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All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

A Medium-/Low-Voltage Joint State Estimator Through Linear Uncertainty Propagation

Traditionally, distribution system state estimations (DSSE) are challenged by the lack of measurements at both primary and secondary sides of the system. The widely available cable television (CATV) voltage sensors installed in low-voltage (LV) networks bring opportunities to achieve higher quality DSSE covering a broader area of the distribution network. This study proposes a medium-/low-voltage (MV/LV) joint distribution system state estimation approach using the untapped CATV measurements. It aims at addressing the need for system situational awareness at the grid edge while improving the estimation accuracy at both the primary and secondary sides compared to its disjointed counterpart. Linearized measurement functions and boundary condition uncertainty propagation rules are derived to ensure the computational efficiency and accuracy of the joint state estimator. Numerical experiments are conducted on an IEEE test feeder to demonstrate the efficacy of the proposed method and the value of CATV measurements.

joint state estimation↗
DOE OSTI2026

CONUS-wide Projected Flood Frequency and Uncertainty Estimates, Version 1.0

This dataset presents a large-ensemble of CONUS-wide projected flood frequency and uncertainty estimates across ~2.7 million NHDPlusV2 river reaches over the CONUS. The framework producing this dataset leverages a multi-model, uncertainty-aware modeling framework that allows evaluating shifts in flood frequences at the stream reach level across the CONUS. CONUS-wide ensemble streamflow projections generated from hydrologic simulations driven by downscaled and bias-corrected Coupled Model Intercomparison Project Phase 6 (CMIP6) outputs are used to derive these flood frequency and uncertainty estimates over the period 1980 - 2099. A spatially consistent regional L-moment algorithm is applied across clusters defined by the US Hydrologic Unit Code Subregions (HUC4s and HUC8s) and NHDPlusV2 stream orders to estimate flood frequencies. The dataset also includes at-site based flood estimates that allow for the comparison between local and regional approach-based estimates, assess projected changes, and characterize their uncertainties. For more reliable estimation of rare flood frequencies such as 500 and 1000-year return periods, super-ensemble based estimates are also included in the dataset. This dataset is derived to support the "Impact-Informed Dam Safety Risk Assessment for Securing Hydropower Assests" project for the US Department of Energy (DOE) Hydropower and Hydrokinetic Office (H2O). For further details, refer to Kao et al. (2022), Ghimire et al. (2023), Ghimire et al. (2025), and Hosking and Wallis (1997).

Ghimire, Ganesh [ORNL] (ORCID:0000000242843941)↗
DOE OSTI2022

Machine Learning Based Network Parameter Estimation Using AMI Data

The expansion of distribution power system and the growing penetration of distributed energy resources present new challenges for situational awareness. Calibrating the extended system model with sensor measurements and maintaining the usability is critical for utilities. This paper presents a distribution network parameter estimation (DNPE) approach using machine learning (ML) and metering data that improve the quality of extended distribution power system modeling. The reliability model can improve the ability of endpoint data to be translated into network-level situational awareness in real time and help distribution system operators (DSOs) solve branch flow and voltage problems. In addition, a data analytic and automate processing scheme is proposed to improve the sensor data quality and prevent misleading information. The effectiveness of the proposed method is verified with actual advanced metering infrastructure (AMI) data on a real utility feeder model, while considering the higher penetration of photovoltaic power generation. The test of DNPE and study results are demonstrated in this paper.

Parameter estimation, machine learning, power dist↗
DOE OSTI2024

Preventive Power Outage Estimation Based on a Novel Scenario Clustering Strategy

The increasing occurrence of extreme weather events is challenging power grid operation. For extreme weather events, the system operator is responsible for estimating the power outages and scheduling the restoration resources. This paper proposes an outage evaluation framework to identify the possible unserved load profiles, vulnerable areas, and mobile energy adequacy. The outputs of an outage prediction model tool are used to generate numerous faulted line scenarios. Next, each scenario's nodal unserved load profile is obtained by solving a three-phase restoration model that considers repair crews and mobile energy resources (MERs). Then, a novel scenario clustering strategy is developed to cluster the unserved load profiles into multiple representative profiles which the system operator can focus on. Finally, case studies on a distribution system evaluate the damage caused by an extreme weather event and verify the effectiveness of the proposed scenario clustering strategy.

MATHEMATICS AND COMPUTING,POWER TRANSMISSION AND D↗
NASA NTRS1995

Global precipitation estimates based on a technique for combining satellite-based estimates, rain gauge analysis, and NWP model precipitation information

The 'satellite-gauge model' (SGM) technique is described for combining precipitation estimates from microwave satellite data, infrared satellite data, rain gauge analyses, and numerical weather prediction models into improved estimates of global precipitation. Throughout, monthly estimates on a 2.5 degrees x 2.5 degrees lat-long grid are employed. First, a multisatellite product is developed using a combination of low-orbit microwave and geosynchronous-orbit infrared data in the latitude range 40 degrees N - 40 degrees S (the adjusted geosynchronous precipitation index) and low-orbit microwave data alone at higher latitudes. Then the rain gauge analysis is brougth in, weighting each field by its inverse relative error variance to produce a nearly global, observationally based precipitation estimate. To produce a complete global estimate, the numerical model results are used to fill data voids in the combined satellite-gauge estimate. Our sequential approach to combining estimates allows a user to select the multisatellite estimate, the satellite-gauge estimate, or the full SGM estimate (observationally based estimates plus the model information). The primary limitation in the method is imperfections in the estimation of relative error for the individual fields. The SGM results for one year of data (July 1987 to June 1988) show important differences from the individual estimates, including model estimates as well as climatological estimates. In general, the SGM results are drier in the subtropics than the model and climatological results, reflecting the relatively dry microwave estimates that dominate the SGM in oceanic regions.

Huffman, George J.↗
NASA NTRS1986

Crop acreage estimation using a Landsat-based estimator as an auxiliary variable

The problem of improving upon the ground survey estimates of crop acreages by utilizing Landsat data is addressed. Three estimators, called regression, ratio, and stratified ratio, are studied for bias and variance, and their relative efficiencies are compared. The approach is to formulate analytically the estimation problem that utilizes ground survey data, as collected by the U.S. Department of Agriculture, and Landsat data, which provide complete coverage for an area of interest, and then to conduct simulation studies. It is shown over a wide range of parametric conditions that the regression estimator is the most efficient unless there is a low correlation between the actual and estimated crop acreages in the sampled area segments, in which case the ratio and stratified ratio estimators are better. Furthermore, it is seen that the regression estimator is potentially biased due to estimating the regression coefficient from the training sample segments. Estimation of the variance of the regression estimator is also investigated. Two variance estimators are considered, the large sample variance estimator and an alternative estimator suggested by Cochran. The large sample estimate of variance is found to be biased and inferior to the Cochran estimate for small sample sizes.

Chhikara, R. S.↗
NASA NTRS2003

Estimating Uncertainty in GPCP and TRMM Multi-Satellite Precipitation Estimates

One of the high-priority problems in satellite precipitation estimation is developing algorithms for estimating the errors in precipitation retrievals by individual sensors and subsequent multi-satellite combinations. Classically, we distinguish between "random" and "bias" errors, which do and do not, respectively average to zero over a "big enough" time/space sample. The current operational GPCP and TRMM multi-satellite algorithms are nearly unique in estimating random error for the monthly precipitation estimates from individual sensor systems (including gauge), following Huffman, and then making multi-sensor combinations. No routinely operational global precipitation produces estimates of bias error. Subsequently, a similar scheme has been followed to provide random error estimates for the Multi-satellite Precipitation Analysis (MPA) being computed in real time and after real time for TRMM. The Huffman algorithm for random error is briefly reviewed, including a discussion of the limitations imposed on the algorithm by standard monthly precipitation data sets. Starting from a very simple theoretical treatment of the histogram of precipitation samples in a month, an equation is developed that depends on the estimated average precipitation rate for the month, the number of samples in the month, and two constants. The constants are set separately for each source of precipitation estimate (such as "raingauge") by calibration at selected ground sites. We discuss recent work validating the random error estimates to highlight the successes and limitations of this first-generation approach. We then consider what information is needed from the individual sensor algorithms to facilitate additional accuracy in the estimation of random errors across the time/space span of climate regimes which a global estimation system must handle. In addition, the thorny issue of estimating bias is raised. Finally, the role of error estimates (and the qualitative errors!) in creating combinations of precipitation estimates from different individual sensors is discussed. This issue is particularly important when fine scales in space and time are being considered, say the 0.25 x 0.25-deg 3-hourly estimates in the MPA.

Huffman, G. J.↗
DOE OSTI2022

Precise ROCOF estimation algorithm for low inertia power grids

The precise estimation of Rate Of Change Of Frequencies (ROCOFs) in a generation trip event can be helpful on power system inertia estimation, fast system response, and accurate event capturing. However, the ROCOF estimations from the existing Synchronized Measurement Devices (SMDs) are usually simply calculated by the finite difference between two adjacent frequency measurement points. The ineluctable noises, disturbances, and spikes from real-world frequency measurements can bring large dynamics to the ROCOF estimation and thus can result in an inaccurate estimation of the initial ROCOF. This issue becomes more serious when the target power grid has low inertia, and a large amount of distributed energy sources are deployed. To address this issue, a precise ROCOF estimation algorithm is designed based on least square method with flexible window size. In addition, a median filter is also designed and applied on the frequency measurements before using the proposed algorithm. The window size and thresholds in the proposed algorithm are determined with historical event data analysis. The proposed algorithm is deployed in a low cost, flexible, and distribution level universal grid analyzer (UGA) platform. In conclusion, multiple experiments are conducted in both a laboratory and the Hawaiian Islands to verify the effectiveness of the proposed algorithm.

24 POWER TRANSMISSION AND DISTRIBUTION↗
NASA NTRS2014

Modelflow Estimates of Stroke Volume Do Not Correlate With Doppler Ultrasound Estimates During Upright Posture

Orthostatic intolerance affects 60‐80% of astronauts returning from long‐duration missions, representing a significant risk to completing mission‐critical tasks. While likely multifactorial, a reduction in stroke volume (SV) represents one factor contributing to orthostatic intolerance during stand and head up tilt (HUT) tests. Current measures of SV during stand or HUT tests use Doppler ultrasound and require a trained operator and specialized equipment, restricting its use in the field. BeatScope (Finapres Medical Systems BV, The Netherlands) uses a modelflow algorithm to estimate SV from continuous blood pressure waveforms in supine subjects; however, evidence supporting the use of Modelflow to estimate SV in subjects completing stand or HUT tests remain scarce. Furthermore, because the blood pressure device is held extended at heart level during HUT tests, but allowed to rest at the side during stand tests, changes in the finger arterial pressure waveform resulting from arm positioning could alter modelflow estimated SV. The purpose of this project was to compare Doppler ultrasound and BeatScope estimations of SV to determine if BeatScope can be used during stand or HUT tests. Finger photoplethysmography was used to acquire arterial pressure waveforms corrected for hydrostatic finger‐to‐heart height using the Finometer (FM) and Portapres (PP) arterial pressure devices in 10 subjects (5 men and 5 women) during a stand test while simultaneous estimates of SV were collected using Doppler ultrasound. Measures were made after 5 minutes of supine rest and while subjects stood for 5 minutes. Next, SV estimates were reacquired while each arm was independently raised to heart level, a position similar to tilt testing. Supine SV estimates were not significantly different between all three devices (FM: 68+/-20, PP: 71+/-21, US: 73+/-21 ml/beat). Upon standing, the change in SV estimated by FM (‐18+/-8 ml) was not different from PP (‐21+/-12), but both were significantly less than US (‐37+/-16 ml, p<.05). Raising finger BP devices to heart level caused no significant change in SV measured with any of the devices (FM: 1.5+/-19, PP: 1.7+/-26, US: 0.5+/-6), although variability was 3‐6x greater as assessed by both blood pressure devices compared to US. Retrospective analysis of blood pressure data to assess SV in 11 supine subjects revealed significantly different estimates between methods (FM: 95+/-17, US: 75+/-32, p<.05), but the change in SV resulting from HUT was similar between methods (FM: ‐37+/-9, US: ‐40+/-18 ml). However, the correlation coefficient determined from pairs of SV estimated by US and FM was weak (r2=0.03). These data suggest Modelflow cannot be used in lieu of Doppler ultrasound to estimate SV during stand or HUT tests. Further investigation should focus on identifying factors contributing to differences between these measurement techniques in order to make use of a simple method for assessing beat‐by‐beat changes in SV during postural changes, especially during field testing.

Ferguson, Connor R.↗
DOE OSTI2022

Real-Time Highly Resolved Spatial-Temporal Vehicle Energy Consumption Estimation Using Machine Learning and Probe Data

Real-time highly resolved spatial-temporal vehicle energy consumption is a key missing dimension in transportation data. Most roadway link-level vehicle energy consumption data are estimated using average annual daily traffic measures derived from the Highway Performance Monitoring System; however, this method does not reflect day-to-day energy consumption fluctuations. As transportation planners and operators are becoming more environmentally attentive, they need accurate real-time link-level vehicle energy consumption data to assess energy and emissions; to incentivize energy-efficient routing; and to estimate energy impact caused by congestion, major events, and severe weather. This paper presents a computational workflow to automate the estimation of time-resolved vehicle energy consumption for each link in a road network of interest using vehicle probe speed and count data in conjunction with machine learning methods in real time. The real-time pipeline can deliver energy estimates within a couple seconds on query to its interface. The proposed method was evaluated on the transportation network of the metropolitan area of Chattanooga, Tennessee. The volume estimation results were validated with ground truth traffic volume data collected in the field. To demonstrate the effectiveness of the proposed method, the energy consumption pipeline was applied to real-world data to quantify road transportation-related energy reduction because of mitigation policies to slow the spread of COVID-19 and to measure energy loss resulting from congestion.

Severino, Joseph↗
DOE OSTI2026

Data for Clumping Index Estimation With 30°-tilted Cameras in Row Crops: Evaluation of Methods and Segment Size Effects

The clumping index (CI) quantifies the spatial distribution of foliage elements and is essential for accurately estimating the plant area index (PAI), canopy radiative transfer, and photosynthesis. Traditionally, the finite-length averaging method (LX), the gap size distribution method (CC), and a combined approach of CC and LX (CLX) have been applied to instruments like TRAC and digital hemispherical photography to estimate CI. However, a comprehensive evaluation of these methods in row crops remains limited, especially regarding the influence of segment size on CI. Meanwhile, digital cameras offer a cost-effective and user-friendly solution for canopy measurements in row crops, yet their application in this context remains underexplored. In this study, we employed a new approach using a 30°-tilted digital camera to estimate CI in corn and soybean fields, applying the LX, CC, and CLX methods. We systematically assessed the performance of these three methods by combining field measurements in real-world fields with simulations using the LESS 3D radiative transfer model. Our results showed that CLX applied to the whole image and 45° segment offered accurate estimation of CI (bias within ±0.1, RMSE < 0.2) and PAI (bias within ±0.4, RMSE < 1) in real-world fields and LESS simulations. The accuracy of the LX method was highly sensitive to segment size, with the best performance observed at the 15° segment (PAI bias within ±0.4). In contrast, the CC method remained stable across different segment sizes, and its performance was generally comparable to that of LX, except at the 15° segment. Across view zenith angles, CI derived from CC generally showed a continuous increase, while those from LX and CLX followed a rising trend at small zenith angles but began to decline at 68°, likely due to an increasing proportion of no-gap segments. Seasonally, LX tended to show decreasing CI during early growth stages but increased as the canopy matured, whereas CC and CLX showed gradually increasing CI before plateauing at peak PAI. The 30°-tilted camera effectively captured CI variations across different angles and growth stages, making it a practical and robust instrument for row crop canopy structure analysis. Applying these CI methods to digital cameras offers a low-cost and accessible CI estimation alternative, improving canopy structure monitoring accuracy in row crops.

Modeling↗
NASA NTRS2001

TRMM: Status of Precipitation Estimates, Science Highlights, and 3-Hour Global, Tropical Precipitation Estimates

The Tropical Rainfall Measuring Mission (TRMM) has completed more than three years in orbit. A summary of research highlights will be presented focusing on application of TRMM data to topics ranging from climate analysis, through improving forecasts, to microphysical research. Examples and plans for operational use of TRMM data in tropical cyclone and other applications will be given. The status of precipitation estimates from different instruments and algorithms will be described. Monthly surface rainfall estimates over the ocean based on different instruments on TRMM currently differ by 20% in overall mean. In addition, time changes in global ocean rainfall between El Nino and La Nina conditions show a difference in sign between the active and passive microwave products. These differences are not surprising considering the different type of observations available for the first time from TRMM with both the passive and active microwave sensors. Resolving the differences will strengthen the validity and utility of ocean rainfall estimates. The TRMM rainfall estimates are intercompared among themselves and with other estimates, including those of the standard, monthly Global Precipitation Climatology Project (GPCP) analysis. The GPCP analysis agrees roughly in magnitude with the passive microwave-based TRMM estimates which is not surprising considering GPCP over-ocean estimates are based on passive microwave observations. A three-year TRMM rainfall climatology is presented, including anomaly fields related to the changing ENSO situation during the mission. Results of using TRMM information to calibrate other passive microwave observations and geosynchronous infrared rainfall estimates and then merging them those estimates into a global, tropical 3-hour time resolution analysis will also be described.

Adler, Robert↗
NASA NTRS2019

Multilevel Monte Carlo Estimation of Unbiased Expectation via Sample Reuse and the Low Variance Estimation of Asymptotic Rates

A new variant of the multilevel Monte Carlo estimator [5, 3, 9, 12] is presented for the estimation of expectation statistics that utilizes sample reuse in specified levels, explicitly removes approximation error bias associated with numerically computed output quantities of interest that have an asymptotic limit behavior, and permits a low variance estimate of the asymptotic rate of convergence to that limit. In addition, it is shown that this new multilevel Monte Carlo variant can yield a computational cost savings. A review of Monte Carlo and multilevel Monte Carlo estimators is presented that includes analysis of expected value, expected mean squared error, and the calculation of optimized multilevel sample size parameters. The multilevel Monte Carlo estimator produces estimates of expectation for numerically approximated output quantities of interest that are biased by approximation error. When the quantity of interest can be modeled as the asymptotic limit of numerically approximated output quantities of interest, it is theoretically possible to remove this approximation error bias in the multilevel Monte Carlo estimator. In actual implementations, however, this procedure is unreliable due to statistical variability and inaccuracy in estimating the needed asymptotic limit. Analysis and numerical experiment show that the proposed variant of the multilevel Monte Carlo method greatly reduces (in some cases eliminates) the statistical variability in this limit estimation.

Barth, Timothy↗
DOE OSTI2021

Estimating uncertainty: A Bayesian approach to modelling photosynthesis in C3 leaves

The Farquhar-von Caemmerer-Berry (FvCB) model is extensively used to model pho-tosynthesis from gas exchange measurements. Since its publication, many methods have been developed to measure, or more accurately estimate, parameters of this model. Here, we have created a tool that uses Bayesian statistics to fit photosyn-thetic parameters using concurrent gas exchange and chlorophyll fluorescence mea-surements whilst evaluating the reliability of the parameter estimation. We have tested this tool on synthetic data and experimental data from rice leaves. Our results indicate that reliable parameter estimation can be achieved whilst only keeping one parameter, Km, that is, Michaelis constant for CO2 by Rubisco, prefixed. Additionally, we show that including detailed low CO2 measurements at low light levels increases reliability and suggests this as a new standard measurement protocol. By providing an estimated distribution of parameter values, the tool can be used to evaluate the quality of data from gas exchange and chlorophyll fluorescence measurement proto-cols. Compared to earlier model fitting methods, the use of a Bayesian statistics-based tool minimizes human interaction during fitting, reducing the subjectivity which is essential to most existing tools. A user friendly, interactive Bayesian tool script is provided.

Bayesian statistics, leaf photosynthesis, mesophyl↗
DOE OSTI2021

Pooling Data Improves Multimodel IDF Estimates over Median-Based IDF Estimates: Analysis over the Susquehanna and Florida

Traditional multimodel methods for estimating future changes in precipitation intensity, duration, and frequency (IDF) curves rely on mean or median of models’ IDF estimates. Such multimodel estimates are impaired by large estimation uncertainty, shadowing their efficacy in planning efforts. Here, assuming that each climate model is one representation of the underlying data generating process, i.e., the Earth system, we propose a novel extension of current methods through pooling model data: (i) evaluate performance of climate models in simulating the spatial and temporal variability of the observed annual maximum precipitation (AMP), (ii) bias-correct and pool historical and future AMP data of reasonably performing models, and (iii) compute IDF estimates in a nonstationary framework from pooled historical and future model data. Pooling enhances fitting of the extreme value distribution to the data and assumes that data from reasonably performing models represent samples from the “true” underlying data generating distribution. Through Monte Carlo simulations with synthetic data, we show that return periods derived from pooled data have smaller biases and lesser uncertainty than those derived from ensembles of individual model data. We apply this method to NA-CORDEX models to estimate changes in 24-h precipitation intensity–frequency (PIF) estimates over the Susquehanna watershed and Florida peninsula. Our approach identifies significant future changes at more stations compared to median-based PIF estimates. The analysis suggests that almost all stations over the Susquehanna and at least two-thirds of the stations over the Florida peninsula will observe significant increases in 24-h precipitation for 2–100-yr return periods.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

Single reader between-cases AUC estimator with nested data

The area under the receiver operating characteristic curve (AUC) is widely used in evaluating diagnostic performance for many clinical tasks. It is still challenging to evaluate the reading performance of distinguishing between positive and negative regions of interest (ROIs) in the nested-data problem, where multiple ROIs are nested within the cases. To address this issue, we identify two kinds of AUC estimators, within-cases AUC and between-cases AUC. We focus on the between-cases AUC estimator, since our main research interest is in patient-level diagnostic performance rather than location-level performance (the ability to separate ROIs with and without disease within each patient). Another reason is that as the case number increases, the number of between-cases paired ROIs is much larger than the number of within-cases ROIs. We provide estimators for the variance of the between-cases AUC and for the covariance when there are two readers. We derive and prove the above estimators’ theoretical values based on a simulation model and characterize their behavior using Monte Carlo simulation results. We also provide a real-data example. Moreover, we connect the distribution-based simulation model with the simulation model based on the linear mixed-effect model, which helps better understand the sources of variation in the simulated dataset.

Health Care Sciences & Services↗
DOE OSTI2022

Reducing the cost of energy estimation in the variational quantum eigensolver algorithm with robust amplitude estimation

Quantum chemistry and materials is one of the most promising applications of quantum computing. Yet much work is still to be done in matching industry-relevant problems in these areas with quantum algorithms that can solve them. Most previous efforts have carried out resource estimations for quantum algorithms run on large-scale fault-tolerant architectures, which include the quantum phase estimation algorithm. In contrast, few have assessed the performance of near-term quantum algorithms, which include the variational quantum eigensolver (VQE) algorithm. Recently, a large-scale benchmark study [Gonthier et al. 2020] found evidence that the performance of the variational quantum eigensolver for a set of industry-relevant molecules may be too inefficient to be of practical use. This motivates the need for developing and assessing methods that improve the efficiency of VQE. In this work, we predict the runtime of the energy estimation subroutine of VQE when using robust amplitude estimation (RAE) to estimate Pauli expectation values. Under conservative assumptions, our resource estimation predicts that RAE can reduce the runtime over the standard estimation method in VQE by one to two orders of magnitude. Despite this improvement, we find that the runtimes are still too large to be practical. These findings motivate two complementary efforts towards quantum advantage: 1) the investigation of more efficient near-term methods for ground state energy estimation and 2) the development of problem instances that are of industrial value and classically challenging, but better suited to quantum computation.

Johnson, Peter D.↗
NASA NTRS1978

On-line parameter estimation using a high sensitivity estimator

An on-line parameter identification method is presented. The method is based on a recursive formulation of the maximum likelihood method, with a significant modification on the gains of the state estimator. In the conventional maximum likelihood method, the Kalman gains are used in the state estimator. This produces unbiased, minimum variance parameter estimates in the presence of process noise and measurement noise, but it also slows the convergence rate when the identification is done on-line. Here we suggest choosing the gains to maximize a measure of the sensitivities of the state estimates to parameter variations. One such criterion is to minimize the trace of the inverse information matrix. This increases the convergence rate significantly. After one or two time constants, the gains are switched to the Kalman values to assure unbiased, minimum-variance estimates. The state estimate will initially be nonoptimal, and may not be adequate for control purposes. In this case, a parallel Kalman filter which uses the identifier's parameter estimates can be used. This method is applied here for the identification of a simple first-order system, and for the identification of short-period stability derivatives of an F-8 aircraft from simulated data.

Mishne, D.↗