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At least 1,603 records · Page 89
All sourcesNASA NTRSDOE OSTI
NASA NTRS2016

Application of Design of Experiments and Surrogate Modeling within the NASA Advanced Concepts Office, Earth-to-Orbit Design Process

Decisions made during early conceptual design have a large impact upon the expected life-cycle cost (LCC) of a new program. It is widely accepted that up to 80% of such cost is committed during these early design phases.1 Therefore, to help minimize LCC, decisions made during conceptual design must be based upon as much information as possible. To aid in the decision making for new launch vehicle programs, the Advanced Concepts Office (ACO) at NASA Marshall Space Flight Center (MSFC) provides rapid turnaround pre-phase A and phase A concept definition studies. The ACO team utilizes a proven set of tools to provide customers with a full vehicle mass breakdown to tertiary subsystems, preliminary structural sizing based upon worst-case flight loads, and trajectory optimization to quantify integrated vehicle performance for a given mission.2 Although the team provides rapid turnaround for single vehicle concepts, the scope of the trade space can be limited due to analyst availability and the manpower requirements for manual execution of the analysis tools. In order to enable exploration of a broader design space, the ACO team has implemented an Advanced Design Methods (ADM) based approach. This approach applies the concepts of Design of Experiments (DOE) and surrogate modeling to more exhaustively explore the trade space and provide the customer with additional design information to inform decision making. This paper will first discuss the automation of the ACO tool set, which represents a majority of the development e ort. In order to t a surrogate model within tolerable error bounds a number of DOE cases are needed. This number will scale with the number of variable parameters desired and the complexity of the system's response to those variables. For all but the smallest design spaces, the number of cases required cannot be produced within an acceptable timeframe using a manual process. Therefore, automation of the tools was a key enabler for the successful application of an ADM approach to an ACO design study. Following the overview of the tool set automation, an example problem will be given to illustrate the implementation of the ADM approach. The example problem will first cover the inclusion of Ground Rules and Assumptions (GR&A) for a study. The GR&A are very important to the study as they determine the constraints within which a trade study can be conducted. These trades must ultimately reconcile with the customer's desired output and any anticipated \what if" questions. The example problem will then illustrate the setup and execution of a DOE through the automated ACO tools. This process is accomplished more efficiently in this work by splitting the tools into two separate environments. The first environment encompasses the structural optimization and mass estimation tools, while the second is focused on trajectory optimization. Surrogate models are t to the outputs of each environment and are integrated via connection of the surrogate equations. Throughout this process, checks are implemented to compare the output of the surrogates to the output of manually run cases to ensure that the error of the final surrogates is at an acceptable level. The conclusion of the example problem demonstrates the utility of the ADM based approach. Using surrogate models gives the ACO team the ability to visualize vehicle sensitivities to various design parameters and identify regions of interest within the design space. The ADM approach can thus be used to inform concept down selection and isolate promising vehicle configurations to be explored in more detail through the manual design process. In addition it provides the customer with an almost instantaneous turnaround on any \what if" questions that may arise within the bounds of the surrogate model. This approach ultimately expands the ability of the ACO team to provide its customer with broad and rapid turnaround trade studies for launch vehicle conceptual design. The ability to identify a selection of designs which can meet the customer requirements will have the potential to lower LCC of launch vehicle designs originating from ACO.

Zwack, Mathew R.↗
NASA NTRS2016

Application of Design of Experiments and Surrogate Modeling within the NASA Advanced Concepts Office, Earth-to-Orbit Design Process

Decisions made during early conceptual design have a large impact upon the expected life-cycle cost (LCC) of a new program. It is widely accepted that up to 80% of such cost is committed during these early design phases. Therefore, to help minimize LCC, decisions made during conceptual design must be based upon as much information as possible. To aid in the decision making for new launch vehicle programs, the Advanced Concepts Office (ACO) at NASA Marshall Space Flight Center (MSFC) provides rapid turnaround pre-phase A and phase A concept definition studies. The ACO team utilizes a proven set of tools to provide customers with a full vehicle mass breakdown to tertiary subsystems, preliminary structural sizing based upon worst-case flight loads, and trajectory optimization to quantify integrated vehicle performance for a given mission. Although the team provides rapid turnaround for single vehicle concepts, the scope of the trade space can be limited due to analyst availability and the manpower requirements for manual execution of the analysis tools. In order to enable exploration of a broader design space, the ACO team has implemented an advanced design methods (ADM) based approach. This approach applies the concepts of design of experiments (DOE) and surrogate modeling to more exhaustively explore the trade space and provide the customer with additional design information to inform decision making. This paper will first discuss the automation of the ACO tool set, which represents a majority of the development effort. In order to fit a surrogate model within tolerable error bounds a number of DOE cases are needed. This number will scale with the number of variable parameters desired and the complexity of the system's response to those variables. For all but the smallest design spaces, the number of cases required cannot be produced within an acceptable timeframe using a manual process. Therefore, automation of the tools was a key enabler for the successful application of an ADM approach to an ACO design study. Following the overview of the tool set automation, an example problem will be given to illustrate the implementation of the ADM approach. The example problem will first cover the inclusion of ground rules and assumptions (GR&A) for a study. The GR&A are very important to the study as they determine the constraints within which a trade study can be conducted. These trades must ultimately reconcile with the customer's desired output and any anticipated "what if" questions. The example problem will then illustrate the setup and execution of a DOE through the automated ACO tools. This process is accomplished more efficiently in this work by splitting the tools into two separate environments. The first environment encompasses the structural optimization and mass estimation tools, while the second is focused on trajectory optimization. Surrogate models are fit to the outputs of each environment and are "integrated" via connection of the surrogate equations. Throughout this process, checks are implemented to compare the output of the surrogates to the output of manually run cases to ensure that the error of the final surrogates is at an acceptable level. The conclusion of the example problem demonstrates the utility of the ADM based approach. Using surrogate models gives the ACO team the ability to visualize vehicle sensitivities to various design parameters and identify regions of interest within the design space. The ADM approach can thus be used to inform concept down selection and isolate promising vehicle configurations to be explored in more detail through the manual design process. In addition it provides the customer with an almost instantaneous turnaround on any ''what if" questions that may arise within the bounds of the surrogate model. This approach ultimately expands the ability of the ACO team to provide its customer with broad and rapid turnaround trade studies for launch vehicle conceptual design. The ability to identify a selection of designs which can meet the customer requirements will help ensure lower LCC of launch vehicle designs originating from ACO.

Zwack, Mathew R.↗
DOE OSTI2021

Estimating Leaf Area Index in Row Crops Using Wheel-Based and Airborne Discrete Return Light Detection and Ranging Data

Leaf area index (LAI) is an important variable for characterizing plant canopy in crop models. It is traditionally defined as the total one-sided leaf area per unit ground area and is estimated by both direct and indirect methods. This paper explores the effectiveness of using light detection and ranging (LiDAR) data to estimate LAI for sorghum and maize with different treatments at multiple times during the growing season from both a wheeled vehicle and Unmanned Aerial Vehicles. Linear and nonlinear regression models are investigated for prediction utilizing statistical and plant structure-based features extracted from the LiDAR point cloud data with ground reference obtained from an in-field plant canopy analyzer (indirect method). Results based on the value of the coefficient of determination ( R 2 ) and root mean squared error for predictive models ranged from ∼0.4 in the early season to ∼0.6 for sorghum and ∼0.5 to 0.80 for maize from 40 Days after Sowing to harvest.

59 BASIC BIOLOGICAL SCIENCES↗
NASA NTRS2018

Greedy Sampling and Incremental Surrogate Model-Based Tailoring of Aeroservoelastic Model Database for Flexible Aircraft

This paper presents a data analysis and modeling framework to tailor and develop linear parameter-varying (LPV) aeroservoelastic (ASE) model database for flexible aircrafts in broad 2D flight parameter space. The Kriging surrogate model is constructed using ASE models at a fraction of grid points within the original model database, and then the ASE model at any flight condition can be obtained simply through surrogate model interpolation. The greedy sampling algorithm is developed to select the next sample point that carries the worst relative error between the surrogate model prediction and the benchmark model in the frequency domain among all input-output channels. The process is iterated to incrementally improve surrogate model accuracy till a pre-determined tolerance or iteration budget is met. The methodology is applied to the ASE model database of a flexible aircraft currently being tested at NASA/AFRC for flutter suppression and gust load alleviation. Our studies indicate that the proposed method can reduce the number of models in the original database by 67%. Even so the ASE models obtained through Kriging interpolation match the model in the original database constructed directly from the physics-based tool with the worst relative error far below 1%. The interpolated ASE model exhibits continuously-varying gains along a set of prescribed flight conditions. More importantly, the selected grid points are distributed non-uniformly in the parameter space, a) capturing the distinctly different dynamic behavior and its dependence on flight parameters, and b) reiterating the need and utility for adaptive space sampling techniques for ASE model database compaction. The present framework is directly extendible to high-dimensional flight parameter space, and can be used to guide the ASE model development, model order reduction, robust control synthesis and novel vehicle design of flexible aircraft.

numerical analysi↗
DOE OSTI2023

Size-dependent errors in real-time electron density propagation

Real-time (RT) electron density propagation with time-dependent density functional theory (TDDFT) or Hartree–Fock (TDHF) is one of the most popular methods to model the charge transfer in molecules and materials. However, both RT-TDHF and RT-TDDFT within the adiabatic approximation are known to produce inaccurate evolution of the electron density away from the ground state in model systems, leading to large errors in charge transfer and erroneous shifting of peaks in absorption spectra. Given the poor performance of these methods with small model systems and the widespread use of the methods with larger molecular and material systems, here we bridge the gap in our understanding of these methods and examine the size-dependence of errors in RT density propagation. We analyze the performance of RT density propagation for systems of increasing size during the application of a continuous resonant field to induce Rabi-like oscillations, during charge-transfer dynamics, and for peak shifting in simulated absorption spectra. We find that the errors in the electron dynamics are indeed size dependent for these phenomena, with the largest system producing the results most aligned with those expected from linear response theory. The results suggest that although the RT-TDHF and RT-TDDFT methods may produce severe errors for model systems, the errors in charge transfer and resonantly driven electron dynamics may be much less significant for more realistic, large-scale molecules and materials.

Chemistry↗
DOE OSTI2021

Assessing the predictive impact of factor fixing with an adaptive uncertainty-based approach .

Despite widespread use of factor fixing in environmental modeling, its effect on model predictions has received little attention and is instead commonly presumed to be negligible. We propose a proof-of-concept adaptive method for systematically investigating the impact of factor fixing. The method uses Global Sensitivity Analysis methods to identify groups of sensitive parameters, then quantifies which groups can be safely fixed at nominal values without exceeding a maximum acceptable error, demonstrated using the 21-dimensional Sobol’ G-function. Furthermore, three error measures are considered for quantities of interest, namely Relative Mean Absolute Error, Pearson Product-Moment Correlation and Relative Variance. Results demonstrate that factor fixing may cause large errors in the model results unexpectedly, when preliminary analysis suggests otherwise, and that the default value selected affects the number of factors to fix. To improve the applicability and methodological development of factor fixing, a new research agenda encompassing five opportunities is discussed for further attention.

54 ENVIRONMENTAL SCIENCES↗
NASA NTRS1994

A measurement system for large, complex software programs

This paper describes measurement systems required to forecast, measure, and control activities for large, complex software development and support programs. Initial software cost and quality analysis provides the foundation for meaningful management decisions as a project evolves. In modeling the cost and quality of software systems, the relationship between the functionality, quality, cost, and schedule of the product must be considered. This explicit relationship is dictated by the criticality of the software being developed. This balance between cost and quality is a viable software engineering trade-off throughout the life cycle. Therefore, the ability to accurately estimate the cost and quality of software systems is essential to providing reliable software on time and within budget. Software cost models relate the product error rate to the percent of the project labor that is required for independent verification and validation. The criticality of the software determines which cost model is used to estimate the labor required to develop the software. Software quality models yield an expected error discovery rate based on the software size, criticality, software development environment, and the level of competence of the project and developers with respect to the processes being employed.

Rone, Kyle Y.↗
DOE OSTI2024

Energy-conserving physics for nonhydrostatic dynamics in mass coordinate models

Abstract. Motivated by reducing errors in the energy budget related to enthalpy fluxes within the Energy Exascale Earth System Model (E3SM), we study several physics–dynamics coupling approaches. Using idealized physics, a moist rising bubble test case, and the E3SM's nonhydrostatic dynamical core, we consider unapproximated and approximated thermodynamics applied at constant pressure or constant volume. With the standard dynamics and physics time-split implementation, we describe how the constant-pressure and constant-volume approaches use different mechanisms to transform physics tendencies into dynamical motion and show that only the constant-volume approach is consistent with the underlying equations. Using time step convergence studies, we show that the two approaches both converge but to slightly different solutions. We reproduce the large inconsistencies between the energy flux internal to the model and the energy flux of precipitation when using approximate thermodynamics, which can only be removed by considering variable latent heats, both when computing the latent heating from phase change and when applying this heating to update the temperature. Finally, we show that in the nonhydrostatic case, for physics applied at constant pressure, the general relation that enthalpy is locally conserved no longer holds. In this case, the conserved quantity is enthalpy plus an additional term proportional to the difference between hydrostatic pressure and full pressure.

58 GEOSCIENCES↗
NASA NTRS1998

Application of the Shuttle Laser Altimeter in an Accuracy Assessment of Global 1-Kilometer Digital Elevation Data

Shuttle Laser Altimeter (SLA) data have been used to evaluate the accuracy of GTOPO30, the first comprehensive, 1 km resolution, global topographic data set. GTOPO30 was developed by the USGS Eros Data Center (EDC), in part, to address NASA's needs for a global topographic model in support of remote sensing instruments aboard the Earth Observing System AM-1 spacecraft. SLA flew as a part of the STS-72 mission in January, 1996 observing the latitude band from +/- 28.5 deg, and on STS-85 in August, 1997 extending the observations to +/- 57 deg. Combining the SLA ranging data with shuttle position and pointing knowledge yields surface elevation data of very high vertical accuracy in an Earth-centered, absolute reference frame (2.8 m rms difference for SLA-01 with respect to ocean reference surface). Use of the well-determined mean sea surface reference for calibration allows propagation of high accuracy altimetry onto the continents. 436,635 SLA-01 land elevations were compared to the GTOPO30 grid after conversion to a mean sea level vertical datum using the Earth Geoid Model 96, jointly developed by Goddard and NIMA. The comparison reveals systematic elevation biases in southern Asia, Africa, Australia, and south America on the order 10's to 100 meters in the GTOPO30 compilation on spatial scales of 100's to 1000's of kilometers. These biases are likely due to vertical datum errors in the topographic source materials used to compile GTOPO30, which primarily consist of Defense Mapping Agency (DMA) digital elevation and topographic map products. These biases imply that elevation corrections applied to land gravity measurements using these DMA source materials will be biased, leading to errors in geoid models incorporating these land gravity data.

Harding, David J.↗
DOE OSTI2025

Mapping tree height in complex terrain of northern China using ultra-high-resolution images

Tree height is a key parameter for estimating forest biomass and carbon sequestration. In recent years, notable progress has been made in mapping tree height using satellite imagery. However, existing tree height products show low accuracy in mountainous and complex terrains, and few studies typically addressed tree height estimations in mountain areas. This study examines the Mentougou district of Beijing, China, characterized by complex terrain and mountainous landscapes. We analyzed two methods for estimating tree height: one using only spectral features and another combining spectral features with topographic factors (elevation, slope, aspect). We used 3-m resolution PlanetScope 8-band multispectral imagery, with 710 field-measured individual tree heights averaged to obtain 471 pixel-level tree height values as ground-truth, to develop tree height prediction models using eXtreme Gradient Boosting (XGBoost), Random Forest (RF), and Gradient Boosting Machine (GBM) models. The results show that the XGBoost model consistently presented the highest accuracy for both methods evaluated. Specifically, the XGBoost model that combined spectral data with elevation and slope variables with an R² of 0.75 and an RMSE of 2.69 m. Using the XGBoost model, we generated the tree height map for the Mentougou area at 3 m resolution, showing tree heights ranging from 0.5 to 30.4 m, and the model’s prediction error standard deviations ranged from 2.50 to 4.71 m, indicating reliable performance across varied terrain. Additionally, we compared and evaluated the global tree height products, identifying limitations in the accuracy within complex terrains. This study demonstrates the potential for accurately predicting tree heights by combining high-resolution multispectral satellites with a terrain factor modeling approach.

Complex terrain↗
NASA NTRS1983

Amalthea - Implications of the temperature observed by Voyager

The temperature profile of Amalthea's surface layer is modeled as a function of location and time of day, assuming a triaxial ellipsoid shape and thermal properties similar to those of the lunar soil. Although the major heat source is direct insolation, temperatures are slightly increased by thermal radiation from Jupiter, sunlight reflected from the planet, and charged particle bombardment. Possible sources of error in the model are discussed in detail, including satellite shape effects, unusually low emissivity, uncommonly rough surface, abnormal thermal inertia, charged particle flux variability, and Joule heating. Voyager 1 IRIS observations suggest that the Amalthean surface has an emissivity near unity, but cannot put any useful limits on the thermal inertia of the Amalthean surface layer.

Simonelli, D. P.↗
DOE OSTI2020

Skill of Mesoscale Models in Forecasting Springtime Macrophysical Cloud Properties at the Savannah River Site in the Southeastern US

Predicting boundary layer clouds is important for the accurate modeling of pollutant dispersion. Higher resolution mesoscale models would be expected to produce better forecasts of cloud properties that affect dispersion. Using ceilometer observations, we assess the skill of two operational mesoscale models (RAMS and WRF) to forecast cloud base altitude and cloud fraction at the Savannah River Site in the southeastern US during the springtime. Verifications were performed at small spatial and temporal scales necessary for dispersion modeling. Both models were unreliable with a 50% (RAMS) and a 46% (WRF) rate of predicting clouds observed by the ceilometer which led to low cloud fraction predictions. Results indicated that WRF better predicted daytime cloud bases from convection that occurred frequently later in the period and RAMS better predicted nighttime cloud bases. Using root mean squared error (RMSE) to score the forecast periods also highlighted this diurnal dichotomy, with WRF scores better during the day and RAMS scores better at night. Analysis of forecast errors revealed divergent model cloud base biases—WRF low and RAMS high. A hybrid solution which weighs more heavily the RAMS nighttime forecasts and WRF daytime forecasts will likely provide the best prediction of cloud properties for dispersion.

54 ENVIRONMENTAL SCIENCES↗
NASA NTRSDate not supplied

Data Fusion of In-Flight Aerothermodynamic Heating Measurements Using Kalman Filtering

On February 18th, 2021, the Mars 2020 entry system successfully delivered the Perserverance rover to the surface of Mars at Jezero Crater. The entry capsule carried a set of instrumentation installed on the heat shield and backshell, named the Mars 2020 Entry, Descent, and Landing Instrumentation. The instruments include pressure transducers, thermocouples, heat flux gauges, and radiometers to measure the aerodynamic and aerothermodynamic performance of the entry vehicle. Three of these sensors, a thermocouple plug, heat flux gauge, and a radiometer, are located in close proximity on the backshell. Each sensor is exposed to roughly the same environment, but measured these environments in different ways, each with its own set of modeling and measurement error complications. This paper develops a method for blending each of these measurements together in a single algorithm to produce estimates of the aerothermodynamic environments at that location on the backshell. The approach makes use of the Kalman filter methodology for solving state estimation problems. The filter has a predictor/corrector structure in which simplified process models are used to propagate the aeroheating states forward through time, which are then updated based on the measurement data. The method has been tested on simulated datasets.

Christopher D. Karlgaard↗
NASA NTRS1980

Geodetic positioning using a global positioning system of satellites

Geodetic positioning using range, integrated Doppler, and interferometric observations from a constellation of twenty-four Global Positioning System satellites is analyzed. A summary of the proposals for geodetic positioning and baseline determination is given which includes a description of measurement techniques and comments on rank deficiency and error sources. An analysis of variance comparison of range, Doppler, and interferometric time delay to determine their relative geometric strength for baseline determination is included. An analytic examination to the effect of a priori constraints on positioning using simultaneous observations from two stations is presented. Dynamic point positioning and baseline determination using range and Doppler is examined in detail. Models for the error sources influencing dynamic positioning are developed. Included is a discussion of atomic clock stability, and range and Doppler observation error statistics based on random correlated atomic clock error are derived.

Fell, P. J.↗
NASA NTRS1988

Interference by rain scatter

The data from Japan and the U.S. (the Virginia Precipitation Scatter Experiment) show excellent agreement between the two-component rain scatter model predictions and bistatic scatter measurements. In employing the model, all the scattering geometries should be classified as backscattering as defined by Crane (1974). The forward scatter model should only be used for great circle paths with both antennas pointed at the horizon and at each other in a typical troposcatter communication system geometry. The forward scatter model can also be used for main-lobe, side-lobe coupling when one antenna is pointed toward the other along the great circle path. The forward scatter observations made over the Prospect Hill - Mt Tug path show that the two-component model is incomplete. Much stronger signals were observed at Ku-band than expected based on simultaneous C-band measurements. The discrepancies may be due to: (1) scattering by ice/snow at height (posssible in April) at the 1 km height of the scattering volume), (2) the coherent effects of turbulent fluctuations in the hydrometeor number densities and (3) errors in the modeling of the statistical relationship between attenuation along the path and scattering in the common volume.

Crane, Robert K.↗
NASA NTRS1999

The Role of Data Assimilation in the Study of Regional and Global Variability of the Hydrological Cycle

In the coming years, researchers will have at their disposal a host of new observations from advanced space-based sensors (e.g. the Tropical Rainfall Measuring Mission, EOS Terra and PM missions) providing, among other things, a more complete and accurate description of various components of the Earth's hydrological cycle. Also, increasingly more sophisticated and comprehensive geophysical models will provide researchers better tools for simulating the hydrological cycle, and for carrying out mechanistic studies of the role of moist processes in the climate system. In addition, new data sets generated with global four-dimensional data assimilation (4DDA) systems will provide comprehensive and complete information on both the state and forcing of the climate system. Ideally, the 4DDA systems optimally incorporate all relevant information from the observations together with a first guess from a state-of-the-art geophysical model to produce a "best" estimate of the climate state. Furthermore, to the extent that the assimilating models are realistic and are constrained by the observations, they should provide reliable estimates of the associated physical processes or climate forcing fields. While operational weather centers now have a considerable history of providing reliable estimates of the basic atmospheric state variables, the associated processes or diagnostic fields (which are less well constrained by the observations and sensitive to errors in the model's physical parameterizations) are still considered experimental and of uncertain quality. In this study we will examine the current generation of reanalysis products to assess the capabilities of 4DDA systems to represent components of the hydrological cycle. The focus is on the role of the model in providing consistent estimates of moist processes. We will also assess whether current observations provide sufficient constraints on these model- generated fields.

Schubert, Siegfried↗
DOE OSTI2026

Digital quantum magnetism on a trapped-ion quantum computer

Digital quantum matter—realized when discrete quantum gates approximate continuous time evolution—is susceptible to heating into chaotic, structureless states. If digitization errors are adequately suppressed, a long-lived transient regime of approximately energy-conserving dynamics can be observed on gate-based quantum computers. Conservation of energy, in turn, enables the exploration of a wide variety of complex behaviours observed in equilibrium systems, ranging from the non-trivial microscopic origins of thermalization itself to the stabilization of effective models hosting exotic emergent properties. Here we use Quantinuum’s H2 quantum computer to simulate digitized dynamics of the quantum Ising model, suppressing digitization errors well enough to observe thermalization on timescales that severely challenge classical simulation methods. Relaxation of an inhomogeneous state reveals an emergent hydrodynamics owing to approximate energy conservation and we compute the associated diffusion constant. By reprogramming our simulations to take place on a triangular lattice with periodic boundary conditions, we observe thermalization consistent with emergent gauge and topological constraints resulting from lattice frustration. Furthermore, our results were enabled by continued advances in two-qubit gate quality (native partial entangler fidelities of 99.94(1)%) and establish digital quantum computers as powerful tools for studying (effectively) continuous-time dynamics.

Information theory and computation↗
DOE OSTI2020

Rapid data-driven model reduction of nonlinear dynamical systems including chemical reaction networks using ℓ 1 -regularization

We develop a new data-driven paradigm for efficient model reduction of a broad class of nonlinear dynamical systems. Our model reduction method directly enables the interpretation of key components of the dynamical system, unlike traditional projection-based model reduction methods that focus on reducing computational complexity more than interpretability. Our method is not application specific and is simple to implement on nonlinear dynamical systems arising from a variety of different fields. It requires minimal parameterization using a single parameter to trade-off between model complexity and estimation error. We use a data-driven paradigm to formulate model reduction as an efficient convex optimization problem that scales polynomially in the original size of the complex system, enabling systems with as many as thousands of components to be reduced in a matter of minutes.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗