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NASA NTRS1991

Preliminary results from an airdata enhancement algorithm with application to high-angle-of-attack flight

A technique has been developed to improve the fidelity of airdata measurements during dynamic maneuvering. This technique is particularly useful for airdata measured during flight at high angular rates and high angles of attack. To support this research, flight tests using the F-18 high alpha research vehicle were conducted at the NASA Ames Research Center Dryden Flight Research Facility. A Kalman filter was used to combine information from research airdata, linear accelerometers, angular rate gyros, and attitude gyros to determine better estimates of airdata quantities such as angle of attack, angle of sideslip, airspeed, and altitude. This paper briefly develops the state and observation equations used by the Kalman filter and shows how the state and measurement covariance matrices were determined from flight data.

Moes, Timothy R.↗

NASA NTRS1992

Performance and state-space analyses of systems using Petri nets

The goal of any modeling methodology is to develop a mathematical description of a system that is accurate in its representation and also permits analysis of structural and/or performance properties. Inherently, trade-offs exist between the level detail in the model and the ease with which analysis can be performed. Petri nets (PN's), a highly graphical modeling methodology for Discrete Event Dynamic Systems, permit representation of shared resources, finite capacities, conflict, synchronization, concurrency, and timing between state changes. By restricting the state transition time delays to the family of exponential density functions, Markov chain analysis of performance problems is possible. One major drawback of PN's is the tendency for the state-space to grow rapidly (exponential complexity) compared to increases in the PN constructs. It is the state space, or the Markov chain obtained from it, that is needed in the solution of many problems. The theory of state-space size estimation for PN's is introduced. The problem of state-space size estimation is defined, its complexities are examined, and estimation algorithms are developed. Both top-down and bottom-up approaches are pursued, and the advantages and disadvantages of each are described. Additionally, the author's research in non-exponential transition modeling for PN's is discussed. An algorithm for approximating non-exponential transitions is developed. Since only basic PN constructs are used in the approximation, theory already developed for PN's remains applicable. Comparison to results from entropy theory show the transition performance is close to the theoretic optimum. Inclusion of non-exponential transition approximations improves performance results at the expense of increased state-space size. The state-space size estimation theory provides insight and algorithms for evaluating this trade-off.

Watson, James Francis, III↗

NASA NTRS1999

Behavior of Filters and Smoothers for Strongly Nonlinear Dynamics

The Kalman filter is the optimal filter in the presence of known gaussian error statistics and linear dynamics. Filter extension to nonlinear dynamics is non trivial in the sense of appropriately representing high order moments of the statistics. Monte Carlo, ensemble-based, methods have been advocated as the methodology for representing high order moments without any questionable closure assumptions. Investigation along these lines has been conducted for highly idealized dynamics such as the strongly nonlinear Lorenz model as well as more realistic models of the means and atmosphere. A few relevant issues in this context are related to the necessary number of ensemble members to properly represent the error statistics and, the necessary modifications in the usual filter situations to allow for correct update of the ensemble members. The ensemble technique has also been applied to the problem of smoothing for which similar questions apply. Ensemble smoother examples, however, seem to be quite puzzling in that results state estimates are worse than for their filter analogue. In this study, we use concepts in probability theory to revisit the ensemble methodology for filtering and smoothing in data assimilation. We use the Lorenz model to test and compare the behavior of a variety of implementations of ensemble filters. We also implement ensemble smoothers that are able to perform better than their filter counterparts. A discussion of feasibility of these techniques to large data assimilation problems will be given at the time of the conference.

Zhu, Yanqui↗

NASA NTRS2016

A Quantum-Assisted Algorithm for Sampling Applications in Machine Learning

An increase in the efficiency of sampling from Boltzmann distributions would have a significant impact in deep learning and other machine learning applications. Recently, quantum annealers have been proposed as a potential candidate to speed up this task, but several limitations still bar these state-of-the-art technologies from being used effectively. One of the main limitations is that, while the device may indeed sample from a Boltzmann-like distribution, quantum dynamical arguments suggests it will do so with an instance-dependent effective temperature, different from the physical temperature of the device. Unless this unknown temperature can be unveiled, it might not be possible to effectively use a quantum annealer for Boltzmann sampling. In this talk, we present a strategy to overcome this challenge with a simple effective-temperature estimation algorithm. We provide a systematic study assessing the impact of the effective temperatures in the learning of a kind of restricted Boltzmann machine embedded on quantum hardware, which can serve as a building block for deep learning architectures. We also provide a comparison to k-step contrastive divergence (CD-k) with k up to 100. Although assuming a suitable fixed effective temperature also allows to outperform one step contrastive divergence (CD-1), only when using an instance-dependent effective temperature we find a performance close to that of CD-100 for the case studied here. We discuss generalizations of the algorithm to other more expressive generative models, beyond restricted Boltzmann machines.

Perdomo-Ortiz, Alejandro↗

NASA NTRS2021

Cutting out the Middleman: Calibrating and Validating a Dynamic Vegetation Model (ED2-PROSPECT5) Using Remotely Sensed Surface Reflectance

Canopy radiative transfer is the primary mechanism by which models relate vegetation composition and state to the surface energy balance, which is important to light- and temperature-sensitive plant processes as well as understanding land–atmosphere feedbacks. In addition, certain parameters (e.g., specific leaf area, SLA) that have an outsized influence on vegetation model behavior can be constrained by observations of shortwave reflectance, thus reducing model predictive uncertainty. Importantly, calibrating against radiative transfer outputs allows models to directly use remote sensing reflectance products without relying on highly derived products (such as MODIS leaf area index) whose assumptions may be incompatible with the target vegetation model and whose uncertainties are usually not well quantified. Here, we created the EDR model by coupling the two-stream representation of canopy radiative transfer in the Ecosystem Demography model version 2 (ED2) with a leaf radiative transfer model (PROSPECT-5) and a simple soil reflectance model to predict full-range, high-spectral-resolution surface reflectance that is dependent on the underlying ED2 model state. We then calibrated this model against estimates of hemispherical reflectance (corrected for directional effects) from the NASA Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) and survey data from 54 temperate forest plots in the northeastern United States. The calibration significantly reduced uncertainty in model parameters related to leaf biochemistry and morphology and canopy structure for five plant functional types. Using a single common set of parameters across all sites, the calibrated model was able to accurately reproduce surface reflectance for sites with highly varied forest composition and structure. However, the calibrated model's predictions of leaf area index (LAI) were less robust, capturing only 46 % of the variability in the observations. Comparing the ED2 radiative transfer model with another two-stream soil–leaf–canopy radiative transfer model commonly used in remote sensing studies (PRO4SAIL) illustrated structural errors in the ED2 representation of direct radiation backscatter that resulted in systematic underestimation of reflectance. In addition, we also highlight that, to directly compare with a two-stream radiative transfer model like EDR, we had to perform an additional processing step to convert the directional reflectance estimates of AVIRIS to hemispherical reflectance (also known as “albedo”). In future work, we recommend that vegetation models add the capability to predict directional reflectance, to allow them to more directly assimilate a wide range of airborne and satellite reflectance products. We ultimately conclude that despite these challenges, using dynamic vegetation models to predict surface reflectance is a promising avenue for model calibration and validation using remote sensing data.

Alexey N Shiklomanov↗

NASA NTRS1987

Improving stability margins in discrete-time LQG controllers

Some of the problems are discussed which are encountered in the design of discrete-time stochastic controllers for problems that may adequately be described by the Linear Quadratic Gaussian (LQG) assumptions; namely, the problems of obtaining acceptable relative stability, robustness, and disturbance rejection properties. A dynamic compensator is proposed to replace the optimal full state feedback regulator gains at steady state, provided that all states are measurable. The compensator increases the stability margins at the plant input, which may possibly be inadequate in practical applications. Though the optimal regulator has desirable properties the observer based controller as implemented with a Kalman filter, in a noisy environment, has inadequate stability margins. The proposed compensator is designed to match the return difference matrix at the plant input to that of the optimal regulator while maintaining the optimality of the state estimates as directed by the measurement noise characteristics.

Oranc, B. Tarik↗

NASA NTRS2014

The Vacuum-Compacted Regolith Gripping Mechanism and Unmanned Flights via Quad-Rotors

During the course of the Kennedy Space Center Summer Internship, two main experiments were performed: The Vacuum-Compacted Regolith Gripping Mechanism and Unmanned Flights via Quad-copters. The objectives of the Vacuum-Compacted Regolith Gripping Mechanism, often abbreviated as the Granular Gripper, are to exhibit Space Technology, such as a soft robotic hand, lift different apparatuses used to excavate regolith, and conserve energy while executing its intended task. The project is being conducted to test how much weight the Granular Gripper can hold. With the use of an Animatronic Robotic Hand, Arduino Uno, and other components, the system was calibrated before actually conducting the intended weight test. The maximum weight each finger could hold with the servos running, in the order of pinky, ring, middle, and index fingers, are as follows: 1.340N, 1.456 N, 0.9579 N, and 1.358 N. Using the small vacuum pump system, the maximum weight each finger could hold, in the same order, was: 4.076 N, 6.159 N, 5.454 N, and 4.052 N. The maximum torques on each of the fingers when the servos were running, in the same respective order, was: 0.0777 Nm, 0.0533 Nm, 0.0648 Nm, and 0.0532 Nm. The maximum torques on the individual fingers, when the small vacuum pump was in effect, in the same order as above, was: 0.2318 Nm, 0.3032 Nm, 0.2741 Nm, and 0.1618 Nm. In testing all the fingers with the servos running, the total weight was 5.112 N and the maximum torque on the all the fingers was 0.2515 Nm. However, when the small vacuum pump system was used, the total weight was 19.741 N and the maximum torque on the all the fingers was 0.9713 Nm. The conclusion that was drawn stated that using the small vacuum pump system proved nearly 4 times more effective when testing how much weigh the hand could hold. The resistance provided by the compacted sand in the glove allowed more weight to be held by the hand and glove. Also, when the servos turned off and the hand still retaining its position, energy is being saved because the vacuum created the same resistance the running servos did without using power. The Unmanned Flights via Quad-rotors are built because multi-rotor dynamics are an important starting point and fair analog for space craft control systems and they make good terrestrial development platforms for various aspects of control for space crafts. The project is being conducted to see what the thrust response is going to be when a pulse width modulation command is sent to the control system since the quad-rotors are PWM controlled. A simulation environment in constructed so that one can quickly iterate and test different designs such as control systems, PID control vs. LDR control, and state estimation. Using two DIY Quad Kits, APM 2.6, testing apparatus (called a data acquisition system) to test the quad-rotors, and a simulation program such as Simulink, two quad-rotors are built and controlled via a simulation program, which is designed to be as realistic as possible and not idealistic. Due to the quad-rotors not being completely built nor ready for testing, there are no results or conclusions to report.

Granular Gripper↗

NASA NTRS1999

Angular-Rate Estimation Using Star Tracker Measurements

This paper presents algorithms for estimating the angular-rate vector of satellites using quaternion measurements. Two approaches are compared, one that uses differentiated quatemion measurements to yield coarse rate measurements which are then fed into two different estimators. In the other approach the raw quatemion measurements themselves are fed directly into the two estimators. The two estimators rely on the ability to decompose the non-linear rate dependent part of the rotational dynamics equation of a rigid body into a product of an angular-rate dependent matrix and the angular-rate vector itself This decomposition, which is not unique, enables the treatment of the nonlinear spacecraft dynamics model as a linear one and, consequently, the application of a Pseudo-Linear Kalman Filter (PSELIKA). It also enables the application of a special Kalman filter which is based on the use of the solution of the State Dependent Algebraic Riccati Equation (SDARE) in order to compute the Kalman gain matrix and thus eliminates the need to propagate and update the filter covariance matrix. The replacement of the elaborate rotational dynamics by a simple first order Markov model is also examined. In this paper a special consideration is given to the problem of delayed quatemion measurements. Two solutions to this problem are suggested and tested. Real Rossi X-Ray Timing Explorer (RXTE) data is used to test these algorithms, and results of these tests are presented.

Azor, R.↗

NASA NTRS1999

Angular-Rate Estimation using Star Tracker Measurements

This paper presents algorithms for estimating the angular-rate vector of satellites using quaternion measurements. Two approaches are compared, one that uses differentiated quaternion measurements to yield coarse rate measurements which are then fed into two different estimators. In the other approach the raw quaternion measurements themselves are fed directly into the two estimators. The two estimators rely on the ability to decompose the non-linear rate dependent part of the rotational dynamics equation of a rigid body into a product of an angular-rate dependent matrix and the angular-rate vector itself. This decomposition, which is not unique, enables the treatment of the nonlinear spacecraft dynamics model as a linear one and, consequently, the application of a Pseudo-Linear Kalman Filter (PSELIKA). It also enables the application of a special Kalman filter which is based on the use of the solution of the State Dependent Algebraic Riccati Equation (SDARE) in order to compute the Kalman gain matrix and thus eliminates the need to propagate and update the filter covariance matrix. The replacement of the elaborate rotational dynamics by a simple first order Markov model is also examined. In this paper a special consideration is given to the problem of delayed quaternion measurements. Two solutions to this problem are suggested and tested. Real Rossi X-Ray Timing Explorer (RXTE) data is used to test these algorithms, and results of these tests are presented.

Azor, R.↗

NASA NTRS1999

Autonomous Navigation Using Celestial Objects

In the twenty-first century, National Aeronautics and Space Administration (NASA) Enterprises envision frequent low-cost missions to explore the solar system, observe the universe, and study our planet. Satellite autonomy is a key technology required to reduce satellite operating costs. The Guidance, Navigation, and Control Center (GNCC) at the Goddard Space Flight Center (GSFC) currently sponsors several initiatives associated with the development of advanced spacecraft systems to provide autonomous navigation and control. Autonomous navigation has the potential both to increase spacecraft navigation system performance and to reduce total mission cost. By eliminating the need for routine ground-based orbit determination and special tracking services, autonomous navigation can streamline spacecraft ground systems. Autonomous navigation products can be included in the science telemetry and forwarded directly to the scientific investigators. In addition, autonomous navigation products are available onboard to enable other autonomous capabilities, such as attitude control, maneuver planning and orbit control, and communications signal acquisition. Autonomous navigation is required to support advanced mission concepts such as satellite formation flying. GNCC has successfully developed high-accuracy autonomous navigation systems for near-Earth spacecraft using NASA's space and ground communications systems and the Global Positioning System (GPS). Recently, GNCC has expanded its autonomous navigation initiative to include satellite orbits that are beyond the regime in which use of GPS is possible. Currently, GNCC is assessing the feasibility of using standard spacecraft attitude sensors and communication components to provide autonomous navigation for missions including: libration point, gravity assist, high-Earth, and interplanetary orbits. The concept being evaluated uses a combination of star, Sun, and Earth sensor measurements along with forward-link Doppler measurements from the command link carrier to autonomously estimate the spacecraft's orbit and reference oscillator's frequency. To support autonomous attitude determination and control and maneuver planning and control, the orbit determination accuracy should be on the order of kilometers in position and centimeters per second in velocity. A less accurate solution (one hundred kilometers in position) could be used for acquisition purposes for command and science downloads. This paper provides performance results for both libration point orbiting and high Earth orbiting satellites as a function of sensor measurement accuracy, measurement types, measurement frequency, initial state errors, and dynamic modeling errors.

Folta, David↗

NASA NTRS1991

A Dynamic Failure Model for Predicting the Impact that a Program Location has on the Program

This paper presents a dynamic technique for predicting the effect that a "location" of a program will have on the program's computational behavior. The technique is based on the three necessary and sufficient conditions for software failure to occur: (1) a fault must be executed, (2) the fault must adversely affect the data state, and (3) the adverse effect in a data state must affect program output. In order to predict the effect that a location of a program will have on the program's computational behavior, the following characteristics of each program location are estimated: (1) the probability that a location of the program is executed, (2) the probability that a location of the program noticeably affects the program state created by the location, and (3) the probability that the data states created by a location affect the program's output. With estimates of these characteristics for each location in a program, we can predict those locations where a fault can more easily remain undetected during testing, as well as predict the degree of testing necessary to be convinced that a fault is not remaining undetected in a particular location.

Voas, Jeffrey↗

NASA NTRS2017

Filter Tuning Using the Chi-Squared Statistic

The Goddard Space Flight Center (GSFC) Flight Dynamics Facility (FDF) performs orbit determination (OD) for the Aqua and Aura satellites. Both satellites are located in low Earth orbit (LEO), and are part of what is considered the A-Train satellite constellation. Both spacecraft are currently in the science phase of their respective missions. The FDF has recently been tasked with delivering definitive covariance for each satellite.The main source of orbit determination used for these missions is the Orbit Determination Toolkit developed by Analytical Graphics Inc. (AGI). This software uses an Extended Kalman Filter (EKF) to estimate the states of both spacecraft. The filter incorporates force modelling, ground station and space network measurements to determine spacecraft states. It also generates a covariance at each measurement. This covariance can be useful for evaluating the overall performance of the tracking data measurements and the filter itself. An accurate covariance is also useful for covariance propagation which is utilized in collision avoidance operations. It is also valuable when attempting to determine if the current orbital solution will meet mission requirements in the future.This paper examines the use of the Chi-square statistic as a means of evaluating filter performance. The Chi-square statistic is calculated to determine the realism of a covariance based on the prediction accuracy and the covariance values at a given point in time. Once calculated, it is the distribution of this statistic that provides insight on the accuracy of the covariance.For the EKF to correctly calculate the covariance, error models associated with tracking data measurements must be accurately tuned. Over estimating or under estimating these error values can have detrimental effects on the overall filter performance. The filter incorporates ground station measurements, which can be tuned based on the accuracy of the individual ground stations. It also includes measurements from the NASA space network (SN), which can be affected by the assumed accuracy of the TDRS satellite state at the time of the measurement.The force modelling in the EKF is also an important factor that affects the propagation accuracy and covariance sizing. The dominant force in the LEO orbit regime is the drag force caused by atmospheric drag. Accurate accounting of the drag force is especially important for the accuracy of the propagated state. The implementation of a box and wing model to improve drag estimation accuracy, and its overall effect on the covariance state is explored.The process of tuning the EKF for Aqua and Aura support is described, including examination of the measurement errors of available observation types (Doppler and range), and methods of dealing with potentially volatile atmospheric drag modeling. Predictive accuracy and the distribution of the Chi-square statistic, calculated based of the ODTK EKF solutions, are assessed versus accepted norms for the orbit regime.

Covariance Analysis↗

NASA NTRS1975

Estimating maximum instantaneous distortion from inlet total pressure rms and PSD measurements

An inexpensive method is developed to determine the extreme values of instantaneous inlet distortion. This method also provides insight into the basic mechanics of unsteady inlet flow and the associated engine reaction. The analysis is based on fundamental fluid dynamics and statistical methods to provide an understanding of the turbulent inlet flow and quantitatively relate the rms level and power spectral density (PSD) function of the measured time variant total pressure fluctuations to the strength and size of the low pressure regions. The most probable extreme value of the instantaneous distortion is then synthesized from this information in conjunction with the steady state distortion. Results of the analysis show the extreme values to be dependent upon the steady state distortion, the measured turbulence rms level and PSD function, the time on point, and the engine response characteristics. Analytical projections of instantaneous distortion are presented and compared with data obtained by a conventional, highly time correlated, 40 probe instantaneous pressure measurement system.

Melick, H. C., Jr.↗

NASA NTRS1986

COFS 1: Beam dynamics and control technology overview

The Control of Flexible Structures (COFS) 1 Project provides the invaluable opportunity to test, validate, and measure the effectiveness of theories, structural concepts, control systems, and flight certification processes for future missions through a research program focusing on multiple issues in large flexible structures, dynamics, and controls. The COFS 1 Project consists of a series of ground and flight activities building progressively from modeling and dynamic characterization of large space systems to the more complex issues of flexible-body control. The program objectives are to: determine the degree to which theory and ground testing can predict flight performance of next-generation low-frequency structures; evaluate structural fidelity of representative next-generation large deployable precision structure; assess math modeling requirements for large lightweight complex systems on which ground test results are questionable; determine degree to which scale model analysis and tests can be correlated to full-scale performance; evaluate system identification and state estimation algorithms on complex lightweight structures in the space environment; evaluate and verify controls/structures modeling capability; evaluate control laws and control systems; and evaluate damping effects in micro-g environment.

Allen, John L.↗

NASA NTRS1984

Dynamic compression and volatile release of carbonates

Particle velocity profiles upon shock compression and isentropic releases were measured for polycrystalline calcite. The Solenhofen limestone release paths lie, close to the Hugoniot. Calcite 3 to 2 transition, upon release, was observed, but rarefaction shocks were not detected. The equation of state is used to predict the fraction of material devolatilized upon isentropic release as a function of shock pressure. The effect of ambient partial pressure of CO2 on the calculations is demonstrated and considered in models of atmospheric evolution by impact induced mineral devolatilization. The radiative characteristics of shocked calcite indicate that localization of thermal energy occurs under shock compression. Shock entropy calculations result in a minimum estimate of 90% devolatilization upon complete release from 10 GPa. Isentropic release paths from calculated continuum Hugoniot temperatures cross into the CaO (solid) + CO2 (vapor) field at improbably low pressures. It is found that release paths from measured shock temperatures cross into the melt plus vapor field at pressures greater than .5 GPa, which suggests that devolatilization is initiated at the shear banding sites.

Tyburczy, J. A.↗

NASA NTRS1986

Dynamic compression and volatile release of carbonates

Particle velocity profiles upon shock compression and isentropic releases were measured for polycrystalline calcite. The Solenhofen limestone release paths lie close to the Hugoniot. Calcite 3 to 2 transition, upon release was observed, but rarefaction shocks were not detected. The equation of state is used to predict the fraction of material devolatilized upon isentropic release as a function of shock pressure. The effect of ambient partial pressure of CO2 on the calculations is demonstrated and considered in models of atmospheric evolution by impact induced mineral devolatilization. The radiative characteristics of shocked calcite indicate that localization of thermal energy occurs under shock compression. Shock entropy calculations result in a minimum estimate of 90 percent devolatilization upon complete release from 10 GPa. Isentropic release paths from calculated continuum Hugoniot temperatures cross into the CaO (solid) + CO2 (vapor) field at improbably low pressures. It is found that release paths from measured shock temperatures cross into the melt plus vapor field at pressures greater than .5 GPa, which suggests that devolatilization is initiated at the shear banding sites.

Tyburczy, J. A.↗

NASA NTRS1991

A coupled channel study of HN2 unimolecular decay based on a global ab initio potential surface

The unimolecular decay lifetimes of several vibrational states of HN2 are determined on the basis of an accurate coupled channel dynamics study using a global analytical potential surface. The surface reproduces the ab initio points with an rms error of 0.08 kcal/mol for energies below 20 kcal/mol. Modifications to the potential that describe the effect of improving the basis set in the ab initio calculations are provided. Converged coupled channel calculations are performed for the ground rotational state of HN2 to determine the lifetimes of the lowest ten vibrational states. Only the ground vibrational state (000) and first excited bend (001) are found to have lifetimes longer than 1 ps. The lifetimes of these states are estimated at 3 x 10 to the -9th and 2 x 10 to the -10th s, respectively. Variation of these results with quality of the ab initio calculations is not more than a factor of 5.

Koizumi, Hiroyasu↗

NASA NTRS2013

United States Forest Disturbance Trends Observed Using Landsat Time Series

Disturbance events strongly affect the composition, structure, and function of forest ecosystems; however, existing U.S. land management inventories were not designed to monitor disturbance. To begin addressing this gap, the North American Forest Dynamics (NAFD) project has examined a geographic sample of 50 Landsat satellite image time series to assess trends in forest disturbance across the conterminous United States for 1985-2005. The geographic sample design used a probability-based scheme to encompass major forest types and maximize geographic dispersion. For each sample location disturbance was identified in the Landsat series using the Vegetation Change Tracker (VCT) algorithm. The NAFD analysis indicates that, on average, 2.77 Mha/yr of forests were disturbed annually, representing 1.09%/yr of US forestland. These satellite-based national disturbance rates estimates tend to be lower than those derived from land management inventories, reflecting both methodological and definitional differences. In particular the VCT approach used with a biennial time step has limited sensitivity to low-intensity disturbances. Unlike prior satellite studies, our biennial forest disturbance rates vary by nearly a factor of two between high and low years. High western US disturbance rates were associated with active fire years and insect activity, while variability in the east is more strongly related to harvest rates in managed forests. We note that generating a geographic sample based on representing forest type and variability may be problematic since the spatial pattern of disturbance does not necessarily correlate with forest type. We also find that the prevalence of diffuse, non-stand clearing disturbance in US forests makes the application of a biennial geographic sample problematic. Future satellite-based studies of disturbance at regional and national scales should focus on wall-to-wall analyses with annual time step for improved accuracy.

Masek, Jeffrey G.↗