David G. Tarboton

A new method for the determination of flow directions and upslope areas in grid digital elevation models

A new procedure for the representation of flow directions and calculation of upslope areas using rectangular grid digital elevation models is presented. The procedure is based on representing flow direction as a single angle taken as the steepest downward slope on the eight triangular facets centered at each grid point. Upslope area is then calculated by proportioning flow between two downslope pixels according to how close this flow direction is to the direct angle to the downslope pixel. This procedure offers improvements over prior procedures that have restricted flow to eight possible directions (introducing grid bia) or proportioned flow according to slope (introducing unrealistic dispersion). The new procedure is more robust than prior procedures based on fitting local planes while retaining a simple grid based structure. Detailed algorithms are presented and results are demonstrated through test examples and application to digital elevation data sets.

A physical basis for drainage density

Abstract Drainage density, a basic length scale in the landscape, is recognized to be the transition point between scales where unstable channel-forming processes yield to stable diffusive processes. This notion is examined in terms of equations for the evolution of landscapes that include the minimum necessary mathematical complexity. The equations, a version of the equations studied by Smith and Bretherton (1972), consist of convervation of sediment, an assumption that sediment movement is in the steepest downslope direction, and a constitutive relationship which gives the sediment transport rate as a function of slope and upslope area. The difference between processes is embedded in the constitutive relation. Instability to a small perturbation can be determined according to a criteria given by Smith and Bretherton and results when the sediment transport rate is strongly dependent on upslope area, whereas stability occurs if the main dependence is on slope. Where multiple processes are present, the transition from stability to instability occurs at a particular scale. Based on the idea that instability leads to channelization, the transition scale gives the drainage density. This scale can be determined as a maximum, or turn over point in a slope-area scaling function, and can be used practically to determine drainage density from digital elevation data. Fundamentally different scaling behavior, an example of which is the slope-area scaling, is to be expected in the stable and unstable regimes below and above the basic scale. This could explain the scale-dependent fractal dimension measurements that have been reported by others.

Streamflow simulation: A nonparametric approach

In this paper kernel estimates of the joint and conditional probability density functions are used to generate synthetic streamflow sequences. Streamflow is assumed to be a Markov process with time dependence characterized by a multivariate probability density function. Kernel methods are used to estimate this multivariate density function. Simulation proceeds by sequentially resampling from the conditional density function derived from the kernel estimate of the underlying multivariate probability density function. This is a nonparametric method for the synthesis of streamflow that is data-driven and avoids prior assumptions as to the form of dependence (e.g., linear or nonlinear) and the form of the probability density functions (e.g., Gaussian). We show, using synthetic examples with known underlying models, that the nonparametric method presented is more flexible than the conventional models used in stochastic hydrology and is capable of reproducing both linear and nonlinear dependence. The effectiveness of this model is illustrated through its application to simulation of monthly streamflow from the Beaver River in Utah.

Power law distributions of discharge mass and energy in river basins

River networks constitute dissipative systems with many spatial degrees of freedom. Previous work by Mandelbrot (1983) and Bak et al. (1987, 1988, 1990) suggests that such systems will follow power law distributions in their mass and energy characteristics. It is shown that this is the case for river networks where the exponent β in the distribution, P[X > x] ∝ x−β, is approximately equal to 0.45 and 0.90 for discharge and energy respectively in the case of several networks analyzed in North America when these variables are calculated for each individual link throughout the drainage network. An explanation of the values of β is offered based on the fractal structure of rivers and on principles of energy expenditure in river basins.

The Influence of the Spatial Distribution of Snow on Basin-Averaged Snowmelt

Spatial variability in snow accumulation and melt owing to topographic effects on solar radiation, snow drifting, air temperature and precipitation is important in determining the timing of snowmelt releases. Precipitation and temperature effects related to topography affect snowpack variability at large scales and are generally included in models of hydrology in mountainous terrain. The effects of spatial variability in drifting and solar input are generally included only in distributed models at small scales. Previous research has demonstrated that snowpack patterns are not well reproduced when topography and drifting are ignored, implying that larger scale representations that ignore drifting could be in error. Detailed measurements of the spatial distribution of snow water equivalence within a small, intensively studied, 26-ha watershed were used to validate a spatially distributed snowmelt model. These observations and model output were then compared to basin-averaged snowmelt rates from a single-point representation of the basin, a two-region representation that captures some of the variability in drifting and aspect and a model with distributed terrain but uniform drift. The model comparisons demonstrate that the lumped, single-point representation and distributed terrain with uniform drift both yielded poor simulations of the basin-averaged surface water input rate. The two-point representation was a slight improvement, but the late season melt required for the observed stream-flow was not simulated because the deepest drifts were not represented. These results imply that representing the effects of subgrid variability of snow drifting is equally or more important than representing subgrid variability in solar radiation.

On Hack's Law

Hacks law is reviewed, emphasizing its implications for the elongation of river basins as well as its connections with their fractal characteristics. The relation between Hacks law and the internal structure of river basins is investigated experimentally through digital elevation models. It is found that Hacks exponent, elongation, and some relevant fractal characters are closely related. The self-affine character of basin boundaries is shown to be connected to the power law decay of the probability of total contributing areas at any link and to Hacks law. An explanation for Hacks law is derived from scaling arguments. From the results we suggest that a statistical framework referring to the scaling invariance of the entire basin structure should be used in the interpretation of Hacks law.

An integrated system for publishing environmental observations data

Over the next decade, it is likely that science and engineering research will produce more scientific data than has been created over the whole of human history. The successful use of these data to achieve new scientific breakthroughs will depend on the ability to access, integrate, and analyze these large datasets. Robust data organization and publication methods are needed within the research community to enable data discovery and scientific analysis by researchers other than those that collected the data. We present a new method for publishing research datasets consisting of point observations that employs a standard observations data model populated using controlled vocabularies for environmental and water resources data along with web services for transmitting data to consumers. We describe how these components have reduced the syntactic and semantic heterogeneity in the data assembled within a national network of environmental observatory test beds and how this data publication system has been used to create a federated network of consistent research data out of a set of geographically decentralized and autonomous test bed databases.

Sub-grid parameterization of snow distribution for an energy and mass balance snow cover model

Representation of sub-element scale variability in snow accumulation and ablation is increasingly recognized as important in distributed hydrologic modelling, Representing sub-grid scale variability may be accomplished through numerical integration of a nested grid or through a lumped modelling approach. We present a physically based model of the lumped snowpack mass and energy balance applied to a 26-ha rangeland catchment with high spatial variability in snow accumulation and melt. Model state variables are snow-covered area average snow energy content (U), the basin-average snow water equivalence (W a ), and snow-covered area fraction (A f ). The energy state variable is evolved through an energy balance. The snow water equivalence state variable is evolved through a mass balance, and the area state variable is updated according to an empirically derived relationship, A f (W a ), that is similar in nature to depletion curves used in existing empirical basin snowmelt models. As snow accumulates, the snow covered area increases rapidly. As the snowpack ablates, A f decreases as W a decreases. This paper shows how the relationship A f (W a ) for the melt season can be estimated from the distribution of snow water equivalence at peak accumulation in the area being modelled. We show that the depletion curve estimated from the snow distribution of peak accumulation at the Upper Sheep Creek sub-basin of Reynolds Creek Experimental Watershed compares well against the observed depletion data as well as modelled depletion data from an explicit spatially distributed energy balance model. Comparisons of basin average snow water equivalence between the lumped model and spatially distributed model show good agreement, Comparisons to observed snow water equivalence show poorer but still reasonable agreement. The sub-grid parameterization is easily portable to other physically based point snowmelt models. It has potential application for use in hydrologic and climate models covering large areas with large model elements, where a computationally inexpensive parameterization of sub-grid snow processes may be important.

Disaggregation procedures for stochastic hydrology based on nonparametric density estimation

Synthetic simulation of streamflow sequences is important for the analysis of water supply reliability. Disaggregation models are an important component of the stochastic streamflow generation methodology. They provide the ability to simulate multiseason and multisite streamflow sequences that preserve statistical properties at multiple timescales or space scales. In recent papers we have suggested the use of nonparametric methods for streamflow simulation. These methods provide the capability to model time series dependence without a priori assumptions as to the probability distribution of streamflow. They remain faithful to the data and can approximate linear or nonlinear dependence. In this paper we extend the use of nonparametric methods to disaggregation models. We show how a kernel density estimate of the joint distribution of disaggregate flow variables can form the basis for conditional simulation based on an input aggregate flow variable. This methodology preserves summability of the disaggregate flows to the input aggregate flow. We show through applications to synthetic data and streamflow from the San Juan River in New Mexico how this conditional simulation procedure preserves a variety of statistical attributes.

A Nonparametric Wet/Dry Spell Model for Resampling Daily Precipitation

A nonparametric wet/dry spell model is developed for resampling daily precipitation at a site. The model considers alternating sequences of wet and dry days in a given season of the year. All marginal, joint, and conditional probability densities of interest (e.g., dry spell length, wet spell length, precipitation amount, and wet spell length given prior to dry spell length) are estimated nonparametrically using at-site data and kernel probability density estimators. Procedures for the disaggregation of wet spell precipitation into daily precipitation and for the generation of synthetic sequences are proffered. An application of the model for generating synthetic precipitation traces at a site in Utah is presented.