Spectral Analysis of Vertical Temperature Profile Time‐Series Data in Yellowstone Lake Sediments

Abstract

Vertical temperature profile (VTP) data provide a means to quantify heat and mass fluxes across the ground surface in a variety of geological and ecological environments. VTP measurements have long been used to constrain heat fluxes across the Earth's solid surface, and these studies have played an important role in our understanding of the Earth's thermal history (e.g., Bullard, 1945;Kelvin, 1863). VTP measurements can also be used to estimate fluid fluxes. Stallman (1965) and Bredehoeft and Papadopulos (1965) described how temperature-depth profiles in a steady state, porous medium can be used to constrain vertical fluid flux rates. Although this method has been applied in a variety of hydrologic environments (e.g., Cartwright, 1970;Ferguson et al., 2003;Sorey, 1971;Taniguchi, 1993), the steady state thermal assumption is not always satisfied in natural systems. Suzuki (1960) and Stallman (1965) demonstrated that fluid flux rates can be constrained using time-series VTP data if the ground surface interface experiences periodic temperature variations. These studies showed that fluid flow modifies the rate and amplitude decay of downward diffusing thermal signals such that the amplitude ratio and phase lag between vertically offset thermistor pairs can be used to constrain fluid flux rates. Because many natural systems experience periodic temperature variations, the Stallman (1965) method has been widely applied to diverse hydrologic environments, including streams, rivers, coastal oceans, and deep-sea hydrothermal fields (e.g.,

LPMLEn–A Frequency Domain Method to Estimate Vertical Streambed Fluxes and Sediment Thermal Properties in Semi‐Infinite and Bounded Domains

“…For well-posedness, i.e., to obtain a unique solution to (1), two boundary conditions are required. The two choices commonly applied are: (a) the semi-infinite domain where the upper boundary condition at depth 𝐴𝐴 𝐴𝐴𝑈𝑈 is given by the function 𝐴𝐴 𝐴𝐴𝑈𝑈 (𝑡𝑡) and the lower boundary condition is set to approach zero at infinity (Hatch et al, 2006;Luce et al, 2013;Sohn & Harris, 2021;Stallman, 1965;Vandersteen et al, 2015) (Schneidewind et al, 2016). The semi-infinite domain model 𝐴𝐴 𝑆𝑆𝑆𝑆 is given by…”

“…(2016) apply the local polynomial (LP) method to estimate uncertainties and reduce leakage. Where Sohn and Harris (2021) as well as Vandersteen et al. (2015) considered the subsurface as semi‐infinite, Schneidewind et al.…”

“…(2015), Schneidewind et al. (2016), and Sohn and Harris (2021), that solve the 1D heat transport equation after Carslaw and Jaeger (1959) in the frequency domain and explicitly consider multiple frequencies during flux estimation (see Table S1 in Supporting Information for a concise overview).…”

“…Differences in their methods are found in how these uncertainties are determined, how the model domain is delineated and the used information content. For example, Sohn and Harris (2021) use the multitaper method while Vandersteen et al. (2015) and Schneidewind et al.…”

“…TECHNICAL REPORTS et al (2016) apply the local polynomial (LP) method to estimate uncertainties and reduce leakage. Where Sohn and Harris (2021) as well as Vandersteen et al (2015) considered the subsurface as semi-infinite, Schneidewind et al (2016) developed an early bounded (finite) domain model with the idea to estimate vertical flux for distinct streambed sections. This early bounded model (LPMLE3) utilized information from three (noisy) sensors, two for the input (top and bottom boundaries) and the third for the output.…”