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NASA NTRS2021

Fit to be Tied: Embracing Tethered Robots for Exploring Extreme Planetary Environments

Tethers are supportive wires that could provide power, communication, and even science instrument capabilities to planetary rovers, landers, and future human settlements. Whether it be power distribution and optical communication connecting robots exploring extreme, undersea and deep subsurface environments, to your home office on Earth, we live, and will continue to live, in a predominantly ‘wired’ world. Accordingly, as humanity continues to push beyond Earth to exploring extreme planetary terrains like caves, cliffs, craters, and crevasses, and moves towards establishing human settlements on the Moon and Mars, we can expect tethers to play a major role. For exploration purposes, tethers are critical to enabling next-generation, science-focused missions to access high-value, resource-limited targets, where exploring rovers or astronauts lack direct communication or access to viable sunlight for solar power. NASA’s Jet Propulsion Laboratory (JPL) is formulating new mission concepts that could deploy robotic assets into some of the Solar System’s harshest and steepest terrain and, in many cases, tethers are the enabling technology. Examples include i) a rappelling rover to explore vertically along lunar pit walls to determine how the Moon formed, ii) tethered rovers to reach ice deposits on Mars and the Moon, iii) large, tether-deployed antenna structures to probe into the subsurface of a planet and/or look skyward to image the highly red-shifted, early universe, iv) probes that descend into the icy layers of Europa and Enceladus in search of subsurface oceans and, potentially, extant life, and v) science instruments that could be suspended on supportive tethers from balloons on Venus to directly image the surface and sample its clouds. This paper will present a survey of the state-of-the-art for tether related exploration of the solar system and chronicle ongoing work at JPL, which is leading to robust tether designs, high-power and communication transmission over multi-km distances, and highly capable, tether-based rover systems.

McGarey, Patrick↗
NASA NTRS1977

Studies of chemical abundances in the outer solar system

Ground-based observations and the Pioneer 10 mission have led to new discoveries and revisions of previous ideas about the outer solar system. Among these are the discovery of atmospheres on lo and Ganymede, emission from sodium and hydrogen in a cloud around lo, and the presence of acetylene, ethane, and phosphine in the atmosphere of Jupiter. Titan, the largest satellite of Saturn, continues to be an extremely interesting and baffling object, clearly very different in composition from the bodies we are familiar with in the inner solar system; this is also true of Ganymede and Callisto. New data on the abundances of methane and hydrogen in the atmospheres of Uranus and Neptune suggest that the values of C/H in these atmospheres may be much lower than had been previously thought. This result reinforces the apparent compositional differences between these two planets and Jupiter and Saturn, whose atmospheres exhibit a near-solar value for this ratio.

Owen, T.↗
NASA NTRS1995

Interplanetary planar magnetic structures associated with expanding active regions

Planar magnetic structures are interplanetary objects whose magnetic field cannot be explained by Parker's solar wind model. They are characterized by two-dimensional structure of magnetic field that are highly variable and parallel to a plane which is inclined to the ecliptic plane. They appeared independently of interplanetary compression, solar flares, active prominences nor filament disappearances, but the sources often coincided with active regions. On the other hand, it has been discovered by the Yohkoh Soft X-ray telescope that active-region corona expand outwards at speeds of a few to a few tens of km/s near the Sun. The expansions occurred repeatedly, almost continually, even in the absence of any sizable flares. In the Yohkoh Soft X-ray images, the active-region corona seems to expand out into interplanetary space. Solar sources of interplanetary planar magnetic structures observed by Sakigake were examined by Yohkoh soft X-ray telescope. During a quiet period of the Sun from January 6 to November 11, 1993, there found 5 planar magnetic structures according to the criteria (absolute value of Bn)/(absolute value of B) less than 0.1 for planarity and (dB)/(absolute value of B) greater than 0.7 for variability of magnetic field, where Bn, dB, and the absolute value of B are field component normal to a plane, standard deviation, and average of the magnitude of the magnetic field, respectively. Sources of 4 events were on low-latitude (less than 5 degrees) active regions from which loop-like structures were expanding. The coincidence, 80%, is extremely high with respect to accidental coincidence, 7%, of Sakigake windows of solar wind observation with active regions. The last source was on loop-like features which seemed to be related with a mid-latitude (20 degrees) active region.

Nakagawa, Tomoko↗
NASA NTRS2006

Using Quasiparticle Poisoning To Detect Photons

According to a proposal, a phenomenon associated with excitation of quasiparticles in certain superconducting quantum devices would be exploited as a means of detecting photons with exquisite sensitivity. The phenomenon could also be exploited to perform medium-resolution spectroscopy. The proposal was inspired by the observation that Coulomb blockade devices upon which some quantum logic gates are based are extremely sensitive to quasiparticles excited above the superconducting gaps in their leads. The presence of quasiparticles in the leads can be easily detected via the charge states. If quasiparticles could be generated in the leads by absorption of photons, then the devices could be used as very sensitive detectors of electromagnetic radiation over the spectral range from x-rays to submillimeter waves. The devices in question are single-Cooper-pair boxes (SCBs), which are mesoscopic superconducting devices developed for quantum computing. An SCB consists of a small superconducting island connected to a reservoir via a small tunnel junction and connected to a voltage source through a gate capacitor. An SCB is an artificial two-level quantum system, the Hamiltonian of which can be controlled by the gate voltage. One measures the expected value of the charge of the eigenvectors of this quantum system by use of a radio-frequency single-electron transistor. A plot of this expected value of charge as a function of gate voltage resembles a staircase that, in the ideal case, consists of steps of height 2 e (where e is the charge of one electron). Experiments have shown that depending on the parameters of the device, quasiparticles in the form of "broken" Cooper pairs present in the reservoir can tunnel to the island, giving rise to steps of 1 e. This effect is sometimes called "poisoning." Simulations have shown that an extremely small average number of quasiparticles can generate a 1-e periodic signal. In a device according to the proposal, this poisoning would be turned to advantage. Depending on the wavelength, an antenna or other component would be used to couple radiation into the reservoir, wherein the absorption of photons would break Cooper pairs, thereby creating quasiparticles that, in turn, would tunnel to the island, creating a 1-e signal. On the basis of conservative estimates of device parameters derived from experimental data and computational simulations that fit the data, it has been estimated that the noise equivalent power of a device according to the proposal could be as low as 6 10(exp -22) W/Hz(exp 1/2). It has also been estimated that the spectroscopic resolution (photon energy divided by increment of photon energy) of such a device in visible light would exceed 100.

Echternach, Pierre↗
NASA NTRS1997

Effects of volume loading and pressor agents in idiopathic orthostatic tachycardia

BACKGROUND: Idiopathic orthostatic tachycardia (IOT) is characterized by an increase in heart rate (HR) with standing of > or = 30 bpm that is associated with elevated catecholamine levels and orthostatic symptoms. A dynamic orthostatic hypovolemia and alpha1-adrenoreceptor hypersensitivity have been demonstrated in IOT patients. There is evidence of an autonomic neuropathy affecting the lower-extremity blood vessels. METHODS AND RESULTS: We studied the effects of placebo, the alpha1-adrenoreceptor agonist midodrine (5 to 10 mg), the alpha2-adrenoreceptor agonist clonidine (0.1 mg), and I.V. saline (1 L) in 13 patients with IOT. Supine and upright blood pressure (BP) and HR were measured before and at 1 and 2 hours after intervention. Midodrine decreased both supine and upright HR (all HR values are given as bpm) at 2 hours (from 78+/-2 supine to 108+/-5 upright before treatment and from 69+/-2 supine to 95+/-5 upright after treatment, P<.005 for supine and P<.01 for upright). Saline decreased both supine and upright HR (from 80+/-3 supine to 112+/-5 upright before infusion and from 77+/-3 supine to 91+/-3 upright 1 hour after infusion, P<.005 for supine and P<.001 for upright). Clonidine decreased supine HR (from 78+/-2 to 74+/-2, P<.03) but did not affect the HR increase with standing. Clonidine very significantly decreased supine systolic BP (from 109+/-3 at baseline to 99+/-2 mm Hg at 2 hours, P<.001), and midodrine decreased supine systolic BP mildly. CONCLUSIONS: IOT responds best acutely to saline infusion to correct the underlying hypovolemia. Chronically, this can be accomplished with increased salt and water intake in conjunction with fludrocortisone. The response of patients to the alpha1-agonist midodrine supports the hypothesis of partial dysautonomia and indicates that the use of alpha1-agonists to pharmacologically replace lower-extremity postganglionic sympathetics is an appropriate overall goal of therapy. These findings are consistent with our hypothesis that the tachycardia and elevated catecholamine levels associated with IOT are principally due to hypovolemia and loss of adequate lower-extremity vascular tone.

Non-NASA Center↗
NASA NTRS1984

The potential for bulk undercooling as an industrial process

The main focus is on solidification occurring in highly supercooled melts. Solidification rates in such melts are extremely high, an attractive feature from a commercial standpoint. Thus, the reported growth velocities for pure Ni and Co dendrites at a supercooling of 175 K are in excess of 180 km/hr. Rapidly quenched crystalline alloys produced by various atomization processes (e.g., centrifugal atomization or inert gas atomization) or melt spinning are examples of solidification processes, currently being intensively explored commercially, wherein extremely high solidification rates are achieved. Estimated dendrite tip growth rates are about 2 km/hr in a binary Al-4.5 wt % alloy, with a heat transfer coefficient of 6.4x10 sub 5 w/sq cm K or 15 cal/cu cm sK. In the limit, when the solidification rate exceeds a critical value, a glassy microstructure is obtained even in highly alloyed melts, which under normal conditions would solidity to form one or more crystalline phases. Glassy metals, also called metallic glasses, are candidate materials for distribution transformers because of their very low energy losses and are also being used in brazing and soldering applications.

Laxmanan, V.↗
NASA NTRS1986

Volcanic processes and landforms on Venus - Theory, predictions, and observations

The ascent and eruption of magma on Venus in the current Venusian environment is modeled, taking into account the influence of extreme surface temperatures and pressures. Comparison of present predictions with observations obtained from Pioneer Venus, Arecibo, and Venera data support a picture of regional pyroclastic deposits being very rare, magma contents not usually exceeding about 4 wt pct, and the atmospheric pressure having been about the same as the present value over a time period equivalent to the average age of the northern areas of the northern hemisphere. Data suggest that numerous eruptions had effusion rates exceeding common terrestrial rates, and that shield volcanoes are often wide, but are low relative to those on Mars and earth. Implications of the proposed Venusian reduction of the driving density contrast include dike intrusion being very common, and large minimum magma volumes being required to ensure surface eruptions.

Head, James W., III↗
NASA NTRS1993

Derivation of satellite wind model functions using operational surface wind analyses - An altimeter example

One year of global surface wind products from multiple operational numerical weather prediction (NWP) forecast/analysis systems are used as comparison data to derive an empirical wind speed model function for the Geosat altimeter. The resulting model function is nearly identical to the modified Chelton-Wentz model for wind speeds from 4.5 to 15 m/s. Highly skewed distributions at low wind speeds are consistent with specular reflections and antenna mispointing errors hypothesized by others on the basis of extremely limited data. Mesoscale variability in the wind field and synoptic scale errors in the NWP products are shown to account for about 30 percent of the observed scatter of sigma(0) at each wind speed. The remaining scatter is largest at low winds, and decreases to a nearly constant value of about 12 percent at speeds greater than 7 m/s. Model function uncertainty expressed more traditionally in units of wind speed is examined for historical model functions as well as the present NWP-based model. The historical models have significant biases at high wind speeds owing to the lack of comparison in situ data used in their construction. The present study demonstrates that modern operational NWP surface wind products are sufficiently accurate to allow development of fully empirical model functions and associated error analyses.

Freilich, Michael H.↗
NASA NTRS1947

Wind-Tunnel Investigation of Effects of Unsymmetrical Horizontal-Tail Arrangements on Power-on Static Longitudinal Stability of a Single-Engine Airplane Model

A wind-tunnel investigation has been made to determine the effects of unsymmetrical horizontal-tail arrangements on the power-on static longitudinal stability of a single-engine single-rotation airplane model. Although the tests and analyses showed that extreme asymmetry in the horizontal tail indicated a reduction in power effects on longitudinal stability for single-engine single-rotation airplanes, the particular "practical" arrangement tested did not show marked improvement. Differences in average downwash between the normal tail arrangement and various other tail arrangements estimated from computed values of propeller-slipstream rotation agreed with values estimated from pitching-moment test data for the flaps-up condition (low thrust and torque) and disagreed for the flaps-down condition (high thrust and torque). This disagreement indicated the necessity for continued research to determine the characteristics of the slip-stream behind various propeller-fuselage-wing combinations. Out-of-trim lateral forces and moments of the unsymmetrical tail arrangements that were best from consideration of longitudinal stability were no greater than those of the normal tail arrangement.

AIRFOILS - NACA 65(216-012(MOD ) - CHAR & MOMENTS ↗
NASA NTRS1994

Turbidity of a binary fluid mixture: Determining eta

A ground based (1-g) experiment is in progress that will measure the turbidity of a density-matched, binary fluid mixture extremely close to the critical point. By covering the range of reduced temperatures t is equivalent to (T-T(sub c))/T(sub c) from 10(exp -8) to 10(exp -2), the turbidity measurements will allow the critical exponent eta to be determined. No experiment has determined a value of the critical exponent eta, yet its value is significant to theorists in critical phenomena. Interpreting the turbidity correctly is important if future NASA flight experiments use turbidity as an indirect measurement of relative temperature in shuttle experiments on critical phenomena in fluids.

Jacobs, Donald T.↗
NASA NTRS1999

Dust on Mars: An Aeolian Threat to Human Exploration?

The NASA HEDS Program is duly concerned for human explorers regarding the potential hazard posed by the ubiquitous dust mantle on Mars. To evaluate properties of dust that could be hazardous to humans, the MPS 2001 Lander payload will include the Mars Environmental Compatibility Assessment (MECA) experiment. This includes optical and atomic-force microscopy to evaluate soil grains for shape and size, wet chemistry to evaluate toxic substances, electrometry to evaluate triboelectric charging, and test-material palets to evaluate electrostatic and magnetic adhesion, and the hardness/abrasiveness of soil grains; these experimental subcomponents are delivered samples by the camera-equipped robotic arm of the lander which will acquire material from depths of 0.5 to 1.0 m in the soil. Data returned by MECA will be of value to both the BEDS and planetary/astrobiology communities. Dust poses a threat to human exploration because the martian system does not hydrologically or chemically remove fine particles that are being continuously generated by thermal, aeolian, and colluvial weathering, and by volcanism and impact over billions of years. The dust is extremely fine-grained, in copious quantities, ubiquitous in distribution, continually mobile, and a source of poorly-grounded static charges -- a suite of characteristics posing a particulate and electrical threat to explorers and their equipment. Dust is mobilized on global and regional scales, but probably also unpredictably and violently at local scales by dust devils. The latter might be expected in great abundance owing to near surface atmospheric instability (dust devils were detected by Pathfinder during its brief lifetime). Preliminary laboratory experiments suggest that space-suit materials subjected to windblown dust may acquire a uniform, highly adhesive dust layer that is also highly cohesive laterally owing to electrostatic forces. This layer will obscure visibility through the helmet visor, penetrate joints and fabrics, change the thermal properties of the suit, and possibly affect electronic/electrical suit functions. It is paramount that future missions address the issue of interparticle forces, and in particular, the role played by ionizing radiation in affecting these forces on Mars.

Marshall, J.↗
NASA NTRS1999

Dust on Mars: An Aeolian Threat to Human Exploration?

The NASA HEDS Program is duly concerned for human explorers regarding the potential hazard posed by the ubiquitous dust mantle on Mars. To evaluate properties of dust that could be hazardous to humans, the NMS 2001 Lander payload will include the Mars Environmental Compatibility Assessment (MECA) experiment. This includes optical and atomic-force microscopy to evaluate soil grains for shape and size, wet chemistry to evaluate toxic substances, electrometry to evaluate triboelectric charging, and test-material palets to evaluate electrostatic and magnetic adhesion, and the hardness/abrasiveness of soil grains; these experimental subcomponents are delivered samples by the camera-equipped robotic arm of the lander which will acquire material from depths of 0.5 to 1.0 m in the soil. Data returned by MECA will be of value to both the hEDS and planetary/astrobiology communities. Dust poses a threat to human exploration because the martian system does not hydrologically or chemically remove fine particles that are being continuously generated by thermal, aeolian, and colluvial weathering, and by volcanism and impact over billions of years. The dust is extremely fine-grained, in copious quantities, ubiquitous in distribution, continually mobile, and a source of poorly-grounded static charges -- a suite of characteristics posing a particulate and electrical threat to explorers and their equipment. Dust is mobilized on global and regional scales, but probably also unpredictably and violently at local scales by dust devils. The latter might be expected in great abundance owing to near surface atmospheric instability (dust devils were detected by Pathfinder during its brief lifetime). Preliminary laboratory experiments suggest that space-suit materials subjected to windblown dust may acquire a uniform, highly adhesive dust layer that is also highly cohesive laterally owing to electrostatic forces. This layer will obscure visibility through the helmet visor, penetrate joints and fabrics, change the thermal properties of the suit, and possibly affect electronic/electrical suit functions. It is paramount that future missions address the issue of interparticle forces, and in particular, the role played by ionizing radiation in affecting these forces on Mars.

Marshall, J.↗
NASA NTRS2000

LSENS, The NASA Lewis Kinetics and Sensitivity Analysis Code

A general chemical kinetics and sensitivity analysis code for complex, homogeneous, gas-phase reactions is described. The main features of the code, LSENS (the NASA Lewis kinetics and sensitivity analysis code), are its flexibility, efficiency and convenience in treating many different chemical reaction models. The models include: static system; steady, one-dimensional, inviscid flow; incident-shock initiated reaction in a shock tube; and a perfectly stirred reactor. In addition, equilibrium computations can be performed for several assigned states. An implicit numerical integration method (LSODE, the Livermore Solver for Ordinary Differential Equations), which works efficiently for the extremes of very fast and very slow reactions, is used to solve the "stiff" ordinary differential equation systems that arise in chemical kinetics. For static reactions, the code uses the decoupled direct method to calculate sensitivity coefficients of the dependent variables and their temporal derivatives with respect to the initial values of dependent variables and/or the rate coefficient parameters. Solution methods for the equilibrium and post-shock conditions and for perfectly stirred reactor problems are either adapted from or based on the procedures built into the NASA code CEA (Chemical Equilibrium and Applications).

Radhakrishnan, K.↗
NASA NTRS2006

Aerogels for Thermal Insulation of Thermoelectric Devices

Silica aerogels have been shown to be attractive for use as thermal-insulation materials for thermoelectric devices. It is desirable to thermally insulate the legs of thermoelectric devices to suppress lateral heat leaks that degrade thermal efficiency. Aerogels offer not only high thermal- insulation effectiveness, but also a combination of other properties that are especially advantageous in thermoelectric- device applications. Aerogels are synthesized by means of sol-gel chemistry, which is ideal for casting insulation into place. As the scale of the devices to be insulated decreases, the castability from liquid solutions becomes increasingly advantageous: By virtue of castability, aerogel insulation can be made to encapsulate devices having any size from macroscopic down to nanoscopic and possibly having complex, three-dimensional shapes. Castable aerogels can permeate voids having characteristic dimensions as small as nanometers. Hence, practically all the void space surrounding the legs of thermoelectric devices could be filled with aerogel insulation, making the insulation highly effective. Because aerogels have the lowest densities of any known solid materials, they would add very little mass to the encapsulated devices. The thermal-conductivity values of aerogels are among the lowest reported for any material, even after taking account of the contributions of convection and radiation (in addition to true thermal conduction) to overall effective thermal conductivities. Even in ambient air, the contribution of convection to effective overall thermal conductivity of an aerogel is extremely low because of the highly tortuous nature of the flow paths through the porous aerogel structure. For applications that involve operating temperatures high enough to give rise to significant amounts of infrared radiation, opacifiers could be added to aerogels to reduce the radiative contributions to overall effective thermal conductivities. One example of an opacifier is carbon black, which absorbs infrared radiation. Another example of an opacifier is micron- sized metal flakes, which reflect infrared radiation. Encapsulation in cast aerogel insulation also can help prolong the operational lifetimes of thermoelectric devices that must operate in vacuum and that contain SiGe or such advanced skutterudite thermoelectric materials as CoSb3 and CeFe3.5Co0.5Sb12. The primary cause of deterioration of most thermoelectric materials is thermal decomposition or sublimation (e.g., sublimation of Sb from CoSb3) at typical high operating temperatures. Aerogel present near the surface of CoSb3 can impede the outward transport of Sb vapor by establishing a highly localized, equilibrium Sb vapor atmosphere at the surface of the CoSb3.

Sakamoto, Jeffrey↗
NASA NTRS2020

Technologies for Future Venus Exploration

The purpose of this white paper is to provide an overview to the NRC Decadal Survey Inner Planets Sub-Panel on thermal protection system (TPS) technologies required for future Venus exploration missions. It considers the capability of heritage TPS technology used by the Pioneer Venus and Galileo probe missions and identifies new technologies that could enable greater science value and more ambitious missions in the future. A prime conclusion is that there are important advances regarding the availability of forebody TPS required for Venus entry probes. Specifically, the carbon phenolic flown on the Pioneer Venus probes has been deemphasized and a new 3D woven material system, Heatshield for Extreme Entry Environment Technology (HEEET) has replaced carbon phenolic. Along with the development of HEEET, there have been test facility upgrades and design tool improvements. These new material and test methods are enabling for Venus missions. However, without a mission scheduled in the near future, even these new developments are at risk of becoming unavailable. Therefore, we recommend that NASA invest in a cross-cutting technology program that focuses on sustainment of relevant TPS materials, entry systems, test facilities, design tools, and flight instrumentation.

Thermal Protection Systems↗
NASA NTRSDate not supplied

Modeling Provider Creativity in Medical System Design Simulating a Medical MacGyver

This presentation describes a method for modeling provider creativity during the requirements building and risk assessment phase of medical system design. Practicing medicine in isolated, confined, and extreme environments requires providers to use limited supplies in creative ways. This is necessary to handle unexpected medical events or make better use of limited storage capacity by leaving low-use, bulky equipment behind in favor of in situ or dual-use resources. This is a highly-valued skillset in remote environments, however, the ability to simulate this care creativity computationally is difficult.

Brian Lonquich↗
NASA NTRS2015

Extremely Low Passive Microwave Brightness Temperatures Due to Thunderstorms

Extreme events by their nature fall outside the bounds of routine experience. With imperfect or ambiguous measuring systems, it is appropriate to question whether an unusual measurement represents an extreme event or is the result of instrument errors or other sources of noise. About three weeks after the Tropical Rainfall Measuring Mission (TRMM) satellite began collecting data in Dec 1997, a thunderstorm was observed over northern Argentina with 85 GHz brightness temperatures below 50 K and 37 GHz brightness temperatures below 70 K (Zipser et al. 2006). These values are well below what had previously been observed from satellite sensors with lower resolution. The 37 GHz brightness temperatures are also well below those measured by TRMM for any other storm in the subsequent 16 years. Without corroborating evidence, it would be natural to suspect a problem with the instrument, or perhaps an irregularity with the platform during the first weeks of the satellite mission. Automated quality control flags or other procedures in retrieval algorithms could treat these measurements as errors, because they fall outside the expected bounds. But the TRMM satellite also carries a radar and a lightning sensor, both confirming the presence of an intense thunderstorm. The radar recorded 40+ dBZ reflectivity up to about 19 km altitude. More than 200 lightning flashes per minute were recorded. That same storm's 19 GHz brightness temperatures below 150 K would normally be interpreted as the result of a low-emissivity water surface (e.g., a lake, or flood waters) if not for the simultaneous measurements of such intense convection. This paper will examine records from TRMM and related satellite sensors including SSMI, AMSR-E, and the new GMI to find the strongest signatures resulting from thunderstorms, and distinguishing those from sources of noise. The lowest brightness temperatures resulting from thunderstorms as seen by TRMM have been in Argentina in November and December. For SSMI sensors carried on five DMSP satellites examined so far, the lowest thunderstorm-related brightness temperatures have been from Argentina in November - December and from Minnesota in June-July. The Minnesota cases were associated with spotter reports of large hail, significant severe wind, and tornadoes. Those locations have the record-holders for each satellite. The lowest AMSR-E 36.5 GHz brightness temperatures associated with deep convection have been in Argentina; the lowest 89.0 GHz brightness temperatures were from Typhoon Bolaven in the Philippine Sea. This paper will show examples of cases with the lowest brightness temperatures, and map the locations of these and other storms with brightness temperatures nearly as low. The study is largely motivated by the new GMI sensor on the Global Precipitation Mission core satellite, launched in February 2014, with its high resolution expected to reveal unprecedented low brightness temperatures when extreme events are encountered.

Cecil, Daniel J.↗
NASA NTRS1972

Variational solutions by the use of stepwise constant functions. I - Linear case.

Demonstration that accurate solutions to the integral equation or to the extremal function can be easily obtained by the variational method in many cases when a stepwise constant function is used for the trial function. The evaluation procedure is simple and straightforward; integrals of the kernel function can be evaluated analytically; the method provides, when the solution involves singularities, the best mean value across the singularity; the results are accurate in both the detailed physical quantities and their averages; the resultant solution can be further integrated analytically over the parameters associated with the problem; and the method can be readily applied to nonlinear integral equations.

Yoshikawa, K. K.↗