Search for: All records

Abstract Unfolding is an ill-posed inverse problem in particle physics aiming to infer a true particle-level spectrum from smeared detector-level data. For computational and practical reasons, these spaces are typically discretized using histograms, and the smearing is modeled through a response matrix corresponding to a discretized smearing kernel of the particle detector. This response matrix depends on the unknown shape of the true spectrum, leading to a fundamental systematic uncertainty in the unfolding problem. To handle the ill-posed nature of the problem, common approaches regularize the problem either directly via methods such as Tikhonov regularization, or implicitly by using wide-bins in the true space that match the resolution of the detector. Unfortunately, both of these methods lead to a non-trivial bias in the unfolded estimator, thereby hampering frequentist coverage guarantees for confidence intervals constructed from these methods. We propose two new approaches to addressing the bias in the wide-bin setting through methods called One-at-a-time Strict Bounds (OSB) and Prior-Optimized (PO) intervals. The OSB intervals are a bin-wise modification of an existing guaranteed-coverage procedure, while the PO intervals are based on a decision-theoretic view of the problem. Importantly, both approaches provide well-calibrated frequentist confidence intervals even in constrained and rank-deficient settings. These methods are built upon a more general answer to the wide-bin bias problem, involving unfolding with fine bins first, followed by constructing confidence intervals for linear functionals of the fine-bin counts. We test and compare these methods to other available methodologies in a wide-bin deconvolution example and a realistic particle physics simulation of unfolding a steeply falling particle spectrum. 

Current understanding of the dynamic and slow flow paths that support streamflow in mountain headwater catchments is inhibited by the lack of long-term hydrogeochemical data and the frequent use of short residence time age tracers. To address this, the current study combined the traditional mean transit time and the state-of-the-art fraction of young water ( F yw ) metrics with stable water isotopes and tritium tracers to characterize the dynamic and slow flow paths at Marshall Gulch, a sub-humid headwater catchment in the Santa Catalina Mountains, Arizona, USA. The results show that F yw varied significantly with period when using sinusoidal curve fitting methods (e.g., iteratively re-weighted least squares or IRLS), but not when using the transit time distribution (TTD)-based method. Therefore, F yw estimates from TTD-based methods may be particularly useful for intercomparison of dynamic flow behavior between catchments. However, the utility of 3 H to determine F yw in deeper groundwater was limited due to both data quality and inconsistent seasonal cyclicity of the precipitation 3 H time series data. Although a Gamma-type TTD was appropriate to characterize deep groundwater, there were large uncertainties in the estimated Gamma TTD shape parameter arising from the short record length of 3 H in deep groundwater. This work demonstrates how co-application of multiple metrics and tracers can yield a more complete understanding of the dynamic and slow flow paths and observable deep groundwater storage volumes that contribute to streamflow in mountain headwater catchments. 

Catchment‐scale response functions, such as transit time distribution (TTD) and evapotranspiration time distribution (ETTD), are considered fundamental descriptors of a catchment's hydrologic and ecohydrologic responses to spatially and temporally varying precipitation inputs. Yet, estimating these functions is challenging, especially in headwater catchments where data collection is complicated by rugged terrain, or in semi‐arid or sub‐humid areas where precipitation is infrequent. Hence, we developed practical approaches for estimating both TTD and ETTD from commonly available tracer flux data in hydrologic inflows and outflows without requiring continuous observations. Using the weighted wavelet spectral analysis method of Kirchner and Neal [2013] for δ¹⁸O in precipitation and stream water, we calculated TTDs that contribute to streamflow via spatially and temporally variable flow paths in a sub‐humid mountain headwater catchment in Arizona, United States. Our results indicate that composite TTDs (a combination of Piston Flow and Gamma TTDs) most accurately represented this system for periods up to approximately 1 month, and that a Gamma TTD was most appropriate thereafter during both winter and summer seasons and for the overall time‐weighted TTD; a Gamma TTD type was applicable for all periods during the dry season. The TTD results also suggested that old waters, i.e., beyond the applicable tracer range, represented approximately 3% of subsurface contributions to streamflow. For ETTD and using δ¹⁸O as a tracer in precipitation and xylem waters, a Gamma ETTD type best matched the observations for all seasons and for the overall time‐weighted pattern, and stable water isotopes were effective tracers for the majority of vegetation source waters. This study addresses a fundamental question in mountain catchment hydrology; namely, how do the spatially and temporally varying subsurface flow paths that support catchment evapotranspiration and streamflow modulate water quantity and quality over space and time.