Technical note: Uncertainty in multi-source partitioning using large tracer data sets

Alicia Correa,Christian Birkel,Diego Ochoa-Tocachi

doi:10.5194/hess-23-5059-2019

Abstract

Abstract. The availability of large tracer data sets opened up the opportunity to investigate multiple source contributions to a mixture. However, the source contributions may be uncertain and, apart from Bayesian approaches, to date there are only solid methods to estimate such uncertainties for two and three sources. We introduce an alternative uncertainty estimation method for four sources based on multiple tracers as input data. Taylor series approximation is used to solve the set of linear mass balance equations. We illustrate the method to compute individual uncertainties in the calculation of source contributions to a mixture, with an example from hydrology, using a 14-tracer set from water sources and streamflow from a tropical, high-elevation catchment. Moreover, this method has the potential to be generalized to any number of tracers across a range of disciplines.

Highlights

Tracer applications have dramatically increased over recent years across a wide range of disciplines (West et al, 2010)
We propose an alternative methodology based on the first-order Taylor series approximation to estimate the uncertainty of individual end-members or sources to a mixture. We illustrate this application using a multi-tracer data set from Correa et al (2019b), in a three-dimensional space defined by a principal component analysis (PCA)
Our methodology was developed to calculate the contribution of sources to the mixture and its associated uncertainty and was shown to be effective in real application cases

Summary

Introduction

Tracer applications have dramatically increased over recent years across a wide range of disciplines (West et al, 2010). Bayesian methods have been developed to identify multiple (> 3) sources and compute their contributions to a mixture in a two-dimensional mixing space (Parnell et al, 2010; Stock et al, 2018) In this case, a unique solution is not feasible and a higher uncertainty is attributed to the model (Phillips and Gregg, 2001, 2003). We propose an alternative methodology based on the first-order Taylor series approximation to estimate the uncertainty of individual end-members or sources (e.g. precipitation, soil water, groundwater) to a mixture (e.g. streamflow) We illustrate this application using a multi-tracer data set from Correa et al (2019b), in a three-dimensional space defined by a principal component analysis (PCA). The main objective of this technical note is, to explicitly describe the mathematical development that allows the calculation of partial derivatives, degrees of freedom and confidence interval limits for each source fraction contribution and, to provide the code and several examples for their calculation and reproducibility

Uncertainty estimation method development

Study site and data

Uncertainty estimation of water source contributions

Sensitivity of water source uncertainty to input data

Findings

Summary and remarks