Results for “Variational” — Page 79

NASA NTRS2013

Effects of the Observed Meridional Flow Variations since 1996 on the Sun's Polar Fields

The cause of the low and extended minimum in solar activity between Sunspot Cycles 23 and 24 was the small size of Sunspot Cycle 24 itself - small cycles start late and leave behind low minima. Cycle 24 is small because the polar fields produced during Cycle 23 were substantially weaker than those produced during the previous cycles and those (weak) polar fields are the seeds for the activity of the following cycle. The polar fields are produced by the latitudinal transport of magnetic flux that emerged in low-latitude active regions. The polar fields thus depend upon the details of both the flux emergence and the flux transport. We have measured the flux transport flows (differential rotation, meridional flow, and supergranules) since 1996 and find systematic and substantial variation in the meridional flow alone. Here we present experiments using a Surface Flux Transport Model in which magnetic field data from SOHO/MDI and SDO/HMI are assimilated into the model only at latitudes between 45-degrees north and south of the equator (this assures that the details of the active region flux emergence are well represented). This flux is then transported in both longitude and latitude by the observed flows. In one experiment the meridional flow is given by the time averaged (and north-south symmetric) meridional flow profile. In the second experiment the time-varying and north-south asymmetric meridional flow is used. Differences between the observed polar fields and those produced in these two experiments allow us to ascertain the effects of these meridional flow variations on the Sun s polar fields.

Hathaway, David↗

NASA NTRS2013

Effects of the Observed Meridional Flow Variations since 1996 on the Sun's Polar Fields

The cause of the low and extended minimum in solar activity between Sunspot Cycles 23 and 24 was the small size of Sunspot Cycle 24 itself - small cycles start late and leave behind low minima. Cycle 24 is small because the polar fields produced during Cycle 23 were substantially weaker than those produced during the previous cycles and those (weak) polar fields are the seeds for the activity of the following cycle. The polar fields are produced by the latitudinal transport of magnetic flux that emerged in low-latitude active regions. The polar fields thus depend upon the details of both the flux emergence and the flux transport. We have measured the flux transport flows (differential rotation, meridional flow, and supergranules) since 1996 and find systematic and substantial variation in the meridional flow alone. Here we present experiments using a Surface Flux Transport Model in which magnetic field data from SOHO/MDI and SDO/HMI are assimilated into the model only at latitudes between 45-degrees north and south of the equator (this assures that the details of the active region flux emergence are well represented). This flux is then transported in both longitude and latitude by the observed flows. In one experiment the meridional flow is given by the time averaged (and north-south symmetric) meridional flow profile. In the second experiment the time-varying and north-south asymmetric meridional flow is used. Differences between the observed polar fields and those produced in these two experiments allow us to ascertain the effects of these meridional flow variations on the Sun s polar fields.

Hathaway, David H.↗

NASA NTRS2013

Implementation of Speed Variation in the Structural Dynamic Assessment of Turbomachinery Flow-Path Components

During the design of turbomachinery flow path components, the assessment of possible structural resonant conditions is critical. Higher frequency modes of these structures are frequently found to be subject to resonance, and in these cases, design criteria require a forced response analysis of the structure with the assumption that the excitation speed exactly equals the resonant frequency. The design becomes problematic if the response analysis shows a violation of the HCF criteria. One possible solution is to perform "finite-life" analysis, where Miner's rule is used to calculate the actual life in seconds in comparison to the required life. In this situation, it is beneficial to incorporate the fact that, for a variety of turbomachinery control reasons, the speed of the rotor does not actually dwell at a single value but instead dithers about a nominal mean speed and during the time that the excitation frequency is not equal to the resonant frequency, the damage accumulated by the structure is diminished significantly. Building on previous investigations into this process, we show that a steady-state assumption of the response is extremely accurate for this typical case, resulting in the ability to quickly account for speed variation in the finite-life analysis of a component which has previously had its peak dynamic stress at resonance calculated. A technique using Monte Carlo simulation is also presented which can be used when specific speed time histories are not available. The implementation of these techniques can prove critical for successful turbopump design, as the improvement in life when speed variation is considered is shown to be greater than a factor of two.

Brown, Andrew M.↗

NASA NTRS2013

Implementation of Speed Variation in the Structural Dynamic Assessment of Turbomachinery Flow-Path Components

During the design of turbomachinery flow path components, the assessment of possible structural resonant conditions is critical. Higher frequency modes of these structures are frequently found to be subject to resonance, and in these cases, design criteria require a forced response analysis of the structure with the assumption that the excitation speed exactly equals the resonant frequency. The design becomes problematic if the response analysis shows a violation of the HCF criteria. One possible solution is to perform "finite-life" analysis, where Miner's rule is used to calculate the actual life in seconds in comparison to the required life. In this situation, it is beneficial to incorporate the fact that, for a variety of turbomachinery control reasons, the speed of the rotor does not actually dwell at a single value but instead dithers about a nominal mean speed and during the time that the excitation frequency is not equal to the resonant frequency, the damage accumulated by the structure is diminished significantly. Building on previous investigations into this process, we show that a steady-state assumption of the response is extremely accurate for this typical case, resulting in the ability to quickly account for speed variation in the finite-life analysis of a component which has previously had its peak dynamic stress at resonance calculated. A technique using Monte Carlo simulation is also presented which can be used when specific speed time histories are not available. The implementation of these techniques can prove critical for successful turbopump design, as the improvement in life when speed variation is considered is shown to be greater than a factor of two

Brown, Andrew M.↗

NASA NTRS2014

Comparison of the Variations of Sunspot Number, Number of Sunspot Groups, and Sunspot Area, 1875 -2013

Examined are the yearly variations and ratios of sunspot number, the number of sunspot groups, and the total corrected sunspot area for the interval 1875-2013. While yearly sunspot number independently correlates strongly (r = 0.98) with the yearly number of sunspot groups (y = -2 + 11.99x) and the total corrected sunspot area (y = 5 + 0.059x), the strongest correlation (Ry12 = 0.99) is the one based on the bivariate fit of sunspot number against the combined variations of the number of sunspot groups and sunspot area (y = 1 + 5.88x1 + 0.031x2, where y refers to sunspot number, x1 refers to the number of sunspot groups, and x2 refers to the sunspot area). While all cycle minima based on the bivariate fit are concurrent with the observed minimum in sunspot number, cycle maxima are sometimes found to differ. For sunspot cycles 12, 19, 20, and 23, cycle maximum is inferred to have occurred in 1884, 1958, 1970, and 2002, respectively, rather than in 1883, 1957, 1968, and 2000, based on the observed sunspot number. Also, cycle 19's maximum amplitude based on observed sunspot number seems too high in comparison to that found using the bivariate fit. During the 139-year interval 1875-2013, the difference between the observed and predicted sunspot number based on the bivariate fit is <1 standard error of estimate (se) (<6.4) for 111 years, between 1 and <2 se (6.4 to <12.8) for 28 years, and ≥2 se (≥12.8) for only 4 years, these years being 1957 (16.6), 1978 (-15.8), 1980 (23), and 1982 (-16.3). For sunspot cycle 24, the difference between observed and predicted values has been only -0.7 and 3.2 (≤0.5 se).

Wilson, Robert M.↗

NASA NTRS2013

Tropical Atlantic Dust and Smoke Aerosol Variations Related to the Madden-Julian Oscillation in MODIS and MISR Observations

In this study, Moderate Resolution Imaging Spectroradiometer (MODIS) fine mode fraction and Multi-angle Imaging SpectroRadiometer (MISR) nonspherical fraction data are used to derive dust and smoke aerosol optical thickness (T(sub dust) and T(sub smoke)) over the tropical Atlantic in a complementary way: due to its wider swath, MODIS has 3-4 times greater sampling than MISR, but MISR dust discrimination is based on particle shape retrievals, whereas an empirical scheme is used for MODIS. MODIS and MISR show very similar dust and smoke winter climatologies. T(sub dust) is the dominant aerosol component over the tropical Atlantic, accounting for 40-70 percent of the total aerosol optical thickness (AOT), whereas T(sub smoke) is significantly smaller than T(sub dust). The consistency and high correlation between these climatologies and their daily variations lends confidence to their use for investigating the relative dust and smoke contributions to the total AOT variation associated with the Madden-Julian Oscillation (MJO). The temporal evolution and spatial patterns of the τdus anomalies associated with the MJO are consistent between MODIS and MISR: the magnitude of MJO-realted T(sub dust) anomalies is comparable to or even larger than that of the total T, while the T(sub smoke) anomaly represents about 15 percent compared to the total, which is quite different from their relative magnitudes to the total T on the climatological time scale. This suggests that dust and smoke are not influenced by the MJO in the same way. Based on correlation analysis, dust is strongly influenced by the MJO-modulated trade wind and precipitation anomalies, and can last as long as one MJO phase, whereas smoke is less affected.

Madden-Julian Oscillation↗

NASA NTRS2013

Interannual Variations of MLS Carbon Monoxide Induced by Solar Cycle

More than eight years (2004-2012) of carbon monoxide (CO) measurements from the Aura Microwave Limb Sounder (MLS) are analyzed. The mesospheric CO, largely produced by the carbon dioxide (CO2) photolysis in the lower thermosphere, is sensitive to the solar irradiance variability. The long-term variation of observed mesospheric MLS CO concentrations at high latitudes is likely driven by the solar-cycle modulated UV forcing. Despite of different CO abundances in the southern and northern hemispheric winter, the solar-cycle dependence appears to be similar. This solar signal is further carried down to the lower altitudes by the dynamical descent in the winter polar vortex. Aura MLS CO is compared with the Solar Radiation and Climate Experiment (SORCE) total solar irradiance (TSI) and also with the spectral irradiance in the far ultraviolet (FUV) region from the SORCE Solar-Stellar Irradiance Comparison Experiment (SOLSTICE). Significant positive correlation (up to 0.6) is found between CO and FUVTSI in a large part of the upper atmosphere. The distribution of this positive correlation in the mesosphere is consistent with the expectation of CO changes induced by the solar irradiance variations.

carbon monoxide↗

NASA NTRS2014

Correlation of ISS Electric Potential Variations with Mission Operations

Orbiting approximately 400 km above the Earth, the International Space Station (ISS) is a unique research laboratory used to conduct ground-breaking science experiments in space. The ISS has eight Solar Array Wings (SAW), and each wing is 11.7 meters wide and 35.1 meters long. The SAWs are controlled individually to maximize power output, minimize stress to the ISS structure, and minimize interference with other ISS operations such as vehicle dockings and Extra-Vehicular Activities (EVA). The Solar Arrays are designed to operate at 160 Volts. These large, high power solar arrays are negatively grounded to the ISS and collect charged particles (predominately electrons) as they travel through the space plasma in the Earth's ionosphere. If not controlled, this collected charge causes floating potential variations which can result in arcing, causing injury to the crew during an EVA or damage to hardware [1]. The environmental catalysts for ISS floating potential variations include plasma density and temperature fluctuations and magnetic induction from the Earth's magnetic field. These alone are not enough to cause concern for ISS, but when they are coupled with the large positive potential on the solar arrays, floating potentials up to negative 95 Volts have been observed. Our goal is to differentiate the operationally induced fluctuations in floating potentials from the environmental causes. Differentiating will help to determine what charging can be controlled, and we can then design the proper operations controls for charge collection mitigation. Additionally, the knowledge of how high power solar arrays interact with the environment and what regulations or design techniques can be employed to minimize charging impacts can be applied to future programs.

Willis, Emily M.↗

NASA NTRS2014

L-Band Brightness Temperature Variations at Dome C and Snow Metamorphism at the Surface

The Antarctic Plateau is a promising site to monitor microwave radiometers' drift, and to inter-calibrate microwave radiometers, especially 1.4 GigaHertz (L-band) radiometers on board the Soil Moisture and Ocean Salinity (SMOS), and AquariusSAC-D missions. The Plateau is a thick ice cover, thermally stable in depth, with large dimensions, and relatively low heterogeneities. In addition, its high latitude location in the Southern Hemisphere enables frequent observations by polar-orbiting satellites, and no contaminations by radio frequency interference. At Dome C (75S, 123E), on the Antarctic Plateau, the substantial amount of in-situ snow measurements available allows us to interpret variations in space-borne microwave brightness temperature (TB) (e.g. Macelloni et al., 2007, 2013, Brucker et al., 2011, Champollion et al., 2013). However, to analyze the observations from the Aquarius radiometers, whose sensitivity is 0.15 K, the stability of the snow layers near the surface that are most susceptible to rapidly change needs to be precisely assessed. This study focuses on the spatial and temporal variations of the Aquarius TB over the Antarctic Plateau, and at Dome C in particular, to highlight the impact of snow surface metamorphism on the TB observations at L-band.

l-band↗

NASA NTRS2014

Aquarius Brightness Temperature Variations at Dome C and Snow Metamorphism at the Surface

The Antarctic Plateau is a promising site to monitor microwave radiometers' drift, and to inter-calibrate microwave radiometers, especially 1.4 GHz (L-band) radiometers on board the Soil Moisture and Ocean Salinity (SMOS), and AquariusSAC-D missions. The Plateau is a thick ice cover, thermally stable in depth, with large dimensions, and relatively low heterogeneities. In addition, its high latitude location in the Southern Hemisphere enables frequent observations by polar-orbiting satellites, and no contaminations by radio frequency interference. At Dome C (75S, 123E), on the Antarctic Plateau, the substantial amount of in-situ snow measurements available allows us to interpret variations in space-borne microwave brightness temperature (TB) (e.g. Macelloni et al., 2007, 2013, Brucker et al., 2011, Champollion et al., 2013). However, to analyze the observations from the Aquarius radiometers, whose sensitivity is 0.15 K, the stability of the snow layers near the surface that are most susceptible to rapidly change needs to be precisely assessed. This study focuses on the spatial and temporal variations of the Aquarius TB over the Antarctic Plateau, and at Dome C in particular, to highlight the impact of snow surface metamorphism on the TB observations at L-band.

l-band↗

NASA NTRS2014

When a Standard Candle Flickers: Hard X-ray Variations in the Crab Nebula

In the first two years of science operations of the Fermi Gamma-ray Burst Monitor (GBM), August 2008 to August 2010, an approximately 7% (70 mcrab) decline was discovered in the overall Crab nebula flux in the 15 - 50 keV band, measured with the Earth occultation technique. This decline was independently confirmed with four other instruments: the RXTE/PCA, Swift/BAT, INTEGRAL/IBIS, and INTEGRAL/SPI. The pulsed flux measured with RXTE/PCA from 1999-2010 was consistent with the pulsar spin-down, indicating that the observed changes were nebular. From 2001 to 2010, the Crab nebula flux measured with RXTE/PCA was particularly variable, changing by up to approximately3.5% per year in the 15-50 keV band. These variations were confirmed with INTEGRAL/SPI starting in 2003, Swift/BAT starting in 2005, and Fermi GBM starting in 2008. Before 2001 and since 2010, the Crab nebula flux has appeared more stable, varying by less than 2% per year. At higher energies, above 50 keV, the Crab flux appears to be slowly recovering to its 2008 levels. I will present updated light curves in multiple energy bands for the Crab nebula, including recent data from Fermi GBM, Swift/BAT, INTEGRAL, MAXI, and NuSTAR and a 16-year long light curve from RXTE/PCA. We will compare these variations to higher energies as well, e.g. Fermi LAT.

Wilson-Hodge, Colleen↗

NASA NTRS2011

On the Non-Monotonic Variation of the Opposition Surge Morphology with Albedo Exhibited by Satellites' Surface

We used well know phase functions of satellites and rings around the giant planets of our Solar System to study the morphology of the opposition effect (at phase angles alpha < 20 degrees. To avoid the effect of the variable finite size of the Sun, we use a deconvolution morphological model to retrieve the morphological parameters of the surge (A and HWHM). These parameters are found to have a non-monotonic variation with the single scattering albedo, similar to that observed in asteroids, which is unexplained so far. The non-monotonic variation is discussed in the framework of the coherent backscattering and shadow hiding mechanisms.

solar system↗

NASA NTRS2010

Monthly Variations of Low-Energy Ballistic Transfers to Lunar Halo Orbits

The characteristics of low-energy transfers between the Earth and Moon vary from one month to the next largely due to the Earth's and Moon's non-circular, non-coplanar orbits in the solar system. This paper characterizes those monthly variations as it explores the trade space of low-energy lunar transfers across many months. Mission designers may use knowledge of these variations to swiftly design desirable low-energy lunar transfers in any given month.

Mission Design↗

NASA NTRS2015

OSOAA: A Vector Radiative Transfer Model of Coupled Atmosphere-Ocean System for a Rough Sea Surface Application to the Estimates of the Directional Variations of the Water Leaving Reflectance to Better Process Multi-angular Satellite Sensors Data Over the Ocean

In this study, we present a radiative transfer model, so-called OSOAA, that is able to predict the radiance and degree of polarization within the coupled atmosphere-ocean system in the presence of a rough sea surface. The OSOAA model solves the radiative transfer equation using the successive orders of scattering method. Comparisons with another operational radiative transfer model showed a satisfactory agreement within 0.8%. The OSOAA model has been designed with a graphical user interface to make it user friendly for the community. The radiance and degree of polarization are provided at any level, from the top of atmosphere to the ocean bottom. An application of the OSOAA model is carried out to quantify the directional variations of the water leaving reflectance and degree of polarization for phytoplankton and mineral-like dominated waters. The difference between the water leaving reflectance at a given geometry and that obtained for the nadir direction could reach 40%, thus questioning the Lambertian assumption of the sea surface that is used by inverse satellite algorithms dedicated to multi-angular sensors. It is shown as well that the directional features of the water leaving reflectance are weakly dependent on wind speed. The quantification of the directional variations of the water leaving reflectance obtained in this study should help to correctly exploit the satellite data that will be acquired by the current or forthcoming multi-angular satellite sensors.

Ocean surface↗

NASA NTRS2016

Monitoring Interannual Variation in Global Crop Yield Using Long-Term AVHRR and MODIS Observations

Advanced Very High Resolution Radiometer (AVHRR) and Moderate Resolution Imaging Spectroradiometer (MODIS) data have been extensively applied for crop yield prediction because of their daily temporal resolution and a global coverage. This study investigated global crop yield using daily two band Enhanced Vegetation Index (EVI2) derived from AVHRR (1981-1999) and MODIS (2000-2013) observations at a spatial resolution of 0.05deg (approx.5 km). Specifically, EVI2 temporal trajectory of crop growth was simulated using a hybrid piecewise logistic model (HPLM) for individual pixels, which was used to detect crop phenological metrics. The derived crop phenology was then applied to calculate crop greenness defined as EVI2 amplitude and EVI2 integration during annual crop growing seasons, which was further aggregated for croplands in each country, respectively. The interannual variations in EVI2 amplitude and EVI2 integration were combined to correlate to the variation in cereal yield from 1982-2012 for individual countries using a stepwise regression model, respectively. The results show that the confidence level of the established regression models was higher than 90% (P value < 0.1) in most countries in the northern hemisphere although it was relatively poor in the southern hemisphere (mainly in Africa). The error in the yield predication was relatively smaller in America, Europe and East Asia than that in Africa. In the 10 countries with largest cereal production across the world, the prediction error was less than 9% during past three decades. This suggests that crop phenology-controlled greenness from coarse resolution satellite data has the capability of predicting national crop yield across the world, which could provide timely and reliable crop information for global agricultural trade and policymakers.

Crop phenology↗

NASA NTRS2015

Comparison of Surface and Column Variations of CO2 Over Urban Areas for Future Active Remote CO2 Sensors

High resolution in-situ CO2 measurements were recorded onboard the NASA P-3B during the DISCOVER-AQ (Deriving Information on Surface Conditions from Column and Vertically Resolved Observations Relevant to Air Quality) Field Campaign, to investigate the ability of space-based observations to accurately assess near surface conditions related to air quality. This campaign includes, Washington DC/Baltimore, MD (July 2011), San Joaquin Valley, CA (January - February 2013), Houston, TX (September 2013), and Denver, CO (July-August 2014). Each of these campaigns consisted of missed approaches and approximately two hundred vertical soundings of CO2 within the lower troposphere (surface to about 5 km). In this study, surface (0 - 1 km) and column-averaged (0 - 3.5 km) CO2 mixing ratio values from the vertical soundings in the four geographically different urban areas are used to investigate the temporal and spatial variability of CO2 within the different urban atmospheric emission environments. Tracers such as CO, CH2O, NOx, and NMHCs are used to identify the source of CO2 variations in the urban sites. Additionally, we apply nominal CO2 column weighting functions for potential future active remote CO2 sensors operating in the 1.57-microns and 2.05-microns measurement regions to convert the in situ CO2 vertical mixing ratio profiles to variations in CO2 column optical depths, which is what the active remote sensors actually measure. Using statistics calculated from the optical depths at each urban site measured during the DISCOVER-AQ field campaign and for each nominal weighting function, we investigate the natural variability of CO2 columns in the lower troposphere; relate the CO2 column variability to the urban surface emissions; and show the measurement requirements for the future ASCENDS (Active Sensing of CO2 Emissions over Nights, Days, and Seasons) in the continental U.S. urban areas.

Choi, Yonghoon↗

NASA NTRS2016

Small vs. Large Convective Cloud Objects from CERES Aqua Observations: Where are the Intraseasonal Variation Signals?

During inactive phases of Madden-Julian oscillation (MJO), there are plenty of deep but small convective systems and far fewer deep and large ones. During active phases of MJO, a manifestation of an increase in the occurrence of large and deep cloud clusters results from an amplification of large-scale motions by stronger convective heating. This study is designed to quantitatively examine the roles of small and large cloud clusters during the MJO life cycle. We analyze the cloud object data from Aqua CERES observations for tropical deep convective (DC) and cirrostratus (CS) cloud object types according to the real-time multivariate MJO index. The cloud object is a contiguous region of the earth with a single dominant cloud-system type. The size distributions, defined as the footprint numbers as a function of cloud object diameters, for particular MJO phases depart greatly from the combined (8-phase) distribution at large cloud-object diameters due to the reduced/increased numbers of cloud objects related to changes in the large-scale environments. The medium diameter corresponding to the combined distribution is determined and used to partition all cloud objects into "small" and "large" groups of a particular phase. The two groups corresponding to the combined distribution have nearly equal numbers of footprints. The medium diameters are 502 km for DC and 310 km for cirrostratus. The range of the variation between two extreme phases (typically, the most active and depressed phases) for the small group is 6-11% in terms of the numbers of cloud objects and the total footprint numbers. The corresponding range for the large group is 19-44%. In terms of the probability density functions of radiative and cloud physical properties, there are virtually no differences between the MJO phases for the small group, but there are significant differences for the large groups for both DC and CS types. These results suggest that the intreseasonal variation signals reside at the large cloud clusters while the small cloud clusters represent the background noises resulting from various types of the tropical waves with different wavenumbers and propagation directions/speeds.

Xu, Kuan-Man↗

NASA NTRS2015

Background Noises Versus Intraseasonal Variation Signals: Small vs. Large Convective Cloud Objects From CERES Aqua Observations

During inactive phases of Madden-Julian Oscillation (MJO), there are plenty of deep but small convective systems and far fewer deep and large ones. During active phases of MJO, a manifestation of an increase in the occurrence of large and deep cloud clusters results from an amplification of large-scale motions by stronger convective heating. This study is designed to quantitatively examine the roles of small and large cloud clusters during the MJO life cycle. We analyze the cloud object data from Aqua CERES (Clouds and the Earth's Radiant Energy System) observations between July 2006 and June 2010 for tropical deep convective (DC) and cirrostratus (CS) cloud object types according to the real-time multivariate MJO index, which assigns the tropics to one of the eight MJO phases each day. The cloud object is a contiguous region of the earth with a single dominant cloud-system type. The criteria for defining these cloud types are overcast footprints and cloud top pressures less than 400 hPa, but DC has higher cloud optical depths (=10) than those of CS (<10). The size distributions, defined as the footprint numbers as a function of cloud object diameters, for particular MJO phases depart greatly from the combined (8-phase) distribution at large cloud-object diameters due to the reduced/increased numbers of cloud objects related to changes in the large-scale environments. The medium diameter corresponding to the combined distribution is determined and used to partition all cloud objects into "small" and "large" groups of a particular phase. The two groups corresponding to the combined distribution have nearly equal numbers of footprints. The medium diameters are 502 km for DC and 310 km for cirrostratus. The range of the variation between two extreme phases (typically, the most active and depressed phases) for the small group is 6-11% in terms of the numbers of cloud objects and the total footprint numbers. The corresponding range for the large group is 19-44%. In terms of the probability density functions of radiative and cloud physical properties, there are virtually no differences between the MJO phases for the small group, but there are significant differences for the large groups for both DC and CS types. These results suggest that the intreseasonal variation signals reside at the large cloud clusters while the small cloud clusters represent the background noises resulting from various types of the tropical waves with different wavenumbers and propagation speeds/directions.

Xu, Kuan-Man↗