Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Mesoscale”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 757 records · Page 42

Impact of Assimilating GOES-Derived Land Surface Variables into the PSU/NCAR MM5

Land surface processes are known to have a profound impact on the overlying atmosphere over a wide range of spatial and temporal scales. Many atmospheric numerical models include special parameterizations to improve the specification and partitioning of surface fluxes which are critical to the accurate prediction of warm season boundary layer behavior, organized mesoscale circulations, and convective precipitation. However, the added degrees of freedom resulting from the inclusion of vegetation and soil schemes require the specification of additional surface parameters such as vegetative resistances, green vegetation fraction, leaf area index, soil physical and hydraulic characteristics, and the vertical distribution of soil moisture. As satellite data have become more readily available in recent years, many investigations have attempted to use these new measurements to infer missing components of the surface energy budget. Sensitivity studies have shown land-skin temperature (LST) tendencies during the mid-morning hours are strongly sensitive to the surface moisture availability (a function of soil wetness and vegetation) and less sensitive to other parameters such as surface roughness. Based upon results from these and other studies, developed a simple technique that dynamically assimilates Geostationary Operational Environmental Satellite (GOES) derived land-surface products into the surface energy budget of a mesoscale model. The purpose of this paper is to demonstrate that assimilating the GOES satellite data has the potential to improve the representation of land surface characteristics within the model without prior knowledge of the land surface characteristics. The assimilation technique is presented in Section 2 and the numerical experiments are detailed in Section 3. Preliminary results and conclusions are presented in Sections 4 and 5, respectively.

Lapenta, William M.↗

Monsoon Convective During the South China Sea Monsoon Experiment: Observations from Ground-Based Radar and the TRMM Satellite

A main goal of the recent South China Sea Monsoon Experiment (SCSMEX) was to study convective processes associated with the onset of the Southeast Asian summer monsoon. The NASA TOGA C-band scanning radar was deployed on the Chinese research vessel Shi Yan #3 for two 20 day cruises, collecting dual-Doppler measurements in conjunction with the BMRC C-Pol dual-polarimetric radar on Dongsha Island. Soundings and surface meteorological data were also collected with an NCAR Integrated Sounding System (ISS). This experiment was the first major tropical field campaign following the launch of the Tropical Rainfall Measuring Mission (TRMM) satellite. These observations of tropical oceanic convection provided an opportunity to make comparisons between surface radar measurements and the Precipitation Radar (PR) aboard the TRMM satellite in an oceanic environment. Nearly continuous radar operations were conducted during two Intensive Observing Periods (IOPS) straddling the onset of the monsoon (5-25 May 1998 and 5-25 June 1998). Mesoscale lines of convection with widespread regions of both trailing and forward stratiform precipitation were observed following the onset of the active monsoon in the northern South China Sea region. The vertical structure of the convection during periods of strong westerly flow and relatively moist environmental conditions in the lower to mid-troposphere contrasted sharply with convection observed during periods of low level easterlies, weak shear, and relatively dry conditions in the mid to upper troposphere. Several examples of mesoscale convection will be shown from the ground (ship)-based and spaceborne radar data during times of TRMM satellite overpasses. Examples of pre-monsoon convection, characterized by isolated cumulonimbus and shallow, precipitating congestus clouds, will also be discussed.

Cifelli, Rob↗

The Dependence on Grid Resolution of Numerically Simulated Convective Cloud Systems Using Ice Microphysics

Mesoscale research and forecast models are increasingly being used at horizontal resolutions of 1-8 km to simulate a variety of precipitating systems. When the model is used to simulate convective systems, it is uncertain to what extent the dynamics and microphysics of convective updrafts can be resolved with grids larger than 1 km. In this study, two- and three-dimensional versions of the Goddard Cumulus Ensemble model are used to determine the impact of horizontal grid resolution on the behavior of the simulated storms and on the characteristics of the cloud microphysical fields. It will be shown that as resolution decreases from about 1 km to greater than 3 km, there is a fairly rapid degradation of the storm structure in the form of reduced convective mass fluxes, updraft tilts, and cloud microphysics. A high-resolution simulation of hurricane outer rainbands using the MM5 mesoscale model shows also that there can be a substantial modification of the key microphysical processes that contribute to rainfall as a result of reducing the horizontal resolution.

Braun, Scott A.↗

The Relationship Between the Background and Transient Signals in Schumann Resonances

Two distinct measures of the Earth's Schumann resonances-the background and the transients--are studied through comparisons at the mesoscale and at the continental scale. A rough proportionality is shown between the far more abundant afternoon lightning activity and the larger positive mesoscale discharges that make sprites and simultaneously ring the Earth-ionosphere cavity to levels higher than the integration of all other lightnings.

Williams, E.↗

Evaluation of Bogus Vortex Techniques with Four-Dimensional Variational Data Assimilation

The effectiveness of techniques for creating "bogus" vortices in numerical simulations of hurricanes is examined by using the Penn State/NCAR nonhydrostatic mesoscale model (MM5) and its adjoint system. A series of four-dimensional variational data assimilation (4-D VAR) experiments is conducted to generate an initial vortex for Hurricane Georges (1998) in the Atlantic Ocean by assimilating bogus sea-level pressure and surface wind information into the mesoscale numerical model. Several different strategies are tested for improving the vortex representation. The initial vortices produced by the 4-D VAR technique are able to reproduce many of the structural features of mature hurricanes. The vortices also result in significant improvements to the hurricane forecasts in terms of both intensity and track. In particular, with assimilation of only bogus sea-level pressure information, the response in the wind field is contained largely within the divergent component, with strong convergence leading to strong upward motion near the center. Although the intensity of the initial vortex seems to be well represented, a dramatic spin down of the storm occurs within the first 6 h of the forecast. With assimilation of bogus surface wind data only, an expected dominance of the rotational component of the wind field is generated, but the minimum pressure is adjusted inadequately compared to the actual hurricane minimum pressure. Only when both the bogus surface pressure and wind information are assimilated together does the model produce a vortex that represents the actual intensity of the hurricane and results in significant improvements to forecasts of both hurricane intensity and track.

Pu, Zhao-Xia↗

Rainfall Estimation Over Tropical Oceans: Area Average Rain Rate - 1

Multichannel dual polarization microwave radiometer SSM/I observations over oceans do not contain sufficient information to differentiate quantitatively the rain from other hydrometeors on a scale comparable to the radiometer field of view (approx. 30 km). For this reason we have developed a method to retrieve average rain rate over a mesoscale grid box of approx. 300 x 300 sq km area over the TOGA COARE region where simultaneous radiometer and radar observations are available for four months (Nov. 92 to Feb. 93). The rain area in the grid box, inferred from the scattering depression due to hydrometeors in the 85 Ghz brightness temperature, constitutes a key parameter in this method. Then the spectral and polarization information contained in all the channels of the SSM/I is utilized to deduce a second parameter. This is the ratio S/E of scattering index S, and emission index E calculated from the SSM/I data. The rain rate retrieved from this method over the mesoscale area can reproduce the radar observed rain rate with a correlation coefficient of about 0.85. Furthermore monthly total rainfall estimated from this method over that area has an average error of about 15%.

Cuddapah, Prabhakara↗

Wavelet Analysis of Satellite Images for Coastal Watch

The two-dimensional wavelet transform is a very efficient bandpass filter, which can be used to separate various scales of processes and show their relative phase/location. In this paper, algorithms and techniques for automated detection and tracking of mesoscale features from satellite imagery employing wavelet analysis are developed. The wavelet transform has been applied to satellite images, such as those from synthetic aperture radar (SAR), advanced very-high-resolution radiometer (AVHRR), and coastal zone color scanner (CZCS) for feature extraction. The evolution of mesoscale features such as oil slicks, fronts, eddies, and ship wakes can be tracked by the wavelet analysis using satellite data from repeating paths. Several examples of the wavelet analysis applied to various satellite Images demonstrate the feasibility of this technique for coastal monitoring.

Liu, Antony K.↗

Effect of Combined Spaceborne Microwave and Continuous Lightning Measurements on Precipitation Forecasts of the 1998 Ground-Hog Day Storm

We evaluated the impact of several newly available sources of meteorological data on mesoscale model forecasts of precipitation produced by the extra-tropical cyclone that struck Florida on February 2, 1998. Precipitation distributions of convective rainfall events were derived from Special Sensor Microwave Imager (SSM/I) and Multi-Channel Passive Microwave Sensor (TMI) microwave radiometric data by means of the Goddard PROFiling (GPROF) algorithm. Continuous lightning distributions were obtained from sferics measurements obtained from a network of VLF radio receivers. Histograms of coincident sferics frequency distributions were matched to those of precipitation to derive bogus convective rainfall rates from the continuously available sferics measurements. SSM/I and TMI microwave data were used to derive Integrated Precipitable Water (IPW) distributions. The TMI also provided sea surface temperatures (SSTS) of the Loop Current and Gulf Stream with improved structural detail. A series of experiments assimilated IPW and latent heating from the bogus convective rainfall for six-hours in the MM5 mesoscale forecast model to produce nine-hour forecasts of all rainfall as well as other weather parameters. Although continuously assimilating latent heating only slightly improved the surface pressure distribution forecast, it significantly improved the precipitation forecasts. Correctly locating convective rainfall was found critical for assimilating latent heating in the forecast model, but measurement of the rainfall intensity proved to be less important. The improved SSTs also had a positive impact on rainfall forecasts for this case. Assimilating bogus rainfall in the model produced nine-hour forecasts of radar reflectivity distributions that agreed well with coincident observations from the TRMM spaceborne precipitation radar, ground based radar and spaceborne microwave measurements.

Weinman, James A.↗

ER-2 Doppler Radar (EDOP) Investigations of the Eyewall of Hurricane Bonnie CAMEX-3

A persistent, mesoscale region of intense eyewall convection contained within Hurricane Bonnie on 23 August 1998 is examined from multiple observations synthesized from the NASA ER-2 and DC-8 aircraft. The intense convection occurred late in the day as Bonnie was attaining its minimum central pressure and during a stage when the inner core featured a markedly asymmetric structure. The main purpose of this paper is to describe the internal structure of a convective burst and its relationship to the warm core using a synthesis of high-resolution satellite, aircraft radar, and in situ data. An exceptionally vigorous eyewall tower penetrating to nearly 18 km is described. A second intense eyewall tower, adjacent to the eye, is shown to be associated with a mesoscale subsiding current of air that extends horizontally nearly 20 km into the eye interior. The subsidence occupies a substantial depth within the eye and appears to be a much larger scale feature than the convectively-induced, symmetric overturning which commonly occurs on the upper-level flanks of convective towers in other tropical environments.

Heymsfield, G. M.↗

Atmospheric Constraints on Landing Site Selection

The Martian atmosphere is a significant part of the environment that the Mars Exploration Rovers (MER) will encounter. As such, it imposes important constraints on where the rovers can and cannot land. Unfortunately, as there are no meteorological instruments on the rovers, there is little atmospheric science that can be accomplished, and no scientific preference for landing sites. The atmosphere constrains landing site selection in two main areas, the entry descent and landing (EDL) process and the survivability of the rovers on the surface. EDL is influenced by the density profile and boundary layer winds (up to altitudes of 5 to 10 km). Surface survivability involves atmospheric dust, temperatures and winds. During EDL, the atmosphere is used to slow the lander down, both ballistically and on the parachute. This limits the maximum elevation of the landing site to -1.3 km below the MOLA reference aeroid. The landers need to encounter a sufficiently dense atmosphere to be able to stop, and the deeper the landing site, the more column integrated atmosphere the lander can pass through before reaching the surface. The current limit was determined both by a desire to be able to reach the hematite region and by a set of atmosphere models we developed for EDL simulations. These are based on Thermal Emission Spectrometer (TES) atmospheric profile measurements, Ames Mars General Circulation Model (MGCM) results, and the 1-D Ames GCM radiative/convective model by J. Murphy. The latter is used for the near surface diurnal cycle. The current version of our model encompasses representative latitude bands, but we intend to make specific models for the final candidate landing sites to insure that they fall within the general envelope. The second constraint imposed on potential landing sites through the EDL process is the near surface wind. The wind in the lower approximately 5 km determines the horizontal velocity that the landers have when they land. Due to the mechanics of the landing process, the total velocity (including both the horizontal and vertical components) determines whether or not the landers are successful. Unfortunately, the landing system has no easy way to nullify any horizontal velocity imparted by the wind, so the landing sites selected need to have as little wind as possible. In addition to the mean wind velocity, the landing system is sensitive to vertical wind shear in the lowest kilometer or so. Wind shear can deflect the retro rockets (RADs) from their nominal vertical orientation producing unwanted horizontal spacecraft velocities. Both mean velocity and wind shear are dominated by the the local topography and other surface properties (in particular albedo and thermal inertia which control the surface temperature). This is seen even in simplified 2-D mesoscale models. The effects in a fully 3-D model are expected to he even more topographically dependent. In particular there is potential for wind channeling in canyons and other terrain features. Boundary layer winds and wind shear are currently being modeled based on terrestrial data and boundary layer scaling laws modified for Martian conditions. We hope to supplement this with mesoscale model results (from several sources) once the number of landing sites is reduced to a manageable number.

Kass, David M.↗

Numerical Simulations of Precipitation Processes, Microphysics, and Microwave Radiative Properties of flood Producing Storms in Mediterranean & Adriatic Basins

A comprehensive understanding of the meteorological and microphysical nature of Mediterranean storms requires a combination of in situ data analysis, radar data analysis, and satellite data analysis, effectively integrated with numerical modeling studies at various scales. An important aspect of understanding microphysical controls of severe storms, is first understanding the meteorological controls under which a storm has evolved, and then using that information to help characterize the dominant microphysical processes. For hazardous Mediterranean storms, highlighted by the October 5-6, 1998 Friuli flood event in northern Italy, a comprehensive microphysical interpretation requires an understanding of the multiple phases of storm evolution. This involves intense convective development, Sratiform decay, orographic lifting, and sloped frontal lifting processes, as well as the associated vertical motions and thermodynamical instabilities governing physical processes that effect details of the size distributions and fall rates of the various types of hydrometeors found within the storm environment. This talk overviews the microphysical elements of a severe Mediterranean storm in such a context, investigated with the aid of TRMM satellite and other remote sensing measurements, but guided by a nonhydrostatic mesoscale model simulation of the Friuli flood event. The data analysis for this paper was conducted by my research groups at the Global Hydrology and Climate Center in Huntsville, AL and Florida State University in Tallahassee, and in collaboration with Dr. Alberto Mugnai's research group at the Institute of Atmospheric Physics in Rome. The numerical modeling was conducted by Professor Oreg Tripoli and Ms. Giulia Panegrossi at the University of Wisconsin in Madison, using Professor Tripoli's nonhydrostatic modeling system (NMS). This is a scalable, fully nested mesoscale model capable of resolving nonhydrostatic circulations from regional scale down to cloud scale and below.

Smith, Eric A.↗

A Study of the Role of Clouds in the Relationship Between Land Use/Land Cover and the Climate and Air Quality of the Atlanta Area

The goal of Project ATLANTA is to derive a better scientific understanding of how land cover changes associated with urbanization affect climate and air quality. In this project the role that clouds play in this relationship was studied. Through GOES satellite observations and RAMS modeling of the Atlanta area, we found that in Atlanta (1) clouds are more frequent than in the surrounding rural areas; (2) clouds cool the surface by shading and thus tend to counteract the warming effect of urbanization; (3) clouds reflect sunlight, which might other wise be used to produce ozone; and (4) clouds decrease biogenic emission of ozone precursors, and they probably decrease ozone concentration. We also found that mesoscale modeling of clouds, especially of small, summertime clouds, needs to be improved and that coupled mesoscale and air quality models are needed to completely understand the mediating role that clouds play in the relationship between land use/land cover change and the climate and air quality of Atlanta. It is strongly recommended that more cities be studied to strengthen and extend these results.

Kidder, Stanley Q.↗

TRMM Validation: Integration of TRMM Satellite and Ground-Validation Data to Determine Tropical Heating

Steiner and Houze showed from ground validation data that the Tropical Rain Measuring Mission (TRMM) satellite Precipitation Radar's (PR's) twice daily only sampling should lead to an uncertainty of approximately 20% in rain estimates. They further showed that the uncertainties are smallest at the 5-7.5 km level. Schumacher and Houze used Kwajalein ground validation data to show that the TRMM PR misses only 2.3% of the near surface rainfall but does not see 46% of the area where rain occurs, because of the 17 dBZ PR reflectivity threshold. Houze discusses how the TRMM data extend earlier tropical convective studies to global coverage of the vertical profile of latent heating via the TRMM PR''s ability to distinguish and globally map convective and stratiform precipitation. Process studies carried out under this TRMM grant Yuter and Houze and Yuter et al. studied ship-based radar observations in the tropical eastern Pacific ITCZ. The eastern Pacific precipitation process is different from the western Pacific (the COARE area); rain is heavier but the clouds are not as deep. These process differences may affect the ability to remotely sense precipitation accurately in the two regions. Satellite microwave data were able to detect the precipitation as long as the rain areas exceeded 10 km in dimension. However, the microwave algorithms had difficulty distinguishing light and heavy rain. Satellite IR algorithms only partially detected the rain because the tops of the smaller and more short-lived rain clouds were sometimes not cold enough for the IR algorithms to detect them. Houze et al. focused on the west Pacific precipitating mesoscale convective systems and showed how their precipitation and internal dynamics vary in relation to the slowly varying large-scale heating-driven circulation, which has a structure described by a combination of Kelvin and Rossby wave response to the near-equatorial convective heating constituted by the mesoscale convective systems. Ship and aircraft radar data were used in this study.

Houze, Robert A., Jr.↗

Routine High-Resolution Forecasts/Analyses for the Pacific Disaster Center: User Manual

Enclosed herein is our HWCMO user manual. This manual constitutes the final report for our NASA/PDC grant, NASA NAG5-8730, "Routine High Resolution Forecasts/Analysis for the Pacific Disaster Center". Since the beginning of the grant, we have routinely provided experimental high resolution forecasts from the RSM/MSM for the Hawaii Islands, while working to upgrade the system to include: (1) a more robust input of NCEP analyses directly from NCEP; (2) higher vertical resolution, with increased forecast accuracy; (3) faster delivery of forecast products and extension of initial 1-day forecasts to 2 days; (4) augmentation of our basic meteorological and simplified fireweather forecasts to firedanger and drought forecasts; (5) additional meteorological forecasts with an alternate mesoscale model (MM5); and (6) the feasibility of using our modeling system to work in higher-resolution domains and other regions. In this user manual, we provide a general overview of the operational system and the mesoscale models as well as more detailed descriptions of the models. A detailed description of daily operations and a cost analysis is also provided. Evaluations of the models are included although it should be noted that model evaluation is a continuing process and as potential problems are identified, these can be used as the basis for making model improvements. Finally, we include our previously submitted answers to particular PDC questions (Appendix V). All of our initially proposed objectives have basically been met. In fact, a number of useful applications (VOG, air pollution transport) are already utilizing our experimental output and we believe there are a number of other applications that could make use of our routine forecast/analysis products. Still, work still remains to be done to further develop this experimental weather, climate, fire danger and drought prediction system. In short, we would like to be a part of a future PDC team, if at all possible, to further develop and apply the system for the Hawaiian and other Pacific Islands as well as the entire Pacific Basin.

Roads, John↗

Numerical Study of the Role of Shallow Convection in Moisture Transport and Climate

The objective of this investigation was to study the role of shallow convection on the regional water cycle of the Mississippi and Little Washita Basins of the Southern Great Plains (SGP) using a 3-D mesoscale model, the PSU/NCAR MM5. The underlying premise of the project was that current modeling of regional-scale climate and moisture cycles over the continents is deficient without adequate treatment of shallow convection. At the beginning of the study, it was hypothesized that an improved treatment of the regional water cycle can be achieved by using a 3-D mesoscale numerical model having high-quality parameterizations for the key physical processes controlling the water cycle. These included a detailed land-surface parameterization (the Parameterization for Land-Atmosphere-Cloud Exchange (PLACE) sub-model of Wetzel and Boone), an advanced boundary-layer parameterization (the 1.5-order turbulent kinetic energy (TKE) predictive scheme of Shafran et al.), and a more complete shallow convection parameterization (the hybrid-closure scheme of Deng et al.) than are available in most current models. PLACE is a product of researchers working at NASA's Goddard Space Flight Center in Greenbelt, MD. The TKE and shallow-convection schemes are the result of model development at Penn State. The long-range goal is to develop an integrated suite of physical sub-models that can be used for regional and perhaps global climate studies of the water budget. Therefore, the work plan focused on integrating, improving, and testing these parameterizations in the MM5 and applying them to study water-cycle processes over the SGP. These schemes have been tested extensively through the course of this study and the latter two have been improved significantly as a consequence.

Seaman, Nelson L.↗

Development and Testing of Coupled Land-surface, PBL and Shallow/Deep Convective Parameterizations within the MM5

The objective of this investigation was to study the role of shallow convection on the regional water cycle of the Mississippi and Little Washita Basins using a 3-D mesoscale model, the PSUINCAR MM5. The underlying premise of the project was that current modeling of regional-scale climate and moisture cycles over the continents is deficient without adequate treatment of shallow convection. It was hypothesized that an improved treatment of the regional water cycle can be achieved by using a 3-D mesoscale numerical model having a detailed land-surface parameterization, an advanced boundary-layer parameterization, and a more complete shallow convection parameterization than are available in most current models. The methodology was based on the application in the MM5 of new or recently improved parameterizations covering these three physical processes. Therefore, the work plan focused on integrating, improving, and testing these parameterizations in the MM5 and applying them to study water-cycle processes over the Southern Great Plains (SGP): (1) the Parameterization for Land-Atmosphere-Cloud Exchange (PLACE) described by Wetzel and Boone; (2) the 1.5-order turbulent kinetic energy (TKE)-predicting scheme of Shafran et al.; and (3) the hybrid-closure sub-grid shallow convection parameterization of Deng. Each of these schemes has been tested extensively through this study and the latter two have been improved significantly to extend their capabilities.

Stauffer, David R.↗

Microphysics, Meteorology, Microwave and Modeling of Mediterranean Storms: The M(sup 5) Problem

Comprehensive understanding of the microphysical nature of Mediterranean storms requires a combination of in situ meteorological data analysis and radar-passive microwave data analysis, effectively integrated with numerical modeling studies at various scales, particularly from synoptic scale down to mesoscale. The microphysical properties of and their controls on severe storms are intrinsically related to meteorological processes under which storms have evolved, processes which eventually select and control the dominant microphysical properties themselves. Insofar as hazardous Mediterranean storms, highlighted by the September 25-28/1992 Genova flood event, the October 5-7/1998 Friuli flood event, and the October 13-15/2000 Piemonte flood event (all taking place in northern Italy), developing a comprehensive microphysical interpretation requires an understanding of the multiple phases of storm evolution and the heterogeneous nature of precipitation fields within the storm domains. This involves convective development, stratiform transition and decay, orographic lifting, and sloped frontal lifting proc esses. This also involves vertical motions and thermodynamical instabilities governing physical processes that determine details of the liquid/ice water contents, size distributions, and fall rates of the various modes of hydrometeors found within the storm environments. This paper presents detailed 4-dimensional analyses of the microphysical elements of the three severe Mediterranean storms identified above, investigated with the aid of SSM/I and TRMM satellite measurements (and other remote sensing measurements). The analyses are guided by nonhydrostatic mesoscale model simulations at high resolution of the intense rain producing portions of the storm environments. The results emphasize how meteorological controls taking place at the large scale, coupled with localized terrain controls, ultimately determine the most salient features of the bulk microphysical properties of the storms. These results have bearing on precipitation remote sensing from space, and the role of modeling in designing precipitation retrieval algorithms.

Smith, Eric A.↗

Regional Data Assimilation Using a Stretched-Grid Approach and Ensemble Calculations

The global variable resolution stretched grid (SG) version of the Goddard Earth Observing System (GEOS) Data Assimilation System (DAS) incorporating the GEOS SG-GCM (Fox-Rabinovitz 2000, Fox-Rabinovitz et al. 2001a,b), has been developed and tested as an efficient tool for producing regional analyses and diagnostics with enhanced mesoscale resolution. The major area of interest with enhanced regional resolution used in different SG-DAS experiments includes a rectangle over the U.S. with 50 or 60 km horizontal resolution. The analyses and diagnostics are produced for all mandatory levels from the surface to 0.2 hPa. The assimilated regional mesoscale products are consistent with global scale circulation characteristics due to using the SG-approach. Both the stretched grid and basic uniform grid DASs use the same amount of global grid-points and are compared in terms of regional product quality.

Fox-Rabinovitz, M. S.↗