# Predicting Water Cycle Characteristics from Percolation Theory and Observational Data

^{1}

^{2}

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## Abstract

**:**

_{0}/P, with P being the precipitation and ET

_{0}being the potential evapotranspiration. Existing work was able to predict the global fractions of P represented by Q and ET through an optimization of plant productivity, in which downward water fluxes affect soil depth, and upward fluxes plant growth. In the present work, based likewise on the concepts of percolation theory, we extend Budyko’s model, and address the partitioning of run-off Q into its surface and subsurface components, as well as the contribution of interception to ET. Using various published data sources on the magnitudes of interception and information regarding the partitioning of Q, we address the variability in ET resulting from these processes. The global success of this prediction demonstrated here provides additional support for the universal applicability of percolation theory for solute transport as well as guidance in predicting the component of subsurface run-off, important for predicting natural flow rates through contaminated aquifers.

## 1. Introduction and General Background

_{2}and organic acids represents by far the largest sink for atmospheric carbon [12]. In contrast, [13] hypothesized that the silicate weathering’s share to atmospheric CO

_{2}sink has been overestimated, indicating the presence of other sources of atmospheric CO

_{2}. Due to rapid kinetics, carbonate weathering in the short term might be significant in controlling the climate (even though the net drawdown by this pathway is zero). Vegetation growth and production of plant material represents a much more rapidly varying sink for atmospheric carbon, although with far less total sequestration [12]. Weathering of rocks to their products typically enables physical erosion as well [14,15,16,17,18] thus placing constraints on the rock mineralogical cycle, deep water cycle, and deep carbon cycle. Water importance in all these processes should not be underestimated. Since the rate of each of these processes depends on the magnitude of the associated water fluxes [19,20,21,22,23], understanding the partitioning of water into these fundamental fluxes will help future modeling efforts to generate reliable predictions of both these fluxes and their associated processes. In an operational perspective, the soil–water balance can be represented simply in terms of a mass conservation law. However, it is desirable to determine what fraction of the precipitation follows each specific pathway through the environment. A more challenging problem is to predict how these precipitation fractions depend on climate along various gradients. Such a prediction may be adaptable to employment in climate change models as well as to understanding current results.

## 2. Objectives

_{0}/P ≫ 1, while, in the current paper, we address partitioning of ET into evaporation and transpiration for ET

_{0}/P ≪ 1. Division into components of evaporation and surface run-off, which do not intersect the carbon cycle meaningfully, and transpiration and subsurface run-off, which do, would make our treatment more compatible with research into the carbon cycle and its interactions with the water cycle under various scenarios of climate change.

_{0}/P values together with a short discussion of how the solution can be extended to give a predicted range of ET/P values for any particular value of the ET

_{0}/P parameter. Next, we present the extension to address the partitioning of Q as well as ET.

## 3. Budyko Theory

_{0}is known as the potential evapotranspiration and corresponds to how much water could be evaporated if all the radiant energy falling on the Earth’s surface caused water evaporation. If ET

_{0}= ⟨P〉, it is possible for all the precipitation falling to the Earth to be evaporated and simultaneously for all radiant energy to be consumed in the process of evaporating water. Thus, both H in Equation (4) and ⟨Q〉 in Equation (3) could be zero. Such an idealization is, of course, never realized, as it would require ET

_{0}= ⟨P〉 in essentially every time period. With an increase in ⟨P〉 (or decrease in ET

_{0}), ⟨Q〉 must be non-zero. This condition is known as energy-limited. With a decrease in ⟨P〉 (or increase in ET

_{0}), H/L must be non-zero. The corresponding condition is referred to as water-limited.

_{0}/⟨P〉 ≫ 1, there may be far less water available for evaporation than could actually be evaporated. ET

_{0}/⟨P〉 is accordingly known as the aridity index. Thus, for large aridity index, H/L in Equation (4) is likely to be large and ⟨Q〉 near zero, whereas, in the limit that the aridity index is much smaller than 1, H/L is likely to be near zero and Q large.

_{0}/⟨P〉. While the functional form cannot be rigorously derived by any procedure, many models have been constructed that lead to results similar, or equal, to the Budyko phenomenological result. This interest owes to the relatively good job that the phenomenology does in predicting actual values of ⟨ET〉/⟨P〉 as a fraction of ET

_{0}/⟨P〉, meaning that the relationship that Budyko proposed can be used as a first estimate for ecosystems and drainage basins with unmeasured values of ⟨ET〉/⟨P〉. Moreover, Budyko [28] also found reasonable limits for ⟨ET〉/⟨P〉 in the extreme cases of water-limited ecosystems (ET

_{0}/⟨P〉 ≫ 1) and energy-limited ecosystems (ET

_{0}/⟨P〉 ≪ 1). In the former case, ⟨ET〉/⟨P〉 must approach 1, whereas, in the latter, ⟨ET〉/⟨P〉 should not exceed ET

_{0}/⟨P〉. The dependence of ⟨ET〉/⟨P〉 on the aridity index, ET

_{0}/⟨P〉, has elicited significant attention in the hydrologic sciences [49,50], and a wide variety of means to address this problem have been proposed [2,3,49,51,52,53,54,55].

## 4. Data Sources

## 5. Concepts from Percolation Theory

^{1.21}. When root tip extension rates remain constant, the rate of lengthening of RRE diminishes in time. The preference for the two-dimensional context for selecting the optimal paths exponent is the relative shallow depth of tree roots in forests, often assumed and/or documented to be on the order of 1 m [72,73], or even less [30,74], on account of the shallow depth of most nutrient sources (Lynch, 1995). For length scales of 1–40 m, RRE and tree height are nearly identical [75,76,77]. Thus, slowing of tree growth in time is due to a reduction (in time) in the rate of access of nutrients from root growth along optimal flow paths.

^{1.87}. Thus, the travel distance after a time t, x (t), is proportional to t

^{1/1.87}. The important consequence is that the solute velocity, v

_{s}, diminishes in time according to another power law, i.e., vs. ∝ t

^{−0.87/1.87}. Both the soil production rate and the chemical weathering rate are then proportional to the solute velocity [79]. The high degree of universality of percolation exponents makes the results applicable for most systems at most length and time scales [27].

_{0}, to a time scale, t

_{0}. Pore structures in soils may be represented as pore networks. For the soil formation, the network scale, x

_{0}, was proposed in [23] to be the pore separation (taken to be the median particle diameter, d

_{50}), while the rate x

_{0}/t

_{0}was proposed [8] to be ratio of the deep infiltration rate (or subsurface run-off), ⟨Q

_{sub}〉, to the porosity. ⟨Q

_{sub}〉 turns out to be the major portion of the total run-off ⟨Q〉. Then, the soil production rate, R

_{s}(represented as a function of soil depth), which is given as the derivative of the solute transport distance, is proportional to ⟨Q

_{sub}〉. In particular, ${R}_{s}\propto \u27e8{Q}_{sub}\rangle {\left(x/{x}_{0}\right)}^{-0.87}$. To define steady-state conditions, the soil production rate is then set equal to the denudation rate, D. The solution is,

_{sub}〉 to a pore-scale velocity. ⟨Q

_{sub}〉, the subsurface run-off, is equal to ⟨P – ET – Q

_{surf}〉, where ⟨Q

_{surf}〉 is the surface run-off. Note that Equation (5), by this partitioning of run-off into surface and subsurface, is a basin-scale equation, in accordance with Budyko theory. At the local level, the infiltration rate, or vertical flow through the soil, which drives the chemical weathering and contributes to subsurface flow, would include a term with the negative of the divergence of the surface water flux. Further, D could be slope-dependent, although many studies assume that D takes on a single value throughout a given drainage basin.

_{0}can serve, at least approximately, as a length scale for both plant growth and soil development. Transpiration has been approximated in [22] as well as in most data collections, as ET. However, in [80] it was shown that the length and time scales of the mean growing season transpiration, T

_{g}, and growing season length, t

_{g}, can be used in the scaling relationship for RRE = T

_{g}(t/t

_{g})

^{1/1.21}. More importantly, since the yearly increase in RRE turns out to equal the transpiration [80], the root volume, which increases as the RRE to the mass fractal dimension, d

_{f}, of the root system, should thus be proportional to ${T}_{g}^{{d}_{f}}$. Knowing d

_{f}is, thus, a critical input towards understanding the dependence of NPP on transpiration.

_{f}was taken from percolation theory and general knowledge about the architecture of roots and the soil (see Figure 2 for details). It has often been noted that the bulk of root mass is found in the top 2 m of soil [72], or the top 1 m [73]. or even the top 0.5–0.68 m [30,74]. The top 1 m of the Earth’s surface is typically taken up by the soil [81] and the reason for the predominance of the roots in this layer is argued to trace to the concentration of nutrients in the top 1 m or so of the soil [73] Since the optimal paths exponent from percolation theory in two dimensions describes the effect of root tortuosity on the slowing in time of the increase of the root radial extent (RRE) (and thus tree height) [22,69,75,76,77], it was reasonably conjectured [22] that the mass fractal dimensionality of percolation theory in two dimensions should describe their mass as a function of a critical linear dimension (RRE). The mass fractal dimensionality of large clusters near the percolation threshold in two dimensions is 1.9; thus, d

_{f}= 1.9. It is noted that, although measured plant root fractal dimensionalities [56] tend to converge to this number (as shown in [22]), the spread in their values is quite wide. As has also been shown [26], while the percolation prediction generates nicely the global average for ET/P, measured values of d

_{f}generate the observed variability of ET/P for climatic conditions equivalent to those under which the plants were grown.

_{0}is a reference area that does not depend on the fluxes and can thus be ignored in the subsequent optimization procedures.

_{sub}〉 approximated as ⟨Q〉, gives 〈ET〉 = 0.623 ⟨P〉 [26]. If 1.15 is approximated as 1 and 1.9 is approximated as 2, the result is ⟨ET〉 = (2/3) ⟨P〉. The optimization is accomplished by setting d⟨NPP〉/d⟨ET〉 = 0 and leaving the denudation rate D in the denominator as an unknown input, independent of ET. The extension of theory presented here addresses the fact that the denudation rate is also a function of the hydrologic fluxes. It is important that putting together two already verified predictions (in different contexts) in terms of universal parameters yields, without adjustable parameters, the global average ⟨ET〉. One of these parameters, D

_{b}= 1.87, describes the time-length scaling of solute transported by advection through porous media with fundamental pore-scale connectivity in 3D.

_{sub}〉 = ⟨Q〉 and equating transpiration with ⟨ET〉, two additional results already obtained should be addressed. One relates to the variability of ⟨ET〉 at a particular value of the aridity index, while the other focuses on the variation of ⟨ET〉 with aridity index. Both issues are illustrated in Figure 3.

_{f}for the predicted fractal dimensionality of the root system, the three-dimensional value, i.e., 2.5, should be used (Figure 2). However, if ⟨ET〉

^{2.5}is applied, then dimensional analysis requires replacement of the factor x by ${x}^{0.5}={x}^{3-{d}_{f}}$. In the work by Hunt [26], these changes were made in Equation (6) to address ecosystems in regions of arid climate. Further, the area covered by plants, which is known to be a decreasing function of aridity index, was assumed to be proportional to ⟨P〉/ET

_{0}, while ⟨ET〉, attributed to pure evaporation, was assumed equal to ⟨P〉 between plants. Substitution of d

_{f}= 2.5 into Equation (6), but using the factor x

^{0.5}yields ⟨Q〉 = 0.187 ⟨P〉, implying ⟨ET〉 = 0.813 ⟨P〉 in areas covered by plants. Then, the total ⟨ET〉 was written as (0.813) ⟨P〉 (P/ET

_{0}) + ⟨P〉 (1 − ⟨P〉/ET

_{0}) = ⟨P〉 − (0.183) ⟨P〉 (⟨P〉/ET

_{0}). Division by ⟨P〉 yields ⟨ET〉/⟨P〉 = 1 − 0.183 (⟨P〉/ET

_{0}). Consistent with the structure of this calculation, the fraction of ⟨ET〉 represented by evaporation from the ground surface would be (1 − ⟨P〉/ET

_{0})/[1 − ⟨P〉/ET

_{0}+ 0.817(⟨P〉/ET

_{0})] = (1 − ⟨P〉/ET

_{0})/(1 − 0.183⟨P〉/ET

_{0}). Clearly, such a result cannot be useful when ⟨P〉/ET

_{0}= 1, for which it delivers a fraction of ⟨ET〉 due to pure evaporation identically 0. In fact, its usefulness will be restricted more severely. Given that the global average of ⟨E〉/⟨T〉 has been suggested to be 0.36–0.39 [34,83] and that a quick estimate from the interception bounds in [61] alone generate a range of ⟨E〉/⟨T〉 values of between 18% and 73%, we infer that the use of this expression is probably limited to values of ET

_{0}/⟨P〉 for which it yields fractions greater than ca. 0.4. In other words, we consider evaporation of bare ground to be the primary contribution to E under conditions complementary to those for which evaporation off the foliage is key. For ⟨E〉/⟨T〉 = 0.4, ET

_{0}/⟨P〉 from the above expression is 1.575.

## 6. Extending Budyko Theory

_{surf}〉, whereas the physical erosion has been asserted to be proportional to ⟨Q

_{surf}〉 or, equivalently [84], to ⟨P〉. The first proportionality is necessary from any result that the chemical weathering rate is proportional to the flow through the soil into the subsurface. A consequence is that the chemical erosion rate is also proportional to the solute velocity, itself proportional to the flow. While it is possible to define these proportionalities, the term representing physical erosion is much less certain, making it currently inefficient to address this subject more deeply. In any case, if one makes this assumption, D = a (⟨P – EP − Q

_{surf}〉) + b ⟨Q

_{surf}〉, where a and b are unknown constants of proportionality. The subsurface component of run-off is typically larger than the surface component [63]. However, the fluxes in rivers from chemical weathering are typically smaller than those from physical erosion processes. Thus, b must normally be considerably larger than a (b ≫ a). Exceptions are low relief carbonate substrates, such as the Canadian shield and the St. Lawrence River that drains it.

_{surf}〉, rather than just ⟨P〉. However, Equation (8) makes it clear that retention of the first term by itself will lead to an underestimation of ⟨ET〉. Moreover, if surface run-off is included in the analysis, then so should the process of interception, < I

_{t}>, be included, since this water never makes it to the soil to be partitioned into transpiration and run-off.

_{t}〉 is then a separate input. Define ⟨P〉 as the precipitation reaching the treetops. Then, it is possible to use the derivations above (Equations (7)–(9)) by substituting for ⟨P〉, ⟨P′〉 ≡ ⟨P〉 − ⟨l

_{t}〉, while also adding ⟨l

_{t}〉 as a separate term to evapotranspiration. Thus, to lowest order,

_{surf}〉 and ⟨l

_{t}〉. In [45] it was estimated that 65% of ⟨P〉 = 834 mm is lost to ⟨ET〉 with 35% left for run-off. There [45] it was also indicated that 11% of the total precipitation is lost to deep infiltration. Accordingly, ⟨Q

_{surf}〉 = 0.24 P. In [61], [59] was cited as reporting a range of interception from 12% to 48% of precipitation. To predict to lowest order the effect of interception on total ⟨ET〉, one could use the midpoint (30%) of the range of ⟨l

_{t}〉 values cited above. However, [62] tabulated interception ratios from 17 studies, and determined that the mean and standard deviation of ⟨l

_{t}〉 were 18% and 10%, respectively. Using a midpoint of the range of ⟨l

_{t}〉 values cited by [61], and using Lvovitch’s value for surface run-off, Equation (10) yields 59–60% for ⟨ET〉. However, [63] reported a considerably higher fraction of streamflow traceable to groundwater (“most streamflow derives from groundwater discharges, for most rivers, most of the time”), with mean values as high as 80%. Thus, we use a range of subsurface flows that constitute between 40% and 80% of the total run-off. Fractions of ⟨P〉 lost to interception range [61] from 12% to 48% (a simple mean of 30%). These numbers are generally in accordance with the estimate from [34] that the worldwide average of evaporation is 39% of ⟨ET〉 or about 25% of ⟨P〉. Using 40%, 60%, and 80% for the fraction of run-off traveling through the subsurface, we report the corresponding variability in ⟨ET〉 due to the variability in interception in Table 1. As can be seen, the predicted ⟨ET〉/⟨P〉 values have an average of 0.645. Alternatively, we can average all 41 values for the interception component taken from [57,58,60,62]. This method gives ⟨l

_{t}〉 = 0.214⟨P〉 and standard deviation 0.115⟨P〉. Combined with the 60% run-off midpoint estimate from [63], use of a 0.214 ⟨P〉 value for the interception generates a mean ET/P = 0.610 ± 0.045. In this case, the variability from uncertainty in interception is not addressed, since it would merely duplicate the values already tabulated. Our best estimates for evapotranspiration thus range from 0.61 to 0.645, depending on which assessment of interception values we base the calculation on, provided we choose slightly over half (60%) of the run-off as subsurface, compatible with [63]. These values compare with the average of the pre-1995 ⟨ET〉/⟨P〉 estimates of 0.645 or the average post-1995 ET/P estimates of 0.623, for example.

_{surf}as generated from the discussion in [63], as well as from cited interception uncertainty. The first three entries are generated from the general bounds on interception cited by [61]. The last entry is generated from the mean and standard deviation of the data summarized in [57,58,60,62]. Note, however, that one should probably expect a correlation between surface fluxes and interception, which means that uncorrelated Gaussian statistical analyses should not be relied on for quantitative predictions.

## 7. Discussion: Variability and Discrepancies

^{2}values that result are: 0.06

^{2}/(0.06

^{2}+ 0.08

^{2}+ 0.16

^{2}) = 0.08, 0.08

^{2}/(0.06

^{2}+ 0.08

^{2}+ 0.16

^{2}) = 0.15, and 0.162/(0.6

^{2}+ 0.08

^{2}+ 0.16

^{2}) = 0.76 for their contributions to the variability in ⟨ET〉. Such an analytical estimation is possible only if other potential inputs are neglected and the statistics are Gaussian. The run-off characteristics will relate most strongly to variability in soil type and depth, as well as climatic variables that affect precipitation intensity and regularity, while interception characteristics appear to relate most strongly to seasonality of precipitation and tree canopy physical structure. Root architecture appears to be the most important single variable, when plant species are considered individually. Presumably, when entire ecosystem response is addressed, however, variability in root structure from assemblages of plants would be reduced relative to individual plant species, at least when those ecosystems are not disturbed.

_{surf}〉 in the optimization of ⟨NPP〉. ⟨Q

_{surf}〉 is one of the smaller fluxes in magnitude, but its effect on the landscape is one of the most intense. Furthermore, in contrast to the other hydrologic fluxes, its effect is highly variable, in both space and time. Knowledge of this variability is important for a wide range of scientific and practical applications: evaluation of the surface characteristics, such as local slope, infiltration, groundwater recharge, soil hydraulic conductivity, antecedent moisture contents, and vegetation health, as well as because a very large fraction of the denudation of the landscape takes place during intense weather events. Thus, ⟨Q

_{surf}〉 has a disproportionately large role in landscape evolution, while its input is also highly variable as well as quantitatively unpredictable. This combination of factors should lead to the treatment of ⟨Q

_{surf}〉 as a stochastic input process, with potentially important non-linear contribution. In the present context, these complex attributes of surface water fluxes make it difficult to characterize their effects on the role of vegetation in the landscape, particularly as concerns an optimization of productivity.

## 8. Time Scales for Ecosystem Adjustment to Changes in External Parameters, Such as in Climate or Land-Use

_{I}= τ

_{A}/τ

_{R}[10,64], is useful for assessing the relative importance of solute transport and reaction kinetics to reaction rates in porous media. As long as Da

_{I}≫ 1, solute transport is limiting; the other limit, when Da

_{I}≪ 1, is kinetics limited. When a reaction is limited by the kinetics, to lowest order (neglecting changes in the nature of mineral surfaces), the reaction rate is constant. When soil formation is limited by a reaction rate that is constant, the soil formation rate should be constant and the soil depth increase linearly in time. The details in the calculation of Da

_{I}make this subject beyond the present scope (for further details, see [10,23]). However, the consequences of a soil formation rate that is constant can be seen in Figure 4, where, at short time scales, some alpine soil depths increase linearly in time and more slowly than predicted purely on the basis of transport limitations. Note that estimations of Da

_{I}for these cases [23] suggested that its value increased from less than 1 to greater than 1 at length scales of about 10 cm, roughly in accordance with the data, which exhibit a cross-over to transport limitations at a compatible length scale. In the following, since transport limitations appear dominant in the formation of most soils, we discuss biological and denudation time scales in comparison with time scales derived from transport limitations.

_{D}, from the ratio of the soil depth to the denudation rate, x/D. The time scale for soil formation to a depth x is ${t}_{x}={t}_{0}{\left(x/{d}_{50}\right)}^{{D}_{b}}$. Equating t

_{x}with t

_{0}= d

_{50}Φ/Q

_{sub}, leads almost to Equation (5), but without the factor 1.87 in the denominator. This small discrepancy arises from the fact that Equation (5) is derived by setting the denudation and soil formation rates equal, which involves the derivative of the soil depth as a function of time. Use of such time scales as defined in this section, however, allows determination of the time required to achieve steady state depth, although not necessarily steady-state soil characteristics. Consider that the thicknesses of natural soils are virtually always within an order of magnitude of 1 m. t

_{D}ranges can be easily calculated from the ratio of 1 m to the denudation rate, D, a quantity which ranges from a minimum of about 1 m/Myr in arid continental interiors, such as in Australia, or deserts such as the Atacama, to about 1000 m/Myr in tectonically active regions with high precipitation, such as the New Zealand Alps or the Himalayan mountain range. Thus, t

_{D}ranges from about 1 Myr to 1 kyr. For soils of a few decimeter thickness, this time scale can be as small as about 200 years. Attainment of steady-state soil depths at compatible time scales can be seen in a range of environments from the California Central Valley to Gongga Mountain in China in the work of [8], marked by cessation of deepening of soils. In the work of [23], denudation rates between 50 m/Myr and 20 m/Myr correspond to calculated time scales for attainment of steady-state from about 20 kyr to about 50 kyr, respectively. In [23], alpine soils require about 20 kyr, but Mediterranean soils closer to 100 kyr than 50 kyr to reach steady state depth.

## 9. Conclusions and Recommendations

_{2}drawdown from the atmosphere along climate gradients, which would not have been possible when these results were lumped together into evapotranspiration.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Rates of weathering in the lab [65] and in the field [21] as a function of throughflow. Field measurements were of silicates. Lab experiments were performed on magnesite, MgCO

_{3}, whose rapid reaction kinetics allows the approximately linear dependence on flow rate to extend to higher flow rates than in silicates, whose kinetic limitations are orders of magnitude smaller.

**Figure 2.**Illustration of the effects of the difference in root properties across a climate gradient in response to varying ET

_{0}/P (aridity index). Under more arid conditions, water is a more common factor limiting vegetation growth than are nutrients. Water is found more deeply in arid zones, while roots in forests are confined more nearly at the surface, suggesting that distinct values of the mass fractal dimensionality from percolation theory should be utilized in the two cases.

**Figure 3.**Results of ET/P plotted against aridity index. “Out of phase” and “In phase” refer to the synchronicity between maximum irradiance and precipitation. All data for ET were digitized from [30]. The theoretical bound was determined by the methods here. Predictions for forbs and grasses (the dominant vegetation of the northern Great Plains, [82]) were made using experimental data for root fractal dimensionality [56], rather than the percolation prediction. At large values of the aridity index, plant separation may be quite large [74], requiring an areal estimate for plant coverage.

**Figure 4.**After [23]. A demonstration of the ability of percolation concepts to predict actual soil depths. Note, however, the distinction between alpine (

**a**) and Mediterranean (

**b**) sites. In the former case, soil depths at time scales up to about 200 years may be overpredicted. The interpretation is that in some cases reaction kinetics provided a stronger limitation on the weathering process than transport. note that the upper limit of these soil depths has slope approximately one, rather than one-half, consistent with a time-independent weathering rate. Figure 4a thus illustrates a time scale that defines equal influences on chemical weathering from reaction kinetics and solute transport (just over 100 years).

**Figure 5.**Prediction of chemical weathering rates for strongly heterogeneous media using flow rate variability ±1 order of magnitude (after [11]).

Subsurface Fraction of Run-Off | Predicted ⟨ET〉/⟨P〉 | Variability from Interception |
---|---|---|

0.4 | 0.59 | 0.068 |

0.6 | 0.642 | 0.068 |

0.8 | 0.691 | 0.068 |

0.6 | 0.610 | 0.045 |

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**MDPI and ACS Style**

Hunt, A.; Faybishenko, B.; Ghanbarian, B.; Egli, M.; Yu, F. Predicting Water Cycle Characteristics from Percolation Theory and Observational Data. *Int. J. Environ. Res. Public Health* **2020**, *17*, 734.
https://doi.org/10.3390/ijerph17030734

**AMA Style**

Hunt A, Faybishenko B, Ghanbarian B, Egli M, Yu F. Predicting Water Cycle Characteristics from Percolation Theory and Observational Data. *International Journal of Environmental Research and Public Health*. 2020; 17(3):734.
https://doi.org/10.3390/ijerph17030734

**Chicago/Turabian Style**

Hunt, Allen, Boris Faybishenko, Behzad Ghanbarian, Markus Egli, and Fang Yu. 2020. "Predicting Water Cycle Characteristics from Percolation Theory and Observational Data" *International Journal of Environmental Research and Public Health* 17, no. 3: 734.
https://doi.org/10.3390/ijerph17030734