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NASA NTRS2004

Characterization of Asian Dust Properties Near Source Region During ACE-Asia

Asian dust typically originates in desert areas far from polluted urban regions. During transport, dust layers can interact with anthropogenic sulfate and soot aerosols from heavily polluted urban areas. Added to the complex effects of clouds and natural marine aerosols, dust particles reaching the marine environment can have drastically different properties than those from the source. Thus, understanding the unique temporal and spatial variations of Asian aerosols is of special importance in regional-to-global climate issues such as radiative forcing, the hydrological cycle, and primary biological productivity in the mid-Pacific Ocean. During ACE-Asia campaign, we have acquired ground- based (temporal) and satellite (spatial) measurements to infer aerosol physical/optical/radiative properties, column precipitable water amount, and surface reflectivity over this region. The inclusion of flux measurements permits the determination of aerosol radiative flux in addition to measurements of loading and optical depth. At the time of the Terra/MODIS, SeaWiFS, TOMS and other satellite overpasses, these ground-based observations can provide valuable data to compare with satellite retrievals over land. In this paper, we will demonstrate new capability of the Deep Blue algorithm to track the evolution of the Asian dust storm from sources to sinks. Although there are large areas often covered by clouds in the dust season in East Asia, this algorithm is able to distinguish heavy dust from clouds over the entire regions. Examination of the retrieved daily maps of dust plumes over East Asia clearly identifies the sources contributing to the dust loading in the atmosphe~~. We have compared the satellite retrieved aerosol optical thickness to the ground-based measurements and obtained a reasonable agreement between these two. Our results also indicate that there is a large difference in the retrieved value of spectral single scattering albedo of windblown dust between different sources in East Asia.

Tsay, Si-Chee↗

NASA NTRS2012

Estimating Climatological Bias Errors for the Global Precipitation Climatology Project (GPCP)

A procedure is described to estimate bias errors for mean precipitation by using multiple estimates from different algorithms, satellite sources, and merged products. The Global Precipitation Climatology Project (GPCP) monthly product is used as a base precipitation estimate, with other input products included when they are within +/- 50% of the GPCP estimates on a zonal-mean basis (ocean and land separately). The standard deviation s of the included products is then taken to be the estimated systematic, or bias, error. The results allow one to examine monthly climatologies and the annual climatology, producing maps of estimated bias errors, zonal-mean errors, and estimated errors over large areas such as ocean and land for both the tropics and the globe. For ocean areas, where there is the largest question as to absolute magnitude of precipitation, the analysis shows spatial variations in the estimated bias errors, indicating areas where one should have more or less confidence in the mean precipitation estimates. In the tropics, relative bias error estimates (s/m, where m is the mean precipitation) over the eastern Pacific Ocean are as large as 20%, as compared with 10%-15% in the western Pacific part of the ITCZ. An examination of latitudinal differences over ocean clearly shows an increase in estimated bias error at higher latitudes, reaching up to 50%. Over land, the error estimates also locate regions of potential problems in the tropics and larger cold-season errors at high latitudes that are due to snow. An empirical technique to area average the gridded errors (s) is described that allows one to make error estimates for arbitrary areas and for the tropics and the globe (land and ocean separately, and combined). Over the tropics this calculation leads to a relative error estimate for tropical land and ocean combined of 7%, which is considered to be an upper bound because of the lack of sign-of-the-error canceling when integrating over different areas with a different number of input products. For the globe the calculated relative error estimate from this study is about 9%, which is also probably a slight overestimate. These tropical and global estimated bias errors provide one estimate of the current state of knowledge of the planet's mean precipitation.

Global Precipitation Climatology Project (GPCP)↗

NASA NTRS1990

A network flow model for load balancing in circuit-switched multicomputers

In multicomputers that utilize circuit switching or wormhole routing, communication overhead depends largely on link contention - the variation due to distance between nodes is negligible. This has a major impact on the load balancing problem. In this case, there are some nodes with excess load (sources) and others with deficit load (sinks) and it is required to find a matching of sources to sinks that avoids contention. The problem is made complex by the hardwired routing on currently available machines: the user can control only which nodes communicate but not how the messages are routed. Network flow models of message flow in the mesh and the hypercube were developed to solve this problem. The crucial property of these models is the correspondence between minimum cost flows and correctly routed messages. To solve a given load balancing problem, a minimum cost flow algorithm is applied to the network. This permits one to determine efficiently a maximum contention free matching of sources to sinks which, in turn, tells one how much of the given imbalance can be eliminated without contention.

Bokhari, Shahid H.↗

NASA NTRS1993

A network flow model for load balancing in circuit-switched multicomputers

Bokhari, Shahid H.↗

NASA NTRS1994

Optimized design of a hypersonic nozzle

Conventional procedures for designing nozzles involve the design of an inviscid contour (using the method of characteristics) that is corrected with a displacement thickness calculated from boundary-layer theory. However, nozzles designed using this classical procedure have been shown to exhibit poor flow quality at Mach numbers characteristic of hypersonic applications. The nozzle to be designed will be a part of the NASA HYPULSE facility which is being used for hypervelocity flight research. Thus, the flow quality of the nozzle is a critical question that needs to be addressed. Design of nozzles for hypersonic applications requires a proper assessment of the effects of the thick boundary layer on the inviscid flowfield. Since the flow field is largely supersonic, the parabolized form of the Navier-Stokes (PNS) equations can be used. The requirement of a uniform flow at the exit plane of the nozzle can be used to define an objective function as part of an optimization procedure. The design procedure used in this study involves the coupling of a nonlinear (least-squares) optimization algorithm with an efficient, explicit PNS solver. The thick boundary layers growing on the walls of the nozzle limit the extent of the usable core region (region with uniform flow) for testing models (especially rectangular). In order to maximize the region of uniform flow, it was decided to have the exit plane of this nozzle to be (nearly) rectangular. Thus, an additional constraint on the nozzle shape resulted, namely the nozzle will have a shape transitioning from a circular one at the inlet to that of a rectangle at the exit. In order to provide for a smooth shape transition, the cross sectional contour of the nozzle is defined by a superellipse. The nozzle is taken to be a meter in length. The axial variations of the major and minor radii of the superellipse are governed by cubic splines. The design parameters are the coefficients of the splines associated with the local nozzle wall slopes. Extensive calculations have been made (with a three-dimensional Euler code) to understand the effects of various parameters such as location of the knot points of the spline function, different ways of characterizing the uniformity of the flow in the exit plane, as well as the effect of constraining the area of the nozzle to be invariant. Turbulent flow (measurements indicate that the flow at the nozzle inlet is turbulent) calculations are now being performed (with the inviscidly designed nozzle contours) to assess the flow quality.

Krishnamurthy, Ramesh↗

NASA NTRS2017

The Sensitivity of Ground-Reflected GNSS Signals to Near-Surface Soil Moisture, as Recorded by Spaceborne Receivers

Spatial and temporal variations in near-surface soil moisture are important to measure for climate studies, numerical weather forecasts, and drought monitoring. Several previous studies have shown success in using ground-reflected Global Navigation Satellite System (GNSS) signals as a form of bistatic radar to sense soil moisture. However, the ability of this type of data to sense soil moisture variations from space is still a nascent field of study. In the past two years, three satellites have been launched that were either designed to capture ground-reflected GNSS signals or have been modified to record these signals. The data provided by these satellites are giving scientists an unprecedented opportunity to investigate their ability to detect changes in Earth’s land surface, including but certainly not limited to near-surface soil moisture. This paper will present spaceborne observations of ground-reflected GNSS signals and evaluate their sensitivity to near-surface soil moisture. This sensitivity will be compared to empirical and theoretical sensitivities of monostatic L-band radar measurements to soil moisture. We will also comment on possibilities for retrieval algorithm development, using techniques employed for monostatic radar as a guide.

Chew, C.↗

DOE OSTI2025

Counterdiabatic Driving with Performance Guarantees

Counterdiabatic (CD) driving has the potential to speed up adiabatic quantum state preparation by suppressing unwanted excitations. However, existing approaches either require intractable classical computations or are based on approximations that do not have performance guarantees. We propose and analyze a nonvariational, system-agnostic CD expansion method and analytically show that it converges exponentially quickly in the expansion order. In finite systems, the required resources scale inversely with the spectral gap, which we argue is asymptotically optimal. To extend our method to the thermodynamic limit and suppress errors stemming from high-frequency transitions, we leverage finite-time adiabatic protocols. In particular, we show that a time determined by the quantum speed limit is sufficient to prepare the desired ground state, without the need to optimize the adiabatic trajectory. Numerical tests of our method on the quantum Ising chain show that our method can outperform state-of-the-art variational CD approaches.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2023

Field Measurement and Analysis of Frequency and RoCoF for Low-Inertia Power Systems

The frequency measurement and rate of change of frequency (RoCoF) calculations are important for active power, frequency control, and frequency-related protection in power system operations. With the rapid decrease in inertia under modern power systems, however, multiple power variations introduced by variable renewable energy bring large RoCoF and introduce oscillations in frequency measurements. To shed light on this issue, this article proposes a frequency measurement and RoCoF analysis technology, as well as the sensitivity analysis for the real low-inertia power grid. First, a real-world event in an island power grid is analyzed as an example, and an event numerical model is estimated considering four uncertainties, including event RoCoF, oscillation magnitude, oscillation frequency, and underfrequency load-shedding threshold. Then, a sensitivity analysis is performed using the Sobol indices method and quasi-Monte Carlo simulations to identify the dominant uncertainties based on the event model. Here, the dominant uncertainties are compared to provide a guideline for the frequency estimation algorithm selection under different event cases. This algorithm is compared with an enhanced zero-crossing algorithm and gives a 3.41% better RoCoF error rate. Finally, an experiment is conducted using universal grid analyzers to validate the simulation results and the theoretical analysis.

24 POWER TRANSMISSION AND DISTRIBUTION↗

NASA NTRS2012

Polarimetric Retrievals of Surface and Cirrus Clouds Properties in the Region Affected by the Deepwater Horizon Oil Spill

In 2010, the Goddard Institute for Space Studies (GISS) Research Scanning Polarimeter (RSP) performed several aerial surveys over the region affected by the oil spill caused by the explosion of the Deepwater Horizon offshore platform. The instrument was deployed on the NASA Langley B200 aircraft together with the High Spectral Resolution Lidar (HSRL), which provides information on the distribution of the aerosol layers beneath the aircraft, including an accurate estimate of aerosol optical depth. This work illustrates the merits of polarization measurements in detecting variations of ocean surface properties linked to the presence of an oil slick. In particular, we make use of the degree of linear polarization in the glint region, which is severely affected by variations in the refractive index but insensitive to the waviness of the water surface. Alterations in the surface optical properties are therefore expected to directly affect the polarization response of the RSP channel at 2264 nm, where both molecular and aerosol scattering are negligible and virtually all of the observed signal is generated via Fresnel reflection at the surface. The glint profile at this wavelength is fitted with a model which can optimally estimate refractive index, wind speed and direction, together with aircraft attitude variations affecting the viewing geometry. The retrieved refractive index markedly increases over oil-contaminated waters, while the apparent wind speed is significantly lower than in adjacent uncontaminated areas, suggesting that the slick dampens high-frequency components of the ocean wave spectrum. The constraint on surface reflectance provided by the short-wave infrared channels is a cornerstone of established procedures to retrieve atmospheric aerosol microphysical parameters based on the inversion of the RSP multispectral measurements. This retrieval, which benefits from the ancillary information provided by the HSRL, was in this specific case hampered by prohibitive variability in atmospheric conditions (very inhomogeneous aerosol distribution and cloud cover). Although the results presented for the surface are essentially unaffected, we discuss the results obtained by typing algorithms in sorting the complex mix of aerosol types, and show evidence of oriented ice in cirrus clouds present in the area. In this context, polarization measurements at 1880 nm were used to infer ice habit and cirrus optical depth, which was found in the subvisual/threshold-visible regime, confirming the utility of the aforementioned RSP channel for the remote sensing of even thin cold clouds.

Polarization↗

NASA NTRS1993

Investigation for improving Global Positioning System (GPS) orbits using a discrete sequential estimator and stochastic models of selected physical processes

GEODYNII is a conventional batch least-squares differential corrector computer program with deterministic models of the physical environment. Conventional algorithms were used to process differenced phase and pseudorange data to determine eight-day Global Positioning system (GPS) orbits with several meter accuracy. However, random physical processes drive the errors whose magnitudes prevent improving the GPS orbit accuracy. To improve the orbit accuracy, these random processes should be modeled stochastically. The conventional batch least-squares algorithm cannot accommodate stochastic models, only a stochastic estimation algorithm is suitable, such as a sequential filter/smoother. Also, GEODYNII cannot currently model the correlation among data values. Differenced pseudorange, and especially differenced phase, are precise data types that can be used to improve the GPS orbit precision. To overcome these limitations and improve the accuracy of GPS orbits computed using GEODYNII, we proposed to develop a sequential stochastic filter/smoother processor by using GEODYNII as a type of trajectory preprocessor. Our proposed processor is now completed. It contains a correlated double difference range processing capability, first order Gauss Markov models for the solar radiation pressure scale coefficient and y-bias acceleration, and a random walk model for the tropospheric refraction correction. The development approach was to interface the standard GEODYNII output files (measurement partials and variationals) with software modules containing the stochastic estimator, the stochastic models, and a double differenced phase range processing routine. Thus, no modifications to the original GEODYNII software were required. A schematic of the development is shown. The observational data are edited in the preprocessor and the data are passed to GEODYNII as one of its standard data types. A reference orbit is determined using GEODYNII as a batch least-squares processor and the GEODYNII measurement partial (FTN90) and variational (FTN80, V-matrix) files are generated. These two files along with a control statement file and a satellite identification and mass file are passed to the filter/smoother to estimate time-varying parameter states at each epoch, improved satellite initial elements, and improved estimates of constant parameters.

Goad, Clyde C.↗

NASA NTRS2021

Error Propagation in Microwave Soil Moisture and Vegetation Optical Depth Retrievals

Satellite soil moisture and vegetation optical depth (VOD; related to the total vegetation water mass per unit area) are increasingly being used to study water relations in the soil-plant continuum across the globe. However, soil moisture and VOD are typically jointly estimated, where errors in the optimization approach can cause compensation between both variables and confound such studies. It is thus critical to quantify how satellite microwave measurement errors propagate into soil moisture and VOD. Such a study is especially important for VOD given limited investigations of whether VOD reflects in-situ plant physiology. Furthermore, despite new approaches that constrain (or regularize) VOD dynamics to reduce soil moisture errors, there is limited study of whether regularization reduces VOD errors without obscuring true vegetation temporal dynamics. Here, we find that, across the globe, VOD is less robust to measurement error (more difficult for optimization methods to find the true solution) than soil moisture in their joint estimation. However, a moderate degree of regularization (via time-constrained VOD) reduces errors in VOD to a greater degree than soil moisture and reduces spurious soil moisture-VOD coupling. Furthermore, despite constraining VOD time dynamics, regularized VOD variations on sub-weekly scales are both closer to simulated true VOD time series and have global VOD post-rainfall responses with reduced error signatures compared to VOD retrievals without regularization. Ultimately, we recommend moderately regularized VOD for use in large scale studies of soil-plant water relations because it suppresses noise and spurious soil moisture-VOD coupling without removing the physical signal.

Vegetation optical depth↗

NASA NTRS1973

Nonlinear problems of the theory of heterogeneous slightly curved shells

An account if given of the variational method of the solution of physically and geometrically nonlinear problems of the theory of heterogeneous slightly curved shells. Examined are the bending and supercritical behavior of plates and conical and spherical cupolas of variable thickness in a temperature field, taking into account the dependence of the elastic parameters on temperature. The bending, stability in general and load-bearing capacity of flexible isotropic elastic-plastic shells with different criteria of plasticity, taking into account compressibility and hardening. The effect of the plastic heterogeneity caused by heat treatment, surface work hardening and irradiation by fast neutron flux is investigated. Some problems of the dynamic behavior of flexible shells are solved. Calculations are performed in high approximations. Considerable attention is given to the construction of a machine algorithm and to the checking of the convergence of iterative processes.

Kantor, B. Y.↗

NASA NTRS2012

Physics of Acoustic Radiation from Jet Engine Inlets

Numerical simulations of acoustic radiation from a jet engine inlet are performed using advanced computational aeroacoustics (CAA) algorithms and high-quality numerical boundary treatments. As a model of modern commercial jet engine inlets, the inlet geometry of the NASA Source Diagnostic Test (SDT) is used. Fan noise consists of tones and broadband sound. This investigation considers the radiation of tones associated with upstream propagating duct modes. The primary objective is to identify the dominant physical processes that determine the directivity of the radiated sound. Two such processes have been identified. They are acoustic diffraction and refraction. Diffraction is the natural tendency for an acoustic wave to follow a curved solid surface as it propagates. Refraction is the turning of the direction of propagation of sound waves by mean flow gradients. Parametric studies on the changes in the directivity of radiated sound due to variations in forward flight Mach number and duct mode frequency, azimuthal mode number, and radial mode number are carried out. It is found there is a significant difference in directivity for the radiation of the same duct mode from an engine inlet when operating in static condition and in forward flight. It will be shown that the large change in directivity is the result of the combined effects of diffraction and refraction.

Tam, Christopher K. W.↗

NASA NTRS1988

Variability in surface ECG morphology: signal or noise?

Using data collected from canine models of acute myocardial ischemia, we investigated two issues of major relevance to electrocardiographic signal averaging: ECG epoch alignment, and the spectral characteristics of the beat-to-beat variability in ECG morphology. With initial digitization rates of 1 kHz, an iterative a posteriori matched filtering alignment scheme, and linear interpolation, we demonstrated that there is sufficient information in the body surface ECG to merit alignment to a precision of 0.1 msecs. Applying this technique to align QRS complexes and atrial pacing artifacts independently, we demonstrated that the conduction delay from atrial stimulus to ventricular activation may be so variable as to preclude using atrial pacing as an alignment mechanism, and that this variability in conduction time be modulated at the frequency of respiration and at a much lower frequency (0.02-0.03Hz). Using a multidimensional spectral technique, we investigated the beat-to-beat variability in ECG morphology, demonstrating that the frequency spectrum of ECG morphological variation reveals a readily discernable modulation at the frequency of respiration. In addition, this technique detects a subtle beat-to-beat alternation in surface ECG morphology which accompanies transient coronary artery occlusion. We conclude that physiologically important information may be stored in the variability in the surface electrocardiogram, and that this information is lost by conventional averaging techniques.

NASA Discipline Cardiopulmonary↗

NASA NTRS1991

A model for the disc Lyman alpha emission of Uranus

A new efficient radiative transfer algorithm for nonhomogeneous model atmospheres has been applied to the Uranian atmosphere. The contribution of the scatter solar Lyman-alpha to the Uranain emission is of the order of 300 R, and the Rayleigh contribution may reach 450 R for small values of the eddy diffusion coefficient (EDC). The total solar contribution may then reach about 750 R for a solar flux of 2.5 x 10 to the 11th photons/sq cm/s/A. A level of up to 400 R is confirmed in some directions for the interstellar wind contribution. The values of the atmospheric EDC necessary to mimic the observations are 50-100 sq cm/s. A small additional source located on the dayside Uranian atmosphere seems necessary correctly to fit the shape of the limb to limb intensity variation, especially near the limbs. Its contribution to the emergent intensity would range from 100 to 500 R.

Ben Jaffel, L.↗

NASA NTRS2004

Application of Remote Sensing to Assess the Impact of Short Term Climate Variability on Coastal Sedimentation

The purpose of this joint University of Wisconsin (UW) and Louisiana State University (LSU) project has been to relate short term climate variation to response in the coastal zone of Louisiana in an attempt to better understand how the coastal zone is shaped by climate variation. Climate variation in this case largely refers to variation in surface wind conditions that affect wave action and water currents in the coastal zone. The primary region of focus was the Atchafalaya Bay and surrounding bays in the central coastal region of Louisiana. Suspended solids in the water column show response to wind systems both in quantity (through resuspension) and in the pattern of dispersement or transport. Wind systems associated with cold fronts are influenced by short term climate variation. Wind energy was used as the primary signature of climate variation in this study because winds are a significant influence on sediment transport in the micro-tidal Gilf of Mexico coastal zone. Using case studies, the project has been able to investigate the influence of short term climate variation on sediment transport. Wind energy data, collected daily for National Weather Service (NWS) stations at Lake Charles and New Orleans, LA, were used as an indicator of short term climate variation influence on seasonal time scales. A goal was to relate wind energy to coastal impact through sediment transport. This goal was partially accomplished by combining remote sensing and wind energy data. Daily high resolution remote sensing observations are needed to monitor the complex coastal zone environment, where winds, tides, and water level all interact to influence sediment transport. The NASA Earth Observing System (EOS) era brings hope for documenting and revealing response of the complex coastal transport mosaic through regular high spatial resolution observations from the Moderate resolution Imaging Spectrometer (MODIS) instrument. MODIS observations were sampled in this project for information content and should continue to be viewed as a resource for coastal zone monitoring. The project initialized the effort to transfer a suspended sediment concentration (SSC) algorithm to the MODIS platform for case 2 waters. MODIS enables monitoring of turbid coastal zones around the globe. The MODIS SSC algorithm requires refinements in the atmospheric aerosol contribution, sun glint influence, and designation of the sediment inherent optical properties (IOPs); the framework for continued development is in place with a plan to release the algorithm to the MODIS direct broadcast community.

Menzel, W. Paul↗

NASA NTRS2008

On-Orbit Noise Characterization for MODIS Reflective Solar Bands

Since launch, the Moderate Resolution Imaging Spectroradiometer (MODIS) has operated successfully on-board the NASA Earth Observing System (EOS) Terra and EOS Aqua spacecraft. MODIS is a passive cross-track scanning radiometer that makes observations in 36 spectral bands with spectral wavelengths from visible (VIS) to long-wave infrared. MODIS bands 1-19 and 26 are the reflective solar bands (RSB) with wavelengths from 0.41 to 2.2 micrometers. They are calibrated on-orbit using an on-board solar diffuser (SD) and a SD stability monitor (SDSM) system. For MODIS RSB, the level 1B calibration algorithm produces top of the atmosphere reflectance factors and radiances for every pixel of the Earth view. The sensor radiometric calibration accuracy, specified at each spectral band's typical scene radiance, is 2% for the RSB reflectance factors and 5% for the RSB radiances. Also specified at the typical scene radiance is the detector signal-to-noise ratio (SNR), a key sensor performance parameter that directly impacts its radiometric calibration accuracy and stability, as well as the image quality. This paper describes an on-orbit SNR characterization approach developed to evaluate and track MODIS RSB detector performance. In order to perform on-orbit SNR characterization, MODIS RSB detector responses to the solar illumination reflected from the SD panel must be corrected for factors due to variations of the solar angles and the SD bi-directional reflectance factor. This approach enables RSB SNR characterization to be performed at different response levels for each detector. On-orbit results show that both Terra and Aqua MODIS RSB detectors have performed well since launch. Except for a few noisy or inoperable detectors which were identified pre-launch, most RSB detectors continue to meet the SNR design requirements and are able to maintain satisfactory short-term stability. A comparison of on-orbit noise characterization results with results derived from pre-launch calibration and characterization are also provided.

Xiong, X.↗

DOE OSTI2026

Latent Twins

Over the past decade, scientific machine learning has transformed the development of mathematical and computational frameworks for analyzing, modeling, and predicting complex systems. From inverse problems to numerical partial differential equations (PDEs), dynamical systems, and model reduction, these advances have pushed the boundaries of what can be simulated. Yet they have often progressed in parallel, with representation learning and algorithmic solution methods evolving largely as separate pipelines. With Latent Twins, we propose a unifying mathematical framework that creates a hidden surrogate in latent space for the underlying equations. Whereas digital twins mirror physical systems in the digital world, Latent Twins mirror mathematical systems in a learned latent space governed by operators. Through this lens, classical modeling, inversion, model reduction, and operator approximation all emerge as special cases of a single principle. We establish the fundamental approximation properties of Latent Twins for both ordinary differential equations (ODEs) and PDEs and demonstrate the framework across three representative settings: (i) canonical ODEs, capturing diverse dynamical regimes; (ii) a PDE benchmark using the shallow-water equations, contrasting Latent Twin simulations with deep operator network and forecasts with a four-dimensional variational method baseline; and (iii) a challenging real-data geopotential reanalysis dataset, reconstructing and forecasting from sparse, noisy observations. Latent Twins provide a compact, interpretable surrogate for solution operators that evaluate across arbitrary time gaps in a single-shot, while remaining compatible with scientific pipelines such as assimilation, control, and uncertainty quantification. Looking forward, this framework offers scalable, theory-grounded surrogates that bridge data-driven representation learning and classical scientific modeling across disciplines.

Latent Twins↗