David A. Woolhiser

Linearity of basin response as a function of scale in a semiarid watershed

Linearity of basin runoff and peak response as a function of watershed scale was examined for a set of 29 nested semiarid watersheds within the U.S. Department of Agriculture–Agricultural Research Service Walnut Gulch Experimental Watershed, located in southeastern Arizona. Watershed drainage areas range from 1.83 × 103 to 1.48 × 108 m2 (0.183–14800 ha), and all stream channels are ephemeral. Observations of mean annual runoff, database-derived 2- and 100-year peak runoff rates, ephemeral channel area, and areal rainfall characteristics derived from 304 events were examined to assess the nature of runoff response behavior over this range of watershed scales. Two types of distributed rainfall-runoff models of differing complexity were applied to a subset of the watersheds to further investigate the scale-dependent nature of the collected data. Contrary to the conclusions of numerous studies in more humid regions, it was found that watershed runoff response becomes more nonlinear with increasing watershed scale, with a critical transition threshold area occurring roughly around the range of 3.7 × 105 to 6.0 × 105 m2 (37–60 ha). The primary causes of increasingly nonlinear response are the increasing importance of ephemeral channel losses and partial storm area coverage. The modeling results indicate that significant error will result in model estimates of peak runoff rates when rainfall inputs from depth area-frequency relationships are applied beyond the area of typical storm coverage. For runoff modeling in Walnut Gulch and similar semiarid environments, explicit treatment of channel routing and transmission losses from channel infiltration will be required for watersheds larger than the critical drainage area.

Disaggregation of Daily Rainfall

Abstract A parameter-efficient model for disaggregating daily rainfall into individual storms is presented. This model allows simulation of the number of rainfall events (storms) in a day, and the amount, duration, and starting time of each event, given only the total rainfall on that day and on the preceding and following days. Twenty-three years of data for July and August, from a gage on the Walnut Gulch Experimental Watershed, were used to find the appropriate model structure and to estimate parameters. Statistical tests indicate that simulated sequences of storms compare favorably with observed sequences, and that the disaggregation model structure and parameters identified for one gage provide a satisfactory fit for three stations within a 121 km radius where elevation differs by as much as 244 m, and mean annual rainfall differs by up to 76 mm.

Effect of storm rainfall intensity patterns on surface runoff

Abstract The adoption of physically-based infiltration models as components of watershed models has been impeded because they require rainfall data with high temporal and spatial resolution, and the spatial variability of infiltration model parameters must be accounted for. Techniques have been developed to disaggregate daily rainfall into the intermittent shower process within the day, and further disaggregate significant showers into short-period rainfall hyetographs. A simple model describing infiltration and unsteady flow over a plane and a single channel is used to investigate the sensitivity of derived distributions of runoff volume and peak flow rates to input differences due to different rainfall disaggregation methods and parameters and to spatial variability of infiltration parameters. The Woolhiser and Osborn disaggregation scheme is superior to simpler forms of disaggregation for all but a highly damped system. For the elementary watershed considered, the channel has little effect on basis response in comparison to overland flow characteristics. A significant interaction between climate and spatially variable infiltration and its effects on response was also discovered and is discussed.

Southern oscillation effects on daily precipitation in the southwestern United States

The effect of the Southern Oscillation on daily precipitation in the southwestern United States is examined by using the Southern Oscillation Index (SOI) to perturb parameters of a stochastic daily precipitation model. Daily precipitation is modeled with a Markov chain-mixed exponential model and seasonal variability of model parameters is described by Fourier series. The hypothesized linkage between the SOI and the model parameters is of the form Gi(N, t) = Gi(t) + biS(N, t − τi) where Gi(N, t) is the perturbed parameter i for day t of year N, Gi(t) is the annually periodic parameter i for day t, bi is a coefficient, S is the SOI, and τi is a lag in days. Daily precipitation data for 27 stations in California, Nevada, Arizona, and New Mexico were analyzed. Perturbations of the logits of the dry-dry transition probabilities resulted in statistically significant improvements in the log likelihood function for 23 stations and perturbations of the mean daily rainfall resulted in significant increases for 18 stations. The most common lag identified was 90 days, suggesting the possibility of conditional simulations of daily precipitation. Seasonal effects were detected, confirming the results of previous analysis with groups of stations.

Estimation of spatial recharge distribution using environmental isotopes and hydrochemical data, I. Mathematical model and application to synthetic data

Abstract A mathematical model is proposed to estimate the spatial distribution of annual recharge rates into an aquifer using environmental isotopes and hydrochemical data. The aquifer is divided into cells within which the isotopes and dissolved constituents are assumed to undergo complete mixing. For each mixing cell mass-balance equations expressing the conservation of water, isotopes and dissolved chemicals are written. These equations are solved simultaneously for unknown rates of recharge into the various cells by quadratic programming. The degree to which individual dissolved constituents may be considered conservative is tested a-priori by means of a chemical equilibrium model such as wateqf . Constituents which do not pass this test are either disregarded or suitably assigned a small weight in the quadratic program. In Part I, the model is applied to synthetic data corrupted by random noise and its sensitivity to input errors is examined. Part II ∗ describes an application of the model to real data from the Aravaipa Valley in southern Arizona.

Kinematic routing using finite elements on a triangular irregular network

Automated extraction of geometry for hydraulic routing from digital elevation models (DEM) is a procedure that must be easily accomplished for widespread application of distributed hydraulically based rainfall excess-runoff models. One-dimensional kinematic routing on a regular grid DEM is difficult due to flow division and convergence. Two-dimensional kinematic routing on a triangular irregular network (TIN) surmounts many of these difficulties. Because TIN DEMs typically require far fewer points to represent topography than regular grid DEMs, substantial computational economy is also realized. One-dimensional routing using vector contour data overcomes the grid-based routing disadvantages but often requires several orders of magnitude more storage points than a TIN. The methodology presented in this paper represents a compromise between slightly increased computational complexity and the economy of TIN topographic representation. We take the unique approach of subdividing each topographic triangle (TIN facet) into a set of coplanar triangular finite elements, performing routing on a single facet and then routing the resulting excess hydrograph to downstream facets and channels via upstream boundary conditions. Results indicate that shock conditions are readily handled, computed depths match analytic results to within ±3% and volume balances are typically within 1%. This modeling system illustrates the viability of kinematic routing over a TIN DEM derived directly from digital mapping data.

Simplifications of watershed geometry affecting simulation of surface runoff

Abstract In formulating the equations describing the flow of water on the surface of a watershed, geometric simplifications must be made. A geometric simplification is the substitution of a simple geometry for a more complex one. The problem is to examine techniques for and consequences of such simplifications, and thereby develop objective procedures for geometric simplification of complex watersheds. Watershed geometry is represented by a series of planes and channels in cascade. When overland flow and open-channel flow in the cascade are described by the kinematic wave equations, the resulting mathematical model is called the kinematic cascade model. Planes are fitted to coordinate data from topographic maps by a least-squares procedure. Residuals of this fit form a geometric goodness-of-fit statistic as the improvement over using the mean elevation. Channel elements are determined, using Grays method, as the slope of the hypotenuse of a right triangle with the same area as that under the observed stream profile. The ratio of the altitude of this right triangle to the total relief of a stream is the index of concavity, a channel goodness-of-fit statistic. An overall goodness-of-fit statistic is the drainage density ratio, the ratio of drainage density in the cascade of planes and channels to drainage density of the watershed. The mean value of a hydrograph goodness-of-fit statistic, as the improvement over using the mean discharge, increases as the geometric goodness-of-fit statistic increases but also decreases as the drainage density increases. A combined goodness-of-fit statistic, the product of the drainage density ratio and the geometric goodness-of-fit statistic, is related to the degree of distortion in optimal-hydraulic roughness parameters. Distortions in watershed geometry result in optimal roughness parameters smaller than the corresponding empirically derived values for simple watersheds where less distortion is involved. Given rainfall, runoff and topographic data for a small watershed, it is possible to define the simplest kinematic cascade geometry which when used in simulation will, on the average, preserve selected hydrograph characteristics to given degree of accuracy.

Wet-dry cycle effects on warm-season grass seedling establishment.

A series of 14d8y field experiments were conducted to evahtatt seedling estrblishment ch8r8cteristics of Bou&ka, ErogrostLr, md Punicum grass specks with controlled wet-dry w8tering combinations. The objective of the study ~8s to vrlid8te previously published greenhouse d8tr of Frrsier et 8l. (1985) on the effects of the first wet-dry wnitering sequence followhtg pl8nting on seedling emergence urd surviv8i. &ding surviv8i numbers were different between the field 8nd greenhouse experiments but the s8me genenl responses to w8tering sequences were measured. With short wet periods (2 drys), seeds gener8lly did not germimte but survived the subsequent dry period as vi8ble seeds. Most seeds genuin8ted with 5 Wet d8ys urd produced seedlings thit were 8ble to survive drought periods of 5 to 7 d8ys. Fewer seedlings survived with 3 drys wet th8n with either 2 or 5 d8ys wet. High rites of soil moisture ev8pontion in 8 spring fleid experiment mide it difficult to m8int8in adequ8te soil mOi!%turc for seed germin8tion, 8ad seeds which germhmted f&d to produce seedlings. Seedlings were successfully est8blished in 2 experiments conducted Lter in the summer following the onset of summer rrins, which incre8sed the rel8tive humidity 8nd reduced the nte of soil moisture evrporntion. This effect w8s verified in 8 greenhouse study. In both the greenhouse 8nd 5eld experiments, seedlings were est8blished when the reiative humidity exceeded 59% for over one-half of the time during the initirl wet-dry period.

Emergence and Survival Response of Seven Grasses for Six Wet-Dry Sequences

A greenhouse study was conducted to determine seedling emergence and survival responses of 7 warm-season grasses to 6 combinations of initial wet-day and dryday water sequences. Two factors which affected the number of seedlings that survived the first wet-dry watering sequence following planting were: (1) the number of seedlings produced in the first wet period which developed sufficient vigor to survive the subsequent drought or dry period, and (2) the number of ungerminated but viable seeds which remain after the first wet-dry watering sequence. Sideoats grama (Bouteloua curtipendzda (Michx.) Torr.) seedlings emerged within 18 h of the initial wetting, with maximum numbers occurring on days 2 and 3. There was a high seedling mortality during the dry periods. ‘Cochise’ lovegrass (Eragrostis lehmanniana Nees X E. trichophora Coss and Dur.), ‘Catalina’ Boer lovegrass (E. curvula var. conferta (Schrad.) Nees), and ‘A-130’ and ‘SDT’ blue panicgrass (Panicum antidotale Retz) emerged on day 2 or later, and maximum seedling counts occurred on days 4 to 6. ‘A-68’ Lehmann lovegrass (E. lehmanniana Nees) and ‘A-84’Boer lovegrass did not have significant emergence until there were 3 or more consecutive wet days. Seedling mortality, during dry periods of 2 to 7 days following initial wetting, ranged from 0 to 70% of the viable seeds. Survival characteristics of the grasses were not directly affected by total water loss. There were differences within varieties of the same species, and some grasses were better suited for surviving short term droughts during early seedling stages. These studies provided information showing how the survival characteristics of plants to the first wet-dry watering sequence can be used to assist in selecting species for range revegetation. The timing and quantity of precipitation immediately following seeding are 2 factors which significantly affect the success of rangeland revegetation efforts. Cox and Jordan (1983) found that the quantity and frequency of first-year growing-season precipitation was a major factor in explaining plant densities and forage production measured 11 years after planting in southeastern Arizona. McLean and Wikeem (1983), in British Columbia, found that the persistence of available soil moistureat the time of seeding was the most important single factor in seedling establishment. Frasier et al. (1984) showed that the initial germination and seedling survival of sideoats grama (Boureloua curtipendulu (Michx.) Torr.) and ‘Cochise’ lovegrass (Eragrostis lehmanniana Nees X E. trichophoru Coss and Dur.) was directly affected by the relative lengths of the first wet and dry periods following seeding. They demonstrated how a basic understanding of plant-water relations and plant responses could be combined with probability models of the occurrences of natural precipitation-drought combinations to Authorsare research hydraulicengineer, Southwest Rangeland Watershed Research Center; range scientist, Aridland Ecosystem Improvement; and research hydraulic engineer, Southwest Rangeland Watershed Research Center, respectively, USDAARS, 2000 East Allen Road, Tucson Ark 85719. This oaoer was oresented in aart at the 1983 annual meetinn of the Societv for Range kakgeme~t in Albuque;que. N. Mex., February 16 18,1%3 under the t&e of “Drought Effects on the Germination and Seedling Emergence of Selected Warmseason Grasses.” Manuscript accepted January 28, 1985. develop a description of the seedling environment to guide in selecting the optimum time for seeding. Research Hypothesis: To use the concept of combining rainfall occurrence probabilities and seedling survival characteristics, it is first necessary to develop an understanding of the seedling response characteristics of various plant species to the initial wet and dry periods following seeding. This study was to determine if there are differences in the means of live seedlings between species or accessions and watering sequences. There are several possible response alternatives for seed germination, seedling emergence following the initial wet period. If the first wet period is short, the seeds may not germinate and will survive the wet-dry watering sequence as viable seeds. If the wet period is long enough to germinate most seeds, and is followed by a long dry period, many, if not all, of the plants will die. If the first wet period is long enough for the seedlings to develop a root system and plant vigor sufficient to survive a drought-induced quiescence, a high percentage of the plants might survive a long drought period. For a wet period of length LI, followed by a dry period of length LO, there will be a particular response in the number of viable seeds, viable seedlings, and dead seeds or seedlings at the end of the drought period. If N(t), t = 0, I, 2, . . . , 14 signifies the number of live seedlings observed on an area on day t, the outcome of an experiment of length t = 14can be described by the random vector (nt, ns, . . . , n14), N(0) = no = 0; N(t) 5 m where m = number of pure live seeds planted. A completed description of the process N(t) requires that the multivariate distribution function F(nl,nz,...,nlr)=P{N(I)In~,N(2)5n~,...,N(14)1nlr) (1) be specified. With 7 grasses and 6 water treatments, samples are taken from 42 different 14-variate populations. The hypothesis to be examined is that the parent populations are identical. Hypothesis testing in multivariate analysis is confined to the multivariate normal case (Kendall et al. 1983). Distributions within this study are discrete, not multivariate normal, which makes it necessary to modify the approach. The usual procedure is to consider a single random variable and to test hypotheses regarding one particular parameter, such as the mean. This simplification is justified, since the study is primarily exploratory, rather than confirmatory (Tukey 1977). A typical sample function of the process N(t) is shown (Figure 1) for a specified wet-dry sequence oft, wet days followed by a dry period of td tw days. For each experiment, the following discrete random variables can be identified: (1) N,.,, the maximum number of emerged seedlings during, or following, the first wet period, (2) Lx, the number of days from planting till N,, was observed, (3) N,,,h, the minimum number of live seedlings observed after tmax, and (4) N( 14), the number of live seedlings on day 14, the end of the experiment. By inference, L = N,., N,r,, which is the minimum number of seedlings that died during the study, while JOURNAL OF RANGE MANAGEMENT 38(4), July 1985

Hydrologic and Watershed Modeling-State of the Art

P RESENT concern of society for environmental quality requir s con sideration of water as a transporting medium for pollutants. Symbolic hydro logic models provide a quantitative, mathematical description of the trans port processes within a watershed. A conceptual model of a watershed as a continuous system in three space dimen sions is presented. Examples are given of mathematical formulations of this mod el as a distributed system (partial differ ential equations) and as a lumped sys tem (ordinary differential equations). The structure 0f several currently used watershed models is examined briefly. Penman (1961) proposed a concise definition of hydrology in the form of the question: “What happens to the rain?” This is the general question we are trying to answer through the use of material or symbolic watershed models. As society becomes more aware of the environmental problems that may result from man’s activities on a watershed, we must direct our activities toward an swering the questions: “What happens to the fertilizer?” or “What happens to the pesticides?” Because nutrients or pesticides may be carried by running water or may be adsorbed by sediments transported by runoff, the last two questions can only be answered through the use of hydrologic models. A watershed is an extremely compli cated natural system that we cannot hope to understand in all detail. There fore abstraction is necessary if we are to understand or control some aspects of watershed behavior. Abstraction con sists in replacing the watershed under consideration with a model of similar but simpler structure. There are two classes of models: material and symbol ic. (R.osenblueth and Wiener, 1945). A material model is the representation of the real system by another system that is assumed to have similar properties but is not as complicated or difficult to work with. A symbolic model is a mathematical description of an idealized situation that shares some of the struc tural properties of the real system. Material models include the iconic or “look alike” models and analog models. For example, lysimeters or rainfall simu lators can be classified as material mod els. Symbolic or mathematical models are sometimes subdivided into theoreti cal models and empirical models. This is a rather arbitrary subdivision because one man’s empiricism may be another man’s theory. However, the point can be made that an empirical model merely presents the facts—it is a representation of the data. If conditions change, it has no predictive capabilities. The theoreti cal model, on the other hand, has a logical structure similar to the real world system and may be helpful under changed circumstances. All theoretical models simplify and therefore are more or less wrong. All empirical relationships have some chance of being fortuitous and in prin ciple should not be applied outside the range of data from which they were obtained. Both types of models are useful, but in somewhat different cir cumstances. If anyone wishes to develop a model to aid in understanding a process, they should choose the theoretical model. However, if they wish to make a deci sion based upon answers obtained by using the model, the choice is not necessarily obvious. For example, engi neering models contain components de rived from the social science of econom ics as well as physically based compo nents. Because the objective of engineer ing design is stated in economic terms, the physical fidelity of the model com ponents is irrelevant. Net benefits of any project are a function of design costs. Therefore if an empirical compo nent gave equal accuracy at a lower cost, it would be preferred to a theoreti cal model. Symbolic models may be classified further as lumped or distributed, sto chastic or deterministic. In general, a lumped model can be represented by an ordinary differential equation or a series 0f linked ordinary differential equa tions. A distributed model includes spa tial variations in the inputs, parameters and dependent variables and consists of a partial differenital equation or linked partial differential equations. Stochastic models describe processes occurring in time governed by certain probability laWs. A model is determinis tic if when the initial conditions, bound ary conditions and inputs are specified, the output is known with certainty. The purpose of this paper is to briefly review currently used watershed models and to examine how they might be used in understanding and predicting transport of pesticides, plant nutrients or other substances that might affect water quality. Other important aspects of the environment such as scenic beauty are not considered because hydrologic modeling does not seem to be directly involved in their evaluation.