; Sauvageot, 1994; Morrissey, 1994]. Thus, /z may be estimated from radar rainfall data or from the rain gauge networks. One of the major limitations of observational studies, however, is scale, since most of studies have been performed for relatively small areas. Gong et al. [1994] developed a method to estimate/z based on large scale observations (over 1700 precipitation stations) within the contiguous United States. One of the limitations of their method is that it yields a climatological/•, while in practice,/• varies storm by storm. Eventually, it may be possible to derive/• inter- nally within the GCM, based on its atmospheric state variables. For the purpose of this paper, however, it is sufficient to prescribe/z, since the differences evaluated in this study are based solely on sensitivity analysis.

4. Testing of One-Dimensional Statistical Dynamic Model

As described in the previous section, four major assumptions are required to derive the one-dimensional statistical dynamic model. In this section, computer experiments are designed to test these assumptions by comparing the one-dimensional analytical approach with an explicit pixel-based approach over a grid cell (or an area). In addition, the results based on use of a constant precipitation rate over the precipitation-covered area

P(x)

2b

ß e e

P(x)

Figure 2. Schematic representation of the computer experiments for the pixel-based and analytical approaches: (a) 2500 pixels for a grid cell; (b) VIC-2L representation for each pixel; and (c) the analytical approach of VIC-2L representation for the area that receives rainfall.

are compared with the results from the one-dimensional statistical dynamic model. Sensitivity studies are carried out for several values of precipitation coverage/• as well.

The pixel-based approach is a "brute force" experi- ment (see Figures. 2a and 2b) where a grid cell (or an area) is divided into LM pixels. In this study, there are LM-2500 pixels (subgrid elements). At each time, the average precipitation depth of an area (or grid cell) is given by the mean of the exponential precipitation distribution, from which the precipitation rates for the 2500 pixels are drawn randomly. In generating the rainfall field, the variable z in equation (28) is obtained from a uniform random generator, and z is kept con-


stant within each storm for each pixel. When a new storm arrives, a• in equation (28) is randomly (from a uniform distribution) sampled again. By generating the rainfall field in this way, the spatial correlations among pixels are kept the same within a storm, but the rainfall rate at the same pixel changes with time since P,• in equation (28) is a function of time. In other words, if a pixel receives a relatively high rainfall rate compared with other pixels at one time step, it will still receive a relatively high rate compared with other pixels at other time steps during the same storm, but the rainfall rate at each time step for the same pixel is not necessarily the same. Also, the same pixel may receive a relatively small rainfall rate compared with other pixels in a new storm.

The procedure for generating the rainfall field over a grid cell (or an area) in this way would be more appropriate for storms whose centers and shapes do not change much with time during each storm period. Within each pixel, the precipitation rate is assumed to be the same. As for the infiltration capacity, it is assumed that one infiltration capacity distribution charac- terizes the entire area. The infiltration capacity distribution for each pixel is obtained through random sam- pling from the infiltration capacity distribution for the entire area (Figure 2b). These pixel hydrologic properties are kept unchanged throughout each simulation. Since the precipitation rate is assumed to be the same within each pixel, the two-layer VIC model ILianit ei al., 1994] can be applied. By applying the model on a pixel by pixel basis, the direct runoff, evaporation, sen-

sible heat flux, surface temperature and soil moisture are calculated for each pixel and then are aggregated to obtain results for the entire area. The results from the

pixel-based approach are taken as the "truth". The one-dimensional statistical dynamic model de-

scribed in this paper is an analytical approach, and is used to compute the direct runoff, evaporation, sensible heat flux, surface temperature, and soil moisture for the same area as in the pixel-based approach. In the study, the precipitation over the area follows an exponential distribution with the same distribution parameters as used in the pixel-based approach (Figure 2c).

In addition, two variations of an average approach that uses a uniform precipitation rate over the fractional coverage • of the grid cell and over the entire grid cell (i.e., • = 1) were tested. The first average case uses the same • as in the pixel-based and analytical approaches. The second average case uses tz = 1 (uniform precipitation over the entire grid cell).

In the study, one year of Project for Intercompart- son of Land-surface Parameterization Schemes (PILPS) [Henderson-Sellers ei al., 1995] prescribed atmospheric forcings representing two climatic regimes were used' (1) moist, tropical forest; and (2)a midlatitude grassland. The PILPS atmospheric forcing data were obtained from the NCAR CCM1-Oz for a forested grid cell centered at 3ø$, 60øW, and a grassland grid cell centered at 52øN, OøE, respectively [œitrnan et al., 1993a]. They include downward shortwave radiation, downward longwave radiation, precipitation, air temperature, wind speed, surface pressure, and specific hu-

Table 1. VIC-2L Model Parameters at PILPS Grassland and Forest Sites

Name Grass Forest

Fractional coverage of vegetation Depth of upper layer, m Depth of lower layer, m Roots in upper layer, % Roots in lower layer, % Saturated hydraulic conductivity, mm/s Soil wetness exponent Slope Displacement height, m Surface albedo (snow free) Constant soil temperature, øK Architectural resistance, s/m Soil porosity Fraction of water content at which permanent wilting occurs Fraction of water content at which critical point occurs Inœ1tration shape parameter Fraction of maximum subsurface flow

Fraction of lower layer maximum soil moisture Aerodynamic roughness length, m Leaf area index

Minimum stomatal resistance, s/m Soil thermal conductivity, Wm -t øK -t Soil heat capacity, Jm-3øK -t

1.0 1.0

1.0 1.0

9.0 9.0

100 90

0 10

0.45 x 10 -2 0.16 x 10 -2 6.8 9.2

0.17 0.17

0.0 18.0

0.21 0.131

274.6 299.6

2.0 25.0.

0.51 0.6

0.378 0.487

O.7 O.6

0.1 0.03

0.008 0.008

0.9 0.9

0.1 2.0

Monthly LAIs 5.0 200.0 150.0

1.03 0.866

2.085 x 106 1.756 x l0 s

21,412 LIANG ET AL.. SUBGRID SPATIAL VARIABILITY IN VIC-2L MODEL

midity. Pitman et al. [1993a] provide details of the PILPS experiments where the above two sets of forcing data were used by 20 different land surface schemes run to equilibrium. The model parameters for the Manabe bucket, SiB-, and BATS- type models were given for both the grassland and forest sites for PILPS partici- pants. Although the VIC-2L model is not strictly similar to any of these types, there are some common parameters, and the VIC-2L model parameters were determined from the PILPS values and the literature where

necessary. The magnitudes of each parameter of VIC- 2L model used in this study are given in Table 1. Since the present form of VIC-2L does not represent variations in spatial snow properties, only summer month simulations for the grassland are considered. At the forest site, snow does not occur, so the simulations were conducted for an entire year. For simplicity, the precipitation fractional coverage parameter/z is prescribed (time-independent) in each study. All simulations were conducted at an hourly time step.

4.1. Comparison of Different Approaches at a Forest Site

In this section, the results from the three approaches with the same fixed precipitation coverage/z are shown and discussed for the forest site. The results with

/z - 1.0 are discussed in section 4.3. For the average approach discussed here, a uniform precipitation rate was used over the precipitation-covered fraction /z of the grid cell. All approaches start with the same initial soil moisture (50% of the maximum soil water content). The purpose of this study is to test the four assumptions in the one-dimensional statistical dynamic model, and to explore differences among the three approaches.

In January (Figure 3) at the forest site, all three approaches start at the same initial conditions. The analytical and average approaches both overestimate the

average latent heat flux compared with the pixel-based approach by about 7 Wrn -•' out of 40.3 Wrn -•' (see Table 2). Table 2 shows the monthly quantities for the pixel-based approach and the relative biases of the analytical and average approaches. A positive sign in Table 2 represents underestimation of the corresponding pixel- based quantities. Although the three approaches result in similar total soil moisture throughout the month, differences in the upper layer begin on January 21 and continue through the end of the month. There are two sources for the difference in the upper layer soil moisture. First, in the pixel-based approach, more pixels are saturated at the end of January due to precipitation after January 21, thus more water runs off and less is infil- trated. Second, in the pixel-based approach, more water drains to the lower layer from the upper layer in the saturated pixels due to the nonlinear drainage formulation, thus the pixel-based approach has more water in its lower layer on a spatial average basis. In the analytical and average approaches, however, the soil moisture in the upper layer is averaged within each vegetation cover type at the end of each time step (assumption 2), and is also averaged at the onset of next storm over the entire area (assumption 3), thus a smaller portion of the area is saturated under the same precipitation condition, which leads to less saturation overland flow. Therefore, more water goes into the soil, but less goes into the lower layer. This is why the soil moisture difference in the upper layer between the analytical and average, and the pixel-based approaches, is not present in the total soil moisture. In this month, both the analytical and average approaches underestimate runoff, in particular from January 21 onwards as shown in Fig- ure 4. However, the magnitude of the runoff is small; the highest peak is less than 0.4 mm/hr. It should be noted that the runoff results from the average approach with/•- 0.3 are not shown in Figure 4, since they are

'• 2.0 E --- 1.,5

ß • 1.0

•. o.o 295 • I 5 9 13 17 21 25 29 I 5 9 13 17 21 25 29

400 ]

I 5 9 13 17 21 25 29 I 5 9 13 17 21 25 29

E•E 450 '- 400

._c 350 300

E•' 3150 ' 3100'

_.. 3050 3000

I 5 9 13 17 21 25 29 I 5 9 13 17 21 25 29

Figure 3. Comparison of effective surface fluxes, temperature, and soil moisture among the pixel- based (solid), analytical (dotted), and average (dashed) approaches for January with/•- 0.3 for the PILPS forest site.


Table 2. Comparison of Monthly Average Quantities for Fixed p- 0.3 at the Forest Site

Method, Month Latent, Sensible, Surface T, Runoff, Upper W, Total W, R•infall, % Wm -•' Wm -•' øK mm/mo mm mm mm/mo

Pixel- based 1 40.35 84.76 302.86 13.7 344.6 3062.2 239.9

Analytical -17.2 7.4 0 34 -4.1 0 0 Average -17.5 7.6 0 39 -4.1 0 0 Pixel-based 2 51.44 90.27 301.09 344.6 435.1 3277.1 679.3

Analytical -9.3 4.8 0 3.3 -12.6 0 0 Average -9.3 4.8 0 5.3 -12.6 0 0 Pixel-based 3 84.27 60.87 303.04 56.0 439.7 3526.1 222.6

Analytical -1.1 1.3 0 39 -9.1 -0.5 0 Average -1.1 1.4 0 52 -9.1 -0.9 0 Pixel-based 4 50.92 79.70 300.53 105.1 432.4 3629.4 321.6

Analytical -2.3 1.5 0 0.7 -8.9 -0.7 0 Average -2.5 1.5 0 5.4 -8.9 -1.2 0 Pixel-based 5 81.05 40.58 298.09 2.8 416.3 3695.1 67.7

Analytical -2.3 4.3 0 -59 -9.4 -0.6 0 A v. er age - 2.3 4.4 0 -54 -9.4 - 1.1 0 Pixel-based 6 70.85 45.62 296.49 8.1 380.3 3678.4 85.6

Analytical -2.0 29 0 38 -9.9 0 0 Average -2.0 29 0 41 -9.9 -0.9 0 Pixel-based 7 77.21 29.89 296.90 4.0 362.5 3678.7 74.3

Analytical -31 78 0 -3.2 -9.2 0 0 Average -32 78 0 0.8 -9.2 -0.5 0 Pixel- based 8 60.12 59.09 302.10 1.7 323.2 3638.3 27.1

Analytical -47 45 0 -23 -1.5 1.1 0 Average -48 45 0 -22 -1.5 0.5 0 Pixel-based 9 53.99 78.06 301.04 28.3 380.5 3731.0 268.1

Analytical -23 15.1 0 53 -6.3 1.3 0 Average -23 15.2 0 56 -6.3 0.8 0 Pixel-based 10 73.88 65.86 301.28 372.7 437.1 3948.1 633.1

Analytical 0.9 - 1.0 0 8.7 - 10.9 0.7 0 Average 0.9 -1.0 0 10.9 -10.9 0 0 Pixel- based 11 81.68 81.19 302.75 23.0 414.3 4024.2 181.4

Analytical -2.3 2.2 0 24 -11.7 0 0 Average -2.4 2.4 0 33 -11.7 0 0 Pixel- based 12 46.37 82.38 300.54 150.7 465.0 4249.7 466.2

Analytical 9.9 -5.1 0 27 -9.2 0 0 Average 9.9 -5.0 0 34 -9.2 -1.4 0

similar to the results from the analytical approach with p- 0.3. The differences in runoff between the analytical and average approaches with p - 0.3 are shown in Table 2. Figure 4 shows results for the average approach with p- 1.0 (to be discussed in section 4.3).

In February, a few large storms (highest rainfall rate greater than 10 mm/hr) occur at the end of the month, and result in much larger runoff than in January. Fig- ure 4 and Table 2 show that the analytical and average approaches simulate the runoff quite well compared with the runoff from the pixel-based approach (relative errors of 3.3% and 5.3% for the analytical and average approaches, respectively). The total soil moisture is also well simulated. The differences in the upper layer soil moisture between the pixel-based approach, and the analytical and average approaches are the result of a carry over effect from January. The latent heat flux is well simulated by both analytical and average approaches

with a difference of about 5 Wm -2 out of 51.4 Wm -2

(see Table 2). In March, the average latent heat fluxes from the an-

alytical and average approaches are within 1 Wm -2 of the pixel-based simulation of 84.3 Wm -2 (see Table 2). The differences in sensible heat flux and surface tem-

perature are also negligible. More significant differences occur in the runoff and soil moisture. The total March

rainfall and storm intensities are comparable to those in January. The soil is much wetter than in January due to the February storms. The difference between the pixel- based and other approaches in the upper layer soil moisture is still on the same order as in the previous months, but the difference in the total soil moisture is much

larger as shown in Figure 5. The reason is that as the soil becomes wetter, the saturated area becomes larger. However, the growth of saturation area in the pixel- based approach is faster than in either the analytical or

21,414 LIANG ET AL ß SUBGRID SPATIAL VARIABILITY IN VIC-2L MODEL

analytical(mu=0.3) I II ß 0.3 average(mu=l.O) I o.• "' ß 0.1

I 5 9 13 17 21 25 29 I 5 9 13 17 21 25

1.0

0.8 0.6

0.4 0.2 0.0

1

Mar.

5 9 13 17 21 25 29

Apr.

I 5 9 13 17 21 25 29

10

Dec.

I 5 9 13 17 21 25 29 I 5 9 13 17 21 25 29

Figure 4. Comparison of runoff among the pixel-based (solid), analytical (dotted), and average (dashed) approaches with • - 0.3 for the pixel-based and analytical approaches, and • - 1.0 for the average approach for the PILPS forest site.

average approaches. Therefore, less water goes into the soil, but relatively more, as compared with the other two approaches, drains to the lower layer. Thus there is more saturation overland flow. The analytical approach underestimates the total runoff by 39% in this month, while the average approach underestimates by 52% (see Table 2). The analytical approach approximates the soil moisture of the pixel-based approach better than the average approach. The analytical and average approaches give about the same grid average moisture in the upper layer; however, they differ in their totals because the analytical approach has a faster growth of the portion of saturation area over a smaller area than the average approach (although the growth is slower than in the pixel-based approach). Therefore, less water goes

into the soil in the analytical approach, but relatively more water goes to the lower layer. The April results are similar to March, except that both analytical and average approaches simulate the runoff much better, with relative errors of 0.7% and 5.4% for the analytical and average approaches respectively (see Table 2).

From May to August, it is qui[e dry for the tropical forest climate, with monthly precipitation less than 86 mm/mo. Among the 4 months, August is the driest with monthly precipitation less than 28 mm. The results are similar to March and April, except that the differences in latent and sensible heat fluxes are larger. The results for July (Figure 6) are particularly inter- esting. In July, the average latent heat flux is overestimated by about 24 Wm -2 (out of 77.2 Wm -2) by the

I 5 9 13 17 21 25

315

• 310 o 305

/• • 300 29 I 5 9 13 17 21 25 29

6oo[ 400

-•oo 1 1 5 9 13 17 21 25 29 I 5 9 13 17 21 25 29

3560

:oø1 '- • '"'"J I ....... analytical

• 420 ] I 5 9 13 17 21 25 29 I 5 9 13 17 21 25 29

Figure 5. Comparison of effective surface fluxes, temperature, and soi! moisture among the pixel-based/solid), ana!:ytica! (dotted), and average (dashed) approaches for March with/•- 0.3 for the PILPS forest site.


2.0 •" 1.5

._a 1.o ß • 0.5

o. 0.0

300J 295 [ 290 l 2851

I 5 9 13 17 21 25 29 I 5 9 13 17 21 25 29

200

150 100 so

o

4oo i

.2oo I I 5 9 13 17 21 25 29 I 5 9 13 17 21 25 29

3720

3700 •'3680 •3660

3640 I 5 9 13 17 21 25 29 I 5 9 13 17 21 25 29

Figure 6. Comparison of effective surface fluxes, temperature, and soil moisture among the pixel-based (solid), analytical (dotted), and average (dashed) approaches for July with t•- 0.3 for the PILPS forest site.

analytical and average approaches due to more available water in the upper layer as compared to the pixel-based approach. The upper layer soil moisture from the analytical and average approaches is still higher than the pixel-based approach, but the total soil moisture from the analytical approach is lower than the pixel-based approach from the middle of July on. This is because the soil moisture in the lower layer varies less than the upper layer soil moisture, and also is less affected by precipitation during the dry months. During the wet months, the soil moisture in the lower layer is higher in the pixel-based approach than in the analytical and average approaches, thus the total soil moisture becomes larger in the second half of July, due to the larger decrease in the upper layer soil moisture in the analytical approach. In general, during the dry months, the effect of more evaporation results in decreases of soil moisture in both upper and lower layers. From Figure 6, the change in relative soil moisture among the three approaches can be clearly seen in the upper layer and total layer. By the end of August, the upper layer and total layer soil moisture is reduced to about the same level as the initial soil moisture in January. The maximum monthly runoff during the dry months (May to August ) Occurs in June (8.1 mm). The difference in sensible heat fluxes increases during this period, but the surface temperature remains largely unaffected (see Table 2).

From September to December, precipitation increases and the soil becomes wet again. The large differences in latent and sensible heat fluxes 'occurred in the dry period are significantly reduced. The relative positions of the soil moisture distribution in the upper layer and total become similar to those in March and April. In September, the precipitation increases to 268.1 mm/mo, but the runoff is still small compared with March due to the dry antecedent soil moisture. Both the maximum total amount (372.7 mm/mo) and runoff peaks occur in October, with peaks greater than 6 mm/hr. The major annual and monthly quantities from January to December are summarized in Tables 2 and 3.

An index, defined as the ratio of the absolute difference of a given monthly quantity between the average and pixel-based approaches to the absolute difference of the analytical and pixel-based approaches, is used to summarize the performance of the analytical and average approaches. If the ratio is greater than one, it indicates that the analytical approach approximates the pixel-based approach better than the average approach for that variable. For the latent and sensible heat fluxes, surface temperature, and the upper layer soil moisture, the ratio is only slightly greater than one (not shown in Table 4), which indicates that the two approaches give similar results, with the analytical approach slightly better. Only the monthly runoff and the monthly average total soil moisture show significant dif-

Table 3. Comparison of Annual Quantities for Fixed •- 0.3 at the Forest Site

Method Latent, Sensible, Runoff, Upper W, Total W, Wm -:• Wm -•' mm/yr mm mm

Pixel- based 64.34 66.52 1110.6 402.6 3678.2

Analytical 71.97 59.34 977.0 438.0 3674.5 Average 72.04 59.27 935.7 438.2 3693.8


Table 4. Comparison of Monthly Runoff and Total Soil Moisture for Fixed p - 0.3 at the Forest Site

Ratio Jan. Feb. March April May June July Aug. Sept. Oct. Nov. Dec.

Runoff 1.15 1.61 1.33 7.71 0.91 1.08 0.25 0.98 1.05 1.25 1.38 1.23 Moisture 0.93 1.00 1.77 1.71 1.88 2.69 4.94 0.48 0.58 0.09 0.62 2.94

ferences between the two approaches (Table 4). For the monthly runoff, there are only 3 months in which the ratios are smaller than 1, others are all greater than 1. However, those 3 months are the driest months with monthly total runoff less than 4 mm. In fact, it can be seen from Table 2 that the monthly runoff is well simulated for the big storms by both the analytical and average approaches (February, April, and October), but not so well for the small storms. The reason is that for

big storms, the differences among the methods for the portion of saturation area are small, but increase as the storms become smaller. For the monthly average total soil moisture, even though there are 5 months in which the ratio is smaller than 1.0, the maximum relative error over the year is less than 2% (see Table 2). Therefore, the soil moisture is reasonably well simulated by both the analytical and average approaches.

Overall, there are almost no differences for p = 0.3 among the pixel-based, analytical, and average approaches for surface temperature throughout the year (Table 2). The differences in latent and sensible heat fluxes are small in the wet season and are more significant in the dry season in interstorm periods and also during small storms. Runoff and soil moisture are the two variables with the largest differences among the three approaches. The analytical approach underestimates the annual total runoff by 12 %, and the average approach underestimates by 16 %. In general, the analytical approach gives closer results than the average approach compared with the results from the pixel-based approach. However, the differences between the analytical and average approaches are small, particularly

in latent and sensible heat fluxes, and surface temperature.

The case with p = 0.1 was also studied. The differences in simulated latent and sensible heat fluxes, surface temperature, runoff, and soil moisture are qualita- tively similar to those for p - 0.3. Again, the analytical approach approximates the theoretical approach better than the average approach. The only difference is that the assumption made in both analytical and average approaches of averaging the soil moisture (i.e., assumption 3) is less defensible, and hence, results in larger errors than for the p = 0.3 case. Therefore, the latent heat fluxes from the analytical and average approaches tend to be overestimated compared with the pixel-based approach, and the sensible heat fluxes tend to be underes- timated. This is because when soil moisture is averaged, the area that does not receive precipitation in the previous time steps becomes wetter after averaging than it should be, and thus it has more water available for evapotranspiration than from the pixel-based approach.

The monthly differences between p = 0.3 and 0.1 for latent and sensible heat fluxes, surface temperature, runoff, and soil moisture are listed in Table 5 for the analytical approach. The positive sign in the table indicates that the variables for p - 0.1 give smaller values than those for p = 0.3. From the table, it is seen that during the wet months, the change with different precipitation coverages is small in latent and sensible heat fluxes, but relatively large in the runoff. This is because during the wet months, the soil moisture in both cases is wet enough for transpiration to occur. As p decreases, the rainfall rate within the area which re-

Table 5. Monthly Differences Between p - 0.3 and 0.1 for the Analytical Approach at the Forest Site

Month Latent, Sensible, Surface T, Runoff, Upper W, Total W, Wm -2 Wm -2 øK mm/mo mm mm

I 13.4 -12.4 -0.2 -17.7 17.6 -2.6 2 0.4 -0.4 0.0 -130.1 61.7 46.8 3 1.5 -1.4 0.0 -68.1 39.4 182.1 4 1.5 -1.5 0.0 -70.4 57.8 236.8 5 0.0 -0.1 0.0 0.0 58.6 267.9 6 12.0 -11.3 -0.1 -6.9 59.3 263.7 7 28.4 -27.2 -0.2 -0.6 54.0 249.1 8 37.9 -35.3 -0.4 0.2 17.6 208.9 9 6.2 -5.9 -0.1 -43.4 38.2 213.3 10 -0.2 0.1 0.0 -119.4 66.5 305.4

11 1.3 -1.2 0.0 -25.9 58.5 352.7 12 2.0 -1.8 0.0 -119.2 49.9 463.9


Table 6. Comparison of Monthly Average Quantities for Different Fixed p at the Grassland Site

Method, Month Latent, Sensible Surface T, Runoff, Upper W, Total W, Rainfall, % Wrn -•' Wrn -•" øK ram/too mm mm mm/rno

tz = 0.3 Pixel- based 5 27.3 38.3 290.6 3.8 253.3 2548.0 26.9

Analytical -3.3 1.3 0 -32 0 0 0 Average -4.0 1.8 0 -29 0 0 0 Pixel-based 6 47.1 25.0 297.3 4.0 238.8 2538.3 50.3

Analytical -9.9 16.0 0 -55 0 - 1.2 0 Average -10.4 16.4 0 -50 0 -1.1 0 Pixel-based 7 67.4 13.0 298.5 26.1 274.3 2634.3 282.9

Analytical -21 91 0 -30 -11.4 0.7 0 Average -21 91 0 -26 -11.4 0.7 0 Pixel-based 8 48.6 20.3 294.3 25.5 328.4 2891.7 445.6

Analytical -11.7 23 0 -177 -18.0 2.0 0 Average -11.9 24 0 -171 -18.0 2.0 0 Pixel-based 9 31.6 28.0 290.8 19.0 340.9 3165.5 144.6

Analytical -9.7 10.4 0 -56 -9.3 3.0 0 Average -9.7 10.7 0 -52 -9.3 2.9 0

tz = 0.6 Pixel-based 5 29.7 36.6 290.5 3.7 252.7 2547.1 26.9


Analytical -7.2 15.8 0 -47 0 0.6 0 Average -7.4 16.4 0 -44 0 0.7 0 Pixel-based 7 88.4 -4.8 298.0 7.1 290.2 2626.4 282.9


Analytical -9.9 87 0 -570 -14.0 2.4 0 Average - 10.1 89 0 -556 - 14.0 2.4 0 Pixel-based 9 37.2 23.1 290.7 11.8 347.6 3153.4 144.6


tz = 1.0 Pixel-based 5 31.6 35.3 290.5 3.7 252.2 2546.5 26.9


Analytical -5.3 15.5 0 -33 1.4 0 0 Average -5.3 16.1 0 -33 1.5 0 0 Pixel-based 7 110.5 -23.4 297.5 4.8 298.9 2610.0 282.9

Analytical -7.2 28 0 -327 -2.1 0.7 0 Average -8.0 31 0 -310 -2.1 0.7 0 Pixel-based 8 82.2 -8.8 293.6 5.9 385.6 2836.6 445.6



ceives precipitation increases. Therefore, it results in a larger portion of saturated area, and thus more direct runoff results, and less water infiltrates (although relatively more water drains to the lower soil layer). While a decrease in precipitation coverage p may result in an

increase in runoff, it decreases soil moisture. During the dry months, the change of precipitation coverage • results in a large difference in latent and sensible heat fluxes, but small change in runoff. This is because the soil is quite dry, and transpiration occurs under soil


water stress. Also, when the precipitation coverage is increased, the interception is increased, which increases the evaporation.

4.2. Comparison of Different Approaches at a Grassland Site

For the grassland site, three cases (p - 0.3, 0.6, 1.0) were tested for the same three approaches. All of the three approaches start with the same initial soil moisture (50% of the maximum). The monthly values for latent and sensible heat fluxes, surface temperature, runoff, and soil moisture for the pixel-based approach, and the relative biases for the analytical and average approaches are summarized in Table 6 for p-0.3, 0.6, and 1.0. The positive and negative signs have the same meanings as in Table 2. From the table, it is seen that the differences between the pixel-based, and the analytical and average approaches are smallest for p - 1.0, and largest for p - 0.3. All of the relative biases are not large except for the monthly total runoff and the sensible heat flux in July. The large differences in the monthly runoff are because the soil is relatively dry in the grassland site, and it takes time for the pixel- based approach to develop the saturation area for direct runoff. However, in the analytical and average approaches, there is always a portion of saturated area no matter how dry the soil is, due to the structure of the infiltration curve. Therefore, there will be always some direct runoff as long as there is rainfall. The hourly time series for runoff (not shown here) indicate that the analytical and average approaches overestimate the very small runoff (on the order of less than 0.3 mm/hr) relative to the pixel-based approach. For larger runoff under slightly larger precipitation, the analytical and average approaches underestimate the runoff peaks for the same reason as in the forest case. Since there

are many small rainfall occurrences at the grassland site, the monthly total runoff from both the analytical and average approaches are overestimated. However, it should be emphasized that the monthly total runoff is less than 10 mm in most cases (see Table 6), so the absolute errors are small. As for the sensible heat flux

in July, the large relative bias is due to its quite small magnitude.

From Table 6, it is also seen that as the precipitation coverage p increases, the latent heat flux increases, and the sensible heat flux decreases as in the forest case.

This is again due to more intercepted water for evaporation when the precipitation coverage is increased. For the analytical approach, the maximum change occurs in July with a 21 Wm -2 increase from p- 0.3 to 0.6, and 16 Wm -2 increase from p = 0.6 to 1.0 for latent heat flux. The average summer month increase for latent heat flux is 10 and 9 Wm -2 from p -- 0.3 to 0.6 and from 0.6 to 1.0, respectively. The average summer month decrease for sensible heat flux is about 9 Wm-2.

The average surface temperature change is only about 0.2øK from p = 0.3 to 0.6, and about the same for

p - 0.6 to 1.0. The average runoff change is less than 5 mm/mo in both cases. Unlike the forest case, the soil moisture does not change much with different precipitation coverages in the grassland case. This may be the result of smaller precipitation. The upper layer soil moisture average change is less than 4 mm and the total soil moisture average change is less than 15 mm in both cases. The latent and sensible heat fluxes are

most sensitive to the change of precipitation coverage. Also, it is seen that the monthly (Table 6) and the average summer month changes from p - 0.3 to 0.6 is larger than that from p - 0.6 to 1.0 in general, which indicates that the effects of'the change in precipitation coverage is nonlinear.

From the above analysis and discussion for both the forest and grassland sites, it can be concluded that for the same fractional precipitation coverage p, the analytical approach approximates the pixel-based approach better than the average approach, but the differences between the analytical and average approaches are modest. The four assumptions made in deriving the one- dimensional statistical dynamic model seem justifiable in general. Assumption 3, averaging the soil moisture over the fractional area p with the unwetted fractional area i-p at the beginning of next storm, begins to break down for the smallest value of p=0.1 investigated.

4.3. Comparison of Different Approaches- Effects of p- 1.0 in the Average Approach

Most GCMs still use p - 1.0. Therefore, we also studied the effects on the latent and sensible heat fluxes, runoff, surface temperature, and the soil moisture in comparison to different p values for both the pixel-based and analytical approaches at the forest and grassland sites.

At the forest site, hourly results of March, for example, for the latent and sensible heat fluxes, surface temperature, and soil moisture, with p -- 0.3 for the pixel-based and analytical approaches, and p = 1.0 for the average approach, are shown in Figure 7. Also, the runoff results for the same months as shown for

the pixel-based and analytical approaches with p - 0.3 are shown in Figure 4. The comparison of Bowen ratio among the three approaches with both p- 0.3 and 1.0 for the average approach, and p - 0.3 for the pixel- based and analytical approaches is shown in Figure 8. The monthly results, expressed as relative biases to the ones obtained by the pixel-based approach for p - 0.3, for the latent and sensible heat fluxes, surface temperature, soil moisture are summarized in Table 7, where a negative sign indicates that the average approach overestimates the pixel-based approach. In Table 8, annual average latent and sensible heat fluxes for p - 0.3 and 0.6 for both the pixel-based and analytical approaches, and for p -- 1.0 for the average approach are shown. Again, all of the approaches start with the same initial soil moisture (50% of the maximum). From these figures and tables and also by comparing Table ? with

LIANG ET AL.- SUBGRID SPATIAL VARIABILITY IN VIC-2L MODEL 21,419

I 5 9 13 17 21 25 29 I 5 9 13 17 21 25 29

3ooJ .' i , i Ii I I J Alii, I ,I,

I 5 9 13 17 21 25 29 I 5 9 13 17 21 25 29

550 t • 500[ •, 45ø! '"'"-:-.-.., i ................. ' .... ß 400'

37501 / - ' - ......

36501 '// • pixel-based analytical

•' 3600 ! --- average •oo I

I 5 9 13 17 21 25 29 I 5 9 13 17 21 25 29

Figure 7. Comparison of effective surface fluxes, temperature, and soil moisture among the pixel-based (solid), analytical (dotted), and average (dashed) approaches for March with t• - 0.3 for the pixel-based and analytical approaches, p- 1.0 for the average approach for the PILPS forest site.

Table 2, it is seen clearly that taking/• = 1.0 in the average approach results in much higher evaporation, lower sensible heat flux, much smaller runoff, and higher soil moisture than the pixel-based and analytical approaches. This is because when p = 1.0, the precipitation rate within the grid cell (or an area) becomes much smaller than if t• is less than 1. Therefore, much higher latent heat flux is obtained due to higher evaporation from canopy interception, and also due to higher transpiration from more moist soil. In addition, with the decrease of precipitation rate, the saturated area is developed much slower in the average approach than in the pixel-based and analytical approaches, and therefore results in much less saturation overland flow (see Figure 4). Although the evaporation simulated by the average approach is always larger than the pixel-based and analytical approaches, the much smaller runoff from

2.0

0.0

•..• pixel-based analytical avg.(mu=0.3) avg.(mu=1.0)

Months

Figure 8. Comparison of Bowen ratio among the pixel- based, analytical, and average approaches with/• = 0.3 for the pixel-based and analytical approaches, and/•- 0.3 and/• = 1.0 for the average approach for the PILPS forest site.

the average approach results in higher average soil moisture in the average approach than in the other two approaches, and the higher soil moisture resulted in the average approach favors again the transpiration for the dry months.

At the grassland site, which has a drier climate than the forest site, simulations for /• - 0.3, 0.6, and 1.0 for the pixel-based and analytical approaches, but for p- 1.0 for the average approach, were also conducted for the summer months (May to September), with the initial soil moisture half of the maximum. Again, similar results were obtained as those for the forest case.

More details and analysis of the results are given by Lian# [1994].

Compared with the results discussed previously where the average approach uses the same/• as the other two approaches, it is seen clearly that larger differences are obtained for the latent and sensible heat fluxes, surface temperature, runoff, and soil moisture when /• is taken to be 1.0. Such results indicate that for VIC-2L

model, the fluxes are more sensitive to the precipitation coverage parameter/• than to the spatially distributed exponential precipitation. This finding seems to con- flict with the findings by Pitman et al. [1990] that both the exponential precipitation distribution and the precipitation coverage p have significant effects on monthly evaporation, and runoff. On closer examination though, the two findings are consistent. From equation (2), we see that both the precipitation P(a•) and the soil prop-

f* ai- tributed random variables. What this study shows is that for VIC-2L, when the subgrid spatial variability of the soil property is included, the effect of subgrid variability of precipitation on surface fluxes, runoff, and soil moisture will be small if it is not included, as long as the precipitation coverage/• is considered. In other words, in equation (2), it is not necessary to treat both P(z)


Table 7. Relative Biases of Monthly Average Quantities With p - 1.0 in the Average Approach at the Forest Site

Month Latent, Sensible, Surface T, Upper W, Total W, % % % % %

1 -91 40 0 0 0.6

2 -40 21 0 -17.0 0

3 -18.1 24 0 -13.1 -5.9

4 -35 21 0 -16.4 -6.7

5 -13.3 6.9 -0.5 -4.4 -11.1

6 -35 51 0 -9.2 -6.4

7 -54 132 0 -6.0 -5.5

8 -34 32 0 2.6 -4.5

9 -70 46 0 -5.4 -3.9

10 -11.9 12.5 0 -15.6 -8.1

11 -29 28 0 -13.4 -9.8

12 -21 11.5 0 -12.5 -11.4

and f*(z) as spatially distributed random variables. The effect of the spatially distributed precipitation and soil properties seems to be well represented by just considering one of them, which is the soil infiltration variable in this case. That is, the inclusion of the spatial variability of infiltration property seems to be able to represent the major effect of spatial variabilities due to the random variables of P(a•) and/* (a•), since the VIC- 2L model results are quite similar with and without the inclusion of the spatial precipitation distribution. Pit- ma•. et al. [1990] assumed a constant infiltration rate in equation (2), and thus, the subgrid spatial variability is only represented through the spatial distributed precipitation. Therefore, they found that there is large difference in monthly evaporation and runoff with or without considering the spatially distributed exponential precipitation. In other words, what they found is that there is large difference if one of the two variables in equation (2) is considered to be spatially varied ver- sus if both are constant. They did not study the case when both subgrid variabilities are considered in equation (2). Therefore, it is hard to say for BATS if there would be small or large difference if both variables in equation (2) are considered to be spatially distributed.

The experiments performed in this paper indicate that the effect of spatial subgrid-scale variability in precipitation is important and should be included in land surface parameterizations. Among the features

of the spatial subgrid-scale variability in precipitation, the precipitation coverage p is more important than the precipitation distribution over the precipitation-covered area for VIC-2L model parameterization. In addition, the comparison shows that the analytical approach approximates the pixel-based approach reasonably well in terms of surface fluxes, surface temperature, runoff, and soil moisture, and is superior to the average approach, although the differences between the analytical and average approaches are modest.

5. Conclusions

This paper has described a one-dimensional statistical dynamic model using an analytical approach that accounts for the effects of subgrid spatial variability in both rainfall and soil property. The approach is tested against an explicit pixel-based approach, and compared with an average approach which uses spatially averaged precipitation over the precipitation covered area p and also over an entire grid cell. In addition, sensitivity studies on the precipitation coverage parameter p are conducted. The major conclusions from this study are summarized below.

The analytical approach proposed in this paper approximates the pixel-based approach quite well in simulations of surface fluxes, surface temperature, runoff, and soil moisture at both the forest and grassland sites.

Table 8. Comparison of Annual Fluxes With p = 1.0 in the Average Approach at the Forest Site

0.3 /• = 0.6

Latent, Sensible, Latent, Sensible, Wm -2 Wm-2 Wm-2 Wm-•

Pixel-based 64.3 66.5 73.7 57.8

Analytical 72.0 59.3 76.9 54.7 Average 87.7 44.5 87.7 44.5


The differences between the two approaches are that for the forest site, the analytical approach underestimates the annual runoff and sensible heat flux, and overestimates the latent heat flux; and for the grassland site, it overestimates the latent heat flux and annual runoff, and underestimates the sensible heat flux and total soil

moisture. The surface temperature is very robust at both sites.

The effect of spatial subgrid-scale variability in precipitation is important and should be included in land surface parameterizations. Among the features of the spatial subgrid-scale variability in precipitation, the precipitation coverage parameter/• is more important than the type of precipitation distribution over the precipitation-covered area/•, in part due to the representation of the subgrid spatial variability of soil properties included in VIC-2L. For the area where k• value varies largely with time, it is important to obtain a good estimate of precipitation coverage/• for each storm based on radar images or on other methods.

The differences among surface fluxes, runoff, and soil moisture for the pixel-based, analytical and average approaches are modest for VIC-2L model if the precipitation coverage p is the same in each of the three approaches, although the analytical approach approximates the pixel-based approach better than the average approach, especially in the runoff and soil moisture. However, for the sake of simplicity, it is not necessary to include the spatial subgrid variabilities both in precipitation and in soil properties into a land-surface scheme if it has already accounted for the spatial subgrid variability of soil properties. Therefore, the average approach of VIC-2L is recommended for GCM applications, if the precipitation fractional coverage k• is considered.

The latent and sensible heat fluxes are quite sensitive to the amount of water intercepted by the canopy. Therefore, more attention needs to be given to the variables that describe interception in the future. For example, the spatial subgrid variabilities of canopy and/or maximum canopy storages over large areas will eventually be incorporated in a consistent way in our one- dimensional statistical dynamic model so that the effects of spatial variabilities of canopy interceptions could be considered. The work of Eliahir and Bras [1993b] suggests one path that might be followed. In this study, however, the sensitivities of predicted fluxes to precipitation coverage were studied with fixed leaf area index (LAIs) (hence fixed interception capacity).

The four assumptions made in deriving the one-dimensional statistical dynamic model seem justifiable in general. The third assumption of averaging the soil moisture over the fractional coverage t• and the non-rainfall fractional coverage 1-t• at the beginning of next s•orm may lead to significant errors if the precipitation coverage t• is too small (e.g., less than 0.3). Although the second assumption which averages •he soil moisture at each time step within each vegetation cover type may

seem questionable, the results suggest that it is defensible.

It is important to note that the feedback effects from atmosphere are not included in this study, that is, the testing was conducted off-line. Pitman et al. [1993b] found less sensitivity in the response of BATS to the spatial subgrid variability of precipitation in an on-line (coupled land-atmosphere simulations) than did Pitman et al. [1990] in an off-line simulations. However, in the 1993b study the annual precipitation was considerably (30 to 40%) less than that in their 1990 study, which may well affect this conclusion. Nevertheless, atmospheric feedback effects are potentially important and will be a focus of future research.

Future work will also compare our analytical approach with the observations. The availability of observational data is problematic, but intensive field cam- paigns such as Boreal Ecosystem-Atmosphere Study (BOREAS), where tower flux and aircraft flux measurements, and radar precipitation JR. Schnur et al., Spa- tial and temporal analysis of radar-estimated precipitation during the BOREAS summer, 1994 field cam- paigns, submitted to the Journal of Geophysical Re- search, 1996.] are available, offer one possibility for model evaluation.

Acknowledgments. This research was funded in part by the U.S. Department of Energy's (DOE) National Insti- tute for Global Environmental Change (NIGEC) through the NIGEC Western Regional Center (DOE Cooperative Agreement DE-FC03-90ER61010), and in part by the Na- tional Science Foundation under grants EAR-9318896-001 and EAR-9318898-001. The authors would like to thank

the anonymous reviewers for their careful considerations and valuable suggestions.

References

Abdulla, F.A.M., Development of a stochastic single-site seasonal rainfall simulator (for application in Irbid re- gion), M.S. thesis, Jordan Univ. of Sci. and Technol., Irbid, 1987.

Atlas, D., D. Rosenreid, and D.A. Short, The estimation of convective rainfall by area integrals, I, The theoretical and empirical basis, Y. Geoph!Is. Res., 95D, 2153-2160, 1990.

Beven, K.J., and M.J. Kirkby, A physically based variable contributing area model of basin hydrology, H!Idrol. Sci. Bull., œ4(1), 43-69, 1979.

Braud, I., J.D. Creutin, and C. Barancourt, The relation between the mean areal rainfall and the fractional area

where it rains above a given threshold, Y. Appl. Meteorol., 3œ, 193-202, 1993.

Chiu, L.S., Estimating areal rainfall from rain area, in Trop- ical Rainfall Measurements, edited by J.S. Theon and N. Fugono, 528pp., A. Deepak, Hampton, Va., 1988.

Collier, C.G., The application of a continental-scale radar database to hydrological process parameterization within atmospheric general circulation models, Y. H!Idrol., 301-318, 1993.


Dickinson, R.E., A. Henderson-Sellers, P.J. Kennedy, and M.F. Wilson, Biosphere-atmosphere transfer scheme (BA- TS) for the NCAR community climate model, NeAR Tech. Note, TN-•75-1-$TR, 1986.

Dumenil, L., and E. Todini, A rainfall-runoff scheme for use in the Hamburg climate model, in Advances in Theoreti- cal Hydrology, A Tribute to James Dooge, edited by J.P. O'Kane, European Geophys. Soc. Series on Hydrological Sciences, vol. 1, Elsevier, New York, 129-157, 1992.

Eagleson, P.S., The distribution of catchment coverage by stationary rainstorms, Water Resour. Res., •0(5), 581- 590, 1984.

Eagleson, P.S., and Q. Wang, Moments of catchment storm area, Water Resour. Res., •I(8), 1185-1194, 1985.

Eagleson, P.S., N.M. Fennessey, Q. Wang, and I. Rodriguez- Iturbe, Application of spatial Poisson models to air mass thunderstorm rainfall, J. Geophys. Res., 9•(D8), 9661- 9678, 1987.

Eltahir, E.A.B., and R.L. Bras, Estimation of the fractional coverage of rainfall in climate models, J. Clirn., 6, 639- 644, 1993a.

Eltahir, E.A.B., and R.L. Bras, A description of rainfall interception over large areas, J. Clirn., 6, 1002-1008, 1993b.

Entekhabi, D., and P.S. Eagleson, Land surface hydrology parameterization for atmospheric general circulation models including subgrid scale spatial variability, J. Clirn., •, 816-831, 1989.

Famiglietti, J.S., and E.F. Wood, Evapotranspiration and runoff from large land areas; Land surface hydrology for atmospheric general circulation models, in Land Surface- Atmospheric !nteractions •or Climate Modeling: Observa- tions, Models and Analysis, edited by E.F. Wood, 179-204, Kluwer Academic, Norwell, Mass., 1991.

Famiglietti, J.S., E.F. Wood, M. Sivapalan, and D.J. Thongs, A catchment scale water balance model for FIFE, J. Geo- phys. Res., 97(D17), 18,997-19,007, 1992.

Gao, X., and S. Sorooshian, A stochastic precipitation dis- aggregation scheme for GCM applications, J. Clirn., 7, 238-247, 1994.

Gong, G., D. Entekhabi, and G.D. Salvucci, Regional and seasonal estimates of fractional storm coverage based on station precipitation observations, J. Clirn., 7, 1495-1505, 1994.

Henderson-Sellers, A., and A. Pitman, Land-surface schemes for future climate models: Specification, aggregation, and heterogeneity, J. Geophys. Res. 97 (D3), 2687-2696, 1992.

Henderson-Sellers, A., A.J. Pitman, P.K. Love, P. Irannejad, and T.H. Chen, The project for intercomparison of land surface parameterization schemes (PILPS): Phases 2 and 3, Bull. Am. Meteorol. Soc., 76(4), 489-503, 1995.

Liang, X., D.P. Lettenmaier, E.F. Wood, and S.J. Burges, A simple hydrologically based model of land-surface water and energy fluxes for general circulation models, J. Geophys. Res., 99(D7), 14,415-14,428, 1994.

Liang, X., A two-layer variable infiltration capacity land surface representation for general circulation models, Water Resour. Series, TR140, 208 pp., Univ. of Wash., Seattle, 1994.

Morrissey, M.L., The effects of data resolution on the area threshold method, J. Appl. Meteorol. 33, 1263-1270, 1994.

Pitman, A.J., A. Henderson-Sellers, and Z.L. Yang, Sen- sitivity of regional climates to localized precipitation in global models, Nature, 3•46, 734-737, 1990.

Pitman, A.J., et al., Project for intercomparison of land- surface parameterization schemes (PILPS): Results from off-line control simulation (phase la), IGPO Pub. Set., 7, 47pp., WCRP GEWEX, Washington D.C., 1993a.

Pitman, A.J., Z.L. Yang, and A. Henderson-Sellers, Sub-grid scale precipitation in AGCMs: Reassessing the land surface sensitivity using a single column model, Clirn. Dyn., 9, 33-41, 1993b.

Raupach, M.R., The averaging of surface flux densities in heterogeneous landscapes, Proceedings of the Yokohama Symposium, IAHS Publ. •1•, 343-355, 1993.

Rowntree, P.R., and J. Lean, Validation of hydrological schemes for climate models against catchment data, J. Hydrol., 155, 301-323, 1994.

Sauvageot, H., The probability density function of rain rate and the estimation of rainfall by area integrals, J. Appl. Meteorol., 33, 1255-1262, 1994.

Shuttleworth, W.J., Macrohydrology-the new challenge for process hydrology, •. Hydrol., 100, 31-56, 1988.

Sivapaian, M.K., K.Beven, and E.F. Wood, On hydrologic similarity, 2, A scaled model of storm runoff production, Water Resour. Res., •3(12), 2266-2278, 1987.

Thomas, G., and A. Henderson-Sellers, An evaluation of proposed representation of subgrid hydrologic processes in climate models, •. Clirn., •, 898-910, 1991.

Warrilow, D.A., A.B. Sangster, and A. Slingo, Modeling of land-surface processes and their influence on European Climate, Dyn. Clirn. Tech. Note 38, 92pp., U.K. Meteo- rol. Off., Bracknell, Berkshire, England, 1986.

Wigmosta, M., L. Vail, and D.P. Lettenmaier, A distributed hydrology-vegetation model for complex terrain, Water Resour. Res., 30(6), 1665-1679, 1994.

Wolock, D.M., G.M. Hornberger, K.J. Beven, and W.G. Campbell, The relationship of catchment topography and soil hydraulic characteristics to Lake Alkalinity in the Northeastern United States, Water Resour. Res., •5(5), 829-837, 1989.

Wood, E.F., Global scale hydrology: Advances in land surface modeling, U.S. Natl. Rep. Int. Union Geod. Geo- phys. 1987-1990, Rev. Geophys., •9, Suppl., 193-201, 1991.

Zhao, R.J., Y.L. Zhang, L.R. Fang, X.R. Liu, and Q.S. Zhang, The Xinanjiang model, in Hydrological Forecast- ing Proceedings Oxford Symposium, IAHS I•9, 351-356, 1980.

X. Liang and E.F. Wood, Department of Civil En- gineering and Operations Research, Princeton Univer- sity, Princeton, NJ 08544. (xliang•princeton.edu; ef- wood•pucc.princeton.edu)

D.P. Lettenmaier, Department of Civil Engineering, University of Washington, Seattle, WA 98195 (let- tenma•ce. washington. ed u)

(Received August 16, 1995; revised April 16, 1996; accepted April 16, 1996.)

paper number 96jd01448. i--ira [1- (1- a) l/hi] (1)blyon/references/p25.pdf(7), (8), and (9) are all...

Documents