Published online 29 March 2006
Published in Soil Sci Soc Am J 70:728-735 (2006)
DOI: 10.2136/sssaj2005.0173
© 2006 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
Soil Physics
Determining Vertical Root and Microbial Biomass Distributions from Soil Samples
Freeman J. Cooka,* and
Francis M. Kelliherb
a CSIRO Land and Water, 120 Meiers Road, Indooroopilly, QLD 4068, Australia, and The Univ. of Queensland, St. Lucia, QLD 4067, Australia
b Manaaki Whenua–Landcare Research, P.O. Box 69, Lincoln, New Zealand
* Corresponding author (freeman.cook{at}csiro.au)
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ABSTRACT
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When vertical density distributions of root or microbial biomass are calculated using each sampling interval's midpoint as the depth coordinate, the calculated distribution is biased if it is a nonlinear function with depth. In the root biomass literature, distributions are often described by a power function R 
ßz, where ß is a decay coefficient and z is depth. A common alternative formulation is an exponential function, R e–z/Zr, where Zr is a characteristic length scale. These functions are equivalent when Zr = –1/ln ß, so the data according to either function may be unified. The bias can be eliminated by representing the vertical distribution with a continuous function, integrating it over the sampling interval, and using a least squares method to determine the function's parameters. The bias increased by nearly threefold when the sampling interval increased from 0.01 to 1 m. As the sampling interval increases, the bias shifts the function down the z axis. This results in the intercept increasing with increasing sampling interval. When a single profile was sampled at different intervals, the function's intercept and Zr changed. The parameter Zr changed fivefold when the sampling interval increased from 0.1 to 0.5 m, while the calculated fraction of roots above a depth of 0.1 m decreased threefold for the same change in sampling interval. Beneath a tropical forest where root biomass and microbial respiration were sampled throughout the same soil profile, the corresponding microbial and root biomass length scales averaged 0.17 m and differed by only 11%.
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INTRODUCTION
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MANY SOIL PROPERTIES change with depth; such relations are known as vertical distributions. For example, it is a common observation that root biomass density decreases with depth. Analysis often requires a continuous function, but meaningful application also requires the corroboration of measurements that are usually made at discrete depth intervals. We wondered about the consequences of this potentially mismatched combination of requirements. Surprisingly, we found no guidance in the soil science literature. Moreover, although gas requirements vary vertically with root and microbial biomass distributions (Cook and Knight, 2003a, 2003b), data on such distributions were sparse, with root and microbial biomass studies largely published in different, specialist journals. An exception was the study of Veldkamp et al. (2003), whose data we will further analyze.
In this paper we consider the vertical distributions of root and microbial biomass densities in soils. We develop theory for consistent analyses according to comparable power and exponential functions including analogous decay coefficients and length scales. Analyses presented here will quantify and eliminate the bias error in nonlinear relations between average values and depths from sampling intervals. The vertical density distributions of root and microbial biomass represented by nonlinear functions require careful determination (Addiscott and Tuck, 2001). We develop theory for consistent and mathematically correct analysis that eliminate the depth bias caused by averaging depths, and show that decay coefficients in power functions correspond to length scales in exponential functions.
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THEORY
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A soil property's vertical distribution may be written:
 | [1] |
where Y is a soil property, f is a continuous function, and z is the depth given here as a positive downward direction with z = 0 at the soil surface. Measurements are usually based on soil samples, representing a set of average property values (
i) over discrete depth intervals: 
zi = zi+1 – zi, where zi and zi+1 are respectively the depth at the top and bottom of the soil sample, and i is an index variable. These data are often analyzed by plotting i = f(
i), where i = (zi + zi+1)/2. This procedure relates i to the midpoint of the depth interval i. When f(z) is not linear this can yield erroneous values of the parameters.
The vertical distribution, Eq. [1], relates a continuous function to data by
 | [2] |
Bias results because i is not the correct depth corresponding with i, unless f is a linear function of z. Linear functions are unlikely for soil properties, so generally the correct depth (zi*) is given by solving
 | [3] |
for zi*.
We illustrate by considering the vertical distributions of root and microbial biomass densities in soils. Setting aside an erroneous assumption in Gale and Grigal (1987) (see Appendix), the vertical distribution of root biomass density, R, may be estimated by
 | [4] |
where R0 is the root biomass density scaling factor (M L–3), and a decay coefficient ß < 1 normalizes the root data and determines the relation's shape. Equation [4] was used (Jackson et al., 1996, 1997; Jobbagy and Jackson, 2000) to represent vertical root distributions with ß as a discriminating variable that distinguished 11 vegetation types on a functional basis. With these publications, z must be expressed in centimeters; if other units are used for z, then ß must also be converted. We acknowledge that many factors such as oxygen supply, soil freezing, plant species, mechanical impedance, and nutrient limitations can limit root density (Craine et al., 2005).
Alternatively, R can be described by an exponential function that is written (Cook and Knight, 2003a; Dwyer et al., 1988, 1996)
 | [5] |
where R = R0 at z = 0. Equations [4] and [5] are equivalent when the root biomass length scale Zr = –1/ln(ß). This equivalence allows conversion and comparison of data between the two different equations. It also increases the data available for problems like oxygen transport to roots that was analyzed by Cook and Knight (2003a, 2003b).
For the vertical distribution of soil microbial biomass density, M (M L–3), an exponential function that is analogous to Eq. [5] may also be written (Cook, 1995) as
 | [6] |
where M0 is M evaluated at z = 0 and Zm is a soil microbial biomass length scale. To a depth of 1 m, Castellazzi et al. (2004) concluded that an exponential function like Eq. [6] was able to express the results of 12 studies representing a range of soils and vegetation types with the mean value of Zm = 0.21 m and a coefficient of variance of 17%. In studies where the depth of sampling has exceeded 1 m, such as those of Veldkamp et al. (2003), the sampling interval is often increased with depth to maintain the total number of samples at a practical level. This also recognizes that the rate of change in soil properties generally diminish with depth.
Equations [5] and [6] have the same form, and applying Eq. [2] to them results in:
 | [7] |
where Y0 is M0 or R0 and Zx is Zm or Zr. Thus, for a data set of (i, zi, zi+1), the values of Y0 and Zx can be found by minimizing the sum of the squares of the difference between the RHS and LHS of Eq. [7]. This can be done with the Excel (Microsoft Corporation, Redmond, WA) solver function or with SigmaPlot (SPSS Inc., Chicago, IL) as was used here. Care should be taken when using such packages to ascertain that the statistics are correct (McCulloch and Wilson, 1999). We checked our results using Maple v. 7 (2001, Waterloo Maple, Waterloo, ON, Canada).
To obtain zi*, Eq. [5] or [6] may be substituted into Eq. [3], and this yields:
 | [8] |
where 
i = zi/Zx. The bias from relating i+1/2 to i instead of zi* can be expressed
 | [9] |
For a constant i, the numerator in Eq. [9] is constant and hence the curve is displaced along the z axis by this constant amount. Further, for a constant i, Eq. [9] shows that the maximum bias will be in the segment nearest the soil surface, as the functions evaluated here have maximum curvature near the surface. Evaluation of Eq. [9] for the first segment gives
 | [10] |
Because the maximum bias is related solely to 1, we may consider 1 to be the independent variable in Eq. [10]. The function evaluated decreases with depth, so the first segment is most important and has the most weight in any linear fit procedure, as was illustrated by Kimball (1978; see his Eq. [7]).
If Zx is known a priori or has been estimated from preliminary sampling, a sampling regime with constant bias comes from rearranging Eq. [9] to give
 | [11] |
where 
= B/Zx(B – 1) and B is the bias as defined in Eq. [9]. Equation [11] can be solved recursively for i and hence for zi. For example, as the depth increases, Eq. [11] shows that there is an increase in the sampling interval associated with a constant negative 10% bias (Fig. 1
). However, when depth is small or zi is large, the value of zi predicted by Eq. [11] can exceed zi. The utility of Eq. [11] thus includes application limits.
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RESULTS AND DISCUSSION
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For tropical forest, wheat crop and pasture root data, representing a range of vertical distributions, the integral method using Eq. [7] consistently yielded means and standard deviations that were the same or lower than those given by the direct fitting method (Eq. [5]) (Table 1). Variable Zr was calculated consistently from the crop and pasture data with Eq. [5] and [7]. However, for the tropical forest data, Eq. [5] yielded a 4% larger value and slightly lower r2. For R as a function of z from the tropical forest data, this slightly different Zr was only detectable in a log-linear plot (Fig. 2
). The plot also shows that both equations significantly underestimate the data below a depth of 1 m. We acknowledge that a combined linear and exponential function could give a better fit.
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Fig. 2. (a) Relations between root biomass density, R, and depth, z, beneath a tropical forest according to Veldkamp et al. (2003). The lines and data points are explained in the legend. (b) Same data as in (a), plotted as log-linear to highlight differences.
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We realize that vertical distributions are not always understood from variables and functions. As written earlier, f(z) is commonly determined by inspection and portrayed graphically. This can be useful, but no less care is required. For example, Gale and Grigal (1987) reviewed previously published data of cumulative root density with depth to determine values of ß using Eq. [A4]. They then differentiated Eq. [A4] with respect to z and plotted what they considered to be instantaneous root distributions for three values of ß. This seemed misleading to us because of an unstated and unlikely mathematical assumption (see Appendix). For the same three values of ß used by Gale and Grigal (1987), viz. ß = 0.92, 0.95, and 0.97 we calculated the normalized root density (R* = R/R0). As a function of depth, R* is compared with the original data presented by Gale and Grigal (1987) (Fig. 3
). We assert that R* is the correct interpretation of the instantaneous root distribution, not RGG (Appendix) as used by Gale and Grigal (1987). Gale and Grigal (1987) suggested that "instantaneous" vertical root distribution for depths less than 
0.2 m, as related to ß, was 0.97 > 0.95 > 0.92. This interpretation is not correct and the trend in normalized root density (R* = R/R0) related to ß is 0.92 > 0.95 > 0.97, the opposite.
Beneath tropical forest and temperate woodland (Geescoft and Broadbalk soils), the integral method using Eq. [7] yielded consistent values of M0 and Zm compared with the direct fitting method (Eq. [6]) (Veldkamp et al., 2003; Castellazzi et al., 2004; Table 2). Veldkamp et al. provided data for the estimation of root biomass and microbial respiration distributions in their tropical forest. Using Eq. [7], their Zm was within 11% of Zr. This comparison illustrates the potential value of analyzing microbial and root biomass distributions in the same way and, although preliminary, suggests the possibility of equating Zm and Zr values. Compared with the other two soils, the Broadbalk soil's M0 was up to four times larger and Zm was as much as 33% smaller. The Broadbalk soil was thought to have a higher annual return of plant debris than the Geescoft soil. The Broadbalk soil's M0 was 11% greater and Zm was 29% less from Eq. [7] than from Eq. [6], although r2 did not change.
The bias results from relating i to i instead of zi*. When i decreases with depth, the bias means that a vertical distribution function's prediction of depth will be an overestimate, and for a constant zi, Eq. [9] shows that the maximum bias will be in the segment nearest to the soil surface. This maximum bias and C (Eq. [10]) are proportional to 1 (Fig. 4
). For 1 = 0.2, the maximum bias is 6%. As Zx decreases, root or microbial biomass is increasingly concentrated near the surface, so a smaller depth interval is needed to conserve bias in the vertical distribution.
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Fig. 4. Relations between the maximum bias and parameter C determined for a vertical root biomass distribution and 1, a sampling depth interval to length scale ratio, according to Eq. [10]. The maximum bias is denoted by negative values to indicate the required correction.
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The scaled amount by which the curve is displaced along the z axis (C + i/2) decreases with i (Fig. 4), which implies that the displacement will increase with i. This displacement, as shown below, will affect the fitted function's intercept.
For soil properties where i increases nonlinearly with depth, use of the average sampling depth will underestimate rather than overestimate the depth associated with i. Similarly, for an increasing nonlinear function, the bias will increase with depth. An example of a soil property that often increases with depth is bulk density.
We can examine what the bias due to z will mean to the parameters in Eq. [5] or [6]. For a constant value of z, the use of i instead of zi* causes a shift of –Zx(C + i/2) along the z-axis. This does not affect the shape factor Zx, but does affect Y0. The extent of this effect on Y0 can be calculated by equating Eq. [7] and i = Y0*e–i/Zx, where Y0* is the apparent value of Y0:
 | [12] |
This relationship shows that as i increases, Y0*/Y0 increases at a greater rate (Fig. 5
). The error Y0 caused by not correcting the sampling interval size can be substantial. This is illustrated in Fig. 6
using three values of z (0.1, 0.5, and 1.0 m).
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Fig. 5. Relationship between Y0*/Y0, ratio of apparent to actual soil property at the surface, and i, a sampling depth interval to length scale ratio, according to Eq. [11].
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Fig. 6. (a) Relations between root biomass density, R, and depth, z, according to Eq. [5] and [7]. For Eq. [7], values of i were calculated using characteristic length scale (Zr) = 0.184, R0 = 1111 g m–3, and discrete soil sample depth intervals, z, of 0.1 (five solid circles between the surface and z = 0.5 m), 0.5 (2 solid squares shown at z = 0.75 and 1.25 m) and 1.0 m (3 open squares shown at z = 1.5, 2.5, and 3.5 m). The data points were calculated to a depth of 4 m. The three curves were derived by least squares fitting of i and i to Eq. [5] using all the values calculated for a single z, and led to a single value of Zr (= 0.184), but three values of R0 including 1115, 1487, and 3092 g m–3 for z of 0.1, 0.5, and 1.0 m, respectively. (b) Same data plotted as log-linear to better illustrate the intercept shift.
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The discussion above pertains to a constant z throughout the soil profile. When different sampling intervals are used, different shifts along the z axis occur over the soil profile. This may affect R0 and Zr. We constructed a data set with R0 = 1111 g m–3, Zr = 0.1840 m, and three values of z of 0.1, 0.5, and 1.0 m, and applied Eq. [7] for z from 0 to 4 m. Fitting Eq. [5] to the i, i data generated R, z relations showing effects of the z axis shift on R0. We then subsampled from each data set to create composite data of i, i where z was different for different depth segments. These are the data points in Fig. 6. Equation [5] was again fitted to this composite data set, producing a different R0 (1124 g m–3) from that used to generate the data.
The data of Veldkamp et al. (2003) illustrate the effect of different sampling intervals on the R and z relation and its descriptive parameters. Veldkamp et al. (2003) sampled over a 0.1-m interval near the surface increasing up to a 1-m interval between depths of 1 and 4 m. Minimizing the sampling interval will be most effective near the surface where the vertical change in root and microbial biomass tend to be the greatest. However, when different z values occur in the same soil profile, the vertical distribution function becomes a composite and Zr is also changed. For the data of Veldkamp et al. (2003), Zr is overestimated by 3% when Eq. [5] is used instead of Eq. [7] to fit the parameters to the data (Table 1).
The power and exponential functions are equivalent for analyses of root and microbial biomass data. However, for comparisons of vertical distributions, we find the length scale was more meaningful than a dimensionless decay coefficient. Conversion of data from the literature reviews of Jackson et al. (1996, 1997) suggested that Zr had a fivefold range from 0.1 to 0.5 m. Using Eq. [5] and [A4], including conversion to the exponential function, it can be shown that the fraction of root biomass density to a depth of 0.1 m decreased threefold from 0.6 (Zr = 0.1 m) to 0.2 (Zr = 0.5 m).
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CONCLUSIONS
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Vertical distributions of root and microbial biomass densities have usually based on depth-averaged samples, with nonlinear relations between average values and depths determined from sampling intervals. This approach introduces a bias that for a constant sampling interval is greatest in the segment closest to the soil surface. The bias is eliminated by representing the vertical distribution with a continuous function, integrating it over the sampling interval and using a least squares method to determine values of the function's parameters. Analysis showed that the bias error depended on the soil sample depth interval. Increasing the interval from 0.01 to 1 m corresponded with a nearly threefold increase of the error. When different intervals were used in the same soil profile, the distribution became represented by a composite function. This mixture complicated interpretation of the function's parameter values. From the root biomass literature, we analyzed a commonly used power function and its decay coefficient. This revealed an unstated and unlikely mathematical assumption that affected an earlier literature review in its portrayal of root biomass density and depth. Although elementary, because it has apparently been overlooked, we showed that the power and exponential functions are equivalent. The root biomass density length scale had a fivefold range from 0.1 to 0.5 m. Correspondingly, the fraction of roots to a depth of 0.1 m decreased by threefold from 0.6 to 0.2. Beneath a tropical forest where root biomass and microbial respiration were uniquely sampled throughout the same soil profile, the two length scales were similar. Although requiring further tests under a wider range of conditions, we think underlying plant and soil processes support the possibility of interchangeable length scales.
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APPENDIX
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If the root biomass density is given by
 | [A1] |
integration of [A1] with respect to z gives over the limits of 0 to 
, the total root biomass density in the soil profile is
 | [A2] |
and the total root biomass density to any depth z is
 | [A3] |
The fraction of total root biomass above a depth z, Rf, is then
 | [A4] |
Gale and Grigal (1987) used Eq. [A4] to determine values of ß. They integrated Eq. [A4] to produce an "instantaneous" model of the root density, which results in
 | [A5] |
Symbol RGG is used here so that the integration of Eq. [4] is distinctive from R. Comparison of Eq. [A1] and [A5] indicates that A = –ln ß. However, we cannot know the value of A from knowledge of ß in Eq. [4], as A is cancelled in the division in Eq. [A4]. Consequently, the interpretation of Gale and Grigal (1987) from Eq. [A5] and [A1], that A = –ln ß is erroneous. We can, however, define a normalized instantaneous root biomass density function, R*, by writing
 | [A6] |
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ACKNOWLEDGMENTS
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F.M. Kelliher was funded by the New Zealand Foundation for Research, Science and Technology (contract C09X0212). We thank Des Ross and Tim Clough for showing us the papers by Castellazzi et al. and Kimball, respectively. We thank Drs. Mike Trefry and David Rassam for useful comments, especially Mike's comments on the equivalence of Eq. [4] and [5].
Received for publication June 1, 2005.
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REFERENCES
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ABSTRACT
INTRODUCTION
