5.1. Evaluation of End-Member Mixing Analysis Results
 Spatial and temporal variations in the chemical composition of end-members may result in significant uncertainty in hydrograph separations [Genereux, 1998]. In the current study the uncertainty caused by spatial variability of chemical composition in subsurface flow appears to be not significant. Except for the subsurface flow at SSF4, which drains to a stream that is disconnected from Redondo Creek (Figure 1), the U-space projections of subsurface flow at all sites collected at the same time were very close to each other (Figure 5). Subsurface flow was generated from the same soil in the two catchments, the Redondo rubble land association (Figure 1), which may explain why the chemical compositions in subsurface flow were relatively invariant across different locations.
 Though subsurface flow was only sampled once in this study, its temporal variability may be seen from streamflow at La Jara Creek (Figure 2), which was primarily composed of subsurface flow alone as discussed in next section. Similar to the outflow of a ponderosa pine hillslope near Los Alamos [Newman et al., 1998], the solute concentrations in streamflow at La Jara Creek varied over time (Figure 2). A sensitivity test of EMMA was conducted by substituting the time series of chemical compositions in streamflow at La Jara Creek for the single subsurface flow value represented by SSF2 in the two-tracer model for Redondo Creek. The EMMA results (data not presented) using constant and varying values for subsurface flow were almost identical, with a R2 of 0.99 (n = 11, p < 0.001). The estimated mean contribution of subsurface flow to streamflow at Redondo Creek increased only 2%, from 77% to 79%, when time series of chemical compositions in La Jara Creek was used for subsurface flow. It is because of the significant difference of chemical compositions between thermal meteoric water and subsurface flow that it did not cause a significant error in EMMA when the variability of chemical compositions in subsurface flow was ignored or not known.
 The very different EMMA results using all five tracers (Cond, Mg2+, Na+, Cl− and SO42−) versus just two (Mg2+ and Cl−) (Figure 6) appear to be primarily caused by the chemical composition of thermal meteoric water at Spence. The chemistry of this thermal water doesn't fit well the mixing space determined using all five tracers at Redondo Creek, though it is better than thermal meteoric water at San Antonio (Table 2). If only Mg2+ and Cl− are used, however, it does fit the mixing space very well. Thus thermal meteoric water that discharges to Redondo Creek may be very different from thermal meteoric water at Spence for Na+ and SO42− concentrations, but similar for Mg2+ and Cl− concentrations. In fact, the SO42− concentration was lower in the Spence thermal water than in streamflow on 19 April 2004 (Figure 2), suggesting that the thermal meteoric water contributing to Redondo Creek may have a much higher SO42− concentration.
5.2. How Important is Overland Flow Versus Subsurface Flow and Groundwater?
 EMMA results indicate that subsurface flow from hillslopes exerts the major control on Redondo Creek, consistent with a study at an 870-m2 ponderosa pine hillslope near Los Alamos [Wilcox et al., 1997]. Streamflow chemistry at La Jara Creek is primarily controlled by a single flow component (Figure 3). Since chemical compositions at La Jara Creek were very close to those of subsurface flow (Figure 2), this single flow component is deemed to be subsurface flow. Soils in the La Jara Creek catchment are primarily composed of well-drained soils, Redondo rubble land associations (Figure 1), which facilitate snowmelt infiltration. Consistent with Newman et al. [1997, 1998, 2004], subsurface flow at both catchments appears to be generated from a saturated zone in the lower soil horizons above the bedrock in hillslopes, essentially lateral subsurface flow. Lateral subsurface flow from hillslopes is not commonly considered an important agent of runoff generation in semiarid environments [Wilcox et al., 1997], though some previous studies have found pedogenic evidence that it does occur [Thorns, 1983]. This result highlights the importance of lateral subsurface flow in hydrologic and biogeochemical studies in semiarid environments.
 EMMA results indicate that overland flow (either infiltration- or saturation-excess) did not exert a significant control on streamflow during the snowmelt period at either catchment, in agreement with past findings that overland flow rarely occurs in ponderosa pine forests [Dunford, 1954; Heede, 1984; Williams and Buckhouse, 1993]. Field observations also do not support overland flow even during the peak of snowmelt at both catchments. However, Wilcox et al.  found that infiltration-excess overland flow was the second largest flow component (after lateral subsurface flow) at their site, occurring primarily in winter, particularly if soils are frozen before a snowpack develops. Soils in our catchments are primarily composed of well-drained rubble land associations (Figure 1), implying that infiltration rate of snowmelt is high. It is unclear whether or not the soils in our catchments were frozen during WY2005, but catchment size may play a significant role in the difference between the results. Our catchment area is at least 4000 times that of Wilcox et al. . Infiltration-excess overland flow may be generated from one area of a catchment rather than the entire catchment.
 The contribution of thermal meteoric water to streamflow at Redondo Creek was responsive to snowmelt (Figure 6). Thermal meteoric water circulates only in the upper 500 m of the moat zone of the caldera [Goff and Grigsby, 1982]. The release of thermal meteoric water was probably driven by snowmelt infiltration. The infiltration of snowmelt creates a higher hydraulic gradient, displacing thermal meteoric water stored in the fractured rocks, akin to the piston flow mechanism as described by Buttle . In agreement with thermal meteoric water, the discharge measured from a sulfur spring at Sulfur Springs also varied over time, peaked in early summer [Goff and Grigsby, 1982]. The contribution of thermal meteoric water, which has higher chemical concentrations than lateral subsurface flow, explains why the chemical concentrations in streamflow at Redondo Creek were higher than at La Jara Creek (Figure 2).
5.3. Are Mixing Models Still an Effective Tool in Modeling Streamflow Chemistry and Flow Paths?
 Mixing models were often used in a very simple way, e.g., one tracer for two components and two tracers for three components based on mass balance equations of tracers and water, and expressed as linear equations as follows, using two tracers for three components as an example [e.g., Hooper and Shoemaker, 1986].
where f is the fraction of total streamflow discharge due to an end-member; A and B represent compositions of tracers A and B; subscripts 1, 2, 3, and s represent end-members 1, 2, 3, and streamflow. Two key assumptions usually made are that tracers are conservative and the number of end-members is known [e.g., Buttle, 1994]. However, conservative tracers and number of end-members are usually not known a priori. Hydrologists have relied on analysis of watershed hydrology and geology to identify conservative tracers and to acquire the number of end-members. Because of a lack of sufficient information on hydrology and geology in some, if not most, catchments, conservative tracers and the number of end-members are sometime misinterpreted, resulting in significant errors or even an unrealistic presentation of the conceptual model of streamflow generation for a watershed, as illustrated by Burns et al. . This problem is the major reason why the use of mixing models has diminished [Burns, 2002].
 Taking the reverse procedure from the above mixing model (equations (6) and (7)), diagnostic tools of mixing models were used to determine the rank of streamflow chemical data, or in other words, to decompose the data set into a combination of linear equations using eigenvectors of streamflow chemical data. Note, however, that diagnostic tools of mixing models are not used to exactly reconstruct equations (6) and (7). Instead, the tools are used to test if such linear equations can be established (through distribution of residuals between measurements and projections) and if so, how many equations (the number of eigenvectors, which is one less than number of end-members) are needed and which solutes can be expressed linearly (therefore conservative tracers).
 If streamflow chemistry is dominated by chemical interactions between streamwater and rocks and soils, streamflow chemistry cannot be decomposed as linear equations because multivalent ions involve polynomial processes (number of charges serves as power of ionic concentrations in equilibrium constant equation). If streamflow is dominated by subsurface flow from myriad sources with varying residence times due to the heterogeneity of subsurface media, streamflow chemistry follows a power law of fractal contributions of source waters [Kirchner et al., 2000]. In these cases, the residuals of streamflow chemistry between measurements and projections by eigenvectors of PCA should show a structured pattern with pertinent measurements.
 Mixing and fractal analysis of streamflow chemistry complement each other in understanding controls of water quality and (bio)geochemical processes in watershed hydrology. Mixing models apply to watersheds where flows from different hydrologic units have distinct chemical signatures or periods when multiple pathways are activated. In snowmelt-dominated catchments such as Redondo Creek of this study (Figure 2) and Green Lakes Valley in Colorado [Liu et al., 2004], chemical compositions in overland flow or lateral subsurface flow are distinct from groundwater, even though variable over time. In rainfall-dominated catchments such as those in Plynlimon, Wales [Kirchner et al., 2000], there are no flow components that can be defined as end-members. Diagnostic tools of mixing models provide a means to help hydrologists evaluate if end-members can be defined and thus determine if end-member mixing analysis is potentially a useful tool in a study. Note that end-member mixing for conservative tracers depends on there being no chemical reactions once source waters from different hydrologic units converge in streams, and does not explicitly account for geochemical weathering and chemical interactions between water and rocks and soils within each end-member reservoir or hydrologic unit.
 EMMA used in this study is not simply a replacement of traditional mixing models (e.g., two tracers for three components) as a mathematical manipulation, but is part of the process to test a conceptual model of streamflow generation established using chemical tracers. The projection of chemical compositions in end-members using eigenvectors extracted from streamflow chemistry provides an effective means to evaluate if chemical compositions in end-members fit in mixing spaces of streamflow chemistry and thus determine eligibility of end-members (e.g., Spence Spring versus San Antonio Spring for thermal meteoric water at Redondo Creek). Re-creation of streamflow chemistry for conservative tracers using the results of EMMA and chemical compositions in end-member provides a quantitative assessment of model results and ensures that the results are physically meaningful. This re-creation is not recurring because correlations of solutes instead of concentrations are used in PCA to solve for contributions of end-members to streamflow. Particularly, if some tracers are not used in EMMA but used in predictions, it may enhance the evaluations such as the predictions of Cond in the two-tracer model at Redondo Creek (Figure 7).
 The combination of diagnostic tools of mixing models and end-member mixing analysis not only helps to enforce the assumptions of mixing models but also enhances results of mixing models, particularly when chemical data are limited. The combination of these tools reduces uncertainties, particularly in choosing the number of end-members, or conservative tracers, i.e., the so called “model uncertainty” [Joerin et al., 2002]. Also, EMMA uses correlations between tracers other than concentrations of tracers. Ionic concentrations in streamflow significantly varied from snowmelt to summer dry seasons (Figure 2). With several samples collected during snowmelt, however, the data are sufficiently distributed over the year to capture the seasonal variability of ionic concentrations at Redondo Creek (Figure 2). It is assumed that the correlation coefficients between tracers will not be significantly changed if more samples during snowmelt season are added. However, it merits further work in future to evaluate the sensitivity of correlation coefficients to the number of samples, particularly during snowmelt season.