Recent work by Isham et al. and Rodrìguez-Iturbe et al. has characterized the space-time variability of soil moisture through its analytically derived covariance function which depends on soil properties, vegetation structure, and rainfall patterns typical of a region. This paper uses such characterization to address the strategies and methodologies for the sampling of soil moisture fields. The focus is on the estimation of the long-term mean soil moisture and the daily soil moisture averaged over a given area as a function of the network geometry, number of stations, number of sampling days and landscape heterogeneity. It is found that the spatial geometry of the network has a significant impact on the sampling of the average soil moisture over an area in any particular day, while it is much less relevant for the sampling of the long-term mean daily soil moisture over the region. In the latter case, the length of the record is a commanding factor in what concerns the variance of estimation, specially for soils with shallow rooted vegetation. Spatial vegetation heterogeneity plays an important role on the variance of estimation of the soil moisture, being particularly critical for the sampling of the average soil moisture over an area for a given day.
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 One of the most promising strategies for soil moisture monitoring is through remote sensing techniques [e.g., Walker et al., 2001]. These provide estimates of soil moisture over large areas that need to be calibrated with ground-based measurements. Land surface models also require quantitative comparison between simulated soil moisture maps and field measurements. In this context, Robock et al.  evaluated land surface model results from the North American Land Data Assimilation System (NLDAS) [see Mitchell et al., 2004] using in situ observations over the southern Great Plains, but without considering the effects of sampling errors.
 An attempt to estimate sampling errors by using empirically estimated correlation functions is presented by Vinnikov et al.  evaluating the root-mean-square error of the soil moisture networks of Illinois and Oklahoma Mesonet. Using the optimal averaging technique introduced by Kagan , they show that the Mesonet network does not provide a significant improvement in the accuracy respect to the less dense Illinois network. Yoo  adopted the formalism proposed by North and Nakamoto  for the estimation of sampling errors in soil moisture fields using the soil moisture model of Entekhabi and Rodríguez-Iturbe . The procedure has been applied to the Washita 1992 data highlighting the inefficiency of ground-based networks for large-scale observations. Within the Soil Moisture Experiment 2002 (SMEX02), Jacobs et al.  found that number of samples to measure the soil moisture content with a given confidence interval may change according to the field characteristics and to the soil moisture content itself. This suggests that further investigations are necessary to understand the role of spatial heterogeneity of the land surface on the sampling of soil moisture fields.
 In the present paper, the sampling errors are estimated using the space-time soil moisture correlation structures analytically derived by Isham et al.  and Rodríguez-Iturbe et al.  for a model considered appropriated for a water-limited ecosystem. The methodology explicitly accounts for soil characteristics, vegetation patterns, and rainfall dynamics neglecting topographical effects and the upper bound due to saturation. The soil moisture model can be considered representative of a relatively flat landscape under semiarid climate conditions. Both random sampling and stratified random sampling, which are the commonly used schemes over extended areas, are investigated for the case of homogeneous and heterogeneous vegetation. The study adopts the analytical framework introduced by Rodríguez-Iturbe and Mejía  for rainfall sampling that has been extensively used for the design of rainfall networks [e.g., Bras and Rodríguez-Iturbe, 1976].
 It is important to point out that this paper will focus on the estimation of soil moisture integrated over the root zone of the vegetation. No attempt will be made to estimate soil moisture values across the depth of the root profile.
2. Space-Time Soil Moisture Dynamics
 Soil moisture dynamics is modeled through a water balance equation driven by stochastic space-time rainfall forcing as studied by Isham et al.  and Rodríguez-Iturbe et al. . The rainfall model is the same proposed by Cox and Isham  where rainfall occurrences are modeled by a sequence of circular rain cells that occur as a Poisson process of rate λR in space and time. Each cell is characterized by random radius, intensity and duration that are assumed independent of each other and exponentially distributed in order to keep as a minimum the number of parameters. The total intensity of the rainfall process, Y(u, t), is given by the superposition of cells that may overlap at any spatial location u at time t. Although the use of the term cell is customary in this type of model, one should be aware that they are a mathematical construct whose characteristics do not really match those of the usually identified as rain cells in different types of precipitation fields.
 The water balance is written in the simplified form
where the dimensionless random variable S(u, t) represents the relative soil moisture at site u and time t, n is the soil porosity, and Zr is the depth of the root zone. On the right hand side, the term (1 − ϕ)Y(u, t) is the infiltration rate that is determined by the precipitation reduced by (1 − ϕ) representing the vegetation interception. The second term, VS(u, t), is the loss function given by the sum of evapotranspiration and leakage assumed to be a linear function of soil moisture. The water loss coefficient, V, depends on both vegetation and soil properties and the factor ϕ depends on the plant species and condition of vegetation. Typical values of ϕ range between 0.4 and 0.1 according to Lull . Although continuous in time, all results from equation (1) need to be interpreted at the daily timescale since we do not include any dynamics below the daily level neither in the evapotranspiration losses nor in the storm structure [Rodríguez-Iturbe and Porporato, 2004].
 For the purpose of this paper, the use of a linear approximation for the description of soil water losses is reasonable especially for nonhumid conditions. Pan et al.  provide estimates of soil water loss coefficients obtained from the analysis of ground measurements of the Monsoon 1990 and Washita 1992 campaigns relating those values to the soil permeability and the vegetation Leaf Area Index (LAI) [Pan et al., 2003]. They obtain values of V that range between 3 and 16 mm d−1.
 The soil moisture scheme adopted above does not account for the presence of an upper bound due to soil saturation. This limitation was found to be necessary to obtain closed analytical results for the covariance structure of soil moisture fields. Isham et al.  show that the lack of an upper bound is not a restrictive assumption in terms of its impact on the covariance when dealing with arid and semiarid environments.
 With the above modeling scheme, the expected value of the relative soil moisture at a point (e.g., the origin) with given vegetation is
where a0 = [V/(nZr)]0 is the normalized soil water loss, b0 = [(1 − ϕ)/(nZr)]0 is the normalized net rainfall coefficient, μR = 1/ρR is the mean rain cell radius, μD = 1/η is the mean storm duration and μX = 1/β is the mean rainfall intensity of a cell [Isham et al., 2005].
 The expression for the space-time covariance function of the relative soil moisture conditional on the cover of the points A and B, l apart from each other, is [Rodríguez-Iturbe et al., 2006]
where the site (B), characterized by the parameters (aB, bB), is “attached” at the later time and the vegetation cover controls the value of the pairs (aA, bA) and (aB, bB).
 In the case of uniform vegetation, the space-time covariance function of relative soil moisture becomes
 The correlation function of the relative soil moisture is a separable function in its space and temporal components. For the case of uniform vegetation, the space correlation is the same of the rainfall process
and the temporal correlation is
 The case of heterogeneous vegetation is modeled by Rodríguez-Iturbe et al.  by a marked point process that account for the presence of two functionally different vegetation types in the landscape (e.g., grasses and trees). The spatial structure of vegetation is described through a matrix of grass with trees located according to a Poisson process in space with rate λT which have random circular crowns with radii exponentially distributed with parameter ρT. This representation may provide a reasonable approximation for the vegetation patterns found, for instance, in savanna ecosystems (further details of the modeling scheme are reported by Rodríguez-Iturbe et al., 2006).
 To derive the unconditional space-time soil moisture covariance, it is necessary to distinguish between points covered and not covered by a tree crown. The subscript c will be used for first ones and the subscript u for the uncovered points.
 The probability that a point A is not covered by a tree is
where λ′T = 2πλT/ρT2 represents the rate of tree crown occurrence at any point.
 The joint probabilities of occurrence of the different combinations of vegetation cover at two points A and B at distance l in space are given by
 Using the above probabilities, Rodríguez-Iturbe et al.  give the space-time covariance for the relative soil moisture between two points as
where point B corresponds to the later time and the conditional covariances CAB are given by equation (3).
 The variance of the relative soil moisture in a heterogeneous landscape is obtained from equation (12) as
where CF = 1 − exp(−λ′T) represents the tree cover fraction over the landscape.
Figure 1 shows an example of the correlation function of the relative soil moisture in a heterogeneous landscape. Two characteristic regimes are observed arising from the strong separation of scales between the rainfall forcing and the vegetation patterns. The first sharp reduction of correlation is related to the structure of vegetation and, as one may observe in Figure 1, to the density of tree coverage which is 18%, 45%, and 87%, respectively in this particular example. The second reduction is due to the rainfall forcing that imposes a large scale structure in the spatial correlation of the soil moisture field. The control of the atmosphere on the correlation structure of soil moisture fields at large spatial scales has been also point out by Vinnikov et al.  and Entin et al. .
 The mean value of the soil moisture in the heterogeneous vegetation case may be obtained as the sum of the means of covered soil and uncovered soil weighted by their relative probabilities
When the entire landscape is covered by tree or grass, the mean assumes the same expression of the uniform vegetation case.
3. Soil Moisture Sampling in Time and Space
 Two descriptors of the soil moisture process are particularly relevant in hydrology: (1) the long-term mean soil moisture for a given time interval during a given season (e.g., daily soil moisture during month of June) at any point of a statistically homogeneous region, mS, and (2) the mean soil moisture over an area SA = S(xi)da, where S(xi) is the soil moisture at a site xi and represents a realization of the soil moisture process over a region assumed statistically homogeneous. S(xi) may represent an instantaneous value or an average value over a particular time interval (e.g., mean soil moisture for a given day). The last one may be used for water resources management and ecohydrological analyzes. The first one is important for the calibration of remote sensing techniques and global circulation model (GCM). The estimation of mS is described in the following, while that of SA is presented in section 3.3. In both cases we will assume that daily soil moisture is the point variable one wishes to estimate either in a long-term basis or integrated over an area for a particular day.
 Assuming N sample points in space operating during T days of the same statistically homogeneous season, mS is estimated through as given by
 It is assumed that the individual samples, S(xi, t), adequately represent the average daily soil moisture at site xi and day t. The goodness of the estimation is measured through the variance of ,
Equation (16) is a measures of the magnitude of the estimation errors, − mS, that are random variables with zero mean. The variance of the estimate of mS is equivalent to the point variance, σS2, if there is only one sample in space and time (T = 1, N = 1). Any additional samples have the effect of reducing this variance, and the degree of that reduction is a subject of this paper.
 The variance of is a function of the space-time correlation function of the process, the number of sampling points, the geometry of the sampling scheme and the length of operation of the stations. Following Rodríguez-Iturbe and Mejía , the variance of can be written as
where f(xi, t) = (S(xi, t) − mS).
Equation (17) may be written as the product of two reduction factors affecting the variance of the daily soil moisture at a point, σS2,
 The reduction factor F1(T) can be derived explicitly from equation (19) for the case of uniform vegetation using the correlation function given in equation (7),
In this case, the temporal reduction factor, F1(T), depends on the parameters η and a that are related to the physical characteristics of the soil (a) and the dynamics of rainfall (η). The inverse of each of these parameters represents a characteristics timescale; the former is the characteristic time of drying, equal to nZr/V, while the latter represents the mean storm duration, or the characteristic time of wetting. When η ≫ a, the temporal correlation function of soil moisture may be simplified to exp(−ah), implying that the second term in the parenthesis on the right hand side of equation (21) becomes negligible. It is important to remark here that the lack of dependence of the correlation structure on the frequency of storm arrivals is due to the assumption of a Poisson process for rainfall occurrences.
Figure 2 shows two examples of the reduction factor F1(T) as a function of the number of days of sampling, T, ranging from 1 to 200 days. Although the duration of a season through which the atmospheric forcing may be considered statistically homogeneous is likely to be less than 90 days, one could always sample repeatedly over that season throughout many years as long as the initial condition at the beginning of the season is not too close to saturation. In such a case, we should account for the saturation bound and the actual model cannot be applied. The graphs show the improvement in the estimation of mS achieved by increasing the number of sampling days, T, for different values of nZr and η. Figure 2a shows that the increase of daily measurements drastically improves the estimation of mS. Shallow root soils that are more susceptible to temporal fluctuations induced by rainfall and soil losses will quickly benefit from the temporal sampling and 100 days of measurements will accomplish a dramatic reduction of the variance of the long-term mean daily soil moisture with respect to the point variance of the process. On the other hand, soils with deeper roots require much longer daily data to approach comparable reductions in the variance of the long-term mean daily soil moisture. One can also observe that F1(T) is not sensitive to the changes on the mean duration of the rain cells (see Figure 2b).
 With regard to the spatial factor, F2(N), its calculation depends on the spatial geometry of the sampling scheme. The two cases considered herein (random spatial sampling and stratified spatial random sampling) are discussed in the following.
where E[r(xi − xi′)∣A] is the expected value of the correlation between two points randomly located in the area considered. For the case of a rectangular region, Rodríguez-Iturbe and Mejía  obtain E[r(xi − xi′)∣A] as
where f(l) is the probability density function of the distance between two randomly chosen points in the region (see Appendix A) and R represents the diagonal of the rectangular area. For the results of this paper we will assume the areas of interest are approximated by a square region.
 Since the spatial correlation of the relative soil moisture in the case of homogeneous vegetation (equation (6)) is solely a function of the dimensionless product lρR, one can obtain the factor F2(N) as function only of AρR2 as shown in Figure 3 where it is plotted for different values of N. F2(N) decreases monotonically with N and AρR2. The decrease in F2(N) is more pronounced for increases in N when the number of stations is still small (e.g., from 1 to 5 stations). Also depending on the values of AρR2 one may obtain a relatively small or a very large decrease of F2(N) when increasing the number of stations. From a practical point of view, the scales of greater interest for hydrologists are frequently those with AρR2 ≤ 1 (see Figure 3 inset). A value AρR2 = 1 generally implies an area of less than 104 km2 given that typical values of ρR range between 10−1 and 10−2 km−1. At those scales, we observe relatively minor changes in the function F2(N) whose role is clearly less important that the one played by the temporal reduction factor F1(T). Thus for the estimation of the long-term mean daily soil moisture in a region with homogeneous vegetation the length of operation of the network appears to be the commanding factor and trade of time versus space using more stations over a shorter period of measurements is not a realistic option.
3.2. Stratified Spatial Random Sampling
 The variance reduction factor due to spatial sampling, F2(N), is now obtained by dividing the area A in non overlapping strata of area a. Following Rodríguez-Iturbe and Mejía , F2(N) may be written as
 When there is only one station for each stratum F2(N) is
Equation (25) is plotted with a dotted line in Figure 3 where it can be observed that the use of stratified spatial random sampling does not provide significant improvements for the estimation of mS respect to those obtained through a simple random sampling.
3.3. Mean Soil Moisture Over an Area for Any Given Day
 Estimates of the daily soil moisture averaged over a given area, SA, are required in numerous cases like those aimed to provide input or validation of land surface models. Moreover, the use of remotely sensed images provides spatial and temporal description of soil moisture fields, but such techniques also need to be calibrated with ground based approaches.
 Following Rodríguez-Iturbe and Mejía , in the case of a spatial random sampling, the performance of a network for the estimation of SA = S(xi)da may be described by the variance of the sample mean, , using N points in space as
while in the case of stratified random sampling with the assumption of one station per strata, the variance reduction factor due to the spatial sampling is
 In contrast to mS, the variance of the estimate of SA will only be as large as the point variance, σS2, if one has a single sample over an infinitely large area. Thus, for smaller areas (e.g., especially areas smaller than the correlation scale of the rain cells), the variance of the estimate will be less than the point variance. If the area is very small (i.e., smaller than the vegetation and rainfall correlation scales), then a single measurement will be adequate to determine the spatial average. Also, if the averaging area is very large (much larger than any correlation scales), then the variance of the estimate of SA decreases as 1/N as confirmed by Figure 4.
Figure 4 shows the reduction factor F2(N) computed using equations (26) and (27) with the correlation function of soil moisture for the case of a region with homogeneous vegetation. It is seen that the stratified random sampling may produce significant reductions on the errors of estimation of the areal soil moisture with respect to those occurring with a random design. The difference between the two sampling schemes becomes more significant with the increase of N. For cases where AρR2 ≤ 1, the reduction factor F2(N) assumes relatively small values even with only few stations located in the region.
3.4. Soil Moisture Fields With Heterogeneous Vegetation
 The effects of vegetation heterogeneity on the soil moisture sampling are now addressed analyzing the impact of different vegetation covers resulting from varying the rate of occurrence of tree centers in space (λT).
 We will deal first with the estimation of the long-term mean soil moisture, mS. In the case of heterogeneous vegetation cover, the algebraic expression for F1(T) is more complex and is not reported herein, but its numerical computation is straightforward using the correlation structure obtained from equations (12) and (13) in equation (19). Some examples are reported in Figure 5 that shows the variance reduction factor due to temporal sampling, F1(T), for different heterogeneous vegetation cover and with rainfall parameters estimated, during the period May/August, using a data set of 18 stations belonging to the state of Oklahoma (USA). In this graph, the solid lines represent the two extreme cases one with uniform grass vegetation (e.g., λT = 0) and the other for full coverage of trees. In general, the presence of a heterogeneous vegetation tends to increase the values of F1(T) with respect to those corresponding to a homogeneous cover. In the other hand, spatial sampling, as will be explained in the following, provides a stronger reduction of the variance of estimation than that obtained in the homogeneous vegetation case under comparable conditions. A larger value of F1(T) and a smaller F2(N) make trade of time versus space more appealing for the case of heterogeneous vegetation than for an homogeneous vegetation cover when estimating the long-term mean daily soil moisture over a region.
Figure 6 shows an example of variance reduction factor due to the spatial sampling, F2(N), as a function of the dimensionless parameter AρR2 and for different values of N. The graph displays F2(N) for two types of sampling designs and for a region with a heterogeneous vegetation cover with the parameters given in Table 1. This example shows the effect of heterogeneous vegetation on the parameter F2(N) that specially for the small values of AρR2 (see inset of Figure 6) is considerably smaller than F2(N) for the case of homogeneous vegetation shown in Figure 3. Similar to what was obtained for F2(N) for the case of the long-term mean daily soil moisture in a region with homogeneous vegetation, the stratified random sampling offers little advantage over the simple random sampling.
Table 1. Parameters Adopted to Characterize the Two Functionally Different Vegetation Typesa
The percentage of tree cover is 45% in this case.
VT [mm d−1]
VG [mm d−1]
 The impact of vegetation heterogeneity on the estimation of the long-term mean daily soil moisture over a region is more clear in Figure 7, where the variance reduction factor, F2(N), for the case of N = 10 is given for different cases of vegetation structure. The results are a direct consequence of the characteristic shape of the correlation function of relative soil moisture (Figure 1). In particular, the presence of vegetation heterogeneity at small scales produces a reduction of the factor F2(N) for small values of AρR2 which is not present for the cases when λT is either very small or very large.
 We now address the impact of heterogeneous vegetation in the estimation of the mean soil moisture over an area for a particular day (SA). Figure 8 shows the variance reduction factor, F2(N), for random and stratified random geometries. The subdivision of the domain in strata provides substantial improvement in the mean soil moisture estimation as the product AρR2 becomes smaller. The presence of an heterogenous vegetation cover leads to larger values of F2(N) relative to those obtained for the homogeneous vegetation case (Figure 4). This implies an increase in the number of stations in order to maintain comparable reduction factor. To evaluate the goodness of estimation, one needs to multiply the reduction factor times the variance of the soil moisture process, σS2, as in equation (26). The value of σS2 is larger for the heterogeneous vegetation case than for homogeneous vegetation case for comparable rainfall and soil conditions. The impact of heterogenous cover is now the opposite of the one found in F2(N) for the estimation of the long-term daily soil moisture over a region. This is particularly evident in Figure 9 where F2(N = 10) is shown for different values of the parameter λT.
 The space-time sampling of soil moisture fields has been quantitatively explored using a stochastic soil moisture model dependent on rainfall characteristics, soil properties and vegetation structure. The parameter a = V/(nZr) has been identified as a key factor controlling of the temporal correlation function of the soil moisture process. Thus it strongly affects the sampling of the long-term mean soil moisture. In particular, an increase of a leads to a higher variability of the soil moisture dynamics and, as shown in Figure 2, for small values of nZr leads to a drastic decrease in the value of the variance reduction factor due to temporal sampling. This implies that networks with relatively short period of measurements will nevertheless provide significant reductions in the variance of estimation of the long-term mean daily soil moisture in soils with shallow roots, high permeability and high evapotranspiration rates.
 One may also notice, from equation (5), that the variance of the soil moisture is proportional to (1 − ϕ)2/(nZrVη) when η ≫ a. Thus, if from one side the variance reduction factor is smaller with smaller nZr, from the other side the actual variance is larger. On the contrary, the increase of the soil water losses, described through the parameter V, leads to a decrease of both the variance and F1(T). Consequently, the estimate of the long-term mean daily soil moisture will be more accurate in soils with high permeability and high evapotranspiration rates.
 In the case of heterogeneous vegetation cover the role of F1(T) is less commanding than in the homogeneous case but temporal sampling still remains a very important consideration for the estimation of the long-term mean daily soil moisture. The impact of the number of stations in the network depends heavily on the values of the parameter AρR2. In the case of homogeneous vegetation cover for AρR2 ≤ 1, F2(N) plays a minor role in the estimation of the long-term mean daily soil moisture and there is little to gain with sizable increases in the number of stations in the network. This is not the case for regions with heterogeneous cover where F2(N) plays a more important role even for small values of AρR2. For both types of cover it is only important to go beyond N = 5 for cases when AρR2 > 100 showing that the greatest gain in information for the estimation of the long-term mean daily soil moisture in a region is obtained with an initial, relatively small, number of stations.
 The number of stations plays a crucial role, as expected, in the estimation of the average soil moisture over a region for a given day. This role is relatively more effective for the case of an homogeneous vegetation cover which yields values of F2(N) smaller than those obtained for the heterogeneous case in otherwise comparable conditions. This implies the need for a larger number of sampling stations over the region with heterogeneous cover to obtain a value of F2(N) similar to that of the homogeneous vegetation case.
 It is important to point out that although our discussion has been carried out in terms of the variance reduction factors F1(T) and F2(N), the variance of estimation of, both, the long-term mean daily soil moisture and the average soil moisture over an area in any particular day, are given by the decrease that F1(T) and F2(N) have over the point variance of the process (e.g., equations (18) and (26)). Moreover, the variance reduction factors are the only ones controlled by the length of operation and the spatial geometry of the sampling scheme, while the point variance of the process is not affected by those characteristics. The goodness of a sampling network for soil moisture fields should be judged by the values of the variance of and which are a function of the point variance σS2 and which itself depends on the climate, soil, and vegetation characteristics of the area under consideration.
 The network design has been analyzed considering two different sampling schemes: the random sampling and the stratified random sampling. It was found that the role of the network geometry is a secondary one in the estimation of the long-term mean daily soil moisture, while it is significant when one is interested on average soil moisture over an area for a particular day. The first goal is a common one in water resources management and ecohydrological studies while the second estimation is particularly useful to calibrate remote sensing techniques for soil moisture estimates and toward the validation of large-scale models (e.g., global circulation models) in the description of soil moisture dynamics.
Appendix A:: Distribution of Distance of Random Points in a Given Area
 The probability density function (PDF) of the distance, l, between two randomly chosen points in a rectangular area has been derived by Ghosh . Given a rectangular area (A = c × d) with sides c and d with (c > d), the PDF of l has the expression
For a square area of side d, which is the case used in the this paper, the expression simplifies to
normalized net rainfall coefficient that describes the effects of the vegetation interception [mm−1].
c and d
sides of a rectangular area [km].
variance reduction factor due to the temporal sampling [dimensionless].
variance reduction factor due to the spatial sampling [dimensionless].
temporal lag [days].
distance between two points [km].
long-term mean soil moisture for a given time interval during a given season, [dimensionless].
soil porosity [dimensionless].
cell radius [km].
estimator for the long-term mean soil moisture mS [dimensionless].
estimator for the averaged soil moisture over an area in a given day [dimensionless].
mean soil moisture over an area in given day [dimensionless].
relative soil moisture at time t [dimensionless].
water soil loss coefficient [mm d−1].
depth of the root zone [mm].
rate of rain cells per unit time and unit area [d−1 km−2].
rate of tree canopy per unit area [km−2].
mean value of the canopy radius [km−1].
mean value of rainstorm duration (η = 1/μD) [km].
mean cell radius (ρR = 1/μR) [km].
mean rainfall intensity (β = 1/μX) [mm d−1].
interception coefficient [dimensionless].
 I.R.I. and S.M. gratefully acknowledge the support of NOAA under the grant NA17RJ2612, NSF under the National Center for Earth Surface Dynamics (NCED) grant EAR0120914, and Biocomplexity grant DEB0083566. S.M. also acknowledges the support of MIUR (Italian Ministry of Instruction, University and Research) through the research program 2003084552. We are thankful to Andrew Guswa for his thoughtful comments on the manuscript.