We developed a hydrologic model capable of simulating pothole complexes composed of tens of thousands or more individual closed-basin water bodies. It was applied to simulate the hydrologic response of a prairie pothole complex to climatic variability over a 105 year period (1901–2005) in an area of the Prairie Pothole Region in North Dakota. The model was calibrated and validated with a genetic algorithm by comparing the simulated results with observed power law relationships on water area–frequency derived from Landsat images and a 27 year record of water depths from six wetlands in the Cottonwood Lake area. The simulated behavior in water area and water body frequency showed good agreement with the observations under average, dry, and wet conditions. Analysis of simulation results over the last century showed that the power laws changed intra-annually and interannually as a function of climate. Major droughts and deluges can produce marked variability in the power law function (e.g., up to 1.5 orders of magnitude variability in intercept from the extreme Dust Bowl drought to the extreme 1993–2001 deluge). Analyses also revealed the frequency of occurrence of small potholes and puddles did not follow pure power law behavior and that details of the departure from linear behavior were closely related to the climatic conditions. A general equation, which encompasses both the linear power law segment for large potholes and nonlinear unimodal body for small potholes and puddles, was used to build conceptual models to describe how the numbers of water bodies as a function of water area respond to fluctuations in climate.
If you can't find a tool you're looking for, please click the link at the top of the page to "Go to old article view". Alternatively, view our Knowledge Base articles for additional help. Your feedback is important to us, so please let us know if you have comments or ideas for improvement.
 Prairie potholes are water-holding depressions of glacial origin, developed across some 750,000 km2 of the northern Great Plains of North America (Figure 1) [Sloan, 1972]. The exact numbers of lakes and wetlands in the so-called Prairie Pothole Region (PPR) are unknown but could easily be several million [Larson, 1995], with densities in some areas >100 water bodies per km2 [Last and Ginn, 2005]. The PPR is the brooding area for more than half of North America's migratory ducks, making it “one of the most ecologically valuable freshwater resources of the Nation” [Guntenspergen et al., 2006, pp. 1–2]. Moreover, these lakes and wetlands are important for agriculture, recreation, and groundwater recharge; they play important roles in water purification, flood control, and carbon sequestration, as well as preservation of local water quality [Kantrud et al., 1989; Euliss et al., 2006]. Recent research also has suggested that wetlands can influence precipitation patterns both locally and regionally [Taylor, 2010].
 The natural ecosystem of the PPR is vulnerable to climate variability and will likely be impacted by projected changes in climate [Fritz, 1990; LaBaugh et al., 1996; Winter and Rosenberry, 1998; Johnson et al., 2005; Millett et al., 2009]. For example, Sorenson et al.  suggested that global warming could reduce the number of ponds available for breeding ducks from 1.6 million to 0.6 million in the north central United States. Larson  determined that climate change impacts on potholes in the northerly parkland ecoregion of Canada will be even more severe than those projected for the United States. These potential impacts give rise to an important set of science questions in their own right. Before these can be adequately addressed, however, there is a need to better understand how the prairie pothole complex responds to climate.
 There has been recent emphasis on developing a quantitative description of the numbers and size distribution of lakes around the world. What is emerging is a rather robust scaling theory based on power laws [e.g., Birkett and Mason, 1995; Meybeck, 1995; Lehner and Doll, 2004; Downing et al., 2006; Zhang et al., 2009] that provides an ability to examine the function of such lakes in a new and different way. Moreover, power law concepts have proven to be extremely useful in describing the behavior of water bodies in the PPR [Zhang et al., 2009]. Our work, based on Landsat observations, showed how water bodies in the Waubay Lake area of South Dakota follow well-defined power laws that change intra-annually and interannually as a function of climate. These observation-based approaches are a useful complement to more traditional, field-based characterizations on clusters of pothole lakes and wetlands [e.g., Winter and Rosenberry, 1998]. Yet there are limitations with satellite observations given the relatively short observational record, limited resolution, and need for cloud-free images. More generally, satellite-based monitoring approaches, like those of Zhang et al. , cannot quantitatively describe water transfers, and their shortcomings emphasize the need for more quantitative approaches.
 An overlooked area in the study of water body populations is the large number of small ponds and puddles that likely contain a small, but significant, fraction of stored surface waters. These small water bodies are an enigma, however, because their size and numbers defy direct counting and because they are commonly dry during some parts of a year. It could be expected then that power laws will have a lower limit of validity imposed by the behavior of these smaller water bodies that fluctuate on an intra-annual and interannual basis. Knowledge concerning the existence and behavior of such limits is tremendously important in providing a complete quantitative understanding of the volumes of water stored in surface waters.
 Mathematical models have traditionally provided a powerful approach in the analysis of hydrologic systems. They facilitate quantitative explorations of the behavior of systems in terms of underlying processes and parameters and provide a framework for extrapolating system behavior to evaluate future stresses [e.g., Crowe, 1993; Johnson et al., 2005]. Various lumped element models have been remarkably successful in modeling the hydrologic response of lakes and wetlands [e.g., Crowe and Schwartz, 1981a, 1981b, 1985; Su et al., 2000] and hydrologic responses coupled with vegetation dynamics [e.g., Poiani et al., 1996]. Lake-wetland systems can also be simulated using more physically rigorous, distributed hydrologic models [e.g., Yu and Schwartz, 1995, 1998; Sudicky et al., 2005; Panday and Huyakorn, 2004]. These approaches are based on coupled solutions of surface-subsurface flow equations with a physically realistic description of processes. However, their use has been constrained by overall complexity of use, difficulty of calibration, extensive data requirements, and an often rigid geometry imposed by grid blocks. An important next step with modeling potholes is in simulating problems involving tens of thousands or more water bodies, i.e., complexes of wetlands, which are characterized by huge observational databases.
 This paper aims to develop an improved understanding of the dynamic hydrologic response of a pothole complex to variability in climate and to elucidate the behavior and the importance of smaller water bodies (less than 0.01 km2 in area) in the regional hydrologic cycle. It particularly focuses on power law approaches as a basis for organizing the study of complex surface water systems and for presenting results in a straightforward manner. The first objective of this paper is to describe a pothole complex hydrologic model (PCHM) that is capable of simulating the hydrologic behavior of a complex composed of tens of thousands or more water bodies. The second objective is to demonstrate the application and efficacy of such a model in elucidating the behavior of a large complex of potholes along the Missouri Coteau in North Dakota through the twentieth century. The third objective is to investigate power law behavior, especially considering small potholes and puddles. The study results presented here should contribute to a better understanding of the linkages between climate and the prairie pothole complex; fill in important historical data and information gaps, especially concerning the Dust Bowl drought; and provide an opportunity to raise awareness about the role and importance of small water bodies in water resource inventories.
 This study makes use of the newly developed PCHM to simulate hydrologic response of a complex of pothole water bodies with water areas ranging over more than 5 orders of magnitude from lakes to puddles. Here we classify potholes into two groups on the basis of the size of their water area: large potholes and small potholes. Large potholes are the water bodies with area >1 × 104 m2. Pothole lakes are included in this group as bodies with areas ≥1 × 105 m2 and depths >2 m [Sloan, 1972]. Small potholes are the water bodies with area ≤1 × 104 m2. Puddles are distinguished by water areas of <250 m2 and short lifetimes (e.g., 1 or 2 months).
2.1. Pothole Complex Hydrologic Model
 PCHM is derived from a lumped element, lake-watershed model [Crowe and Schwartz, 1981a, 1981b, 1985], capable of simulating the hydrologic responses of a lake-watershed system as a function of precipitation (P), temperature, evapotranspiration (ET), and the physical setting. The model simulates the routing of water through an idealized watershed to a closed-basin lake. As demonstrated in Figure 2, it incorporates key hydrologic processes, evapotranspiration, runoff, infiltration, and groundwater flow to provide a monthly accounting of inflows to and outflows from a single water body. Knowing the geometry of the lake-wetland basin, these fluxes can be interpreted in terms of monthly water area and stage. In the system, water is assumed to be stored in units, which correspond to components of the hydrologic system, such as the soil zone or groundwater zone.
 Precipitation is assumed to occur as rain or snow, depending upon temperature, with snow accumulated during the winter and melted in spring. A small fraction of the rain or snowmelt is able to run off directly into the surface water body. The model distributes precipitation excess (P − ET) among storage, recharge, and surface runoff. Water can be stored in the upper and lower soil zones (Figure 2), which are parameterized in terms of maximum storage capacities. Excess water from the soil zone is available for recharge to groundwater storage. Routing equations, described by Crowe and Schwartz [1981a], determine inflows to the lake or wetland and outflows from the watershed. Groundwater outflow from the water body is simply estimated on the basis of a seepage rate and a seepage area. Surface runoff is developed when the quantity of water available for infiltration exceeds the total amount of water retained in the soil zone and the maximum allowable recharge to the groundwater zone. The code calculates the monthly fluxes of water through the system and changes in storage. The water volume in the surface water body changes in response to the changes in inflows and outflows. Water depth and area are determined by a geometric model that relates volume to water depth and area.
 PCHM represents a major extension of our earlier model. Rather than simulating a single pothole basin, the enhanced PCHM code simulates a complex of basins, comprising numerous individual depressions of different sizes. As an initial condition for the simulation, a complex of potholes is created that properly accounts for the tendency for the smallest basins (containing small potholes or puddles) to be more numerous than the largest (containing large potholes or lakes). This proportioning is based on abundances of water bodies, determined as a power law description for the wettest condition observed. Effectively, this approach provides a starting pothole complex composed of some maximum number of water bodies collectively having the maximum individual water areas. As a practical matter, simulation results are not particularly sensitive to this initial assumption because a relatively long model spin-up time eliminates overall sensitivity to the initial conditions.
 PCHM incorporates a general, self-similar geometric model of a pothole watershed. All potholes in the complex are assumed to share the same symmetric sinusoidal bathymetry, formed by rotating the cross section around the central z axis (Figure 3a). Such an assumption is not unusual [e.g., Johnson et al., 2005]. Mathematically, our bathymetric function is defined as
where H is relief across the watershed, L is watershed diameter, and r is the water body radius of depth h. On the basis of equation (1), area-depth (A-h) and volume-depth (V-h) relations are developed as
 Operationally, water volume is calculated directly in PCHM at each time step, and equation (3) is used to calculate the depth inversely: given the known value of V, h = f′(V).
 Potholes represented in the model share a similar general shape but different scaled, geometric parameters. Each watershed is assumed to contain a single water body, and the watershed diameter L is known. A limitation of this simple model is that it cannot produce water bodies that disaggregate during dry periods or coalesce during wet periods. The total relief H in each watershed is calculated as , where is the shape ratio. The value of was chosen to provide a V-A relation for our geometry model that is comparable to a function from Gleason et al. . Gleason's function was developed from a survey for 186 pothole lakes and wetlands on the Missouri Coteau. Figure 3b shows that with a value of 0.0118, pothole geometries generated by the model satisfactorily reproduced Gleason's V-A results. We assume a small uncertainty in the value of with a range from 0.009 to 0.014 and consider as a parameter for calibration (Table 1).
Table 1. Model Parameters for the GA-Based Calibrationa
Impervious/direct runoff fraction during growing season (April–September)
Impervious/direct runoff fraction during dormant season (October–March)
Maximum allowable moisture storage in the upper soil zone
Maximum allowable moisture storage in the lower soil zone
Lake discharge to groundwater
Maximum allowable groundwater recharge
Storage delay time of a groundwater element
Fraction of maximum water depth to the watershed relief
2.3. Characterization: Missouri Coteau Study Area
 PCHM was applied to simulate pothole systems in a 5396 km2 study area in central North Dakota along the Missouri Coteau (Figure 1). This area was chosen in part to take advantage of the long-term observational record of water levels in wetlands at the Cottonwood Lake area operated by the U.S. Geological Survey [LaBaugh et al., 1996; Winter and Rosenberry, 1998; Swanson et al., 2003]. Glacial retreat and uneven deposition of glacial till created tens of thousands of lakes and wetlands [Euliss et al., 1999]. Major inflows of water to potholes come from snowmelt and summer rainstorms, whereas the greatest loss of water is due to evaporation [Winter and Rosenberry, 1998]. Groundwater inflows and outflows are smaller but still important components of the water budgets. Given the high rates of evaporation in summer and variable precipitation from year to year, the areas, stages, and volumes of pothole systems are sensitive to changes in climate and exhibit significant variability [e.g., Poiani et al., 1996].
 The climatic input variables to PCHM include monthly precipitation, temperature, and potential evapotranspiration (PE). Long-term monthly precipitation and temperature data for a climate division, ND05, and two weather stations (Woodworth and Jamestown, North Dakota) were obtained from the National Climatic Data Center. Division ND05 coincides almost exactly with the study area (Figure 1). Summary data, compiled for this division, were mainly used to simulate the collective behaviors (area-frequency distributions) of the widely distributed pothole complex. The Woodworth and Jamestown stations are located close to the Cottonwood Lake area, and precipitation data from there were applied to the simulation of the six individual wetlands.
 Monthly averages in ND05 were calculated with equal weight given to stations reporting both temperature and precipitation within that division. From 1901 to 2005, the average annual precipitation was 543 mm, with about 16% of the total precipitation falling as snow in winter. Mean monthly mean temperatures ranged from −14.7°C in January to 20.6°C in July.
 Monthly PE was calculated with the Food and Agriculture Organization Penman-Monteith method [Allen et al., 1998] on the basis of monthly maximum and minimum air temperatures, solar radiation, wind speed, and air humidity. Solar radiation data from 1961 to 2005 were obtained by distance weighting values from nearby stations at Jamestown, Devils Lake, Bismarck, Fargo, and Minot. For years without complete solar radiation data, we calculated the PE in three steps: (1) estimating the correlation coefficient between the PE calculation procedures using missing data and complete data, (2) estimating the monthly PE for those missing-data years by using procedures suggested by Allen et al. , and (3) converting the PE obtained from step 2 to the “real” model input PE for missing-data years on the basis of the correlated coefficient. The average annual PE of the study area was estimated to be 950 mm, which was nearly 2 times higher than the precipitation. The annual moisture deficit (precipitation minus evapotranspiration) was more than 400 mm and is comparable to results from other studies [e.g., Laird et al., 1996; Grimm, 2001].
3. PCHM Calibration and Performance Evaluation
3.1. Calibration Method
 Simulation of a lake-wetland complex requires values of parameters relating to climate, soil, land cover, groundwater, and hydrologic processes. Among them, several parameters controlling water–land surface interactions, soil water storage, groundwater inflows and outflows, and lake basin bathymetry are typically uncertain and require calibration. Table 1 defines parameters whose values were determined by calibration and their initial ranges. To ensure a globally optimal set of parameters, those with large uncertainties are given broader initial ranges. Detailed descriptions of parameters and their sensitivities were given by Crowe and Schwartz [1981a].
 Calibration requires one or several appropriate objective functions, which are consistent with the anticipated application of the model. PCHM is developed mainly to simulate water area–frequency relationships. However, as is the case with this study area, measured data on water depths for individual potholes are also available. Therefore, objective functions are defined for both area-frequency and water depth relationships as
subject to xil ≤ xi ≤ xiu (i = 1,2,…,p), where
 Here x1, x2, …, xp are p decision variables (parameters); xil and xiu are the lower bound and upper bound on parameter xi, respectively; wt is the weight factor for data at time t; wi is the weight factor for the data at bin size i; Nit and Nit′ are the observed and simulated numbers of water bodies falling in area bin i from a Landsat image at some time t, respectively; and Lkt and Lkt′ are the observed and simulated water levels of wetland k at some time t, respectively.
Equation (4) provides the objective-oriented fitness function, aiming to provide the best set of model parameters (x1, x2, …, xp)optimum that minimizes differences between the simulated results and the observational data. Nine parameter values (i.e., p = 9) need to be determined (Table 1). The value (fitness) of equation (4) is defined as the sum of two objectives, f1 and f2, to capture data on both area-frequency distributions and water depths. Objective f1 is set to minimize the errors between simulated and observed numbers of water bodies from eight sets of power laws. Objective f2 is set to minimize the errors between simulated and observed water depths of six (m′ = 6) individual wetlands. In each of the f1 or f2 functions, weights were assigned to data points to offset the data bias [Sorooshian et al., 1983; Freedman et al., 1998], which is caused by the uneven nature of power law binning and uneven data coverage for average and extreme climatic conditions.
 With numerous data points (water areas, depths, and counts of all individual potholes in the complex) and complicated relationships among model inputs, parameters, and outputs, model calibration is only feasible with an automated scheme. Here a genetic algorithm (GA) approach was implemented. It involves finding the global optimum by searching a wide solution space [Holland, 1975]. The securGA implementation [Carroll, 2001] used here features a powerful “small-elitist-creeping-uniform-restart” algorithm [Yang et al., 1998].
3.2. Observational Data
 One set of calibration data was composed of eight different water area–frequency relationships, essentially snapshots of the lake-wetland complex from 1986 to 1997, provided by Landsat thematic mapper images [Zhang et al., 2009] with zero cloud coverage. Four other similar data sets from 1998 and 2003 were used for validation (Figure 4). Images were selected to be representative of different climatic conditions. Power laws thus provide a simple way to summarize the observational data and to compare these results with similar monthly frequency distribution on water areas calculated by the model.
 A second set of calibration data comprised sets of conventional water depth measurements [Winter and Rosenberry, 1998] for three permanent and three seasonal wetlands from the Cottonwood Lake area (Figure 5). The 27 year record from 1979 to 2005 included a significant drought from 1988 to 1992 and a significant deluge from 1993 to 2001.
3.3. Parameter Estimation
 GA-based calibration involved searching the global optimum with multiple searching points by improving the fitness of equation (4). Calibration began by randomly generating five chromosomes or individuals (an individual represents a solution combining the values of nine parameters) and continued with the application of a set of evolution operators (e.g., selection, crossover, and mutation) to the individuals in the population in order to generate new individuals. In every generation, each of the five individuals was evaluated to determine its suitability (fitness) on the basis of equation (4). The individuals evolved from generation to generation until the parameters converged to constant values.
 The history of parameter changes through the GA optimization are illustrated in Figure 6, with the x axis representing the GA generations and the y axis representing the value of fitness (Figure 6a) or parameters (Figures 6b–6j). The parameter values, searched in each generation, are plotted as gray points. The wide scatter of points for each parameter (Figures 6b–6j) indicates that the GA sampled across the search space to find the best set of parameters. The best fitness and the best set of parameters found by GA in each generation were linked by the solid lines in Figure 6. The line in Figure 6a shows that the fitness value improved markedly after the first 200 generations and clearly had converged. Figures 6b–6j also show that all nine parameters converged to constant values after a few hundred generations. The calibration was stopped after 1820 generations with a total of 9100 estimates of equation (4) accomplished (1820 generations × 5 estimates per generation). Clearly, an optimal set of parameters was obtained.
3.4. Evaluation of PCHM Performance
Figure 4 provides a visual comparison of the water area–frequency power laws derived from Landsat data (solid squares) and comparable results from the model (open circles). PCHM was successful in simulating the population dynamics of large and small water bodies during average (1986), dry (1988 and 1991), and wet periods (1997). The validation tests for spring and summer/fall of 1998 and 2003 (Figure 4 bottom) yielded good agreement between observations and simulations. A slight tendency to overestimate the numbers of small water bodies was evident in the results for October 1986 and April 1991 (Figure 4).
 The performance of the model in simulating dynamic changes in water areas was also evaluated quantitatively using the coefficient of determination R2. The mean R2 value for all eight pairs of observed and simulated power laws used in calibration is 0.881 (Table 2). Values of R2 range from 0.841 for dry years to 0.905 for wet years to 0.936 for average years. Although the dry-year coefficient is somewhat lower, the correlation is good, indicating strong model performance. The mean R2 value for the validation is 0.884. We attribute this somewhat lower correlation for dry conditions to the inherent nonuniformity in the calibration data (regional lake numbers versus actual lake depths). Dry-time calibrations are probably being influenced more by Cottonwood Lake data, where drought conditions are prominently represented in the record. Drought conditions in the satellite record are underweighted in the calibration and less likely to influence the global calibration. Work is in progress to examine factors affecting calibration with nonuniform data and the overall uncertainty that this might create in calibration. Nevertheless, the calibrated parameter set in PCHM successfully simulated observed water area–frequency relations.
Table 2. Quantitative Evaluation of PCHM Calibrationa
and where o indicates observed value; s indicates simulated value, overbar indicates mean, R2 is coefficient of determination, and NSE is Nash-Sutcliffe efficiency index. Performance ratings [Moriasi et al., 2007] are as follows: NSE ≤ 0.50, unsatisfactory; 0.50 < NSE ≤ 0.65, satisfactory; 0.65 < NSE ≤ 0.75, good; 0.75 < NSE ≤ 1.00, very good.
Mean R2 is 0.936 for an average year (1986), 0.841 for two dry years (1988 and 1981), and 0.905 for a wet year (1997).
 Simulating water depths was not a primary goal of this study. However, the exceptional record of water depth measurements at the Cottonwood Lake area provided an important opportunity for calibration and validation. Figure 5 compares water depths calculated by PCHM with calibrated parameters to the observed water depths in the six individual wetlands. The model performed well in simulating water depths in the semipermanent wetlands P1, P2, and P6 (Figure 5, top). These semipermanent wetlands mostly dried out during the severe drought from 1988 to 1992. Water depths increased markedly after 1993 and reached a maximum around 1999. PCHM clearly was able to simulate the significant transition from drought to deluge in 1993.
 The model also successfully captured the behavior of the small seasonal wetlands T5, T6, and T8 (Figure 5, bottom). Their behavior featured a transition from water depth of less than 1 m in spring to dryness during the hot summer months. Although the measured and predicted water depths matched much of the time, the model underestimated the water depths for seasonal wetlands in 1981 and 1987. One possible explanation is that for these smaller water bodies, single point-in-time measurements on occasion may not be directly equivalent to monthly averages from the model. Another possibility is that the quite idealized model for the pothole geometry does not appropriately capture the actual character of these small water bodies.
 The overall R2, comparing observed and simulated water depths for all six wetlands, is 0.937, with a value of 0.953 for the semipermanent wetlands and 0.705 for the seasonal wetlands. The Nash-Sutcliffe efficiency index (NSE) [Nash and Sutcliffe, 1970], a commonly used criterion for model performance, was also applied to provide a quick overview of PCHM's accuracy. The NSE has a range of −∞ to 1, with 1 indicating a perfect match between the measured and simulated data. The NSE is 0.943 for the three semipermanent wetlands (Table 2) and is considered to be very good [Moriasi et al., 2007]. The NSE value for the seasonal wetlands is 0.691 and is considered to be good, according to the ratings (described in Table 2). Thus, the model was able to capture the wide variety of hydrologic responses in wetlands at the Cottonwood Lake area.
4. Results and Discussion
4.1. Climate-Driven Variability of Pothole Systems
 The modeling approach was used to examine the behavior of pothole water bodies through the twentieth century. Of particular interest were the climatic extremes, two of which are noteworthy: the Dust Bowl drought of the 1930s and the deluge from 1993 to 2001. The simulations covered the period from 1901 to 2005 with a monthly time step. This analysis tracked the hydrologic behavior of a pothole complex consisting of 22,770 variously sized potholes with an initial distribution representative of the North Dakota setting. Thirty years of average climatic conditions were used to spin up the model so that the actual simulation could begin in 1901 with a plausible set of initial conditions rather than poor guesses.
4.1.1. Intra-annual Analysis
 In nearby South Dakota, Zhang et al.  found quite marked changes in water areas in any given year as the pothole complex transitioned from wet spring to hotter and drier conditions in summer. We applied the PCHM model to the study area in North Dakota to reexamine this intra-annual variability more systematically through the twentieth century.
Figure 7 describes the simulated seasonal shift in the size structure of the pothole complex in 1985, a year with average precipitation. A linear power law was maintained from month to month. However, the slopes of the lines for April, June, and August decreased as conditions shifted from wet to drier conditions. This behavior in power law slopes indicated a preferential impact on the frequency distributions of the potholes with small water areas (<1 × 104 m2). Generally, with the increased evaporation from spring to summer, the areas of smaller water bodies declined to a much greater extent than the larger potholes. The sensitivity of the small potholes was also related to water depth in relation to the particular shape of the pothole basin [Zhang et al., 2009]. Effectively, then, these changes were manifested by a reduction in the numbers of small potholes of a given size. However, as Figure 7 shows, the numbers of large water bodies approaching the size of lakes (∼1 × 105 m2) remained constant. The decline in the numbers of small potholes and the stability in the number of large potholes produced the observed slope decline in power law lines in the spring-to-summer transition.
4.1.2. Interannual Analysis
 The climate often deviates from average conditions for several or more years at a time with evident periods of drought and deluge. Figure 8a shows the variation in the Palmer hydrological drought index (PHDI) for ND05 over the twentieth century. PHDI, like the familiar Palmer drought severity index, takes into account precipitation, evapotranspiration, and soil moisture but better reflects the hydrological impacts of drought. A value >0 is indicative of conditions wetter than average, while a value <0 is indicative of conditions drier than average. The Dust Bowl drought from 1930 to 1940 is shown clearly, along with a number of preceding wet years. The deluge beginning in 1993 also stands out (Figure 8). Such climatic extremes in the PPR are known to produce extraordinary impacts on the water area systematics for pothole systems. In South Dakota, Zhang et al.  found that these same periods of drought and deluge provided the lower and upper limits for fluctuations in the power law lines over the twentieth century.
 Our simulation results from 1901 to 2005 provide a basis for describing these effects in more detail. Shown in Figures 8b and 8c are power law lines that illustrate how water area relationships changed for periods preceding and through these two extreme events. In order to minimize the impact of intra-annual variability to some extent, results are presented for August of each year. Through the 1930s, persistent and intense drought conditions dramatically reduced water areas and decreased the number of pothole water bodies of a given size. Power law lines (Figure 8b) declined year after year, reaching the lower limit in August 1940. However, the drought's hydrologic impact was variable. By August 1930, a large number of water bodies had already disappeared, with a marked reduction in the numbers of small potholes (<1 × 104 m2), as seen by comparing the lines of August 1928 and August 1930. The flattening of slopes of power law lines through the early 1930s indicates a rapid decline in the population of smaller water bodies. As the drought intensified in 1934 and in 1936 (Figure 8a), the numbers of large potholes of a given size declined markedly with the reduction in their water areas. During the latter years of the drought, declines were evident but less marked (Figure 8b).
 The deluge from 1993 to 2001 began with several rainstorms in July 1993 that produced nearly 30 cm of rainfall and broke the drought of 1988–1992. For the smaller potholes (<1 × 104 m2), the power law lines flipped nearly instantaneously from a condition indicative of drought to one of deluge (Figure 8c). For the larger water bodies (>1 × 104 m2), there was a lag time of 3 or 4 years (Figure 8c) when they increased their size and stage according to the wetter conditions.
 Simulation data for the month of August in various years were used to create a power law envelope that is bounded by lines from the driest of the Dust Bowl times to the wettest times in the deluge of 1993 to 2001 (Figure 8d). There is a surprising range in the numbers of water bodies reflected by this envelope. The bounding power law lines for these two extreme events are separated by an order of magnitude and by even more for small water bodies (Figure 8d).
 The results of the simulations confirmed basic ideas presented by Zhang et al.  that the frequency distributions of water areas of large prairie potholes are usually well described by a pure power law (Figures 7 and 8). The numbers exhibited a consistent size structure interannually and intra-annually as a function of climate. The impacts of climatic fluctuations on water bodies were remarkable, with a tendency for slopes of power law lines to flatten under drier conditions and for intercepts to be lower under drought conditions. Large and small pothole water bodies, however, showed different rates of response to the climate variations. The numbers of smaller water bodies were more sensitive and fluctuated more rapidly than larger water bodies.
4.2. On the Behavior of Small Water Bodies
4.2.1. Significant Volume Fluctuations
 There are hints in Figures 7 and 8 that the flattening of power law lines indicates a preferential impact of drought on small water bodies (e.g., <1 × 104 m2). In this section, we examine the behavior of small water bodies in more detail. Figure 9a shows how the volume of water in surface water bodies, averaged over the twentieth century, varies from spring through summer. Small water bodies (900 to 1 × 104 m2) contained about 9%–14% of the surface water. However, about 33% of the total loss in surface water from April to August came from these small bodies (Figure 9b). Through the Dust Bowl drought, we estimated an 89% loss in total surface water volume from 78.9 × 106 m3 in 1928 before the drought to 8.5 × 106 m3 at the end of the drought in 1940. The loss attributed to the small water bodies was 15%, the largest among all different sizes of surface water bodies (Figure 9c). These remarkable fluctuations in water storage of the small water bodies are thus quite significant in accounting for the transfer of water through the hydrologic cycle.
4.2.2. Deviation From Pure Power Law Behavior
 Generally, not much is known about the frequency distribution in water areas for the small pothole water bodies. They are difficult to observe remotely because of their small size and because they are often dry during hot summer months. The smallest of these water bodies are essentially puddles (<250 m2) and dry most of the time. The model, however, is sufficiently general that it can simulate the changes in water stored in these small water bodies, including completely dry conditions. Here we examine how the intra-annual and interannual variability in climate especially influences small potholes and puddles.
 PCHM was applied to simulate the behavior of a hypothetical pothole-puddle complex consisting of 55.7 million water bodies. The complex was generated, and water bodies were counted with a greater resolution in bin size (10 m2). With this resolution, the behaviors of water bodies as small as several square meters could be examined.
 The strategy for this analysis was to examine both intra-annual and interannual behaviors. From spring through summer in a single dry year, the greatest number of these smallest water bodies should be associated with the spring snowmelt period or large rainfalls when there is an excess of water. Figures 10a–c illustrate the simulated behavior of the pothole-puddle complex from spring to summer for an average year (1985), a dry year (1992), and a wet year (1998), respectively. Although the details are different, the general patterns of behavior are similar. Pure power law behavior that is evident with the larger potholes broke down when the smaller water bodies (<1000 m2) were included. The power law curves have several important features. First, the point of departure from pure power law behavior changes depending upon wetness or dryness conditions, which are generally related to the time of year. For example, because spring was wetter than summer, the departure points for April in 1985 and 1992 were associated with much smaller water areas than the following months. In the wet year, 1998, several rainy months (i.e., June and August) exhibited pure power law behavior, while other drier months (i.e., July and September) exhibited more complex, nonlinear behaviors. The pure power law behavior had the greatest range in spring or wet months, usually extending to water bodies as small as puddles (<250 m2). By summer, the pure power law behavior only held for the somewhat larger pothole bodies (>1000 m2).
 Second, these nonlinear curves are characterized by some maximum number (with outliers excluded) of water bodies in one or several bins. For example, in Figure 10a, the maximum number of water bodies was about 400,000 in April, the May maximum was about 70,000 water bodies, and the July maximum was about 3000 water bodies. Clearly, the maximum number of water bodies fluctuated markedly from month to month, depending upon the moisture excesses or deficiencies.
 Another feature of these functions is the tendency for the water area–frequency curve to simply terminate as the number of water bodies abruptly drops to zero or sharply increases. The numbers of the smallest water bodies (i.e., puddles) generally decreased in a nonlinear manner as water areas declined. However, there were occasional conditions when the numbers of puddles actually began to increase again. This departure from the general trend has an unsystematic character, e.g., the data point in the 10 m2 bin in May 1985 and the 20 m2 bin in July 1985 (Figure 10a). Thus, puddles can behave unsystematically in a single month. With these small water bodies, individual storms in a month can influence their numbers without appreciably influencing larger bodies, producing quite interesting and complex behavior.
 The interannual variability in the behavior of small water bodies was also examined. Figure 10d compares the small pothole and puddle behaviors in August for an average year (1986) and for years characterized by the extreme events (1938, 1992, and 1997; dry, dry, and wet years, respectively). As expected, pure power law behavior extended to water bodies of smaller sizes during wetter times than was the case with droughts. The water area limit for pure power law behavior was about 3000 m2 for August 1938, 1000 m2 for August 1992, 400 m2 for August 1986, and as small as 200 m2 for August 1997. The simulated water areas in the midst of the deluge from 1993 to 2001 exhibited linear behavior down to quite small bodies. During the two major droughts, a nonlinear power law developed.
 The maximum number of water bodies associated with each of the curves is also highly dependent on conditions of drought and deluge (Figure 10d). The maximum number of water bodies (peak value, with outliers excluded) decreased steadily from a deluge value of 48,170 in August 1997 to about 620 during the Dust Bowl drought in August 1938. Effectively, during droughts, the point of termination of the water area function was associated with larger and larger water bodies.
4.3. Conceptual Models on the Response of a Pothole Complex to Climate
 Pure power laws provide a convenient and simple graphical approach for describing the behavior of a pothole water body complex. They are limited to the extent that they are weakly coupled to the dynamic character of the water bodies because no history of behaviors is incorporated there. Moreover, linear power law relationships only hold for the somewhat larger water bodies observable by Landsat or in aerial photographs. Similar deviations from pure power law behavior have also been observed, for example, with document sizes on the Web [Crovella and Bestavros, 1997], Web page link distribution [Pennock et al., 2002], and actor collaboration [Barabási and Albert, 1999].
 The area-frequency distribution for pothole-puddle systems should be describable with a general functional form that effectively combines linear (e.g., Figures 7 and 8) and curvilinear (e.g., Figure 10) components. Several functions were examined to determine their applicability including lognormal, discrete Gaussian exponential (DGX) distribution [Bi et al., 2001], polynomial, power, etc. The most satisfactory function applied to potholes and puddles was a modification of a function described by Pennock et al. :
where N is the total number of water bodies in the bin, whose mean size value is A. The parameters governing the shape of the function (k, , and m) were explained in detail by Pennock et al. . This function consists of a unimodal body and linear power law tail. The parameter k essentially scales the function, controlling the maximum value of the unimodal body, which occurs at and the parameter controls the slope of the linear portion of the function.
 As the water area varied with climate, so did the shape of the function. The function in equation (5) was fitted to the ensemble of results for large water bodies (e.g., data shown in Figure 8) and small water bodies (e.g., data shown in Figure 10). A genetic algorithm was used to fit the function by optimizing k, , and m.Figure 11 demonstrates the fitting results for an intra-annual pattern (Figure 11a) and an interannual pattern (Figure 11b). Clearly, all simulated distributions (points) were fitted extremely well. The fitted function could be examined to determine the slope of the linear portion and the maximum number of water bodies in a bin. The point of termination was empirically determined as the last bin in which the number of water bodies dropped to zero or there was a significant deviation from this function. To finally confirm validity of the functional relationships suggested by this model will require observational data that captures this small pothole behavior.
 To help in understanding behaviors of the water bodies, two conceptual models have been developed. They encompass the intra-annual (Figure 11c) and interannual (Figure 11d) behaviors of water bodies varying in size from small puddles to large potholes. For illustrative purposes, a single solid line with circles is shown and considered to represent some average condition. Over a few months, only the puddles and small potholes respond to the seasonal variability in precipitation and evaporation (Figure 11c). Excess water from snowmelt in spring maximizes the extent of the linear power law, the number of water bodies, and the overall range of systematic behavior. Evaporation through the hot summer months shifts the function, reducing the linear portion of the power law, the maximum number of water bodies, and the range in systematic behavior (Figure 11c). Large potholes (>1 × 104 m2) are relatively unaffected by short-term climatic variations. The conceptual model for interannual variations (Figure 11d), while similar to Figure 11c, differs in the pattern of behavior of the large potholes. With a significant drought and deluge, the position of the function changes not only at the small water body end but also at the large end (Figure 11d).
 The conceptual model separates the water area–frequency distribution into two parts: a systematic part and an unsystematic, or random, part. The term systematic implies a behavior characterized by regular functional form. The unsystematic part, on the one hand, is associated with the small puddles. At some lower threshold, the numbers of small puddles would become erratic, fluctuating from no puddles to many puddles on a monthly basis. This behavior was likely driven by small irregular variability in monthly rainfall. On the other hand, large potholes or lakes exhibited unsystematic behavior as a result of the relatively limited number of lakes (also bins with no lakes) in certain large-size classes in our study area.
 The boundaries between the systematic and unsystematic parts and division between linear and nonlinear segments of the power law are not fixed. The peak in the numbers of water bodies is directly proportional to the parameter k, and the location of the peak is directly proportion to parameter m. Values of k, m, and are highly correlated to the climatic conditions. Testing of these correlations is one of our future study topics.
 There are three key conclusions. First, it is feasible to simulate the behavior of a pothole complex using the PCHM described in this paper. The calibrated model was successful in simulating the hydrologic response of a pothole complex to the broad range of climatic variability in North Dakota. Second, climate-driven variability in the numbers of large potholes (>1 × 104 m2) as a function of water area can be described well by a pure power law. Major droughts and deluges create marked variability in the power law function. The time required for surface waters to adjust to drier or wetter conditions is proportional to the size of the water body. Thus, increased evaporation in late spring and summer in a given year mostly would affect the small potholes in the complex. Changes to large potholes would be associated with droughts and deluges that span a number of years.
 Third, deviation from pure power law behavior should be expected for small potholes and puddles. As a group, these water bodies exhibited tremendous variability in terms of water volume and area. A significant fraction of the gains or losses in total surface water storage is associated with these water bodies. For this reason, understanding the behavior of the small water bodies is critical to quantitatively describing the evaporative fluxes. Empirical results and theoretical mathematical results suggest that much of the behavior across much of the spectrum of water bodies can be described by a general equation that includes the parameters k, m, and , controlling the shape of the unimodal body and power law tail. At some points the functions would terminate when water bodies became dry under the particular climatic conditions; in a few cases, they behaved in an unsystematic manner.
 This study provides a better understanding of the linkages between climate and wetland-lake systems in the PPR. Such knowledge is important for policy development, water resources management, wetland restoration, and wildlife conservation. Our findings continue to show the potential for space-based monitoring that ultimately might lead to regular monitoring of the behavior of millions of water bodies across the PPR. Such capabilities are essential to better assessing global climate change impacts here. For the first time our modeling approach provides a way to fully account for the spatial and temporal distribution of surface waters. Such knowledge is essential for developing a more realistic understanding of the feedbacks between climate and hydrologic systems and elucidating the role of coupled processes like precipitation recycling. Moreover, understanding the behavior of wetlands as nodes in a complex wetland habitat network is crucial for defining critical habitats and establishing the role of habitat fragmentation as a driver for species biodiversity in the PPR [Wright, 2010].
 The next step is applying these approaches to the PPR more broadly. Although this study primarily focused on pothole basin geometries appropriate for the Missouri Coteau, the PCHM is sufficiently general that it should be applicable to pothole lakes and wetlands in other regions of the PPR, such as southern Saskatchewan, Minnesota, and the Prairie Coteau of South Dakota. Such a comprehensive spatial and temporal investigation on the scale of the PPR opens the possibility for closing the water cycle over this important region and better understanding its implications on regional problems like droughts, deluges, and waterfowl populations.
 This study was supported by the National Science Foundation, NSF award EAR-0440007 and the Ohio State University through the Climate, Water, and Carbon (CWC) Program. Tom C. Winter of the U.S. Geological Survey kindly provided investigational data from the Cottonwood Lake area. The manuscript benefited from the careful and constructive comments of three anonymous reviewers and the Associate Editor, A. Wüest.