Scaling Theory of Floods for Developing a Physical Basis of Statistical Flood Frequency Relations
Summary and Keywords
Prediction of floods at locations where no streamflow data exist is a global issue because most of the countries involved don’t have adequate streamflow records. The United States Geological Survey developed the regional flood frequency (RFF) analysis to predict annual peak flow quantiles, for example, the 100-year flood, in ungauged basins. RFF equations are pure statistical characterizations that use historical streamflow records and the concept of “homogeneous regions.” To supplement the accuracy of flood quantile estimates due to limited record lengths, a physical solution is required. It is further reinforced by the need to predict potential impacts of a changing hydro-climate system on flood frequencies. A nonlinear geophysical theory of floods, or a scaling theory for short, focused on river basins and abandoned the “homogeneous regions” concept in order to incorporate flood producing physical processes. Self-similarity in channel networks plays a foundational role in understanding the observed scaling, or power law relations, between peak flows and drainage areas. Scaling theory of floods offers a unified framework to predict floods in rainfall-runoff (RF-RO) events and in annual peak flow quantiles in ungauged basins.
Theoretical research in the course of time clarified several key ideas: (1) to understand scaling in annual peak flow quantiles in terms of physical processes, it was necessary to consider scaling in individual RF-RO events; (2) a unique partitioning of a drainage basin into hillslopes and channel links is necessary; (3) a continuity equation in terms of link storage and discharge was developed for a link-hillslope pair (to complete the mathematical specification, another equation for a channel link involving storage and discharge can be written that gives the continuity equation in terms of discharge); (4) the self-similarity in channel networks plays a pivotal role in solving the continuity equation, which produces scaling in peak flows as drainage area goes to infinity (scaling is an emergent property that was shown to hold for an idealized case study); (5) a theory of hydraulic-geometry in channel networks is summarized; and (6) highlights of a theory of biological diversity in riparian vegetation along a network are given.
The first observational study in the Goodwin Creek Experimental Watershed, Mississippi, discovered that the scaling slopes and intercepts vary from one RF-RO event to the next. Subsequently, diagnostic studies of this variability showed that it is a reflection of variability in the flood-producing mechanisms. It has led to developing a model that links the scaling in RF-RO events with the annual peak flow quantiles featured here.
Rainfall-runoff models in engineering practice use a variety of techniques to calibrate their parameters using observed streamflow hydrographs. In ungagged basins, streamflow data are not available, and in a changing climate, the reliability of historic data becomes questionable, so calibration of parameters is not a viable option. Recent progress on developing a suitable theoretical framework to test RF-RO model parameterizations without calibration is briefly reviewed.
Contributions to generalizing the scaling theory of floods to medium and large river basins spanning different climates are reviewed. Two studies that have focused on understanding floods at the scale of the entire planet Earth are cited.
Finally, two case studies on the innovative applications of the scaling framework to practical hydrologic engineering problems are highlighted. They include real-time flood forecasting and the effect of spatially distributed small dams in a river network on real-time flood forecasting.
Introduction to the Scaling Theory of Floods
Flood is a geophysical phenomenon. Floods are also a natural hazard because they can result in major loss of life and substantial damage to properties and civil infrastructure. Accurate estimates of the magnitude and frequency of flood flows are needed for the design of bridges and roads, water-use and water-control projects, and for floodplain definition and management (Dawdy, Griffis, & Gupta, 2012). The United States Geological Survey (USGS) developed the regional flood frequency (RFF) equations to estimate quantiles (e.g., the 100-year flood) for annual peak stream discharges, or floods, in ungauged basins, where no streamflow data exist. RFF equations are pure statistical characterizations that use historical streamflow records to carry out regressions. They do not include the physical processes that produce floods in rainfall-runoff (RF-RO) events (Furey, Troutman, Gupta, & Krajewski, 2016). A geophysical understanding of annual flood quantiles is a long-standing unsolved problem. The scaling theory of floods explained here developed a physical framework to make flood predictions in ungauged basins.
To supplement the accuracy of flood quantile estimates due to limited record lengths—typically less than 30 years—a physical solution to the estimation of flood quantiles at annual time scale is required. The RFF analysis assumes a stationary hydro-climate, which means that the future would be statistically similar to the past. Greenhouse gases and other first-order human influences are changing the hydro-climate system (Pielke et al., 2009). Consequently, future flood frequencies are not expected to be statistically similar to the past (Dawdy, 2007). A solution to this problem requires that annual flood quantiles be understood in terms of physical mechanisms producing floods.
Mandelbrot’s book (1982) on the fractal geometry of nature offered self-similarity as a new scientific paradigm. Scaling arises in self-similar systems. A large number of studies had observed scaling, or power laws, in annual flood quantiles versus drainage areas in river basins (Linsley, Kohler, & Paulhus, 1982). Gupta and Dawdy (1995) systematically investigated the presence of power laws between flood quantiles and drainage areas using the RFF analyses of the USGS in three states. A geophysical understanding of how slopes and intercepts can be predicted from physical processes required understanding the scaling in RF-RO events in river basins and led to Gupta, Castro, and Over (1996). It abandoned the concept of “homogeneous regions” and identified the fundamental role of self-similarity in channel network topology and geometry.
Ogden and Dawdy (2003) observed scaling relations in peak discharges for 226 RF-RO events that spanned hourly to daily time scales in the 21 km2 Goodwin Creek Experimental Watershed (GCEW) in Mississippi. They found that scaling slopes and intercepts vary from one event to another. Furey and Gupta (2007) conducted a diagnostic analysis to understand the variability in terms of physical processes generating floods. Two overview papers (Gupta, Troutman, & Dawdy, 2007; Gupta & Waymire, 1998) and subsequent research have identified many areas of science and engineering this article addresses, for example, hydrologic and hydraulic engineering, hydrologic science, non-linear geophysics, fractal geometry, fluvial geomorphology, applied probability, fluid mechanics, biology, and landscape ecology.
A Brief History of the Scaling Theory of Floods
Kirkby (1976) and Lee and Delleur (1976) first demonstrated the unique role of channel network width function in establishing a connection between network geometry and streamflow hydrographs. For any given distance from the outlet, the width function is defined as the number of streams at that distance. If rainfall is deposited instantaneously and uniformly over a network and travels at a constant velocity to the outlet, then the flow hydrograph at the outlet has the same shape as the width function. The width function is constructed from the link-magnitude classification of a network that Shreve (1966, 1967) introduced. It is called geomorphologic instantaneous unit hydrograph (GIUH). Subsequently, several papers contributed to this line of thinking; for example, Troutman and Karlinger (1984, 1998), and Gupta and Mesa (1988). Rodriguez-Iturbe and Valdes (1979) and Gupta, Waymire, and Wang (1980) proposed another version of a GIUH that was constructed from Horton-Strahler ordering and Horton laws. A substantial literature developed in this line of research (Rodriguez-Iturbe & Rinaldo, 1997). The above body of published research clearly demonstrated that channel network topology and geometry have a fundamental role to play in the future developments of river basin hydrology.
Mandelbrot (1982) published a book on the fractal geometry of nature, and a union session of the American Geophysical Union was organized around the book in the fall meeting in 1982. Fractal geometry was based on the hypothesis of self-similarity, a new form of invariance under a change of scale. Its impact became evident through many published papers and books with applications to physics, geophysics, geology, and hydrology (Feder, 1988; Rodriguez-Iturbe & Rinaldo, 1997; Turcott, 1997).
Scaling, or a power law, arises in self-similar systems, which offered a new scientific paradigm to a large number of studies that showed scaling in annual floods in river basins (Linsley et al., 1982, p. 370). Gupta and Dawdy (1995) systematically investigated the presence of power laws between flood quantiles and drainage areas using the RFF analyses of the USGS. However, a physical basis of scaling in RFF required understanding scaling relations in individual RF-RO events, because floods are generated at the scales of events. It was a major shift in focus from annual time scale to event scales of hours and days, and from homogeneous regions to river basins.
Gupta et al. (1996) took the first step to develop a physical basis of scaling in peak flows for RF-RO events in a Peano basin, an idealized self-similar channel network. It was analytically shown that the peak flows exhibit a power law relationship with respect to drainage area with a flood-scaling exponent, . The parameter is a fractal dimension of the spatial regions of a Peano basin that contribute to peak flows at successively larger drainage areas. Gupta and Waymire (1998) introduced an equation of continuity for a channel network in the first review paper on the topic. Troutman and Over (2001) generalized the results on scaling exponents of peak flows to a general class of networks generated by a deterministic self-similar algorithm, such as Tokunaga networks. Other theoretical papers were subsequently published on the topic (Menabde & Sivapalan, 2001). In order to make further progress, it became apparent that a careful observational study on scaling in peak flows for RF-RO events was needed.
Ogden and Dawdy (2003) observed scaling relations in peak discharges for 226 individual RF-RO events that spanned hourly to daily time scales in the 21 km2 GCEW, in Mississippi. They discovered that the scaling slopes and intercepts vary from one event to another. The mean of 226 event slopes (0.82) was close to the common slope (0.77) of mean annual and 20-year return peak discharge quantiles. Furey and Gupta (2007) carried out a diagnostic analysis of Ogden and Dawdy (2003) and found that event-to-event variability in the scaling slopes and intercepts is due to a combination of temporal rainfall variability, spatial variability in infiltration, and discharge velocity on a hillslope and in a channel link. Gupta et al. (2007) wrote a review paper on the scaling theory of floods in which key developments up to 2006 were reviewed. Scaling theory connects several disconnected fields for the purpose of understanding floods and solving related problems. The research is firmly rooted in the peer-reviewed literature. It represents a new paradigm in hydrologic science and engineering. In the remainder of this section, foundational and notable advances as well as current perspectives are discussed.
In 1956 David Dawdy was assigned by the USGS as chief assistant to Manuel Benson in developing the USGS RFF Analysis Program. After that he was assigned to develop the USGS physically based RF-RO model for extending RFF records to larger scales than individual states and stations. As state RFF reports began to appear he continued his interest in RFF analysis. Dawdy obtained the base data for each of these state reports and became convinced there was more information contained in the data than just those reported in the state reports, but he could not figure out how to use the data on a larger scale. It was his feeling that information was being overlooked, which led him to consult with Vijay Gupta, in the late 1980s. He found out that Gupta and Edward Waymire were collaborating on fractals and scaling in hydrology. Dawdy felt that scaling might help in making progress. Gupta and Dawdy met on several occasions in the early 1990s and took the first steps. Subsequently, a framework for solving the problem slowly developed (Dawdy, 2016). Brent Troutman played a substantive role through published research on the topic and co-authored the overview paper (Gupta et al., 2007). He also collaborated with Gupta’s graduate students on scaling in floods. He was a mathematical statistician in the USGS National Research Program from 1975 until his retirement 2011. Troutman’s research addressed the application of statistical methods, probability, and stochastic processes to the prediction and modeling of surface-water flows, particularly questions of scale and spatial/temporal variability of processes influencing flows (Troutman, 2016). Waymire and Gupta collaborated in understanding the mathematics of fractals and scaling in the 1980s and 1990s with applications to hydrology. Waymire co-authored the first overview paper on scaling in floods (Gupta & Waymire, 1998) and wrote a key paper on channel networks (Burd, Waymire, & Winn, 2000; Waymire, 2016).
Dynamic Formulations for Rainfall-Runoff Events in River Basins
Let , , be a space-time field representing river discharge, or volume of flow per unit time, through the downstream end of link e in a channel network . Let denote the runoff intensity into link e from the two adjacent hillslopes, and let denote the combined areas of the two hillslopes. may also include runoff depletion per unit time due to channel infiltration or evaporation from channel surface in link e. Finally, let denote storage in link e at time t. Then the equation of continuity for the system of hillslopes links can be written as (Gupta et al., 2007):(1)
The terms and on the right-hand side represent discharges from the two upstream links joining link e. denotes the runoff from the two hillslopes draining into link e. Solutions of Eq. (1) depend on the branching and geometric structure of a channel network through these two terms. Equation (1) as written is based on an assumption that the channel network is binary. It may easily be generalized, however, to non-binary networks, or those for which more than two upstream links can flow into a junction, for example, a Peano network. However, this situation does not arise in actual river networks. Equation (1) assumes that no loops are present in the network.
A specification of requires a hillslope-scale model for representing runoff generation from a very large number of hillslopes in a network. For example, GCEW has 544 hillslopes. It presents a great scientific challenge because most of the focus on model runoff generation has been in a single hillslope (Corradini, Govindaraju, & Morbidelli, 2002; Duffy, 1996; Freeze, 1980). Such approaches, called “bottom-up,” use a combination of well-known unsaturated-saturated flow continuum equations. By contrast, Furey, Gupta, and Troutman (2013) developed and tested a “top-down” model in GCEW, which is needed to model runoff generation from a large number of hillslopes in Eq. (1).
A functional relationship between link storage and link discharge is needed to express Eq. (1) in terms of one dependent variable. It can be obtained from the definition of storage and discharge and a specification of the link velocity, from a momentum balance equation governing three-dimensional velocity field in an open channel at the sub-link scale. Kean and Smith (2005) made progress toward solving this problem in the context of predicting a theoretical relationship between flow depth and discharge known as a rating curve. They developed a fluid-mechanical model that resolves boundary roughness elements from field measurements over a natural channel reach and calculated the cross-sectional average velocity to predict theoretical rating curve. Their research establishes a scientific framework to specify for a link. Jordan and Kean (2010) applied the model to estimate rating curves at ten ungagged stream reaches, ranging from third to sixth order channels, in the Whitewater basin in Kansas (Mantilla & Gupta, 2005).
Let, denote a link length, the mean link width, and the mean link depth. Then,(2)
In view of the hydraulic-geometric (H-G) relations for a network (Gupta & Mesa, 2014; Leopold & Miller, 1956), width, depth, and velocity may be expressed as functions of discharge, or , where f(.) is an arbitrary function. Substituting the storage-discharge relation into Eq. (1) gives(3)
The functions and in Eq. (3) vary from one hillslope-link pair to the other. Even small basins like GCEW contain a large number of hillslope-link pairs. A specification of the physical parametes in all of these pairs is required for solving Eq. (3). Gupta (2004a) stated that “statistical scaling is an emergent property of a complex system which a-priori is not built into the physical equations.” Scaling can be used to test physical hypotheses (Gupta, 2004b). Once a large number of parameters are specified, Eq. (3) can be solved iteratively to obtain runoff hydrograph, , , , at the bottom of every link, and therefore in all sub-basins of a river basin. Its solutions produce streamflow hydrographs at all junctions in a channel network. They have been used to obtain results on spatial scaling statistics of floods under idealized physical conditions in idealized channel networks, namely Mandelbrot-Vicsek (Menabde & Sivapalan, 2001).
Self-Similar River Networks
Gupta et al. (2007) reviewed 20 years of progress on self-similarity in real channel networks that included details of the Horton-Strahler ordering and the Horton laws. A brief overview of some of the key topics and progress since then is given here.
Horton Relationships for River Networks
Horton-Strahler ordering is defined as follows. A channel with no upstream inflows is given order one. When two stream segments of the same order merge, the order of the out-flowing stream increases by one. In case two merging streams have different orders, the out-flowing stream has the higher of the two orders. The three most widely studied Horton laws are for the number of Strahler streams of order , , the average length of streams of order , , and the average upstream area of streams of order , . It has been observed that the ratios of these quantities in successive orders tend to be independent of order, that is, , , and , which are known as the classical Horton laws. are called Horton ratios. Self-similar channel network models have shown that Horton laws hold in the limit as network order increases to infinity (McConnell & Gupta, 2008; Veitzer & Gupta, 2000).
Peckham and Gupta (1999) generalized the classical Horton laws for mean drainage areas and mean channel lengths to full probability distributions. Specifically, they gave observational and some theoretical arguments for the well-known random model (Shreve, 1967) to show that probability distributions of variables rescaled by their means collapse into a common probability distribution of some random variable, , that is independent of order. It is written as(4)
where is the Horton ratio, and denotes equality in probability distributions of random variables on both sides. The relation in Eq. (4) is known as a generalized Horton law. Thus, not only the mean value, but any finite moment, if it exists, obeys Horton laws. As an example, Figure 1 shows the classical and the generalized Horton laws for drainage areas in the Whitewater basin in Kansas (Mantilla & Gupta, 2005).
Tokunaga Self-Similar Model
Tokunaga (1966) introduced a deterministic model of river networks that was based on Horton-Strahler ordering rather than on link magnitude as is the case for the random topology model. It was characterized by the mean number of side tributaries. For a stream of order , let denote average number of side tributaries of order k, known as generators (Tokunaga, 1966, 1978). Generators are self-similar if they obey the constraint, . It means that the generators only depend on the difference independent of . Under the additional constraint that is a constant, generators have the form (Tokunaga, 1966). The number of streams of different Horton-Strahler order, obey a recursion equation (Tokunaga, 1978),(5)
McConnell and Gupta (2008) rigorously proved the Horton law of stream numbers as an asymptotic result from the solution of Eq. (5) and extended it to a Horton law for magnitude . Tokunaga also showed that the generators of the random topology model (Shreve, 1967) exhibit Tokunaga self-similarity with parameters . The solution of Eq. (5), with these values of parameters, predicts , which is the same that Shreve (1967) obtained for the random model. Gupta et al. (2007) reviewed other key results for Tokunaga networks.
Random Self-Similar Network Model
The geomorphologic and hydrologic thinking were greatly influenced by the random model of channel network based in a link-magnitude classification (Shreve, 1966, 1967). Jarvis and Woldenberg (1984) reproduced the key papers from 1945 to 1976 in a book. The papers were divided into four parts, and each part included editorial overview, which makes it an excellent reference.
The fluvial geomorphologic research took a new turn during the 1990s. Detailed empirical analysis of large river basins was greatly aided by the availability of fine-resolution digital elevation models (DEMs). Observations from large basins showed that random model predictions deviate substantially from empirical values (Peckham, 1995). Burd et al. (2000) proved that the random model is the only one among finite, binary Galton-Watson stochastic branching trees that exhibits the mean self-similar topology of Tokunaga networks. These findings opened the door to develop a new class of statistical channel network models called random self-similar networks (RSNs), which do not belong to the class of binary Galton-Watson stochastic branching processes (Veitzer & Gupta, 2000). RSNs are constructed recursively from random generators, which are essentially simple order-2 networks with a random number of interior nodes. The formulation allows for different generator probability distributions for the replacement of interior and exterior links. Veitzer and Gupta (2000) showed that the generalized Horton law, or distributional simple scaling, holds for the number of links per stream, magnitude, and stream numbers as network order increases. The mean side tributary structure of a subset of RSNs were shown to obey Tokunaga self-similarity.
Progress on Random Self-Similar Network Model
Troutman (2005) developed an algorithm that allows a unique extraction of generators from actual networks under certain restrictions. He gave results showing that a geometric distribution for the number of side tributaries in both interior and exterior generators fit reasonably well the data for the Flint River basin in Georgia. Mantilla, Troutman, and Gupta (2010) tested the RSN model in 30 basins from diverse climatic and geographic settings in the United States. Self-similarity in a statistical sense was found to hold in 26 of the 30 basins, and geometric distribution showed good agreement with data in all cases.
An important consideration in the analysis of scaling properties of river basins and the channel networks that drain them is spatial embedding, which is also necessary for simulating spatially distributed rainfall intensity fields on river basins. The idealized basins, Peano and Mandelbrot-Vicsek, are spatially embedded, which enabled to consider a spatial distribution of rainfall intensities on these basins (Gupta et al., 1996; Menabde & Sivapalan, 2001). For networks generated by the random topology model, deterministic self-similar networks like Tokunaga, RSNs, the branching structure is defined but how the networks fit into a two- or a three-dimensional space is not. Mantilla, Gupta, and Troutman (2012) presented an algorithm for embedding RSNs in a given spatial region.
Numerical Studies on Random Self-Similar Channel Network Models with Applications to Floods
Mantilla, Gupta, and Mesa (2006) simulated peak flows in the Walnut Gulch basin, Arizona, and analyzed the scaling properties of flow hydrographs by solving the mass conservation shown in Eq. (3) under a constant flow velocity assumption. They observed that the scaling exponents for peak flows are larger than the maxima of the width functions (Veitzer & Gupta, 2001). This property for a real network contradicts the previous findings for idealized self-similar networks; for example, Mandelbrot-Vicsek, Mantilla, Gupta, and Troutman (2011) tested this hypothesis in simulated RSNs using geometric distributions, with parameters and corresponding to interior and exterior generators, respectively. They compared the numerical results with the analytic expressions that Troutman (2005) had obtained, which served as benchmark for the accuracy of results from numerical simulations. Their results supported the finding of Mantilla et al. (2006).
Analytical Results for Hydraulic-Geometry in Tokunaga Self-Similar Channel Networks and Implications for Species Richness in Riparian Vegetation
In a classic paper, Leopold and Miller (1956) discovered Horton laws for H-G variables in drainage networks, thereby extending the Horton laws from topological and geometric variables to H-G variables (stream discharge Q, width W, depth D, velocity U, slope S, and Manning’s friction n’). Motivated by the need to theoretically understand Horton laws for the H-G variables, a long-standing unsolved problem, Gupta and Mesa (2014) developed an analytical theory in self-similar Tokunaga networks. The H-G data sets in channel networks from three published studies and one unpublished study were used to test theoretical predictions. Gupta and Mesa (2014) used dimensional analysis and the Buckingham Pi theorem to identify six independent dimensionless river-basin numbers. A mass conservation equation in terms of Horton bifurcation and discharge ratios in Tokunaga networks was derived. Under the assumption that the H-G variables are homogeneous and self-similar functions of stream discharge, it was shown that the functions are of a widely assumed power law form. Asymptotic self-similarity of the first kind, or SS-1 (Barenblatt, 1996), was applied to predict the Horton laws for W, D, and U as asymptotic relationships. Predictions of the exponents agreed with those previously predicted for the optimal channel network (OCN) model. Predicted exponents of width and the Reynolds number were tested against three field data sets. Ashley basin showed deviations from theoretical predictions. Tests of other predicted exponents suggested that H-G in networks does not obey SS-1. It fails because one of the dimensionless river-basin numbers, slope goes to 0 as network order increases, but it cannot be eliminated from the asymptotic limit. Therefore, a generalization of SS-1, based on the asymptotic self-similarity of the second kind, or SS-2 (Barenblatt, 1996), was considered. It introduced two anomalous scaling exponents as free parameters, which enabled them to show the existence of Horton laws for W, D, U, S, and n’. The two anomalous scaling exponents were not predicted. Instead, they used the observed exponents of D and S to predict the exponent of n’ and to test it against exponents from three field studies mentioned above. The Ashley basin showed some deviation from theoretical predictions. A physical reason for this deviation was given, which identified an important topic for research. Finally, how the two anomalous scaling exponents could be estimated from the transport of suspended sediment load and the bed load was briefly sketched. Statistical variability in the Horton laws for the H-G variables was also discussed. Both are important open problems for future research.
Hydro-ecological Theory of Intermittent Riparian Diversity on Stream Networks
Dunn, Milne, Mantilla, and Gupta (2011) generalized the scaling framework to include ecological variables coupled to water. Based on digital land cover maps, they were able to discern scaling and Horton laws in the areas occupied by riparian vegetation in the Whitewater basin. Milne and Gupta (2017a) are developing a new line of research to understand biological diversity and water balance in self-similar Tokunaga networks and test the developments in the Whitewater basin. The idea originated to understand deviation in the Ashley basin from theoretical prediction mentioned in the previous section. It requires that the Tokunaga generator be modified to incorporate intermittency in runoff generation and the growth of riparian vegetation on hillslopes, which is evident from the digital maps of riparian vegetation. Consequently, the rigorous mathematical results for Tokunaga networks (McConnell & Gupta, 2008) are used to derive Horton laws for species richness. A “law of hydro-ecology” is postulated that treats water balance throughout the network as a consequence of self-similar non-local interactions. In a companion paper Milne and Gupta (2017b) developed the entropy-Horton framework and demonstrated how it informs questions of biodiversity, resilience to perturbations in water supply, changes in potential evapotranspiration, and land use changes that move ecosystems away from optimal entropy with concomitant loss of productivity and biodiversity.
Physical Basis of Annual Flood Quantiles in River Basins
Building on Ogden and Dawdy (O-D), Furey et al. (2016) conducted the first rigorous analysis toward a physical understanding of annual flood quantiles in GCEW. They stated two hypotheses based on the results in O-D: (1) scaling slopes of annual peak (AP) quantiles are the same for all return periods and (2) the mean scaling slope of stream discharge peaks from RF-RO events equals the mean slope of AP quantiles. Here, a “mean” refers to an average over exceedance probabilities. Hypothesis 1 is a formal statement of simple scaling. It stands in contrast to multiscaling where scaling slopes depend on return periods (Gupta, Mesa, & Dawdy, 1994). Hypothesis 2 pertains to only the mean slopes both for events and AP quantiles. Therefore, a rejection of hypothesis 1 does not imply a rejection of hypothesis 2.
Results in O-D support hypothesis (1), while support for hypothesis (2) is unclear without further analysis. Given that hypothesis (1) holds, the mean slope of AP quantiles in hypothesis (2) will coincide with the common AP quantile slope in hypothesis (1). Let denote the common slope of AP quantiles and denote the mean scaling slope for RF-RO events. Results in O-D provide the estimates and given in Table 1. To test hypothesis (2), Furey et al. (2016) evaluated the statistical significance of the difference, . They found that the estimated standard error (SE) of the difference obeys SE>0.039, yielding a p-value > 0.15. It indicates that is not significant at the 5% level. Thus, O-D results do not reject Hypothesis (2).
Results in O-D represent only one dataset and required some approximations in the above calculations due to lack of information. Thus, to further test the two hypotheses, Furey et al. (2016) assembled a second dataset for 148 RF-RO events in GCEW. Examining data from a single basin is necessary to obtain a physical understanding of scaling in discharge quantiles. As one goes to larger basins, rainfall estimation requires radar-rainfall data that introduce remote-sensing errors not found in the rain gauge data in GCEW. A standard linear regression model was used to evaluate scaling slopes for RF-RO events and for AP quantiles. Figure 1 shows the scaling slopes for 148 events. The mean and standard deviation of slopes, 0.78 and 0.16, is similar to the corresponding values in O-D (Figure 3).
A Weibull plotting position formula was used to assign probabilities to AP values. For each gauge, the quantiles with empirical probabilities closest to several target probabilities were identified. The smallest probability evaluated, p = 0.071, represents a return period of 14 years, one year beyond the record length. The other value was p = 0.5. The results in Figure 2 show two different scaling slopes, suggesting the possibility of multiscaling (Gupta et al., 1994). It means that slopes of AP quantile scaling relations are different for different return periods. If such a property holds, the mean of the slopes for a given set of return periods can be denoted by . Then hypothesis 2 is an equality between and μb. Table 1 shows that and . These values denote the mean slope of the quantile relationships presented in Figure 2 and the mean slopes of events given in Figure 1. Assuming that hypothesis 1 holds, the significance of the difference, , can be assessed as done earlier for the O-D results. A p-value of ≥ 0.83 was obtained, indicating that the difference is not significant at the 5% level, again supporting hypothesis 2. The same results are found if quantile relationships for other return periods are considered. In summary, results from the second data set support hypothesis 2. To test for simple versus multiscaling in AP quantiles, indicated in Figure 2, data from GCEW that include additional years beyond 1994 are needed.
Table 1. Mean Scaling Slopes of RF-RO Events and Annual Peak Quantiles
Source: Furey et al. (2016).
The above results clearly demonstrate that a connection exists between scaling in events and in AP quantiles. Furey et al. (2016) characterized such a connection within a single physical-statistical model called a nested mixed-effects linear (NMEL) model. The NMEL model characterizes event-to-event variability in scaling relationships between stream discharge peaks and drainage areas. The NMEL model leads to scaling relationships for discharge peak quantiles and annual peak (AP) quantiles that are the basis for RFF equations. Since event-based scaling relationships can be connected to physical processes, it implies that annual peak quantiles can be connected as well.
To summarize, test results of model assumptions were supportive of NMEL model structure, though some discrepancies were found. While the model links events to quantiles, results show that there are important differences between quantile-based scaling statistics using APs and related event-based statistics. In particular, quantile scaling allows for event mixing, using data from different events to determine a scaling relationship and provides slope values having a significantly narrower range than that found with events. The latter result means that AP-quantile scaling relationships and RFF relations could underestimate flooding potential. Given that flood characterization across a river basin should be tied to physical processes and conditions that include pre-event soil moisture, infiltration governing runoff generation from each hillslope in a basin, and space-time variable hydraulic-geometry governing runoff dynamics in a network. The findings in Furey et al. (2016) show that methods for flood characterization based on event scaling are more informative and robust than traditional RFF methods based on analysis of AP quantiles. A larger sample study from GCEW and other watersheds is needed to further test, and possibly refine, these ideas. Test of multiscaling is an important issue that needs to be addressed in future research.
Horton Laws for Diagnosing a Rainfall-Runoff Model Without Calibration of Its Parameters
The measurement and collection of hydrologic data in channel networks is tedious and expensive. Gupta and Mesa (2014) illustrated two field studies as examples (Ibbitt, McKercher, & Duncan, 1998; McKercher, Ibbitt, Brown, & Duncan, 1998). As a result, the discovery of the Horton laws for hydrologic variables has greatly lagged behind geomorphology. The hydrologic data in Gupta, Ayalew, Mantilla, and Krajewski (2015) came from the 32,400 km2 Iowa River basin located in eastern Iowa before it joins the Mississippi River (Figure 4). The basin is continuously monitored by 34 USGS gauging stations. The top four annual maximum peak discharges observed at the basin outlet over the past 112 years occurred in the previous 7 years (Smith, Baeck, Villarini, Wright, & Krajewski, 2013). Figure 4 shows the Iowa River basin and the location of the USGS gauging sites. A Strahler order was assigned to each channel link where the USGS stream gauges are located, and the drainage areas draining into the gauges were estimated. The observed peak flows were also assigned the same order. This information was used to test the Horton laws for drainage areas and for peak flows.
Ayalew, Krajewski, and Mantilla (2015) selected RF-RO events that occurred in the 12-year period from 2002 to 2013. They observed a scale-invariant peak discharge when the entire basin got a runoff-generating rainfall event at some point during a time window equivalent to the basin’s concentration time (defined as the time required for a parcel of water to travel from the most remote hillslope in the basin to the outlet). The concentration time is about 15 days for the Iowa River basin (see Ayalew et al., 2015, for a detailed discussion of the RF-RO event selection and for the criteria used to define peak discharge events). Based on these strict criteria, 51 RF-RO events were identified whose resulting peak discharges exhibit scaling with respect to drainage area. Figure 5 shows a RF-RO event during which the entire basin did not receive rainfall at some point during the 15-day travel time window. Figure 6 illustrates a RF-RO event during which the entire basin received rainfall at some point during the 15-day travel time window.
A Consistent Theoretical Framework to Estimate Horton Ratios
Furey and Troutman (2008) developed a consistent statistical framework that resolved important problems with the interpretation and use of traditional Horton regression statistics. Their approach agrees with distributional simple scaling or the generalized Horton law illustrated in Figure 1. It is used in the data analysis of peak flows.
To understand the key concepts, let be a random variable that represents a geomorphologic property such as drainage area for basins of order ω. Assume that(6)
Here, a and b are constants and is a zero-mean random variable. Equation (6) is a linear regression model. It uses individual sample values rather than the sample mean that is the standard practice in the classical Horton analysis. Furey and Troutman (2008) took the expectation on both sides in Eq. (6) to get the expression , and(7)
Substituting where is the geometric mean, into Eq. (7) gives(8)
Equation (8) gives the expression for the Horton ratio, , as(9)
Equation (9) is based on geometric rather than arithmetic mean and serves as an alternative expression to estimate Horton ratio.
Gupta et al. (2015) estimated the Horton area ratio, , using drainage area information for 471,101 complete order streams using CUENCAS (Mantilla & Gupta, 2005). Two independent methods were used to estimate . The first method uses the regression Eq. (6) and estimates the slope,, which gives . The second method uses Eq. (7) and calculates geometric means for different order streams. for orders 7, 8, and 9 was ignored due to small sample sizes. The remaining estimates of showed statistical variability. To compare these estimates with the value from method 1, the mean of estimates for orders 2, 3, 4, 5, and 6 was computed, which gave 4.66. The results showed that the information from the 34 streamflow-gauging stations can reasonably predict the value that is the same as that estimated using detailed drainage network information. It provided confidence in the estimation of Horton ratios from the 34 gauging stations and also gave a template to estimate Horton ratio for peak flows, where sample size is a more serious issue than for drainage areas.
Classical Horton Law for Peak Flows in Rainfall-Runoff Events
Consider a river basin with stream gauges. Let denote a gauge and its sub-basin, and let i=1,2, … denote a RF-RO event. Data show that peak discharges at the event time scale obey a scaling relationship first observed in GCEW (Ogden & Dawdy, 2003) and later in the medium-sized Iowa River basin (Ayalew et al., 2015; Gupta, Mantilla, Troutman, Dawdy, & Krajewski, 2010). An ordinary linear regression (OLR) model can characterize this relationship (Furey et al., 2016):(10)
Here, Qi,j denotes peak discharge at gauge j for event i, a(i) and b(i) are OLR coefficients, Aj is the drainage area draining gauge j, and is a mean-zero random variable that represents the deviation of peak discharge at gauge j from the average linear relation, a(i)+b(i)Aj. Randomness in peak flows is entirely due to the error term. Figure 7 shows peak discharge for four RF-RO events as examples from the 51 events that Ayalew et al. (2015) analyzed. All of the terms in Eq. (10) change from one RF-RO event to another due to changes in the physical processes that produce peak flows (Ayalew, Krajewski, & Mantilla, 2014a; Ayalew et al., 2015; Ayalew, Krajewski, Mantilla, & Small, 2014b; Furey & Gupta, 2005, 2007). It can be seen that the R2 values are high, which supports the use of OLR to detect scaling in peak flow data. It is also evident that the scatter of peak discharges around the regression line changes from event to another event, which can be quantified by the standard deviation of the error term in Eq. (10). Physically, it is a manifestation of the natural variability of rainfall and other catchment physical variables that control the generation of peak discharges in space and time.
To test the validity of the classical Horton law for peak flows, four different events shown in Figure 7 were selected. Three different methods were used to test the statistical sensitivity of the estimates of the Horton ratio for peak flows, , i=1,2,3 … Since the USGS gauges are not located at the end of complete Strahler streams, which is necessary for the Horton analysis, it was assumed that each USGS stream gauge of a given order is located at the end of a complete Strahler stream of that order. The observed peak flows at these gauges for a given RF-RO event constitute a random sample of peak flows for that order. The result of this assumption produces peak flows for different Strahler streams as shown in Figure 8 with grey circles. The Horton ratio, RQ(i), for i = 1,2,3,4 was calculated for the same four events shown in Figure 7. In method 1, Eq. (9) was applied to compute RQ as the ratio of the geometric means of the peak flows at consecutive orders. The results are shown in Figure 8. In method 2, peak flow data was arranged according to the Strahler order, and OLR, given in Eq. (7), was used to compute . In method 3, the relation derived in Gupta et al. (2015) was used. The appearance of the classical Horton laws for peak flows in RF-RO events shown in Figure 8 was a new hydrologic discovery. Three estimation methods to address the small sample size issue for the peak flow data need further investigation and constitutes an important area of future research.
Generalized Horton Law for Peak Flows in Rainfall-Runoff Events
Do peak discharges at the event time scale obey a generalized Horton law? Gupta et al. (2015) investigated this issue and derived(11)
Equation (11) denotes that peak flows divided by its mean for a RF-RO event, i = 1,2,3, …, and Strahler order, . It looks like a generalized Horton law in so far as the rescaled random variable on the right is independent of the Strahler order, . It was explained in the context of geomorphology (Eq. (4)), but the probability distribution of changes with each event, . This issue does not arise for geomorphologic random variables like drainage areas. In this sense, the feature in Eq. (11) is new and is purely hydrologic in nature. Gupta et al. (2015) hypothesized that the probability distribution of is common for all the events and tested their hypothesis using data.
The cumulative distribution functions (CDF) for the normalized peak discharges for each of the 51 events was plotted, as shown by the thin lines in Figure 8. Two authors have described k-sample tests: the Kolmogorov–Smirnov (KS) (Conover, 1999) and a test based on likelihood statistics (Zhang & Wu, 2007). The condition to perform a k-sample KS test applies, since each point in a CDF corresponds to a stream gauge, which doesn’t change with events. However, the test is more powerful than the KS, and therefore it was adopted. The results indicate that there is insufficient evidence to reject the hypothesis that , which is the CDFs shown in Figure 8 by thin lines, come from a common distribution that is independent of events, , as well as of the Strahler stream order, . This is a new hydrologic discovery with important future applications with respect to flood prediction in gauged and ungauged basins (Gupta et al., 2015). One application is briefly explained next.
Applications of Horton Laws for Testing a Rainfall-Runoff Model Without Calibration of Parameters
Consider a statistical-mechanical system for drawing a formal analogy with physical processes in a river network. In a statistical-mechanical system, there are two scales: microscopic, with a very large number of degrees of freedom (NDOF) that cannot be measured, and macrosopic, with a few degrees of freedom that can be measured. A well-known example is ideal gas law for a system in equilibrium, which has only four parameters: pressure, number of molecules per unit volume, temperature, and Boltzman constant (Reif, 1965). The branch of statistical-mechanics derives macroscopic laws from assumptions about molecular dynamics at the microscopic scale.
The hydrology problem under discussion represents a similar situation, but it is far more complex than an ideal gas. In order to present the formal analogy between RF-RO processes in a river basin and the aforementioned statistical-mechanical system, Gupta (2004a, 2004b) provides a context. Solutions of the coupled ordinary differential equations representing mass and momentum conservation given in Eq. (3) require specification of dynamic parameters describing runoff generation in hillslopes and water flow dynamics in channel links. These parameters vary spatially due to differences in geometric, hydraulic, and biophysical properties among individual channel links and hillslopes, which increase with the area (size) of a drainage basin. Therefore, the number of different values that the physical parameters can take also increases as a basin becomes larger. Gupta (2004a) called it NDOF in order to underscore the idea that the hydrological complexity in a river basin can be compared to the complexity of a statistical-mechanical system that has a very large NDOF. To illustrate that the NDOF in the RF-RO system is very large, Gupta (2004a, 2004b) considered an idealized bucket-type representation of runoff generation and transport processes for a single hillslope-link system. Four physical parameters are required to specify the runoff generation for each hillslope and three parameters governing transport dynamics in a link. These dynamic parameters vary from one hillslope-link pair to another. Typical hillslope has an area of the order of 0.05 km2. Therefore, a 1 km2 basin can be partitioned into 1/0.05 = 20 hillslopes and 10 links, because each link is drained by two adjacent hillslopes. The number of different spatial values of the seven dynamic parameters is 20 · 4 + 10 · 3 = 110. This simple calculation leads to the formula NDOF = 110·A. It shows that the NDOF increases linearly with the area of a basin, A. For the Iowa River basin, NDOF ~ 3.5 million. The flood problem is far more complex than a statistical-mechanical system, which is typically in equilibrium. But the RF-RO system is an open system that is not even in a steady state.
Appearance of a large number of physical parameters at the hillslope-link scale raises two issues. First is the need to develop a theoretical approach for specifying a large number of dynamic parameters (see Furey et al., 2013), but more work is needed on this topic. The issue of diagnosis of a physical RF-RO model is considered here, because it is fundamental to flood prediction in ungauged basins. A formal analogy between microscopic and macroscopic scales in a hydrologic system allows us to illustrate the steps required to diagnose dynamic parameterizations in a hydrologic model.
First, for a given RF-RO event, estimate rainfall intensity field in space and time. Select a set of dynamic parameters and run the hydrologic model that is based on the decomposition of a river basin into hillslope and channel-link units (Mantilla, Gupta, & Troutman, 2011). Simulate a discharge hydrograph at each channel junction in a network, which gives simulated peak flows in the complete Strahler streams of order ω = 1, 2, 3 … . Then estimate using the three methods as explained in Gupta et al. (2015). The estimate of scaling slope for this event is given as . Similarly, the simulated peak flow data at each complete order Strahler stream can be used to compute the mean peak flow. The same data set can be used to compute the rescaled distribution representing the generalized Horton law as shown in Figure 9 for the 51 events. These two Horton statistics provide the macroscopic laws for diagnosing parameterizations in a RF-RO model. The final step can be repeated for different events as needed. The Horton statistics can also serve as a diagnostic tool to understand changes in peak flows over annual and longer time scales in different climates.
Climate Variability and the Scaling Theory of Floods in Medium to Large Basins of the World
How can the scaling theory of floods be generalized to global basins spanning different climates at the scale of the continents and the entire planet Earth? Poveda et al. (2007) took an important first step in this direction. Their study involved conducting annual water balances over several drainage basins for the entire country of Colombia, which has widely varying climates ranging from extremely humid to semi-arid and arid. Annual streamflows were predicted from interpolated fields of annual precipitation and evapotranspiration using ground-based and remotely sensed data, finding reasonable agreement with observed streamflows. Among many alternate models to estimate evapotranspiration, the best estimates were given by the well-known Budyko equation (Budyko, 1974). Poveda et al. (2007) showed that power laws describe the relationship between annual flood quantiles and drainage areas, and the flood scaling parameters can be expressed as functions of annual runoff obtained from the water balance. This line of investigation provides a new research direction for making flood predictions in a changing global hydro-climate system due to human influences (Pielke et al., 2009). Gupta et al. (2007) suggested specific ideas for future research.
Lima and Lall (2010) investigated the role of climate variability and change using scaling framework for floods. They developed and applied hierarchical Bayesian models to assess both regional and at-site trends in time in a spatial scaling framework and simultaneously provided a rigorous framework for assessing and reducing parameter and model uncertainties. The models were tested with reconstructed natural inflow series from more than 40 hydropower sites in Brazil with catchments areas varying from 2,588 km2 to 823,555 km2. Both annual maximum flood series and monthly streamflows were considered. Cross-validated results showed that the hierarchical Bayesian models are able to skillfully estimate monthly and flood flow probability distribution parameters for sites not used in model fitting. The models developed can be used to provide record augmentation at sites that have short records or to estimate flows at ungauged sites, even in the absence of an assumption of time stationarity. Since model uncertainties are accounted for, the precision of the estimates can be quantified and hypotheses tests for regional and at-site trends can be formally made.
Viglione, Merz, Viet Dung, Parajka, Nester, and Bloschl (2016) developed a new framework for attributing flood changes due to atmospheric processes (e.g., increasing precipitation), catchment processes (e.g., soil compaction associated with land use change), and river system processes (e.g., loss of retention volume in the floodplains) based on regional analysis. Spatial scaling of flood changes enabled them to investigate attribution of multiple drivers, and Bayesian method allowed estimation of the attribution uncertainties. Flood peak data for 97 river gauges in upper Austria, with areas ranging from 10 km2 to 79,500 km2 and records for at least 40 years after 1950, were used in a real case study to illustrate the framework.
Two studies that focus on a physical understanding of floods on the planetary scale are O’Conner, Grant, and Costa (2002) and Devineni, Lall, Xi, and Ward (2015). O’Conner et al. (2002) dealt with paleo flood hydrology, and scaling in floods is implicit in their article (see Figure 7). Devineni et al. (2015) investigated the scaling of extreme rainfall areas at the planetary scale.
Engineering Applications of the Scaling Theory of Floods
An important need is to apply the scientific framework that the scaling theory has developed for innovative applications to practical hydrologic engineering problems. Two examples from the papers that the Iowa Flood Center (IFC), University of Iowa, has published are discussed here. The IFC was established following the 2008 record floods and is the only facility of its kind in the United States and the world conducting comprehensive scientific and applied research on floods rooted in the scaling framework.
Real-Time Flood Forecasting
Krajewski et al. (2017) developed a real-time flood forecasting and information dissemination system for use by all Iowans. The system complements the operational forecasting issued by the National Weather Service. At its core the IFC forecasting model is a continuous RF-RO model based on landscape decomposition into hillslopes and channel links. Rainfall conversion to runoff is modeled through soil moisture accounting at hillslopes. Channel routing is based on a non-linear representation of water velocity that considers the discharge amount as well as the upstream drainage area. Mathematically, the model represents a large system of ordinary differential equations organized to follow river network topology. None of physical parameters are calibrated. It illustrates the impact of the scientific ideas developed in the scaling theory of floods on engineering applications.
The IFC also developed an efficient numerical solver suitable for high-performance computing architecture. The solver allows the IFC to update forecasts every 15 minutes for more than 1,000 Iowa communities. The input to the system comes from a radar-rainfall algorithm, developed in-house, that maps rainfall every five minutes with high spatial resolution. The algorithm uses Level II radar reflectivity and other polarimetric data from the WSR-88DP radar network. A large library of flood inundation maps and real-time river stage data from over 200 IFC “stream-stage sensors” complement the IFC information system. The system communicates all this information to the general public through a comprehensive browser-based and interactive platform.
Impact of Dams on Real-Time Flood Forecasting
Dams are ubiquitous in the United States, and more than 87,000 of them across the nation influence streamflows. The significant majority of these dams are small and are often ignored in real-time flood forecasting operations and at-site and regional flood frequency estimations. Even though the impact of individual small dams on floods is often limited, the combined flood attenuation effects of a system of such dams can be significant. Ayalew, Krajewski, Mantilla, Wright, and Small (2017) investigated how a system of spatially distributed small dams affect flood frequency across a range of drainage basin scales using the 660 km2 Soap Creek watershed in southeastern Iowa, which contains more than 133 small dams. Results from continuous simulation of the system of small dams indicate that the peak discharges were reduced between 20 and 70% but that this effect decreases as the drainage area increases. Considering that more small dams are being built across watersheds in Iowa and elsewhere in the country, their results highlight how the peak discharge attenuation effects of these dams is an additional factor that invalidates the stationarity assumption that is used in at-site and RFF analysis. In particular, they showed that neglecting the effects of a system of small dams can lead to an overestimation of flood risk. The results show that real-time flood forecasting that does not account for the flood attenuation effects of these dams may suffer from the overestimation of flood threats across a range of spatial scales.
Prediction of floods in ungauged basins (i.e., at locations where no streamflow data exist) is a global issue because most of the countries involved do not have adequate streamflow records. Two approaches have been developed to solve the problem. First is the RFF analysis by the USGS. RFF equations are pure statistical characterizations that use historical streamflow records in “homogeneous regions” to predict annual peak flow quantiles in ungauged basins. The second approach is the scaling theory of floods that abandoned the homogeneous regions concept and adopted river basins instead. It has the explicit goal of incorporating flood-producing physical processes in understanding the observed scaling, or power law relations, between peak flows and drainage areas in river basins. The scaling theory of floods offers a unified framework to predict floods in RF-RO events and in annual peak flow quantiles in ungauged basins.
The topics covered in the first part of the article include: (1) a brief history is presented of the scaling theory of floods; (2) dynamic formulations governing RF-RO events in river basins are described; (3) self-similar river networks (the Tokunaga model and the RSN model) are briefly reviewed; (4) applications of dynamic formulation in RSN are summarized; (5) an analytical theory of H-G in river networks is highlighted and open problems mentioned; and (6) a hydro-ecological theory of intermittent riparian diversity on Tokunaga stream networks in progress is briefly sketched.
The topics in the second part include: (1) highlights of the first paper on a physical basis of annual peak flow quantiles are presented; (2) highlights of the discovery of the classical and the generalized Horton laws for peak flows in the 32,400 km2 Iowa river basin are described. An application of the Horton laws for diagnosing a RF-RO model without calibrating its parameters is summarized; (3) generalization of the scaling theory of floods in medium to large basins and to longer time scales than annual are highlighted (two key references on scaling in rainfall and paleofloods to understand scaling in floods on the planetary scale are included); and (4) two engineering applications of scaling theory of floods in the Iowa River basin are summarized.
Several significant areas of research remain:
1. Comparative studies are needed to test different model parameterizations without calibration using the Horton laws for peak flows.
2. Studies are needed to test analytical solutions of Eq. (3) under physically realistic assumptions in realistic self-similar river networks. Ramirez (National University of Colombia, personal communication, 2017) has obtained analytical results on floods using the physical framework presented here. It is the first effort of its kind, and papers are being prepared for publication.
3. The scaling theory of floods needs to be extended to global basins representing different hydro-climatic regions ranging from very humid to semi-arid and arid.
4. The scaling framework has important relevance to flood prediction in a changing climate, because the self-similarity of river networks is not expected to change. As a result, the power laws in peak discharges at event and annual time scales would remain intact, but scaling slopes and intercepts for RF-RO events and annual flood quantiles would change, which can be predicted from physical processes.
5. Many physical processes govern floods in upstream basins and in flood plains, but all of them cannot be included in a model. A dynamic understanding of scaling can be used to eliminate those physical processes that play a “secondary role” in the overall context.
Several outstanding graduate students helped me sustain the line of research described here over several decades. My journey started in 1980 with the first graduate student, Oscar Mesa, and ended in 2007 with my last graduate student, Ricardo Mantilla. In between this time frame, several others joined the group, Tom over, Sandra Castro, Scott Peckham, Seth Veitzer, Peter Furey, and contributed to the line of research. I express my deep gratitude for their dedicated contributions. The National Science Foundation, the Army Research Office and the National Aeronautics and Space Administration funding over this time period made the research possible.
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