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DIVISION S-1—SOIL PHYSICS

Exponential Distribution Theory and the Interpretation of Splash Detachment and Transport Experiments

Faculty of Earth and Life Sciences, Vrije Univ. Amsterdam, De Boelelaan 1085, 1081 HV Amsterdam, The Netherlands

* Corresponding author (dija{at}geo.vu.nl)

	ABSTRACT

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Soil detachment and transport by rainsplash is usually the first step in soil loss and sediment transport. It can be measured using a variety of approaches, including splash cups, trays, and boards. However, the results of splash experiments are affected by their geometry and not readily translated into generally applicable parameters. In this study, we develop a theory that can be used to interpret splash experiments. It is based on the assumption that the spatial distribution of particles splashed from a point source can be described by an exponential decay function, for which there is considerable support in the literature. The theory is evaluated for the cited experimental techniques, partly with the use of a numerical model. It is made clear that conventional measurements of splash and the true rate of detachment by splash are two different entities that can be linked if the average splash length is known. In principle, the theory is not valid for a sloping surface, but analysis of the magnitude of the error involved indicates that in many cases good estimates of detached amounts can still be obtained.

Abbreviations: FSDF, fundamental splash distribution function

	INTRODUCTION

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THE DETACHMENT AND TRANSPORT of soil particles ensuing from the impact of raindrops, or splash for short, is usually considered an important first step in the chain of processes leading to loss of soil and subsequent sediment transport. Falling raindrops are able to detach much larger amounts of soil particles than unconcentrated overland flow, after which the detached particles may be entrained and carried off by the flowing water (Hudson, 1995). In addition, splash may result in significant net transport of sediment on sloping soils (Moeyersons and de Ploey, 1976; Wan et al., 1996). To quantify splash detachment and transport, a range of techniques has been developed with time.

Sreenivas et al. (1947) were among the first to use round cups or funnels embedded in the soil to catch splashed particles. They considered the mass of collected material divided by the surface area of the cup (with radius R, in m), denoted here by m_R, as an indicator of detachment rate on the surrounding soil. This method has been applied in many subsequent studies, a number of which were reviewed by Poesen and Torri (1988). However, the cup splash rate (m_R) is not independent of cup size, as already put forward by Rose (1960) and Bolline (1975). Furthermore, the relationship between cup size and splash rate is also influenced by the distribution of distances over which the splashed particles travel: Given a particular amount of soil detached per unit of area outside the cup (i.e., detachment rate µ, as mass per unit area) more sediment will end up in the cup as the distance over which splashed particles travel increases. On the basis of the reciprocity principle, the same will be true for a similar experimental technique in which a cup filled with soil is subjected to (artificial) rain and the sediment ejected from the cup is collected (Ellison, 1947; Rose, 1960; Al-Durrah and Bradford, 1981, 1982; Kinnell, 1982; Riezebos and Epema, 1985; Sharma and Gupta, 1989). If the cup is small compared with the average splash length, it can be argued that the bulk of sediment splashed from the cup will end up outside the cup (Rose, 1960). However, the use of very small cups gives rise to practical problems, and therefore cups are normally used with a diameter of a few to >10 cm. Under such conditions, the resulting amount of collected sediment can no longer be equated to detachment rate, as is still commonly practiced.

Alternatives to circular cups include splash boards (Ellison, 1944; Kwaad, 1977) and trays (Quansah, 1981; Savat and Poesen, 1981; Wan et al., 1996), in which case the amount of splash is expressed either per unit tray area or length. The results are again not readily converted to actual detachment rate (µ) because the amounts of sediment ending up in the collectors themselves depend on the distribution of distances over which the splashed particles travel. In addition, unless the dimensions of the tray are several orders of magnitude greater than the average splash length, tray size will also have a significant effect on measured splash rates, as does the size of splash cups.

The problems outlined above complicate the interpretation of experimental results much more than is often acknowledged. Farmer (1973), for example, subjected a soil-filled tray of 46 by 122 cm to artificial rainfall. From the difference in particle size distributions between the original and splashed material, he inferred preferential detachment of certain size classes. A comparable study by Gabriels and Moldenhauer (1978) used trays of 30 by 45 cm. However, as Poesen and Torri (1988) demonstrated, small particles are splashed over greater distances than larger particles. Consequently, a greater fraction of detached particles will end up outside the tray for small size classes than for larger size classes. Thus, wrong conclusions may be drawn from the measurements (regardless of the possibility that smaller particles are indeed detached more easily).

Summarizing, the (apparent) detachment rates derived from cup, tray, or splash board experiments must be considered ill-defined, experiment-specific, and not applicable to field situations, unless the geometry of the experiment and the spatial distribution of the splashed particles around their source are taken into account. Recognizing this, Farrell et al. (1974) advanced a theory describing the influence of experimental geometry on splash measurements. The practical application of their theory was limited initially, because little information existed about the spatial distribution of splashed particles. However, the spatial distribution of splash has been measured in later studies, such as those of Poesen and Savat (1981), Savat and Poesen (1981), Riezebos and Epema (1985), and Torri et al. (1987).

The objective of the current paper is to develop a mathematical distribution theory that can be used to assess the influence of geometry and splash length in the type of experiments described above. In doing so, it is initially assumed that the spatial distribution of particles splashed from a point of raindrop impact can be described by an exponential function, which will be called the fundamental splash distribution function (FSDF). It will be shown that there is considerable support for this in the literature. On the basis of the FSDF, the relationship between observed amounts of splash and actual detachment rates will be evaluated in terms of a characteristic splash length and the geometry of the experiment. In some cases, this is done analytically; in others, a numerical model is used.

	THEORY

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Exponential Fundamental Splash Distribution Function
Let the rate of detachment µ (in g m^-2) be defined as the mass per unit source area that has been detached by rainsplash. Consider the splash from a small source area onto a small target area at radial distance r; let the splash density m_point(r) (in g m^-4) be the splashed mass per unit of source area, per unit of target area. It is subsequently assumed that m_point(r) is described by the following exponential function:

[1]

where (in m) is the average splash length, that is, the mass-weighted average radial distance over which the particles are splashed. Within the context of the theory presented in this paper, Eq. [1] may be called the FSDF, since all other equations are derived from it by integration.

There is empirical support for the assumption that the FSDF conforms to an exponential equation. Riezebos and Epema (1985) conducted laboratory experiments using a soil cup surrounded by concentric rings at 5-cm intervals that collected sediment splashed from the cup. Individual artificially produced water drops fell from different heights on the center of the soil-filled cup. In all cases, the experimental results for the mass splashed beyond a distance r were described very well by an exponential function (Fig. 1) . It can be shown that this conforms to the assumed FSDF by considering the amount of material M_ring(r) (in g m^-3) that is splashed from a small source area at the center of impact, onto a very narrow ring at a distance r from the source, per unit of radial distance and per unit of source area:

[2]

View larger version (17K): [in this window] [in a new window]   Fig. 1. The mass of soil splashed beyond a radial distance r from its point source (data from Riezebos and Epema, 1985). Exponential functions are fitted to the data, which represent splash resulting from artificial drops falling from indicated heights.

In principle, the amount of material detached (splash mass, M, in g) by a single impacting drop can be calculated once the area of impact (a, in m²) is known:

[3]

which demonstrates that the use of µ in Eq. [1] is consistent. In reality, the exact definition of a and µ may create some problems. One reason is that Eq. [3] is only truly valid if the area of impact a is of infinitely small size. However, a falling raindrop will create an impact crater of up to a few millimeters in diameter and then break up into droplets that carry soil particles (Ghadiri and Payne, 1988). Therefore, the highest density of transported particles will occur around the perimeter of the impact crater. This does not pose a major obstacle to using Eq. [1], provided that the average splash length is not of the same order of magnitude as the size of the impact crater. Assuming, for example, that the impact of a raindrop leaves a crater with a radius of 2 mm and the average splash length () has a value of 0.1 m (a representative value, see below), then the amount of material implicitly assumed by Eq. [1] to be inside the crater would be only 2% of the total amount of splashed material. Another point to be made in this respect is that splash detachment itself is difficult to determine and even to define. Some soil particles may be detached but returned to their original position, while others may be pushed away to the perimeter of the impact crater without actually leaving the soil surface, as has been demonstrated in splash cup experiments (Kinnell, 1974, 1982) and in numerical simulations of raindrop impact (Huang et al., 1982, 1983). For pragmatic reasons, therefore, detachment is defined here as the amount of soil that appears to have been detached based on measurements of splash. In some cases, this may include some pushed material as well (Kinnell, 1974, 1982).

Splash Distribution in One Dimension
The distribution theory developed above can be applied to a situation in which material is splashed across an (effectively) infinitely long straight boundary. This is important to interpret measurements made with, for example, splash trays and splash boards. To this end, the radial form of Eq. [1] has to be converted to a two-dimensional coordinate system first and eventually to a single dimension. One way of doing this is to imagine an array of very small unit source areas, forming a narrow source strip. First, the splash density pattern resulting from one such source strip will be considered. Taking the y-axis along the source strip, it is easy to see that splash density will be a function of the distance |x| (perpendicular to the source strip) alone. This function can be determined by considering an element of the source strip (having length dy and width dx) positioned at (0, y) and a target element positioned at (x, 0). The contribution of the source element to the splash density in (x, 0) is then given by m_point(r)dxdy, where r is the Euclidean distance (Eq. [1]). The contribution of all elements forming the source strip to the splash density at (x,0) will be denoted by v_strip(x)dx. Because this involves the entire length of the strip, v_strip(x) has the dimension of mass per unit width of strip per unit target area (i.e., g m^-3). It can be found by integration along the length of the strip:

[4]

Thus, v_strip(x) is the one-dimensional equivalent of the radial splash density function represented by Eq. [1]. It can be shown that the solution of Eq. [4] involves a zero-order modified Bessel function denoted by K₀ (Eq. [9.6.24] in Abramowitz and Stegun, 1965):

[5]

This result implies that when the exponential FSDF is applied to a one-dimensional strip source, it does not retain its exponential form. However, it does resemble an exponential function at larger distances from the source (Fig. 2) . Another characteristic of Eq. [5] is that, like Eq. [1], it attains an infinite value for x = 0. Mathematically, this is not a problem since integration across a distance larger than zero will always yield a finite value. In further calculations, it is important to know what amount of material splashed from the source strip ends up beyond (a boundary at) a distance x. This can be found by integration of Eq. [5], between x and infinity. The result may be called the transport function, denoted by v_beyond(x). It represents the splashed mass ending up beyond a boundary line at distance x, per unit width and per unit of boundary length. Therefore, v_beyond(x) has the dimension of mass per area (in g m^-2). The integration is written as:

[6]

where x' is the integration variable. There is no simple analytical solution to this integral but its numerical solution is relatively straightforward, except for very small values of x (Abramowitz and Stegun, 1965, Eq. [11.1.9]; see Fig. 2). Taking things one step further, a source area of soil is considered that is infinitely large at one side of a boundary. The amount of material transported across this boundary may be called the rate of transport, denoted by q and expressed in mass per unit boundary length (g m^-1). It represents the cumulative amounts of v_beyond(x) for the entire assembly of possible source strips positioned between zero and an infinitely large distance from the boundary. The value of q may be found by single integration of Eq. [6] between 0 and infinity:

[7]

where x'' is defined by x' = x''- x. The solution of this complicated integration is surprisingly simple (Abramowitz and Stegun, 1965, Eq. [11.4.22]):

[8]

View larger version (14K): [in this window] [in a new window]   Fig. 2. One-dimensional distribution of particles splashed from a source strip expressed as a density function of perpendicular distance x {vstrip(x); Eq. [5]} and as a transport function indicating the fraction of detached material splashed beyond distance x {vbeyond(x); Eq. [6]}. All variables have been made dimensionless; dashed lines represent approximate exponential functions. = average splash length.

However, the source area of an experimental tray is not infinitely large. Momentarily assuming a tray of (effectively) infinite length but of finite width (w in m), the transport rate over one side of the tray [q(w), in g m^-1] can still be obtained using Eq. [6], which is now integrated across 0 < x < w, resulting in:

[9]

For comparative purposes, it will prove useful to express q(w) as a fraction F_w of the infinite splash transport rate (q), by isolating the latter in Eq. [8] and substitution into Eq. [9], after which the resulting equation can be simplified (reversing the order of integration and rewriting; Eq. [11.3.27] in Abramowitz and Stegun, 1965) to:

[10]

where the integration variable is equal to (x/). F_w may be regarded as a correction factor that may be used to correct measurements of splash from a tray of limited width but infinite length. Equation [10] is still not readily solved analytically, but it can be solved numerically (Abramowitz and Stegun, 1965; Eq. [11.1.9] and [11.1.18]). The resulting solution is not an exponential function itself, but F_w can be approximated very well by the following exponential function:

[11]

This expression agrees within 1% with values calculated using Eq. [10] for 1 < w/ < 100. For 0.2 w/ 1, the difference is still <10%. In principle, the detachment rate can be calculated from the results of soil tray experiments using the geometrical correction factor given by Eq. [11], but the average splash length () will need to be known. Its value may be derived from measurements of splash over a range of distances, as will be described below. Of course, Eq. [11] still assumes an infinitely long tray. This problem will be approached with a numerical model further on.

A situation analogous to the one described above concerns splash transport from an infinitely large source area on one side of a boundary, onto the area between distance 0 and X from it, denoted here by q(<X). Because of the mathematical reciprocity of this situation compared to that of a tray of limited width, Eq. [8] and [11] can also be used here, substituting X for w in the latter. In a similar manner, equations can be derived for splash beyond a distance X and splash between distances X₁ and X₂, denoted by q(>X) and q(X), respectively. The resulting equations read:

[12a]

[12b]

[12c]

Clearly, these equations too are based on an approximation of the actual geometrical correction factor (Eq. [11] vs. Eq. [10]), and therefore should not be used at very small distances from the boundary. The practical application of the equations above is that through a plot of distance on the x-axis vs. the amount of material splashed beyond or before X on the y-axis, an exponential function can be drawn, the exponent of which yields the average splash length (; Eq. [12a] or [12b]). Once is known, the detachment rate (µ) can be calculated from the coefficient of the fitted function.

Examples of experimental results that can be interpreted with Eq. [12c] include those of Savat and Poesen (1981), who measured splash from a soil-filled tray (100 x 20 cm) onto a series of 10-cm wide strips next to the long side of the tray (Poesen and Savat, 1981). The results are shown in Fig. 3 ; the average splash lengths () derived from the exponential curves that were fitted to the data were between 0.11 and 0.15 m. However, strictly speaking, Eq. [12c] could not be used in this case because the tray width was limited to 20 cm and numerical calculation is therefore needed. Cursory analysis using Eq. [11] in combination with the determined values suggests that total splash transport from the experimental tray would have been about 8 to 18% less than that from an infinitely large area. Similarly, the results of soil tray experiments (size 200 by 50 cm) by Torri et al. (1987) were described very well by exponential functions, the application of which yielded average splash lengths of 0.10 to 0.12 m. In this case, application of Eq. [11] suggested an underestimation of true transport rate (q) of <0.5%. The fact that simple exponential functions fitted the measurements so well in both cases offers further support to the theory outlined above, although of course the distribution of splash very close to the source remained unknown in these experiments.

View larger version (16K): [in this window] [in a new window]   Fig. 3. Rates of splash transport [q(X)] onto 10-cm wide strips, placed at increasing distances from a soil tray subjected to artificial rainfall and filled with different soils (approximate median grain size indicated; data from Savat and Poesen, 1981) with Eq. [12c] fitted to the data.

	APPLICATIONS

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View larger version (32K): [in this window] [in a new window]   Fig. 4. Illustration of (a) the matrix used to model the distribution of particles splashed from a rectangular area (the meaning of the areas labelled SCREEN and CORNER is explained in the text); (b) a soil tray bordered by a very wide collecting screen having the same length; (c) a soil tray bordered by an infinitely large horizontal collecting screen; and (d) a soil tray bordered by a collector with vertical splash guards on three sides.

The matrix model was used to find relationships between A/ and A/B on the one hand, and transport rate on the other, depending on the boundary conditions of the problem. For example, consider a tray of soil with a horizontal collecting screen on one side, having the same length as the tray and a width of five times the average splash length in the direction of the tray (numerically, this will be virtually equal to a screen that effectively extends infinitely in the direction of the tray; see Fig. 4b). The total amount of material splashed onto this screen (as a ratio to detachment rate) is obtained by summing the amounts for all elements in the area labeled "SCREEN" in Fig. 4a. This total was calculated for a range of A/ and A/B values. The results were then compared with transport rates (q) from an infinitely large area of soil (Eq. [8], again using unit detachment rate). The ratios of the two values listed in Table 1a for various combinations of A/ and A/B are shown in Fig. 5a . These ratios can be used to correct the results of experiments involving splash trays and screens for their geometry.

View this table: [in this window] [in a new window]   Table 1. Geometrical correction factors for splash transport measured in tray experiments, for a range of tray dimensions (A and B) and average splash lengths, . The length A of the considered side of the tray is made dimensionless through division by average splash length , while tray dimensions are expressed in the aspect (A/B). The geometrical correction factor represents measured transport as a fraction of transport from an infinitely large area. Factors are calculated for the situation of a tray equipped with (a) a very wide screen having the same length as the side A of the tray and (b) an infinitely large screen.

View larger version (24K): [in this window] [in a new window]   Fig. 5. Splash transport onto (a) a wide screen having the same length as a side A, and (b) an infinitely large screen, both placed along a side A of a tray with dimensions A and B, as a function of the dimensionless length of that side, A/, and for a number of different values of the aspect (A/B) of the tray. Splash transport was made dimensionless by expressing it as a fraction of the transport from an infinitely large area of soil.

Splash from or into a Round Cup
In studies with splash cups, it is important to know the relationship between the cup splash rate m_R (i.e., the amount of material splashed into or out of a cup of radius R per unit area of cup, in g m^-2) and the detachment rate (µ). In principle, the mass (M[>r], in g m^-2) that is splashed beyond a radial distance r of a small unit source area is easily obtained by integration of Eq. [2]:

[13]

However, a circular source area represents a population of individual sources whose distance to the perimeter is a function of direction (except for the very center point), and this complicates the calculation of the distribution of splashed material. Torri and Poesen (1988) attempted a combined analytical and numerical solution to this problem and presented a table listing the factors that relate detachment rate to cup diameter and average splash length. However, similar to the approach followed for the case of a rectangular soil tray, it may be seen that cup radius R and average splash length () can be combined into one dimensionless variable, R/. Furthermore, the cup splash rate (m_R) can be expressed as a function of R/ and detachment rate (µ), to yield the geometrical cup splash correction factor F_r:

[14]

Because of mathematical reciprocity, Eq. [14] is valid for both ejecting and receiving splash cups. However, it proved very difficult to evaluate Eq. [14] in an analytical manner and therefore again a numerical model was constructed, similar to the one described for a rectangular source area. The representation of a circular area in a matrix introduced additional numerical errors, but these can be reduced by increasing the radius of the source area. As a control on these errors, the number of source elements representing the cup area was compared with the theoretical area (R²) for each model run. If the difference was <0.5% the result was accepted, otherwise the values of both R and were doubled and the model was run again. The summed amounts of material deposited outside the circular source area were calculated and divided by the sum for all elements to determine the geometrical correction factor, F_R, for splash from or into a cup of radius R.

When plotted on a semilogarithmic scale, the results indicate an S-shaped function (Fig. 6) . For low values of R/, the results approach those of Eq. [13] and, consequently, F_R approaches unity. For high values of R/, on the other hand, the fact that the area is not infinitely large becomes progressively less important and eventually the situation approaches that of splash across the boundary of an (effectively) infinitely large circular area, A_R. The total length of the boundary equals the circumference of this circle, while transport rate (q) across the boundary is defined by Eq. [8]. In that case, cup splash rate out of or into a cup of radius R (m_R, g m^-2) is given by:

[15]

View larger version (11K): [in this window] [in a new window]   Fig. 6. Relationship between dimensionless cup radius R/ and the amount of soil splashed out of the cup, expressed as a fraction of the total amount of detached sediment (FR).

[16]

Predictions by Eq. [16] agreed with the numerical model results within 4% (Fig. 6). Probably a significant part of this difference relates to numerical errors within the model itself. Predictions of splash transport based on Eq. [16] were compared with the measurements of Poesen and Torri (1988), who used receiving splash cups. As shown in Fig. 7 , Eq. [16] fits the experimental data slightly better than the exponential function proposed by the authors themselves. Whereas they inferred a detachment rate of 790 g m^-2 and an average splash length of 0.060 m (Poesen and Torri, 1988; Torri and Poesen, 1988), Eq. [16] yielded values of 950 g m^-2 and 0.035 m, respectively. It should be noted, however, that the exponential function derived by Torri and Poesen (1988) is not compatible with the theoretical requirement that an inverse relationship (m_R 1/R) is approached for large cup sizes.

View larger version (13K): [in this window] [in a new window]   Fig. 7. Application of the theory for interpreting splash cup data (Eq. [16]) to data collected by Poesen and Torri (1988). The exponential equation proposed by Poesen and Torri (1988) is shown for comparison. mR = cup splash rate, R = radius.

Clearly, extremely detailed information on the distribution of particles in all vertical and horizontal directions as well as the corresponding initial velocities would be needed to develop a mathematical framework for splash on a slope. To make matters worse, the experiments of Ghadiri and Payne (1988) also demonstrated that the effect of slope on the splash process varies, depending on the material involved. This suggests a wide range of possible slope - (net) splash transport relationships, and indeed a considerable number of such relationships has been reported for different soil types (Foster and Martin, 1969; de Ploey, 1969; Mosley, 1973; Moeyersons and de Ploey, 1976; Quansah, 1981; Savat, 1981).

An alternative, more pragmatic approach is to use the equations derived earlier for a horizontal surface to interpret measurements of splash on a slope and to try and obtain an estimate of the error involved. For example, consider measurements made using a series of long compartments, placed perpendicularly to the slope gradient, to measure downslope or upslope splash transport, or along the slope gradient to measure lateral splash transport. By interpreting these measurements using one-dimensional theory (Eq. [12a–c]), different apparent average splash lengths will be obtained for each direction in which transport was measured. This apparent directional splash length represents a mass-weighted average of the respective average directional splash lengths of particles splashed in the various horizontal directions. An error is likely to be introduced if these apparent splash lengths and observed transport rates are used to estimate detachment rate using Eq. [8]. Arguably, the most important source of error lies in the preferential splash of particles downslope. The order of magnitude of this error can be investigated analytically. For the special case that all detached particles are splashed downslope, with equal amounts of particles still being splashed in all downslope directions, it is readily shown that detachment rate estimated with Eq. [8], using measurements of downslope splash, will result in an overestimate of twice the actual value. It can also be shown that detachment rate determined from lateral splash measurements will still be correct. If the radial distribution of particles is even more asymmetrical than this, the associated error will be greater. The extreme situation is represented by a situation in which all particles are splashed at very small angles to the slope gradient; in that case, lateral and upslope transport are virtually and absolutely absent, respectively. Furthermore, it may be assumed that the distribution function for splash from a narrow strip perpendicular to the slope gradient, becomes the one-sided function

[17]

where is the weighted-average downslope splash length, x is the distance downslope from the source area, and M_down(x) (in g m^-3) is the splash mass per unit of distance per unit source area. Following an approach that is more or less similar to the one described with regard to the FSDF (but less complex), the relationship between transport rate (q) and detachment rate (µ) can be evaluated. Eventually, this relationship is given by:

[18]

Results obtained with Eq. [18] can be compared with those of Eq. [8], but it should be noted that if splash has indeed become entirely one-dimensional as described by Eq. [17], then interpreting the splash measurements with radial theory will yield an apparent splash length that differs from the actual average downslope splash length () by a factor of 1.30 (Eq. [12a–c]). Eventually, the maximum relative difference between apparent detachment rate (µ_a) found using Eq. [12a–c] and true detachment rate (µ) based on Eq. [18] is given by:

[19]

It follows that detachment rate estimated from measurements of downslope splash using the radially isotropic theory for splash on a horizontal plane can be up to 2.42 times higher than the actual detachment rate (in contrast, measurements of upslope and lateral splash will yield an underestimate that, in theory, can become infinitely large). Uncertainties from these considerations are less than variations in the detachability of different soils (defined as the detachment per unit erosive energy input), which cover several orders of magnitude (Poesen and Savat, 1981; Poesen and Torri, 1988). Therefore, despite the error involved, measurements of downslope, upslope, and lateral splash transport and the associated apparent splash lengths on a sloping surface may still be interpreted with the theory developed in this paper, and will result in useful (albeit initial) estimates of detachment rate.

	CONCLUDING REMARKS

TOP ABSTRACT INTRODUCTION THEORY APPLICATIONS CONCLUDING REMARKS REFERENCES

 
The present theoretical study shows that experiments to determine detachment and transport by rain splash must take into account the spatial distribution of the splashed particles. There is considerable empirical evidence to suggest that an exponential FSDF adequately describes reality. On the basis of the FSDF, a theory was developed that facilitates the interpretation of various kinds of splash measurements. Several theoretical applications of the FSDF compared very well with results of experiments involving splash cups and splash trays. It is evident from the theory that the rate of splash detachment (µ) and observed rates of splash transport (q) constitute two different entities. However, the relationship between the two proved surprisingly simple and only requires knowledge of the average splash length (). A number of equations and correction factors are presented that can be used to interpret the results of splash experiments on a horizontal surface. The distribution of splashed particles is anisotropic on a slope and, consequently, splash experiments become more difficult to interpret. It was shown that one-dimensional (i.e., downslope, upslope, or lateral) measurements of splash transport may still be interpreted using the proposed theory, but the magnitude of the uncertainty involved increases with slope.

The geometry of splash experiments is shown to have a significant effect on the results. The use of soil trays introduces edge effects, and correction factors for these were proposed. However, the use of these correction factors for tilted trays leads to errors that increase with slope gradient. In such cases, the tray should be sufficiently large to minimize edge effects. Furthermore, the distribution of splash distances needs to be known, for example, through the use of segmented splash collectors, boards, or screens (Savat and Poesen, 1981). Similarly, splash transport rates measurements using receiving splash cups can only be interpreted correctly if cups of different sizes are used (Poesen and Torri, 1988). In the case of ejecting cups, the distribution of material splashed from the cups also needs to be measured; for example, through the use of a series of concentric ring collectors (Riezebos and Epema, 1985).

The practical application of the presently proposed theory to field measurements using splash cups on subhorizontal terrace beds and splash boards on steep, bare terrace risers in Java, Indonesia, as well as experiments involving soil trays similar to those used by Wan et al. (1996) are dealt with in two separate papers (van Dijk et al., 2002a,b).

	ACKNOWLEDGMENTS

 
This work was performed within the framework of the Cikumutuk Hydrology and Erosion Research Project (CHERP, http://www.geo.vu.nl/~geomil) in Malangbong, West Java, Indonesia. A.I.J.M. van Dijk was supported by a grant from the Netherlands Foundation for the Advancement of Tropical Research (WOTRO, grant no. W76-193), which is gratefully acknowledged.

Received for publication May 14, 2001.

	REFERENCES

TOP ABSTRACT INTRODUCTION THEORY APPLICATIONS CONCLUDING REMARKS REFERENCES

 

• Abramowitz, M., and I.A. Stegun. 1965. Handbook of mathematical functions with formulas, graphs and mathematical tables. Dover Publications, New York.
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