Donald L. Phillips

A Statistical-Topographic Model for Mapping Climatological Precipitation over Mountainous Terrain

Abstract The demand for climatological precipitation fields on a regular grid is growing dramatically as ecological and hydrological models become increasingly linked to geographic information systems that spatially represent and manipulate model output. This paper presents an analytical model that distributes point measurements of monthly and annual precipitation to regularly spaced grid cells in midlatitude regions. PRISM (Precipitation-elevation Regressions on Independent Slopes Model) brings a combination of climatological and statistical concepts to the analysis of orographic precipitation. Specifically, PRISM 1) uses a digital elevation model (DEM) to estimate the “orographic” elevations of precipitation stations; 2) uses the DEM and a windowing technique to group stations onto individual topographic facets; 3) estimates precipitation at a DEM grid cell through a regression of precipitation versus DEM elevation developed from stations on the cells topographic facet; and 4) when possible, calculates...

Source partitioning using stable isotopes: coping with too many sources

Stable isotopes are increasingly being used as tracers in environmental studies. One application is to use isotopic ratios to quantitatively determine the proportional contribution of several sources to a mixture, such as the proportion of various pollution sources in a waste stream. In general, the proportional contributions of n+1 different sources can be uniquely determined by the use of n different isotope system tracers (e.g., δ13C, δ15N, δ18O) with linear mixing models based on mass balance equations. Often, however, the number of potential sources exceeds n+1, which prevents finding a unique solution of source proportions. What can be done in these situations? While no definitive solution exists, we propose a method that is informative in determining bounds for the contributions of each source. In this method, all possible combinations of each source contribution (0–100%) are examined in small increments (e.g., 1%). Combinations that sum to the observed mixture isotopic signatures within a small tolerance (e.g., ±0.1‰) are considered to be feasible solutions, from which the frequency and range of potential source contributions can be determined. To avoid misrepresenting the results, users of this procedure should report the distribution of feasible solutions rather than focusing on a single value such as the mean. We applied this method to a variety of environmental studies in which stable isotope tracers were used to quantify the relative magnitude of multiple sources, including (1) plant water use, (2) geochemistry, (3) air pollution, and (4) dietary analysis. This method gives the range of isotopically determined source contributions; additional non-isotopic constraints specific to each study may be used to further restrict this range. The breadth of the isotopically determined ranges depends on the geometry of the mixing space and the similarity of source and mixture isotopic signatures. A sensitivity analysis indicated that the estimated ranges vary only modestly with different choices of source increment and mass balance tolerance parameter values. A computer program (IsoSource) to perform these calculations for user-specified data is available at http://www.epa.gov/wed/pages/models.htm.

Uncertainty in source partitioning using stable isotopes

Stable isotope analyses are often used to quantify the contribution of multiple sources to a mixture, such as proportions of food sources in an animals diet, or C3 and C4 plant inputs to soil organic carbon. Linear mixing models can be used to partition two sources with a single isotopic signature (e.g., δ13C) or three sources with a second isotopic signature (e.g., δ15N). Although variability of source and mixture signatures is often reported, confidence interval calculations for source proportions typically use only the mixture variability. We provide examples showing that omission of source variability can lead to underestimation of the variability of source proportion estimates. For both two- and three-source mixing models, we present formulas for calculating variances, standard errors (SE), and confidence intervals for source proportion estimates that account for the observed variability in the isotopic signatures for the sources as well as the mixture. We then performed sensitivity analyses to assess the relative importance of: (1) the isotopic signature difference between the sources, (2) isotopic signature standard deviations (SD) in the source and mixture populations, (3) sample size, (4) analytical SD, and (5) the evenness of the source proportions, for determining the variability (SE) of source proportion estimates. The proportion SEs varied inversely with the signature difference between sources, so doubling the source difference from 2‰ to 4‰ reduced the SEs by half. Source and mixture signature SDs had a substantial linear effect on source proportion SEs. However, the population variability of the sources and the mixture are fixed and the sampling error component can be changed only by increasing sample size. Source proportion SEs varied inversely with the square root of sample size, so an increase from 1 to 4 samples per population cut the SE in half. Analytical SD had little effect over the range examined since it was generally substantially smaller than the population SDs. Proportion SEs were minimized when sources were evenly divided, but increased only slightly as the proportions varied. The variance formulas provided will enable quantification of the precision of source proportion estimates. Graphs are provided to allow rapid assessment of possible combinations of source differences and source and mixture population SDs that will allow source proportion estimates with desired precision. In addition, an Excel spreadsheet to perform the calculations for the source proportions and their variances, SEs, and 95% confidence intervals for the two-source and three-source mixing models can be accessed at http://www.epa.gov/wed/pages/models.htm.

A niche for isotopic ecology

Fifty years ago, GE Hutchinson defined the ecological niche as a hypervolume in n-dimensional space with environmental variables as axes. Ecologists have recently developed renewed interest in the concept, and technological advances now allow us to use stable isotope analyses to quantify these niche dimensions. Analogously, we define the isotopic niche as an area (in δ-space) with isotopic values (δ-values) as coordinates. To make isotopic measurements comparable to other niche formulations, we propose transforming δ-space to p-space, where axes represent relative proportions of isotopically distinct resources incorporated into an animals tissues. We illustrate the isotopic niche with two examples: the application of historic ecology to conservation biology and ontogenetic niche shifts. Sustaining renewed interest in the niche requires novel methods to measure the variables that define it. Stable isotope analyses are a natural, perhaps crucial, tool in contemporary studies of the ecological niche.

Incorporating concentration dependence in stable isotope mixing models

Stable isotopes are often used as natural labels to quantify the contributions of multiple sources to a mixture. For example, C and N isotopic signatures can be used to determine the fraction of three food sources in a consumers diet. The standard dual isotope, three source linear mixing model assumes that the proportional contribution of a source to a mixture is the same for both elements (e.g., C, N). This may be a reasonable assumption if the concentrations are similar among all sources. However, one source is often particularly rich or poor in one element (e.g., N), which logically leads to a proportionate increase or decrease in the contribution of that source to the mixture for that element relative to the other element (e.g., C). We have developed a concentration-weighted linear mixing model, which assumes that for each element, a sources contribution is proportional to the contributed mass times the elemental concentration in that source. The model is outlined for two elements and three sources, but can be generalized to n elements and n+1 sources. Sensitivity analyses for C and N in three sources indicated that varying the N concentration of just one source had large and differing effects on the estimated source contributions of mass, C, and N. The same was true for a case study of bears feeding on salmon, moose, and N-poor plants. In this example, the estimated biomass contribution of salmon from the concentration-weighted model was markedly less than the standard model estimate. Application of the model to a captive feeding study of captive mink fed on salmon, lean beef, and C-rich, N-poor beef fat reproduced very closely the known dietary proportions, whereas the standard model failed to yield a set of positive source proportions. Use of this concentration-weighted model is recommended whenever the elemental concentrations vary substantially among the sources, which may occur in a variety of ecological and geochemical applications of stable isotope analysis. Possible examples besides dietary and food web studies include stable isotope analysis of water sources in soils, plants, or water bodies; geological sources for soils or marine systems; decomposition and soil organic matter dynamics, and tracing animal migration patterns. A spreadsheet for performing the calculations for this model is available at http://www.epa.gov/wed/pages/models.htm.

Combining sources in stable isotope mixing models: alternative methods

Stable isotope mixing models are often used to quantify source contributions to a mixture. Examples include pollution source identification; trophic web studies; analysis of water sources for soils, plants; or water bodies, and many others. A common problem is having too many sources to allow a unique solution. We discuss two alternative procedures for addressing this problem. One option is a priori to combine sources with similar signatures so the number of sources is small enough to provide a unique solution. Aggregation should be considered only when isotopic signatures of clustered sources are not significantly different, and sources are related so the combined source group has some functional significance. For example, in a food web analysis, lumping several species within a trophic guild allows more interpretable results than lumping disparate food sources, even if they have similar isotopic signatures. One result of combining mixing model sources is increased uncertainty of the combined end-member isotopic signatures and consequently the source contribution estimates; this effect can be quantified using the IsoError model (http://www.epa.gov/wed/pages/models/isotopes/isoerror1_04.htm). As an alternative to lumping sources before a mixing analysis, the IsoSource mixing model (http://www.epa.gov/wed/pages/models/isosource/isosource.htm) can be used to find all feasible solutions of source contributions consistent with isotopic mass balance. While ranges of feasible contributions for each individual source can often be quite broad, contributions from functionally related groups of sources can be summed a posteriori, producing a range of solutions for the aggregate source that may be considerably narrower. A paleohuman dietary analysis example illustrates this method, which involves a terrestrial meat food source, a combination of three terrestrial plant foods, and a combination of three marine foods. In this case, a posteriori aggregation of sources allowed strong conclusions about temporal shifts in marine versus terrestrial diets that would not have otherwise been discerned.

Mixing models in analyses of diet using multiple stable isotopes: a critique

Stable isotope analysis is used frequently to determine the relative contributions of different food sources to an animal’s diet (Hobson 1999). Isotopic ratios for the animal tissues and each of its potential food sources are determined. The similarity of the ratios for the animal tissues with those of individual food sources (after correcting for fractionation during digestion and assimilation) gives an idea of their relative importance in the diet; in other words “you are what you eat” (DeNiro and Epstein 1978). Two food sources can be partitioned using the isotopic ratio for a single element (e.g., δ13C), or three food sources can be partitioned using isotopic ratios for two elements (e.g., δ13C and δ15N) (Kwak and Zedler 1997). A number of recent papers have used geometric procedures to quantify the contributions of three food sources to the diet using δ13C and δ15N (Ben-David et al. 1997a, 1997b; Kline et al. 1993; Szepanski et al. 1999; Whitledge and Rabeni 1997). However, these methods do not provide correct solutions to this three-endmember mixing problem. The purpose of this paper is to point out the shortcomings of these methods and to propose an alternative procedure which avoids them. Figure 1 shows a graphical representation of the analytical situation. The dietary isotopic composition is represented by point D within the triangle bounded by the points for the adjusted food source isotopic compositions A′, B′, and C′. In the geometric procedures, Euclidean distances are calculated for line segments DA′, DB′, and DC′ and are used to compute the dietary contributions. Several variations of this calculation have been utilized. Kline et al. (1993) used the following equation:

A comparison of geostatistical procedures for spatial analysis of precipitation in mountainous terrain

Abstract Spatially distributed measurements or estimates of precipitation over a region are required for modelling of hydrologic processes and soil moisture for agricultural and natural resource management. Simple interpolation methods fail to consider the effects of topography on precipitation and may be in considerable error in mountainous regions. The performance of three geostatistical methods for making mean annual precipitation estimates on a regular grid of points in mountainous terrain was evaluated. The methods were: (1) kriging; (2) kriging elevation-detrended data; and (3) cokriging with elevation as an auxiliary variable. The study area was the Willamette River basin, a 2.9 million hectare region spanning the area between the Coast Range and the Cascade Range in western Oregon. Compared with kriging, detrended kriging and cokriging both exhibited better precision (as indicated by estimation coefficients of variation of 16 and 17% vs. 21%; and average absolute errors of 19 and 20 cm vs. 26 cm) and accuracy (as indicated by average errors of −1.4 and −2.0 cm vs. −5.2 cm) in the estimation of mean annual precipitation. Contour diagrams for kriging and detrended kriging exhibited smooth zonation following general elevation trends, while cokriging showed a patchier pattern more closely corresponding to local topographic features. Detrended kriging and cokriging offer improved spatially distributed precipitation estimates in mountainous terrain on the scale of a few million hectares. Application of these methods for a larger region, the Columbia River drainage in the USA (57 million hectares), was unsuccessful due to the lack of a consistent precipitation-elevation relationship at this scale. Precipitation estimation incorporating the effects of topography at larger scales will require either piecewise estimation using the methods described here or development of a physically based orographic model.

Advancing fine root research with minirhizotrons

Minirhizotrons provide a nondestructive, in situ method for directly viewing and studying fine roots. Although many insights into fine roots have been gained using minirhizotrons, a review of the literature indicates a wide variation in how minirhizotrons and minirhizotron data are used. Tube installation is critical, and steps must be taken to insure good soil/tube contact without compacting the soil. Ideally, soil adjacent to minirhizotrons will mimic bulk soil. Tube installation causes some degree of soil disturbance and has the potential to create artifacts in subsequent root data and analysis. We therefore recommend a waiting period between tube installation and image collection of 6-12 months to allow roots to recolonize the space around the tubes and to permit nutrients to return to pre-disturbance levels. To make repeated observations of individual roots for the purposes of quantifying their dynamic properties (e.g. root production, turnover or lifespan), tubes should be secured to prevent movement. The frequency of image collection depends upon the root parameters being measured or calculated and the time and resources available for collecting images and extracting data. However, long sampling intervals of 8 weeks or more can result in large underestimates of root dynamic properties because more fine roots will be born and die unobserved between sampling events. A sampling interval of 2 weeks or less reduces these underestimates to acceptable levels. While short sample intervals are desirable, they can lead to a potential trade-off between the number of minirhizotron tubes used and the number of frames analyzed per tube. Analyzing fewer frames per minirhizotron tube is one way to reduce costs with only minor effects on data variation. The quality of minirhizotron data should be assessed and reported; procedures for quantifying the quality of minirhizotron data are presented here. Root length is a more sensitive metric for dynamic root properties than the root number. To make minirhizotron data from separate experiments more easily comparable, idiosyncratic units should be avoided. Volumetric units compatible with aboveground plant measures make minirhizotron-based estimates of root standing crop, production and turnover more useful. Methods for calculating the volumetric root data are discussed and an example presented. Procedures for estimating fine root lifespan are discussed.

Estimating the timing of diet shifts using stable isotopes

Stable isotope analysis has become an important tool in studies of trophic food webs and animal feeding patterns. When animals undergo rapid dietary shifts due to migration, metamorphosis, or other reasons, the isotopic composition of their tissues begins changing to reflect that of their diet. This can occur both as a result of growth and metabolic turnover of existing tissue. Tissues vary in their rate of isotopic change, with high turnover tissues such as liver changing rapidly, while relatively low turnover tissues such as bone change more slowly. A model is outlined that uses the varying isotopic changes in multiple tissues as a chemical clock to estimate the time elapsed since a diet shift, and the magnitude of the isotopic shift in the tissues at the new equilibrium. This model was tested using published results from controlled feeding experiments on a bird and a mammal. For the model to be effective, the tissues utilized must be sufficiently different in their turnover rates. The model did a reasonable job of estimating elapsed time and equilibrial isotopic changes, except when the time since the diet shift was less than a small fraction of the half-life of the slowest turnover tissue or greater than 5–10 half-lives of the slowest turnover tissue. Sensitivity analyses independently corroborated that model estimates became unstable at extremely short and long sample times due to the effect of random measurement error. Subject to some limitations, the model may be useful for studying the movement and behavior of animals changing isotopic environments, such as anadromous fish, migratory birds, animals undergoing metamorphosis, or animals changing diets because of shifts in food abundance or competitive interactions.