Michael R. Karlinger

On the expected width function for topologically random channel networks

An idealized river-channel network is represented by a trivalent planted plane tree, the root of which corresponds to the outlet of the network. A link of the network is any segment between a source and a junction, two successive junctions, or the outlet and a junction. For any x ? 0, the width of the network is the number of links with the property that the distance of the downstream junction from the outlet is = x, and the distance of the upstream junction to the outlet is > x. Expressions are obtained for the expected width conditioned on N, (N, M), and (N, D), where N is the magnitude, M the order, and D the diameter of the network, under the assumption that the network is drawn from an infinite topologically random population and the link lengths are random. NETWORKS; BRANCHING PROCESS

Averaging Properties of Channel Networks Using Methods in Stochastic Branching Theory

Methods in branching theory are used to average properties of channel networks, resulting in expressions for the instantaneous unit hydrograph (IUH) in terms of fundamental network characteristics (Z, α, β), where α parameterizes the link (channel segment) length distribution and β is a vector of hydraulic parameters. Several possibilities for Z are considered, including N, (N, D), (N, M), \(\tilde{D}\) , and (N, \(\tilde{D}\)), where N is magnitude (number of first-order streams), D is topological diameter, M is order, and \(\tilde{D}\) is mainstream length. Linear routing schemes, including translation, diffusion, and general linear routing, are used, and it is demonstrated that translation routing leads to an IUH identical to that obtained by use of the width function, where, for a given distance x, the width of a network is defined to be the number of links some point of which lies at channel distance x from the outlet (analogous to population size in branching theory).

Inference for a generaliaed gibbsian distribution channel networks

We introduce a two-parameter (β0 and β1) Gibbsian probability model to characterize the spatial behavior of channel networks. This model is defined for trees draining basins on a square lattice. The probability of a tree s is proportional to exp [−β0H(s, β1)[, where H(s,β1) is taken to be a summation over lattice points ν in the basin of A(ν, s)β1, letting A(ν, s) be the area upstream from (but not including) point ν. Procedures are developed for estimating the parameters given an actual network obtained, for example, from digital elevation data and for using bootstrapping to obtain confidence intervals for these estimates. We also develop several goodness of fit tests for the model. These procedures are applied to a set of data for 50 subnetworks from Willow Creek in Montana. The estimation algorithm converged successfully for 34 of these subnetworks. For these 34 subnetworks the two-parameter model gives in most cases a fit significantly better than a one-parameter model (with β1constrained to be 1) studied in a previous paper. There is still, however, statistically significant lack of fit for as many as 13 subnetworks (depending on which test is applied). The parameter β1 averages 0.75 for the 34 subnetworks. The function H(s, β1) in the Gibbsian model may be given a statistical mechanical interpretation of “energy” associated with a configuration s. Other researchers have employed a similar definition of energy expenditure in drainage systems but have given physical arguments that the exponent β1 should be 0.50. We finally discuss the idea of defining “optimal channel networks” as those which minimize free energy in the Gibbsian model; it is shown that minimization of energy (rather than free energy) as other researchers have proposed is valid only under the assumption of a “low temperature” (i.e., large β0) system. Our results indicate that this assumption would not be justified for Willow Creek. We devote some discussion to the discrepancies between our approach and results and those of other researchers.

Fat fractal scaling of drainage networks from a random spatial network model

An alternative quantification of the scaling properties of river channel networks is explored using a spatial network model. Whereas scaling descriptions of drainage networks previously have been presented using a fractal analysis primarily of the channel lengths, we illustrate the scaling of the surface area of the channels defining the network pattern with an exponent which is independent of the fractal dimension but not of the fractal nature of the network. The methodology presented is a fat fractal analysis in which the drainage basin minus the channel area is considered the fat fractal. Random channel networks within a fixed basin area are generated on grids of different scales. The sample channel networks generated by the model have a common outlet of fixed width and a rule of upstream channel narrowing specified by a diameter branching exponent using hydraulic and geomorphologic principles. Scaling exponents are computed for each sample network on a given grid size and are regressed against network magnitude. Results indicate that the size of the exponents are related to magnitude of the networks and generally decrease as network magnitude increases. Cases showing differences in scaling exponents with like magnitudes suggest a direction of future work regarding other topologic basin characteristics as potential explanatory variables.

Relations Between Igneous and Metamorphic Rock Fracture Patterns and Groundwater Yield from the Variography of Water–Well Yields—Pinardville Quadrangle, New Hampshire

AbstractExploratory variography was used to examine the spatial continuity of water–well yields in the Pinardville 7

A note on subtrees rooted along the primary path of a binary tree

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Unit hydrograph approximations assuming linear flow through topologically random channel networks.

Minute Quadrangle in southern New Hampshire and to link the variography to the characteristics of the fractured igneous and metamorphic rocks within the aquifer system. In addition to the analyses of variograms computed by using data from 939 wells, analyses were performed on subsets of the data that were stratified according to the level of yield and to five rock types. The stratification according to yield was defined by using the industry standard of a high-yield well that produces 40 gallons per minute or more. The stratification of the low-yield wells by rock type was defined by using the classification on the bedrock map for the State of New Hampshire. Although the variability is high in the low-yield wells, as indicated by the large nugget value, overall continuity ranges to 6000 feet in the fracture zones in which these wells have been drilled. This continuity is dominant in the northwesterly direction, as indicated by the directional variography. This result is consistent with the general trend in the larger tectonic configuration of the region. A lack of spatial correlation for 81 high-yield wells is consistent with the geologic interpretation that these wells occur in locally determined configurations of sheeting and steeply dipping fractures. Yield data in only three of the five rock types were sufficient for variography. Within these three, the dominant direction of the correlation structure ranged from northwesterly for the Massabesic Gneiss Complex to northeasterly for the Rangeley Formation to northerly for the Spaulding Granite, where the signature of the continuity in the low-yield wells is predominately attributed to the fracture system.

Surface water network design by regression analysis simulation

The spatial continuity of water-well yields is associated with the fracture properties of the crystalline basement rocks in the Pinardville quadrangle, New Hampshire. The analysis reported in this paper expands upon previous work of a study of 939 wells in the quadrangle. This expanded analysis was performed on updated and new geologic map data, new fracture-set data and analyses, and a set of water-well yields that was expanded by 752 newly located wells. Variogram mapping of the water-well yields formed the basis for the synthesis of the spatial continuity in well yield with the geological data. A variety of associations were determined that directly relate the spatial properties of well yields to the characteristics and structure of the crystalline bedrocks in the quadrangle.