| May 31, 2022

A Mathematical Model to Analyze Shrinking Surface Water Bodies

Surface water—above-ground water bodies such as lakes, rivers, reservoirs, wetlands, and glaciers—constitutes an essential resource for both humans and the environment. While some freshwater ecosystems are protected, people exploit other surface waters for anthropogenic purposes like drinking, irrigation, transportation, and power generation — all of which are necessary to sustain a society’s economic growth [3, 7]. Human settlements have historically developed along rivers, thereby altering water quality, river morphology, and natural flow regime. Urbanization and demographic expansion are currently triggering a progressive loss of surface waters (i.e., the conversion of water to land), with cascading negative consequences on rivers’ connectivity and the overall water cycle — including infiltration, groundwater recharge, evapotranspiration, and surface runoff. Mathematical models that reproduce the spatial influence of built-up areas on shrinking surface waters yield emerging research avenues that quantitatively analyze the complex interactions between water resources and society [10].

Benefits of Remote Sensing Data

Figure 1. Geographical locations of surface water loss (blue) and built-up areas (orange) that we extracted from datasets derived from remote sensing data [4]. Figure courtesy of the authors and created on QGIS. Background courtesy of Google Satellite.

Remotely sensed global Earth observations at high spatial and temporal resolutions are the most relevant source of data for the detection and monitoring of processes that are associated with environmental or anthropogenic changes on Earth’s surface [1, 8, 9]. In our recent work [5], we processed remotely sensed data [2, 6] to examine urbanization’s impact on surface water loss and quantify changes across the contiguous U.S. (CONUS) between 1984 and 2018. To this aim, we created maps that describe surface water loss locations and built-up areas at 30-meter resolution (see Figure 1) [4].

By coupling these maps, we derived the observed frequency of occurrence for surface water loss locations as a function of the Euclidean distance from urban areas \(f(d_{ij})\) across the CONUS. The associated formula is

\[f(d_{ij})=\frac{swl(d_{ij})}{swl_{tot}}.\tag1\]

Here, \(swl(d_{ij})\) is the count of surface water loss locations between distances \(i\) and \(j\) (with \(i\) and \(j\) ranging from \(0\) to \(d_{max}\), the maximum distance) and \(swl_{tot}\) is the total count of surface water loss locations.

As we expected, the observed frequency of occurrence of surface water loss locations consistently decreases as the distance from urban areas increases [5]. Animation 1 depicts this phenomenon.

Building a Distance-decay Model

Based on this result, we defined a distance-decay model to fit the observed decreasing spatial interaction [5]. The probability of occurrence of surface water loss locations as a function of the distance from urban areas \(p(d_{ij})\) is

\[p(d_{ij})=\alpha e^{-\beta d_{ij}}.\tag2\]

Here, \(\alpha \ [-]\) is the frequency of occurrence of surface water loss locations in the initial distance bin and \(\beta \ ( > 0 \ [\textrm{km}^{-1}])\) represents the rate of decline in the spatial interaction. The distance-decay model in \((2)\) is a truncated exponential probability distribution wherein the maximum probability of finding surface water loss occurs near urban agglomerations; the probability declines exponentially with increasing distance — i.e., as \(\beta\) becomes higher, the decrease in \(p(d_{ij})\) with increasing distance from urban areas steepens. We employed the truncated version of this distribution because distances reach a finite maximum value. We then estimated model parameters \(\alpha\) and \(\beta\) in \((2)\) with a nonlinear regression of the probability of occurrence \(p(d_{ij})\) versus the observed frequency of surface water loss locations \(f(d_{ij})\). The truncated exponential distance-decay model \((2)\) successfully fits the observed surface water loss pattern, with Pearson correlation coefficient values that range from 0.676 to 0.999 [5].

We further tested the reliability of the distance-decay model in \((2)\) by comparing the observed and theoretical distances from urban areas wherein surface water loss occurs on average (\(\langle d \rangle\) and \(\langle \hat d\rangle\), respectively) [5]. Specifically, the observed average distance of surface water loss locations from urban areas is

\[\langle d\rangle = \frac{\Sigma_{k=1}^{swl} d_k}{swl_{tot}},\tag3\]

where \(d_k\) is the distance associated with a generic surface water loss location.

We analytically derive the theoretical average distance of surface water loss locations from urban areas (i.e., the expected value of the truncated exponential distribution \(\langle \hat d\rangle\)) from \((2)\), as follows:

\[\langle \hat d\rangle = \frac{1 - e^{-\beta d_{max}}({1 + \beta d_{max})}}{\beta (1 - e^{-\beta d_{max}})}.\tag4\]

We found a high correlation between the observed and theoretical average distances of surface water loss locations from urban areas (p-value \(<< 0.05\)), which again demonstrates our model’s consistency in reproducing the observed spatial pattern of surface water loss occurrence [5]. Furthermore, model parameters \(\alpha\) and \(\beta\) exhibit sensitivity to climate: the model shows the typical localized and concentrated impacts of urban areas on surface water loss in temperate and continental climate regions, whereas areas with arid climates reveal more widespread impacts that are distributed over larger distances from cities.

Animation 1. Application of our distance-decay model to the whole study area of the contiguous U.S., three representative water resource regions, and one of their subregions. Animation courtesy of the authors, data courtesy of [4].

Outlook

The distance-decay model that we developed and applied over the CONUS provides a theoretical characterization and a statistically significant model of the observed declining trend of surface water loss’ spatial distribution with respect to built-up areas. Our model proves that built-up areas place an exponentially increasing amount of stress on surface water resources in their proximity; it also supplies useful indications about the spatial interactions that might exist between these areas and surface water resources in other regions with similar social and climatic conditions. The proposed mathematical approach will boost the analysis of the connections between ecological water resources and human society. Such models could eventually support planning strategies for water resource management that guarantee a balance between population growth, water demands, and the environment’s own needs.

References
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	Irene Palazzoli is a research fellow in the Hydraulic Structures, Maritime Engineering and Hydrology group at Alma Mater Studiorum – Università di Bologna. Her research uses remote sensing data to assess the impacts of human pressure and climate change on water resources.
	Serena Ceola is an assistant professor in the Hydraulic Structures, Maritime Engineering and Hydrology group at Alma Mater Studiorum – Università di Bologna. Her research interests lie in the analysis of water resource sustainability and water-related risks, and focus on anthropogenic and hydrologic interactions through remote sensing data.