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At least 325 records · Page 18

Gridded National Inventory of U.S. Methane Emissions

We present a gridded inventory of US anthropogenic methane emissions with 0.1 deg x 0.1 deg spatial resolution, monthly temporal resolution, and detailed scale dependent error characterization. The inventory is designed to be onsistent with the 2016 US Environmental Protection Agency (EPA) Inventory of US Greenhouse Gas Emissionsand Sinks (GHGI) for 2012. The EPA inventory is available only as national totals for different source types. We use a widerange of databases at the state, county, local, and point source level to disaggregate the inventory and allocate the spatial and temporal distribution of emissions for individual source types. Results show large differences with the EDGAR v4.2 global gridded inventory commonly used as a priori estimate in inversions of atmospheric methane observations. We derive grid-dependent error statistics for individual source types from comparison with the Environmental Defense Fund (EDF) regional inventory for Northeast Texas. These error statistics are independently verified by comparison with the California Greenhouse Gas Emissions Measurement (CALGEM) grid-resolved emission inventory. Our gridded, time-resolved inventory provides an improved basis for inversion of atmospheric methane observations to estimate US methane emissions and interpret the results in terms of the underlying processes.

Methane↗

Predicting Airport Runway Configurations for Decision-Support Using Supervised Learning

One of the most challenging tasks for air traffic controllers is runway configuration management (RCM). It deals with the optimal selection of runways to operate on (for arrivals and departures) based on traffic, surface wind speed, wind direction, other environmental variables, noise constraints, and several other airport-specific factors. It affects the efficiency of the National Airspace System (NAS) and both surface and airspace operations can benefit from better understanding future runway configurations. In this paper, we present a comprehensive implementation of predictive models for runway configuration estimation from large volumes of historical data. Specifically, operational data from two full years (2018 and 2019) is collected, analyzed, and fused together to build the data product used in this work. The data set differs from prior work in the field in terms of its scope, resolution, and variety of factors collected and considered. Meteorological data is collected from two different sources – current weather conditions from METAR (Meteorological Terminal Aviation Routine Weather Report) and forecast weather conditions from Localized Aviation MOS Program (LAMP). Operational data from the Federal Aviation Administration (FAA) Aviation System Performance Metrics (ASPM) related to scheduled and actual number of arrivals and departures, average taxi times, etc. are collected. NASA’s Sherlock Data Warehouse is used to identify critical information such as go-arounds, and other events that might impact RCM decision-making. All data is collected and aggregated over 15-minute intervals throughout the two years. This provides a resolution like the timescales that might be necessary for runway configuration management decision-making. A variety of supervised learning algorithms are tested including Support Vector Machine, Random Forest, Gradient Boosting, etc. including tuning of the model hyperparameters. The modeling process is applied and presented on two representative U.S. airports – Charlotte Douglas International Airport (KCLT) and Denver International Airport (KDEN). The two airports present different levels of complexity in terms of the total number of configurations used and provide a balanced perspective on the generalizability of the developed approach to other airports in the NAS. Initial results are promising (F1 score of 0.91 at KCLT and 0.83 at KDEN) for data in the test set. The final paper will contain a comprehensive comparison between different models and model building strategies as well as further refined results. Most important predictors for each airport will be identified along with a discussion and recommendations on adapting the framework to other scenarios.

Tejas G Puranik↗

Dark energy survey year 3 results: High-precision measurement and modeling of galaxy-galaxy lensing

We present and characterize the galaxy-galaxy lensing signal measured using the first three years of data from the Dark Energy Survey (DES Y3) covering 4132 deg 2 . These galaxy-galaxy measurements are used in the DES Y3 3 × 2 pt cosmological analysis, which combines weak lensing and galaxy clustering information. We use two lens samples: a magnitude-limited sample and the redmagic sample, which span the redshift range ∼ 0.2 – 1 with 10.7 and 2.6 M galaxies, respectively. For the source catalog, we use the metacalibration shape sample, consisting of ≃ 100 M galaxies separated into four tomographic bins. Our galaxy-galaxy lensing estimator is the mean tangential shear, for which we obtain a total SNR of ∼ 148 for maglim ( ∼ 120 for redmagic), and ∼ 67 ( ∼ 55 ) after applying the scale cuts of 6 Mpc / h . Thus we reach percent-level statistical precision, which requires that our modeling and systematic-error control be of comparable accuracy. The tangential shear model used in the 3 × 2 pt cosmological analysis includes lens magnification, a five-parameter intrinsic alignment model, marginalization over a point mass to remove information from small scales and a linear galaxy bias model validated with higher-order terms. We explore the impact of these choices on the tangential shear observable and study the significance of effects not included in our model, such as reduced shear, source magnification, and source clustering. We also test the robustness of our measurements to various observational and systematics effects, such as the impact of observing conditions, lens-source clustering, random-point subtraction, scale-dependent metacalibration responses, point spread function residuals, and B modes.

79 ASTRONOMY AND ASTROPHYSICS↗

Evidential Deep Learning: Enhancing Predictive Uncertainty Estimation for Earth System Science Applications

Abstract Robust quantification of predictive uncertainty is a critical addition needed for machine learning applied to weather and climate problems to improve the understanding of what is driving prediction sensitivity. Ensembles of machine learning models provide predictive uncertainty estimates in a conceptually simple way but require multiple models for training and prediction, increasing computational cost and latency. Parametric deep learning can estimate uncertainty with one model by predicting the parameters of a probability distribution but does not account for epistemic uncertainty. Evidential deep learning, a technique that extends parametric deep learning to higher-order distributions, can account for both aleatoric and epistemic uncertainties with one model. This study compares the uncertainty derived from evidential neural networks to that obtained from ensembles. Through applications of the classification of winter precipitation type and regression of surface-layer fluxes, we show evidential deep learning models attaining predictive accuracy rivaling standard methods while robustly quantifying both sources of uncertainty. We evaluate the uncertainty in terms of how well the predictions are calibrated and how well the uncertainty correlates with prediction error. Analyses of uncertainty in the context of the inputs reveal sensitivities to underlying meteorological processes, facilitating interpretation of the models. The conceptual simplicity, interpretability, and computational efficiency of evidential neural networks make them highly extensible, offering a promising approach for reliable and practical uncertainty quantification in Earth system science modeling. To encourage broader adoption of evidential deep learning, we have developed a new Python package, Machine Integration and Learning for Earth Systems (MILES) group Generalized Uncertainty for Earth System Science (GUESS) (MILES-GUESS) ( https://github.com/ai2es/miles-guess ), that enables users to train and evaluate both evidential and ensemble deep learning. Significance Statement This study demonstrates a new technique, evidential deep learning, for robust and computationally efficient uncertainty quantification in modeling the Earth system. The method integrates probabilistic principles into deep neural networks, enabling the estimation of both aleatoric uncertainty from noisy data and epistemic uncertainty from model limitations using a single model. Our analyses reveal how decomposing these uncertainties provides valuable insights into reliability, accuracy, and model shortcomings. We show that the approach can rival standard methods in classification and regression tasks within atmospheric science while offering practical advantages such as computational efficiency. With further advances, evidential networks have the potential to enhance risk assessment and decision-making across meteorology by improving uncertainty quantification, a longstanding challenge. This work establishes a strong foundation and motivation for the broader adoption of evidential learning, where properly quantifying uncertainties is critical yet lacking.

Schreck, John S.↗

A Strong Anthropogenic Black Carbon Forcing Constrained by Pollution Trends Over China

Estimates of the effective radiative forcing from aerosol-radiation interaction (ERF ari ) of anthropogenic Black Carbon (BC) have been disputable and require better constraints. Here we find a substantial decline in atmospheric absorption of −5.79Wm −2 decade −1 over eastern central China (ECC) responding to recent anthropogenic BC emission reductions. By combining the observational finding with advances from Coupled Model Intercomparison Project phase6 (CMIP6), we identify an emergent constraint on the ERF ari of anthropogenic BC. We show that across CMIP6 models the simulated trends correlate well with simulated annual mean shortwave atmospheric absorption by anthropogenic BC over China. Making use of this emergent relationship allows us to constrain the aerosol absorption optical depth of anthropogenic BC and further provide a constrained range of 2.4–3.0 Wm −2 for its top-of-atmosphere ERF ari over China, higher than existing estimates. Furthermore, our work supports a strong warming effect of BC over China, and highlights the need to improve BC simulations over source regions.

CMIP6 models↗

Area estimation of environmental phenomena from NOAA-n satellite data

A technique for documenting changes in size of NOAA-n pixels in order to calibrate the data for use in performing area calculations is described. Based on Earth-satellite geometry, a function for calculating the effective pixel size, measured in terms of ground area, on any given pixel was derived. The equation is an application of the law of sines plus an arclength formula. Effective pixel dimensions for NOAA 6 and 7 satellites for all pixels between nadir and the extreme view angles are presented. The NOAA 6 data were used to estimate the areas of several lakes, with an accuracy within 5%. Sources of error are discussed.

Tappan, G.↗

Broadband Phase Retrieval for Image-Based Wavefront Sensing

A focus-diverse phase-retrieval algorithm has been shown to perform adequately for the purpose of image-based wavefront sensing when (1) broadband light (typically spanning the visible spectrum) is used in forming the images by use of an optical system under test and (2) the assumption of monochromaticity is applied to the broadband image data. Heretofore, it had been assumed that in order to obtain adequate performance, it is necessary to use narrowband or monochromatic light. Some background information, including definitions of terms and a brief description of pertinent aspects of image-based phase retrieval, is prerequisite to a meaningful summary of the present development. Phase retrieval is a general term used in optics to denote estimation of optical imperfections or aberrations of an optical system under test. The term image-based wavefront sensing refers to a general class of algorithms that recover optical phase information, and phase-retrieval algorithms constitute a subset of this class. In phase retrieval, one utilizes the measured response of the optical system under test to produce a phase estimate. The optical response of the system is defined as the image of a point-source object, which could be a star or a laboratory point source. The phase-retrieval problem is characterized as image-based in the sense that a charge-coupled-device camera, preferably of scientific imaging quality, is used to collect image data where the optical system would normally form an image. In a variant of phase retrieval, denoted phase-diverse phase retrieval [which can include focus-diverse phase retrieval (in which various defocus planes are used)], an additional known aberration (or an equivalent diversity function) is superimposed as an aid in estimating unknown aberrations by use of an image-based wavefront-sensing algorithm. Image-based phase-retrieval differs from such other wavefront-sensing methods, such as interferometry, shearing interferometry, curvature wavefront sensing, and Shack-Hartmann sensing, all of which entail disadvantages in comparison with image-based methods. The main disadvantages of these non-image based methods are complexity of test equipment and the need for a wavefront reference.

Dean, Bruce H.↗

Evaluating and Countering the Insider Threat to the Radioactive Source Supply Chain

The modern supply chain is a global enterprise and little drove this home more than the global COVID-19 pandemic which sent economic shockwaves throughout the world. Many goods became scarce, as products were delayed, in limited supply, or simply not available. The suddenly diminished supply collided with still high demand and led to greatly increased costs. This was particularly true for the radioactive source supply chain. The pandemic introduced extensive delays for construction projects, slowed the transport of radiological materials to facilities, interrupted treatment deliveries, and impaired the mobility of contractors across the industry. All of these concerns not only adversely affected the economy, but also impacted the safety and security of radiological material, potentially raising national security concerns. The vulnerability of the supply chain, a critical element in an increasingly interconnected world, was exposed. One example that challenged the adaptive capacity of the overall supply chain is the Ever Given container ship, which became stuck in the Suez Canal in 2020. This accident immediately shut down shipments that accounted for 12% of global trade, with long-term impacts estimated to be much larger. Developing the ability to anticipate and react in real-time to sudden changes has quickly become a necessity, particularly in industries that deal with the transport of hazardous material. The reaction to these dramatic incidents was to largely focus attention and resources on protecting the supply chain from external threats. However, the threat to the radioactive material supply chain from insiders intimately involved in the process may be even greater and remains a blind spot that requires increased attention. Recent events revealed the blueprint for targeting and disrupting that supply chain, so the potential for a malicious insider—or a manipulated, unwitting insider—to take advantage of this vulnerability is elevated, creating security concerns for radiological industries. This paper examines and analyzes the potential insider threat to the radioactive source supply chain and recommends steps to take to counter this possibility.

Kinney, Justin↗

On the theoretical limitations in estimating thickness of a plate-like structure from a full-field single-tone response Lamb wave measurement

A persistent question in wavenumber analysis in the estimation of thickness from a steady-state wave field is identifying the theoretical sensitivity of the system. Here, a well-known trade-off between spatial frequency/wavenumber resolution and thickness resolution exists. The current work presents a calculation of the Cramer-Rao Lower bound (CRLB), specifically as applied to thickness estimates, for a 2-dimensional multi-mode waveform in Gaussian noise. Cases of near-field and far-field excitation are considered, and transducer position with respect to the scan area is also varied. Additionally, we consider the CRLB in a plate with multiple sources, simulated as sources placed on the boundary of the plate. We conclude by presenting the CRLB values in terms of frequency for various thicknesses, and by presenting optimal excitation frequencies for a nominal thickness, based on the CRLB.

47 OTHER INSTRUMENTATION↗

Long term monitoring of methane in the atmosphere by multiplex gas chromatography

Methane is of interest in the study of the Earth's atmosphere because of its implication in the future global warming of the surface. This warming is produced by the absorption of infrared energy by trace gases. It has been estimated that in the next 40 to 50 years, methane could contribute 20 to 25 pct. as much atmospheric warming as that expected from carbon dioxide increases. Studies to examine sources, sinks, and cycles of methane will require analytical methods capable of continuous unattended measurement with temporal resolution of an hour or less for weeks at a time. Gas chromatography (GC) is one of the most practical methods available to conduct the analysis of air, but limitations in this technique still exist which can be alleviated with multiplex GC (MGC). MGC is a technique where many samples are pseudo-randomly introduced to the chromatograph without regard to the length of time required for an analysis. The resulting data must then be reduced using computational methods such as cross correlation. In the technique reported, a tube packed with silver oxide was used at the inlet of the GC column to create concentration pulses of methane in a sample stream of air. By using only one carrier, i.e., ambient air, an effective and accurate method to monitor the variations in concentration of methane in the atmosphere over long periods of time was developed. Methane in ambient air was monitored for an eight day period and an interesting temporal variability was found. This work has shown the utility of a relatively simple MGC for the analysis of a real environmental sample.

Valentin, Jose R.↗

Terrestrial Carbon Sinks in the Brazilian Amazon and Cerrado Region Predicted from MODIS Satellite Data and Ecosystem Modeling

A simulation model based on satellite observations of monthly vegetation cover from the Moderate Resolution Imaging Spectroradiometer (MODIS) was used to estimate monthly carbon fluxes in terrestrial ecosystems of Brazilian Amazon and Cerrado regions over the period 2000-2004. Net ecosystem production (NEP) flux for atmospheric CO2 in the region for these years was estimated. Consistently high carbon sink fluxes in terrestrial ecosystems on a yearly basis were found in the western portions of the states of Acre and Rondonia and the northern portions of the state of Par a. These areas were not significantly impacted by the 2002-2003 El Nino event in terms of net annual carbon gains. Areas of the region that show periodically high carbon source fluxes from terrestrial ecosystems to the atmosphere on yearly basis were found throughout the state of Maranhao and the southern portions of the state of Amazonas. As demonstrated though tower site comparisons, NEP modeled with monthly MODIS Enhanced Vegetation Index (EVI) inputs closely resembles the measured seasonal carbon fluxes at the LBA Tapajos tower site. Modeling results suggest that the capacity for use of MODIS Enhanced Vegetation Index (EVI) data to predict seasonal uptake rates of CO2 in Amazon forests and Cerrado woodlands is strong.

Potter, C.↗

Estimating the Mean Annual Surface Air Temperature at Armagh Observatory, Northern Ireland, and the Global Land-Ocean Temperature Index for Sunspot Cycle 24, the Current Ongoing Sunspot Cycle

As noted by Gray et al., Sir William Herschel was the first to suggest a possible close connection between the Sun and the Earth’s climate. The Sun, being the source of energy that impacts and drives the Earth’s climate system, displays a variety of changes over both short and long term time scales, the most obvious examples being the somewhat regular waxing and waning of sunspots with time (i.e., the sunspot cycle (SC)), first described by Samuel Heinrich Schwabe, a German apothecary and amateur astronomer who observed the Sun from Dessau, Germany, and the now well established variation of the Sun’s irradiance over the SC. Other factors related to the SC have been linked to changes in climate as well. Some of these other factors include the role of cosmic rays and the solar wind (i.e., the geomagnetic cycle) on climate, as well as the apparent close association between trends in global and northern hemispheric temperature and the length of the SC, although some investigators have described the inferred association between climate and, in particular, SC length as now being weak. More recently, Solheim et al. have reported on the relation between SC length and the average temperature in the same and immediately following SC for a number of meteorological stations in Norway and in the North Atlantic region. They noted that while they found no significant trend (correlation) between SC length and the average temperature when measured for the same cycle, in contrast, they found a significant negative trend when SC length was compared with the following cycle’s average temperature. From this observation, they suggested that average northern hemispheric temperature during the present ongoing SC (SC24) will be lower by about 0.9 °C than was seen in SC23 (spanning 1996–2007, based on yearly averages of sunspot number (SSN), and onset for SC24 occurring in 2008). The purpose of this Technical Publication (TP) is to examine the annual variations of the Armagh surface air temperature (ASAT) and the Global Land-Ocean Temperature Index (GLOTI) in relation to SSN and the SC in order to determine their likely values during SC24. Hence, it may provide insight as to whether solar forcing of global temperature is now lessening as a contributor to global warming, thereby indicating a possible cooling in the near term immediate future that potentially could ameliorate the effect of increased anthropogenic warming.

Wilson, Robert M.↗

Physics informed machine learning for chemistry tabulation

Modeling of turbulent combustion system requires modeling the underlying chemistry and the turbulent transport. Solving both systems simultaneously is computationally prohibitive. Instead, given the difference in scales at which the two sub-systems evolve, the two sub-systems are typically (re)solved separately. Popular approaches such as the Flamelet Generated Manifolds (FGM) use a two-step strategy where the governing reaction kinetics are pre-computed and mapped to a low-dimensional manifold, characterized by a few reaction progress variables (model reduction) and the manifold is then “looked-up” during the run-time to estimate the high-dimensional system state by the turbulent transport system. While existing works have focused on these two steps independently, in this work we show that joint learning of the progress variables and the look-up model, can yield more accurate results. Here, we build on the base formulation and implementation to include the dynamically generated Thermochemical State Variables (Lower Dimensional Dynamic Source Terms). We discuss the challenges in the implementation of this deep neural network architecture and experimentally demonstrate its superior performance.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Regional Geology Capstone Report for the Midwestern Regional Carbon Sequestration Partnership (Volume V)

The Midwest Regional Carbon Sequestration Partnership (MRCSP) was founded in 2003 as part of the U.S. Department of Energy’s (DOE’s) Regional Carbon Sequestration Partnership initiative. Since its founding, MRCSP has made significant strides toward making CCUS a viable option for states in the region. The public/private consortium, funded through the DOE Regional Carbon Sequestration Initiative, brings together nearly 40 industry partners and 10 states. Battelle, as the project lead, oversees research, development and operations and coordinates activities among the partners. The incremental, phased approach has built a valuable knowledge base for the industry and paved the way for commercial-scale adoption of CCUS technologies. From 2008 to 2020, MRCSP Phase III focused on the development of large-scale injection projects. This report is part of a series of reports prepared under the Midwestern Regional Carbon Sequestration Partnership (MRCSP) Phase III (Development Phase). These reports summarize and detail the findings of the work conducted under the Phase III project. The MRCSP regional geology Capstone report summarizes the regional characterization of geologic storage subtasks performed under MRCSP Phase III. The regional characterization work was completed by the state geological surveys of Delaware, Indiana, Kentucky, Maryland, New York, Ohio, Pennsylvania, and West Virginia along with Western Michigan University and Rutgers University. Building on the Phases I and II, the Phase III research was focused on key research topics identified in previous MRCSP efforts: [1] Regional Geologic Cross Sections for Potential Storage and Containment Zones; [2] Appalachian Basin: Enhanced Recovery Opportunities; [3] Michigan Basin: Assessment of Enhanced Oil Recovery Using Carbon Dioxide in Silurian Pinnacle Reefs; [4] Mid-Atlantic Coastal Plain and Adjacent Offshore Region: Characterization of Carbon Storage Targets; [5] Ordovician-Cambrian Units: Hierarchical Evaluation of Geologic Carbon Storage Resource Estimates; [6] Upper Silurian to Middle Devonian Strata of Ohio: Structural Characterization of Potential CO 2 Reservoirs and Adjacent Strata; [7] Triassic Rift Basins: Preliminary Study of Long-Term CO 2 Storage Potential Together, the regional characterization effort developed useful products to support project developers, policy makers and other stakeholders in the MRCSP region seeking to understand where potential storage exists relative to large stationary sources of CO 2 emissions.

01 COAL, LIGNITE, AND PEAT↗

Water on Mars: Inventory, distribution, and possible sources of polar ice

Theoretical considerations and various lines of morphologic evidence suggest that, in addition to the normal seasonal and climatic exchange of H2O that occurs between the Martian polar caps, atmosphere, and mid to high latitude regolith, large volumes of water have been introduced into the planet's long term hydrologic cycle by the sublimation of equatorial ground ice, impacts, catastrophic flooding, and volcanism. Under the climatic conditions that are thought to have prevailed on Mars throughout the past 3 to 4 b.y., much of this water is expected to have been cold trapped at the poles. The amount of polar ice contributed by each of the planet's potential crustal sources is discussed and estimated. The final analysis suggests that only 5 to 15 pct. of this potential inventory is now in residence at the poles.

Clifford, S. M.↗

A Critical Look at Coulomb Counting Approach for State of Charge Estimation in Batteries

In this paper, we consider the problem of state-of-charge estimation for rechargeable batteries. Coulomb counting is a well-known method for estimating the state of charge, and it is regarded as accurate as long as the battery capacity and the beginning state of charge are known. The Coulomb counting approach, on the other hand, is prone to inaccuracies from a variety of sources, and the magnitude of these errors has not been explored in the literature. We formally construct and quantify the state-of-charge estimate error during Coulomb counting due to four types of error sources: (1) current measurement error; (2) current integration approximation error; (3) battery capacity uncertainty; and (4) timing oscillator error/drift. It is demonstrated that the state-of-charge error produced can be either time-cumulative or state-of-charge-proportional. Time-cumulative errors accumulate over time and have the potential to render the state-of-charge estimation utterly invalid in the long term. The proportional errors of the state of charge rise with the accumulated state of charge and reach their worst value within one charge/discharge cycle. The study presents methods for reducing time-cumulative and state-of-charge-proportional mistakes through simulation analysis.

Battery management system↗

Coupled Thermal-Hydraulic Analysis and Species Mass Transport in a Versatile Experimental Salt Irradiation Loop (VESIL) Using CTF

With the resurgence of interest in molten salt reactors, there is a need for new experiments and modeling capabilities to characterize the unique phenomena present in this fluid fuel system. A Versatile Experimental Salt Irradiation Loop (VESIL) is currently under investigation at Idaho National Laboratory to be placed in the Advanced Test Reactor (ATR). One of the key phenomena this proposed experiment plans to elucidate is fission product speciation in the fuel-salt and the subsequent effects this has on the fuel-salt properties, source term generation, and corrosion control. Specifically, noble gases (Xe & Kr) will bubble out to a plenum or off-gas system, and noble metals (Mo, Tc, Te, etc.) will precipitate and deposit in specific zones in the loop. This work extends the mass transfer and species interaction models in CTF (Coolant-Boiling in Rod Arrays—Two Fluids) and applies these models to give a preliminary estimation of fission product behavior in the proposed VESIL design. A noble metal–helium bubble mass transfer model is coupled with the thermal-hydraulic results from CTF to determine the effectiveness of this insoluble fission product (IFP) extraction method for VESIL. Amounts of IFP species extracted to the off-gas system and species distributions in VESIL after a 60-day ATR cycle are reported.

Walker, Samuel A.↗

Inverse scattering for an exterior Dirichlet problem

Scattering caused by a metallic cylinder in the field of a wire carrying a periodic current is studied, with a view to determining the location and shape of the cylinder in light of far field measurements between the cylinder and the wire. The associated direct problem is the exterior Dirichlet problem for the Helmholtz equation in two dimensions, and an improved low frequency estimate for its solution by integral equation methods is shown by inverse scattering calculations to be accurate to this estimate. The far field measurements are related to low frequency boundary integral equations whose solutions may be expressed in terms of a mapping function for the exterior of the unknown curve onto the exterior of a unit disk. The conformal transformation's Laurent expansion coefficients can be related to those of the far field, the first of which leads to the calculation of the distance between the source and the cylinder, while the other coefficients are determined by placing the source in a different location.

Hariharan, S. I.↗