Does the Pareto distribution adequately describe the size-distribution of lakes?

When ; it comes to evaluating lakes at regional and global scales, a key need is accurate estimates of the abundance and size-distribution of lakes, which are usually described with the Pareto distribution. We demonstrate the considerable uncertainty that truncation in the lower tail of the Pareto distribution introduces into lake abundance estimates and the selection of the lake size-distribution. Truncation in the lower tail eliminates lakes below a certain size and is generally performed because small lakes are not accurately represented on maps. When simulated data are truncated to mimic available lake size data, non-Pareto distributions are visually and statistically indistinguishable from the Pareto distribution. The Pareto distribution may be one of many possible forms that mimic the global lake size-distribution in the upper tail, but the fit of the Pareto to the lower tail is uncertain, largely because the abundance of small lakes is uncertain. Some other potential sizedistributions, such as the lognormal distribution, predict abundances of small lakes to be orders of magnitude lower than do the Pareto distribution predictions. Highly resolved regional lake size data for the Adirondack Mountains of New York and the Northern Highland Lake District of Wisconsin do not conform to the Pareto distribution. Lake sizes on Mars also do not conform to the Pareto. Uncertainty in the lake size-distribution seriously limits understanding of the significance of lakes as repositories of organic carbon as well as the calculation of global greenhouse gas emissions from these systems. Recently, limnologists have become interested in developing more accurate estimates of the global distribution of inland aquatic systems (Lehner and Döll 2004; Downing et al. 2006). This interest is driven partly by the need to assess the importance of inland waters in processes such as the global carbon cycle (Cole et al. 2007; Battin et al. 2009; Tranvik et al. 2009) and partly by the need to more fully assess the numbers and sizes of aquatic systems, given global human pressures on inland waters (Wetzel 2001). This work has led to new findings, including the findings that inland lakes and reservoirs cover a much greater portion of the earth’s land surface (, 3%) and that inland waters process substantial amounts of organic carbon, relative to previous estimates (Downing et al. 2006; Cole et al. 2007; Tranvik et al. 2009). At the global scale, Tranvik et al. (2009) estimated that land exports of carbon to inland waters are twice as high as land exports of carbon to the ocean. Most of this carbon is either subsequently exported to oceans (0.9 Pg yr21), is buried (0.6 Pg yr21), or is oxidized and evaporates into the atmosphere (at least 1.4 Pg yr21) (Tranvik et al. 2009). < Lake sediments may contain as much as 820 Pg C (Cole et al. 2007). Globally, lakes are important methane sources, with greater emissions to the atmosphere than are provided by the world’s oceans (Bastviken et al. 2004). As with other global limnological analyses, these estimates of carbon processing and methane emission require an accurate estimate of the abundance and sizedistribution of lakes (Tranvik et al. 2009). These estimates are particularly critical for assessing the abundance of small lakes, which may both contain and process large amounts of carbon per unit area (Kortelainen et al. 2004; Hanson et al. 2007; Telmer and Costa 2007). For instance, the smallest third of lakes in Finland, studied by Kortelainen et al. (2004), contained two thirds of the carbon stored by lakes in the region. Telmer and Costa (2007) found that including lakes measuring # 0.1 km2 increased the amount of carbon accounted for by lakes by as much as 30% on a per–unit area of landscape basis. Hanson et al. (2007) found that omitting small lakes from a regional study caused large biases when estimating mean dissolved inorganic carbon and dissolved organic carbon concentrations, even though total surface area in this study was dominated by large lakes. The abundance of lakes is difficult to quantify because maps generally omit lakes below a certain size (Hamilton et al. 1992; Lehner and Döll 2004; Downing et al. 2006). A number of studies have used the Pareto distribution to estimate the global or regional abundances and surface areas of lakes (Lazzarino et al. 2009; Marotta et al. 2009; Tranvik et al. 2009). This approach fits a log-abundance log-size regression based on the largest lakes. Abundance is the number of lakes greater than or equal to a size (Lehner and Döll 2004). For example, the abundance of lakes greater than or equal to 10 km2 in the global lake size data of Lehner and Döll (2004) is 17,357. Parameters from the regression are then used to estimate the number of small, unobserved lakes using the Pareto distribution probability density function (Downing et al. 2006). For example, Fig. 1 displays the world’s 17,357 lakes measuring . 10 km2 used by Downing et al. (2006) to estimate the global abundance of lakes. These data are well described by the Pareto distribution because of the excellent visual fit and high r2 value. However, many distributions appear linear on a log– log plot when only the largest values are considered (Perline 2005; Gan et al. 2006). This apparently strong fit of the Pareto distribution to the data is a concern when estimating lake abundance because most lake size data are retrieved Limnology limn-56-01-32.3d 30/11/10 10:20:12 1 Cust # 10-125 * Corresponding author: [email protected] Limnol. Oceanogr., 56(1), 2011, 000–000 E 2011, by the American Society of Limnology and Oceanography, Inc. doi:10.4319/lo.2011.56.1.0000