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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 55 records · Page 3
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Cu site differentiation in tetracopper(I) sulfide clusters enables biomimetic N 2 O reduction

Copper clusters feature prominently in both metalloenzymes and synthetic nanoclusters that mediate catalytic redox transformations of gaseous small molecules. Such reactions are critical to biological energy conversion and are expected to be crucial parts of renewable energy economies. However, the precise roles of individual metal atoms within clusters are difficult to elucidate, particularly for cluster systems that are dynamic under operating conditions. Here, we present a metal site-specific analysis of synthetic Cu 4 (μ 4 -S) clusters that mimic the Cu Z active site of the nitrous oxide reductase enzyme. Leveraging the ability to obtain structural snapshots of both inactive and active forms of the synthetic model system, we analyzed both states using resonant X-ray diffraction anomalous fine structure (DAFS), a technique that enables X-ray absorption profiles of individual metal sites within a cluster to be extracted independently. Using DAFS, we found that a change in cluster geometry between the inactive and active states is correlated to Cu site differentiation that is presumably required for efficient activation of N 2 O gas. More precisely, we hypothesize that the Cu δ+ ∙∙∙Cu δ- pairs produced upon site differentiation are poised for N 2 O activation, as supported by computational modeling. These results provide an unprecedented level of detail on the roles of individual metal sites within the synthetic cluster system and how those roles interplay with cluster geometry to impact the reactivity function. We expect this fundamental knowledge to inform understanding of metal clusters in settings ranging from (bio)molecular to nanocluster to extended solid systems involved in energy conversion.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2024

Streamlined Loads Analysis of Floating Wind Turbines With Fiber Rope Mooring Lines

This paper presents an approach for more accurate yet relatively streamlined accounting for the nonlinear characteristics of synthetic fiber rope mooring lines for floating wind turbines. First, we select a minimal set of parameters that can efficiently approximate a mooring line material's quasi-static and dynamic stiffness characteristics. We also develop a set of baseline coefficients for different rope materials based on published product information and research papers. We then expand a quasi-static mooring model to include dynamic mooring line stiffness terms in a way that allows the nonlinear stiffness behavior of fiber ropes to be considered even in quasi-static analyses. For dynamic analysis, we have updated the model MoorDyn, coupled with OpenFAST, to work with the new dynamic mooring line stiffness terms. This includes the implementation of a new viscoelastic approach that allows the tension-strain relation of each mooring line segment to vary between two stiffness values, depending on the strain rate. After presenting the formulation of the approaches for modeling synthetic ropes, they are demonstrated on a floating wind turbine mooring system with taut polyester rope mooring lines. The results are compared with those of other approaches of similar fidelity, including the static-dynamic method and separate simulations with static and dynamic stiffness values. Comparing the results shows the ability of the new method to match the results of the previous methods in a more streamlined manner.

elasticity↗
NASA NTRS1995

High-Resolution Electron-Impact Study of the Far-Ultraviolet Emission Spectrum of Molecular Hydrogen

The emission spectrum of molecular hydrogen produced by electron-impact excitation at 100 eV has been measured in the wavelength range 1140-1690 A. High-resolution, optically thin spectra (delta(lambda) = 0.136 A) of the far-ultraviolet (FUV) Lyman and Werner band systems have been obtained with a newly constructed 3 m spectrometer. Synthetic spectral intensities based on the transition probabilities calculated by Abgrall et al. are in very good agreement with experimentally observed intensities. Previous modeling that utilized Allison & Daigarno band transition probabilities with Hoenl-London factors breaks down when the transition moment has significant J dependence or when ro-vibrational coupling is significant. Ro-vibrational perturbation between upsilon = 14 of the B(sup 1)Sigma(sup +, sub u) state and upsilon = 3 of the C(sup 1)Pi(sub u) state and the rotational dependence of the transition moment in the bands of the Lyman system are examined. Complete high-resolution experimental reference FUV spectra, together with the model synthetic spectra based on the Abgrall transition probabilities, are presented. An improved calibration standard is obtained, and an accurate calibration of the 3 m spectrometer has been achieved.

Liu, Xian-Ming↗
DOE OSTI2022

A machine learning approach to water quality forecasts and sensor network expansion: Case study in the Wabash River Basin, United States

Abstract Midwestern cities require forecasts of surface nitrate loads to bring additional treatment processes online or activate alternative water supplies. Concurrently, networks of nitrate monitoring stations are being deployed in river basins, co‐locating water quality observations with established stream gauges. However, tools to evaluate the future value of expanded networks to improve water quality forecasts remains challenging. Here, we construct a synthetic data set of stream discharge and nitrate for the Wabash River Basin—one of the United States’ most nutrient polluted basins—using the established Agro‐IBIS and THMB models. Synthetic data enables rapid, unbiased and low‐cost assessment of potential sensor placements to support management objectives, such as near‐term forecasting. Using the synthetic data, we established baseline 1‐day forecasts for surface water nitrate at 12 cities in the basin using support vector machine regression (SVMR; RMSE 0.48–3.3 ppm). Next, we used the SVMRs to evaluate the improvement in forecast performance associated with deployment of additional nitrate sensors. We identified the optimal sensor placement to improve forecasts at each city, and the relative value of sensors at each candidate location. Finally, we assessed the co‐benefit realized by other cities when a sensor is deployed to optimize a forecast at one city, finding significant positive externalities in all cases. Ultimately, our study explores the potential for machine learning to make near‐term predictions and critically evaluate the improvement realized by expanding a monitoring network. While we use nitrate pollution in the Wabash River Basin as a case study, this approach could be readily applied to any problem where the future value of sensors and network design are being evaluated.

54 ENVIRONMENTAL SCIENCES↗
NASA NTRS2019

Quantifying the Relative Impact of Model Microphysics Parameterizations and Scattering Models in Simulating Synthetic Radar and Microwave Radiometer Observations

Output from numerical weather models is often used to simulate observations from remote sensing instruments, for purposes ranging from data assimilation, synthetic retrievals of geophysical quantities, and optimization of observing systems. However, when hydrometeors are present, the level of detail provided by the weather model is generally insufficient to fully constrain the input to the radiative transfer model (RTM), and further assumptions must be made by the RTM user in order to produce synthetic observations. Using a hierarchy of models including cloud-resolving, double-moment, bin microphysical, and ice-habit predicting models, along with scattering properties from the OpenSSP, Atmospheric Radiative Transfer Simulator (ARTS) databases, as well as relatively simple geometries (e.g., cylindrical plates and columns), we demonstrate the spread in synthetic observation output and the extent to which it is reduced when microphysics is more strongly constrained by the model. As an intermediate step, an error budget for the RTM simulations was derived and from that we developed and will describe best practices for observation simulation (e.g., optimal number of hydrometeor size bins, truncation of the particle size distribution, angular resolution of scattering phase function). Some statistical comparisons with observations will also be presented.

Munchak, S. Joseph↗
NASA NTRS2013

Modeling X-Ray Binary Evolution in Normal Galaxies: Insights from SINGS

We present the largest-scale comparison to date between observed extragalactic X-ray binary (XRB) populations and theoretical models of their production. We construct observational X-ray luminosity functions (oXLFs) using Chandra observations of 12 late-type galaxies from the Spitzer Infrared Nearby Galaxy Survey (SINGS). For each galaxy, we obtain theoretical XLFs (tXLFs) by combining XRB synthetic models, constructed with the population synthesis code StarTrack, with observational star formation histories (SFHs). We identify highest-likelihood models both for individual galaxies and globally, averaged over the full galaxy sample. Individual tXLFs successfully reproduce about half of oXLFs, but for some galaxies we are unable to find underlying source populations, indicating that galaxy SFHs and metallicities are not well matched and/or XRB modeling requires calibration on larger observational samples. Given these limitations, we find that best models are consistent with a product of common envelope ejection efficiency and central donor concentration approx.. = 0.1, and a 50% uniform - 50% "twins" initial mass-ratio distribution. We present and discuss constituent subpopulations of tXLFs according to donor, accretor and stellar population characteristics. The galaxy-wide X-ray luminosity due to low-mass and high-mass XRBs, estimated via our best global model tXLF, follows the general trend expected from the L(sub X) - star formation rate and L(sub X) - stellar mass relations of Lehmer et al. Our best models are also in agreement with modeling of the evolution both of XRBs over cosmic time and of the galaxy X-ray luminosity with redshift.

Normal Galaxies↗
DOE OSTI2023

Developing and applying quantifiable metrics for diagnostic and experiment design on Z

This project applies methods in Bayesian inference and modern statistical methods to quantify the value of new experimental data, in the form of new or modified diagnostic configurations and/or experiment designs. We demonstrate experiment design methods that can be used to identify the highest priority diagnostic improvements or experimental data to obtain in order to reduce uncertainties on critical inferred experimental quantities and select the best course of action to distinguish between competing physical models. Bayesian statistics and information theory provide the foundation for developing the necessary metrics, using two high impact experimental platforms on Z as exemplars to develop and illustrate the technique. We emphasize that the general methodology is extensible to new diagnostics (provided synthetic models are available), as well as additional platforms. We also discuss initial scoping of additional applications that began development in the last year of this LDRD.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1980

The chromosphere and transition region

The physical processes occurring as a result of the transfer of energy and momentum from the primary solar flare energy release site in the corona to the underlying chromosphere and transition region during the course of the flare are investigated through a comparison of theoretical models and observational data. Static, dynamic and hydrodynamic models of the lower-temperature chromospheric flare are reviewed. The roles of thermal conduction, radiation, fast particles and mass motion in chromosphere-corona interactions are analyzed on the basis of Skylab UV, EUV and X-ray data, and empirical and synthetic models of the chromospheric and upper photospheric responses to flares are developed. The canonical model of chromospheric heating during flares as a result of primary energy release elsewhere is found to be justified in the chromosphere as a whole, although not entirely as the temperature minimum, and a simplified model of horizontal chromospheric flare structure based on results obtained is presented.

Canfield, R. C.↗
NASA NTRSDate not supplied

Computational Characterization and Model Verification for 3D Microstructure Reconstruction of Additively-Manufactured Materials

The goal of this study is to characterize and validate the texture and grain topology of additively-manufactured anisotropic three-dimensional (3D) polycrystalline microstructures. The special focus is on developing methodologies to compare the grain shapes and orientations of two-dimensional (2D) and 3D microstructure representations using the same metric. To generate statistical data, synthetic microstructures are reconstructed from experimental data using Markov random field (MRF). The statistical similarity between the experimental and synthetic microstructures is verified by comparing their grain topologies. A universal measure to compare 2D and 3D grains is portrayed through the concept of image moments that are invariant to shape transformations. The graphical plots developed based on moment invariants to compare the 2D and 3D grains are used to verify the synthetic model

Materials Characterization↗
DOE OSTI2023

The Potential Benefits of Handling Mixture Statistics via a Bi-Gaussian EnKF: Tests With All-Sky Satellite Infrared Radiances

The meteorological characteristics of cloudy atmospheric columns can be very different from their clear counterparts. Thus, when a forecast ensemble is uncertain about the presence/absence of clouds at a specific atmospheric column (i.e., some members are clear while others are cloudy), that column's ensemble statistics will contain a mixture of clear and cloudy statistics. Such mixtures are inconsistent with the ensemble data assimilation algorithms currently used in numerical weather prediction. Hence, ensemble data assimilation algorithms that can handle such mixtures can potentially outperform currently used algorithms. In this study, we demonstrate the potential benefits of addressing such mixtures through a bi-Gaussian extension of the ensemble Kalman filter (BGEnKF). The BGEnKF is compared against the commonly used ensemble Kalman filter (EnKF) using perfect model observing system simulated experiments (OSSEs) with a realistic weather model (the Weather Research and Forecast model). Synthetic all-sky infrared radiance observations are assimilated in this study. In these OSSEs, the BGEnKF outperforms the EnKF in terms of the horizontal wind components, temperature, specific humidity, and simulated upper tropospheric water vapor channel infrared brightness temperatures. This study is one of the first to demonstrate the potential of a Gaussian mixture model EnKF with a realistic weather model. Our results thus motivate future research toward improving numerical Earth system predictions though explicitly handling mixture statistics.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2024

Cyclically conjugated porphyrin trimers linked through benzo[4,5]imidazo[2,1- a ]isoindole bridges

Cyclically conjugated porphyrin trimers were prepared via a concise synthetic method. Zn-Trimer-1 displayed strong exciton coupling, suggesting the presence of effective electronic interactions. UV-Vis absorption and fluorescence spectra obtained through titration studies on the donor–acceptor adduct (Zn-Trimer-1–C60Im) indicate the occurrence of excited state photo-events. Inspired by nature's photosynthetic process of plants and bacteria, electronically interacting multiporphyrinic systems have long been investigated as synthetic models of natural photosynthetic systems. Remarkable successes have been achieved with cyclic porphyrin arrays, where porphyrins are arranged in a wheel-like structure or nanoring as photosynthetic antennae. Considerable efforts have also been devoted to constructing other porphyrin arrays such as molecular wires for potential applications in sensors, solar energy conversion, non-linear optical (NLO) materials, and more. Due to synthetic difficulties, cyclically conjugated porphyrin arrays in which porphyrins are arranged in planar geometry are very rare. Very recently, Anderson and coworkers have reported unprecedented β,β-directly linked porphyrin trimers and tetramers with different geometries, propeller and saddle, respectively, which result in distinct photophysical and electrochemical properties (Fig. 1a). In this paper, we present novel cyclically conjugated porphyrin trimers, which feature planar geometry with porphyrins assembled around the aromatic triphenylene at the center (Fig. 1b).

Han, Ting↗
DOE OSTI2026

3D seismic imaging of a fracture damage zone controlling reservoir compartmentalization at the Raft River EGS using multi-azimuth walkaway VSP

Accurate imaging of steeply dipping fracture zones in crystalline enhanced geothermal systems (EGS) is critical for constraining permeability architecture and guiding stimulation design. However, such structures remain poorly resolved by conventional surface seismic methods. We present a fully three-dimensional (3D) elastic-waveform inversion-migration workflow applied to multi-azimuth walkaway vertical seismic profiling (VSP) data acquired in a deviated borehole at the Raft River EGS. The workflow integrates first-arrival traveltime tomography, multi-scale elastic waveform inversion (EWI), and elastic least-squares reverse-time migration (ELSRTM) to recover high-resolution compressional-and shear-wave velocity models and to image structural discontinuities in the crystalline basement. The results reveal a laterally continuous low-velocity anomaly, with shear-wave velocity reductions of 25-30%, consistent with fractureinduced mechanical weakening. Two steeply dipping discontinuities bound a 50-80 m wide fracture damage zone. Independent constraints from microseismic clustering and geochemical compartmentalization corroborate the geometry and structural significance of this feature. Synthetic modeling further confirms that structures of this scale are resolvable (~30 m). These findings indicate that the Narrows structure is a distributed fracture damage zone rather than a discrete fault plane. By resolving fracture-zone geometry at the tens-of-meters scale using a single borehole, this workflow provides a practical and transferable approach for improving structural characterization, reducing uncertainty in permeability architecture, and supporting reservoir modeling and stimulation design in fractured crystalline EGS reservoirs.

58 GEOSCIENCES↗
DOE OSTI2023

Quantification of oil and gas methane emissions in the Delaware and Marcellus basins using a network of continuous tower-based measurements

Abstract. According to the United States Environmental Protection Agency (US EPA), emissions from oil and gas infrastructure contribute 30 % of all anthropogenic methane (CH4) emissions in the US. Studies in the last decade have shown emissions from this sector to be substantially larger than bottom-up assessments, including the EPA inventory, highlighting both the increased importance of methane emissions from the oil and gas sector in terms of their overall climatological impact and the need for independent monitoring of these emissions. In this study we present continuous monitoring of regional methane emissions from two oil and gas basins using tower-based observing networks. Continuous methane measurements were taken at four tower sites in the northeastern Marcellus basin from May 2015 through December 2016 and five tower sites in the Delaware basin in the western Permian from March 2020 through April 2022. These measurements, an atmospheric transport model, and prior emission fields are combined using an atmospheric inversion to estimate monthly methane emissions in the two regions. This study finds the mean overall emission rate from the Delaware basin during the measurement period to be 146–210 Mg CH4 h−1 (energy-normalized loss rate of 1.1 %–1.5 %, gas-normalized rate of 2.5 %–3.5 %). Strong temporal variability in the emissions was present, with the lowest emission rates occurring during the onset of the COVID-19 pandemic. Additionally, a synthetic model–data experiment performed using the Delaware tower network shows that the presence of intermittent sources is not a significant source of uncertainty in monthly quantification of the mean emission rate. In the Marcellus, this study finds the overall mean emission rate to be 19–28 Mg CH4 h−1 (gas-normalized loss rate of 0.30 %–0.45 %), with relative consistency in the emission rate over time. These totals align with aircraft top-down estimates from the same time periods. In both basins, the tower network was able to constrain monthly flux estimates within ±20 % uncertainty in the Delaware and ±24 % uncertainty in the Marcellus. The results from this study demonstrate the ability to monitor emissions continuously and detect changes in the emissions field, even in a basin with relatively low emissions and complex background conditions.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2020

Molecular-scale origins of wettability at petroleum–brine–carbonate interfaces

Wettability control of carbonates is a central concept for enhanced petroleum recovery, but a mechanistic understanding of the associated molecular-scale chemical processes remains unclear. We directly probe the interface of calcium carbonate (calcite) with natural petroleum oil, synthetic petroleum analogues, and aqueous brines to understand the molecular scale behavior at this interface. The calcite–petroleum interface structure is similar whether or not calcite was previously exposed to an aqueous brine, and is characterized by an adsorbed interfacial layer, significant structural changes within the calcite surface, and increased surface roughness. No evidence for an often-assumed thin-brine wetting layer at the calcite–petroleum interface is observed. These features differ from those observed at the calcite–brine interface, and for parallel measurements using model synthetic petroleum mixtures (consisting of representative components, including dodecane, toluene, and asphaltene). Changes to the interface after petroleum displacement by aqueous brines are also discussed.

02 PETROLEUM↗
DOE OSTI2022

Ecosystems and Networks Integrated with Genes and Molecular Assemblies (ENIGMA). Final report

The goal of the Stahl lab is to use a combination of isotopic, genomic, and activity-based analyses in the field, of selected field isolates, and defined synthetic communities to identify sources and sinks of different nitrogen species in the field, focusing on the FRC at the ORNL. The most general research objective is to characterize variants in the pathway for denitrification among field isolates and synthetic microbial assemblies as a foundation for predicting the biotic and abiotic controls of nitrogen and carbon transformations in the field. Specifically, we have focused on characterizing environmental controls of nitrous oxide emissions, an important greenhouse gas and ozone depleting molecule. Studies have mapped the flux of nitrous oxide and controlling genetic elements within the FRC subsurface, indicating the importance of different enzyme variants in nitrous oxide production and consumption in controlling surface emissions. We have developed methods to quantify and catalog the isotopic signatures imparted on nitrous oxide by different nitrous oxide reductase variants in isolated organisms and model synthetic communities. We have developed methods to quantify and catalog the nitrous oxide affinity of field relevant isolates and their biomass yield when respiring nitrous oxide on different carbon substrates. We continue to expand the collection of field-relevant microbial isolates within ENIGMA by focused enrichment and isolation from field material. These objectives have been facilitated by the development of new technologies, including: 1) the development of a multiplexed automated optical density measurement system for quantifying growth kinetics of microorganisms dependent on gaseous substrates in monoculture or gas exchange in synthetic communities and 2) the development of a microcalorimetric analysis format to quantify the impact of environmental stress on microbial maintenance energy. Since field relevant processes are mediated by assemblies of organisms interacting with each other and their local environment, we have also collaborated with other ENIGMA investigators on the assembly and characterization of synthetic communities. These studies have evaluated the role of organisms lacking a complete pathway for denitrification in the control of field processes, measuring process rates of assemblies of organisms that individually lack the full denitrification pathway but mediate the complete denitrification when growing together. These efforts have so far resulted in the publication of 18 peer reviewed papers ranging from ecological and physiological drivers of pathway optimization to field-scale process characterization.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
NASA NTRS2002

Retrieval of Precipitation Profiles from Multiresolution, Multifrequency, Active and Passive Microwave Observations

In this study, a technique for estimating vertical profiles of precipitation from multifrequency, multiresolution active and passive microwave observations is investigated using both simulated and airborne data. The technique is applicable to the Tropical Rainfall Measuring Mission (TRMM) satellite multi-frequency active and passive observations. These observations are characterized by various spatial and sampling resolutions. This makes the retrieval problem mathematically more difficult and ill-determined because the quality of information decreases with decreasing resolution. A model that, given reflectivity profiles and a small set of parameters (including the cloud water content, the intercept drop size distribution, and a variable describing the frozen hydrometeor properties), simulates high-resolution brightness temperatures is used. The high-resolution simulated brightness temperatures are convolved at the real sensor resolution. An optimal estimation procedure is used to minimize the differences between simulated and observed brightness temperatures. The retrieval technique is investigated using cloud model synthetic and airborne data from the Fourth Convection And Moisture Experiment. Simulated high-resolution brightness temperatures and reflectivities and airborne observation strong are convolved at the resolution of the TRMM instruments and retrievals are performed and analyzed relative to the reference data used in observations synthesis. An illustration of the possible use of the technique in satellite rainfall estimation is presented through an application to TRMM data. The study suggests improvements in combined active and passive retrievals even when the instruments resolutions are significantly different. Future work needs to better quantify the retrievals performance, especially in connection with satellite applications, and the uncertainty of the models used in retrieval.

Grecu, Mircea↗
DOE OSTI2026

Mimicking Extradiol Dioxygenase Reactivity on Iridium

Extradiol dioxygenases catalyze the cleavage of benzenediol (catechol) or vicinal aminophenols via oxygen atom insertion into the 2,3- C–C bond. These reactions are most often proposed to proceed through the migratory rearrangement of a d6 alkylperoxide, generating the corresponding d 6 ring expanded product. However, regiospecific insertion remains a rare outcome among synthetic model complexes. Here, a dioxygenated Ir complex (2) converts to (a) the paramagnetic metallatrioxolane (3) and (b) oxygen atom inserted products (4) and (5); all three complexes are third-row metal analogues of enzymatic intermediates. The conversion of 2 to 3 was triggered by an H• abstraction, generating a third-row metallatrioxolane that is one electron reduced from the canonical d 6 alkylperoxide. Alternatively, photolysis of 2 (467 nm) results in ring-expanded product 4, from which an H• can be abstracted to generate 5. This latter complex is also one electron reduced relative to the canonical ring expansion product. Because extradiol mechanisms were largely defined using Fe(II) metallocofactors, 3 and 5 may be especially relevant to the known Co(II) accepting variants. We propose these states became synthetically accessible via the incorporation of a catechol-like substrate into the larger, multidentate ligand L1. This perturbation enhances the affinity of L1 (and related intermediates) to the metal. The same perturbation may have also been key in characterizing the first κ 2 -bound, dianionic ortho ester ligand – the observed binding mode in the x-ray structure of 5. Broadly, we propose through this case study that connecting abiological dioxygen complexes to non-heme oxygenase-like intermediates might be a “roadmap” towards regiospecific aerobic oxygenations.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
NASA NTRS2023

Gravity Degree-Depth Relationship Using Point Mass Spherical Harmonics

Relationships between the degree of a spherical harmonic model of the gravitational field of a body and the depth of a source expressed as a density contrast can be used to study the structure of features. Here, we show that the gravitational acceleration per spherical harmonic degree of a constant density source has an extremum that depends on the depth of the source. Using the spherical harmonics expansion for a point mass source, we use this to derive a degree-depth relationship. Our relationship resembles an earlier one derived by Bowin (1983), with substantial differences at the lower degrees. We also find that a recent relationship derived by Deng et al. (2022) over-estimates the source depth. The relationship that we derive relates spherical harmonic degree n to depth d for a planet of radius R according to d = (1 - e -1/n+1 )R, which simplifies to d = R/(n + 1) for high degrees. We support our new relationship with synthetic models of a density contrast in a planet. We also show how the differences between our relationship and that of Bowin (1983) affect band-filtered gravity, for example when inspecting the upper 100 km of the Moon. Using point masses in our modeling results in an approximate relationship where in reality sources can be deeper than estimated, since any source contributes to all spherical harmonic degrees. The use of the contribution per individual degree however provides an intuitive relationship between spherical harmonic degree and depth that can be used to place relative bounds on source depths or to determine the bounds on spherical harmonic expansions when band-filtering gravity field models.

Geopotential theory↗