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It is easy to understand heuristically why single-frequency spaceborne radar measurements, by themselves, are not sufficient to solve the inverse problem of retrieving an exact underlying rain-rate profile.
Weather radar measurements from airborne or satellite platforms can be an effective remote sensing tool for examining the three-dimensional structures of clouds and precipitation. This chapter describes some fundamental properties of radar measurements and their dependence on the particle size distribution (PSD) and radar frequency. The inverse problem of solving for the vertical profile of PSD from a profile of measured reflectivity is stated as an optimal estimation problem for single- and multi-frequency measurements. Phenomena that can change the measured reflectivity Z(sub m) from its intrinsic value Z(sub e), namely attenuation, non-uniform beam filling, and multiple scattering, are described and mitigation of these effects in the context of the optimal estimation framework is discussed. Finally, some techniques involving the use of passive microwave measurements to further constrain the retrieval of the PSD are presented.
The paper applies iterated and non-iterated extended Kalman filters to solve two practical meteorological inversion problems. Passive microwave satellite soundings are used to infer vertical temperature profiles and cloud parameters. In both cases it is shown that improvements can be obtained over presently used techniques. Finally, the results suggest that modern multivariate nonlinear recursive estimation techniques based in a Bayesian methodology can be a valuable tool in the area of remote sounding of atmospheric parameters.
A fundamental problem which must be resolved in virtually all non-trivial robotic operations is the well-known inverse kinematic question. More specifically, most of the tasks which robots are called upon to perform are specified in Cartesian (x,y,z) space, such as simple tracking along one or more straight line paths or following a specified surfacer with compliant force sensors and/or visual feedback. In all cases, control is actually implemented through coordinated motion of the various links which comprise the manipulator; i.e., in link space. As a consequence, the control computer of every sophisticated anthropomorphic robot must contain provisions for solving the inverse kinematic problem which, in the case of simple, non-redundant position control, involves the determination of the first three link angles, theta sub 1, theta sub 2, and theta sub 3, which produce a desired wrist origin position P sub xw, P sub yw, and P sub zw at the end of link 3 relative to some fixed base frame. Researchers outline a new inverse kinematic solution and demonstrate its potential via some recent computer simulations. They also compare it to current inverse kinematic methods and outline some of the remaining problems which will be addressed in order to render it fully operational. Also discussed are a number of practical consequences of this technique beyond its obvious use in solving the inverse kinematic question.
We present new interstellar dust models that are consistent with both, the FUV to near-IR extinction and infrared (IR) emission measurements from the diffuse interstellar medium. The models are characterized by different dust compositions and abundances. The problem we solve consists of determining the size distribution of the various dust components of the model. This problem is a typical ill-posed inversion problem which we solve using the regularization approach. We reproduce the Li Draine (2001, ApJ, 554, 778) results, however their model requires an excessive amount of interstellar silicon (48 ppM of hydrogen compared to the 36 ppM available for an ISM of solar composition) to be locked up in dust. We found that dust models consisting of PAHs, amorphous silicate, graphite, and composite grains made up from silicates, organic refractory, and water ice, provide an improved fit to the extinction and IR emission measurements, while still requiring a subsolar amount of silicon to be in the dust. This research was supported by NASA Astrophysical Theory Program NRA 99-OSS-01.
Theoretical methods (computer programs) which are available for analyzing reflector performance are of the 'forward' type. Performance parameters can be calculated after the configuration of the reflector is given. In many practical applications, however, the problem is often of the 'inverse' type. The reflector has to be designed on the basis of a given set of performance parameters. Attention is given to a systematic procedure based on simple formulas for solving such an inverse problem. Because of the simplicity of the formulas, the final results obtained from the procedure do not have great accuracy. Thus, they represent only a preliminary design, which can be refined by repeatedly using the forward-type programs.
Inverse modeling techniques have been applied to the problem of determining the roles of various forces that may drive and resist plate tectonic motions. Separate linear inverse problems have been solved to find the best fitting pole of rotation for finite element grid point velocities and to find the best combination of force models to fit the observed relative plate velocities for the earth's twelve major plates using the generalized inverse operator. Variance-covariance data on plate motion have also been included. Results emphasize the relative importance of ridge push forces in the driving mechanism. Convergent margin forces are smaller by at least a factor of two, and perhaps by as much as a factor of twenty. Slab pull, apparently, is poorly transmitted to the surface plate as a driving force. Drag forces at the base of the plate are smaller than ridge push forces, although the sign of the force remains in question.
This paper is concerned with the implementation and assessment of joint motion management strategies for kinematically redundant robotic manipulators operating in the micro-gravity environment of Space Station. These robots must be capable of conducting experiments and manufacturing processes without disturbing the micro-gravity environment through base reactions/motions. The redundant degrees of freedom of the manipulator permit the inverse kinematic problem to be solved simultaneously with the minimization of a cost function. The cost function in this case is the weighted sum of the squares of the base forces and moments and is minimized over discrete time segments. The Generalized Inverse Method and Rayleigh Ritz technique are used to solve the combined optimization/inverse kinematics problem. Numerical examples include various robotic configurations and degrees of manipulator redundancy.
Knowledge of subsurface characteristics such as permittivity variations and layering structure could provide a breakthrough in many terrestrial and planetary science disciplines. For Earth science, knowledge of subsurface and subcanopy soil moisture layers can enable the estimation of vertical flow in the soil column linking surface hydrologic processes with that in the subsurface. For planetary science, determining the existence of subsurface water and ice is regarded as one of the most critical information needs for the study of the origins of the solar system. The subsurface in general can be described as several near-parallel layers with rough interfaces. Each homogenous rough layer can be defined by its average thickness, permittivity, and rms interface roughness assuming a known surface spectral distribution. As the number and depth of layers increase, the number of measurements needed to invert for the layer unknowns also increases, and deeper penetration capability would be required. To nondestructively calculate the characteristics of the rough layers, a multifrequency polarimetric radar backscattering approach can be used. One such system is that we have developed for data prototyping of the Microwave Observatory of Subcanopy and Subsurface (MOSS) mission concept. A tower-mounted radar makes backscattering measurements at VHF, UHF, and L-band frequencies. The radar is a pulsed CW system, which uses the same wideband antenna to transmit and receive the signals at all three frequencies. To focus the beam at various incidence angles within the beamwidth of the antenna, the tower is moved vertically and measurements made at each position. The signals are coherently summed to achieve focusing and image formation in the subsurface. This requires an estimate of wave velocity profiles. To solve the inverse scattering problem for subsurface velocity profile simultaneously with radar focusing, we use an iterative technique based on a forward numerical solution of the layered rough surface problem. The layers are each defined in terms of a small number of unknown distributions as given above. An a priori estimate of the solution is first assumed, based on which the forward problem is solved for the backscattered measurements. This is compared with the measured data and using iterative techniques an update to the solution for the unknowns is calculated. The process continues until convergence is achieved. Numerical results will be shown using actual radar data acquired with the MOSS tower radar system in Arizona in Fall 2003, and compared with in-situ measurements.
To solve a geophysical inverse problem means applying measurements to determine the parameters of the selected model. The inverse problem is formulated as the Bayesian inference. The Gaussian probability density functions are applied in the Bayes's equation. The CHAMP satellite gravity data are determined at the altitude of 400 kilometer altitude over the South part of the Pannonian basin. The model of interpretation is the right vertical cylinder. The parameters of the model are obtained from the minimum problem solved by the Simplex method.
COSIE (Coronal Spectrographic Imager in the Extreme Ultraviolet) is a proposed solar tracking ISS imaging payload that will help bridge the theoretical gap between the physics of the low corona and the heliosphere. This scientific instrument requires high pointing accuracy, on the order of arc seconds. The instrument is mounted on to a three revolute joint robotic arm in order to track the roll, pitch and yaw motion of the Sun. The goal of this project is to construct a prototype model of the robotic arm and implement the proposed analytical inverse kinematics algorithm. In robotics, the inverse kinematics problem is solving for the set of joint angles that achieve the desired end effect or location and/or orientation. In this case, orientation is the focus. Depending on the configuration, multiple sets of joint angle solutions may exist. Due to the complexity of robotics, typically iterative methods are used to solve for the joint angle solution sets. However, in this case, an analytical solution exists. A small robotic arm representative of the full size hardware was constructed. The inverse kinematics algorithm, originally in MATLAB/Simulink, was converted into C in order to interface with the motors. This C software was implemented on a Windows PC and micro-controller, and serial communication between the two was established, allowing the motors to be directly controlled by the inverse kinematics algorithm. Testing the inverse kinematics on a physical system will allow the validity and accuracy of the analytic solution to be verified.
The theory of Backus and Gilbert gives a technique for solving the general linear inverse problem. Observational error and lack of data are shown to reduce the reliability of the solution in different ways: the former introduces statistical uncertainties in the model, while the latter smooths out the detail. Precision can be improved by sacrificing resolving power, and vice versa, so that some compromise may be made between the two in choosing the best model. Nonlinear inverse problems can be brought into the domain of the theory by linearizing about a typical solution. The inverse problem of electrical conductivity in the mantle is used to illustrate the Backus-Gilbert technique; an example of the tradeoff diagram is given.
A computerized tool, CAPE, (Conduction Analysis Program using Eigenvalues) has been developed to account for lateral heat conduction in wind tunnel models in the data reduction of the phase-change paint technique. The tool also accounts for the effects of finite thickness (thin wings) and surface curvature. A special reduction procedure using just one time of melt is also possible on leading edges. A novel iterative numerical scheme was used, with discretized spatial coordinates but analytic integration in time, to solve the inverse conduction problem involved in the data reduction. A yes-no chart is provided which tells the test engineer when various corrections are large enough so that CAPE should be used. The accuracy of the phase-change paint technique in the presence of finite thickness and lateral conduction is also investigated.
A model consisting of two ring sources was developed to study the direct radiation of jet noise in terms of correlation, coherence, and phase and also to aid in solving the inverse radiation problem of determining the noise source in terms of far-field measurements. The rings consist of discrete sources which are either monopoles or quadrupoles with Gaussian profiles. Only adjacent sources, both within the rings and between rings, are correlated. Results show that from the far-field information can be used to determine when the sources are compact or noncompact with respect to the acoustic wavelength and to distinguish between the types of sources. In addition, from the inverse radiation approach, the center of mass, the location and separation distance of the ring, and the diameters can be recovered.
One of the greatest problems in measuring particle trajectories with an optical or visual detector system, is the reconstruction of trajectories in real space from their recorded images. In the Nagoya cosmic-ray muon spectrometer, muon tracks are detected by wide gap spark chambers and their images are recorded on the photographic film through an optical system of 10 mirrors and two cameras. For the spatial reconstruction, 42 parameters of the optical system should be known to determine the configuration of this system. It is almost impossible to measure this many parameters directly with usual techniques. In order to solve this problem, the inverse transformation method was applied. In this method, all the optical parameters are determined from the locations of fiducial marks in real space and the locations of their images on the photographic film by the non-linear least square fitting.
Powerful methods are now available for solving linear parametric inverse problems. However, many inverse problems which arise in geohysics are nonlinear. Fortunately, it is possible to treat most of these with the air of linear perturbation theory and liner inversion. But a convenient method is needed for assessing the importance of nonlinearity in these quasi-linear problems. The present paper provides such a method. Matsu'ura and Jackson (1984) have presented a simple algorithm for evaluating the asymptotic covariance matrix fo estimation errors. In the present investigation, aspects of linear inversion are discussed, taking into account linear parametric inverse problems, nonuniqueness, prior information, confidence limits, conditional and marginal statistics, the relative importance of the prior and observational data, and standardized variables. Attention is also given to nonlinear inversion, and the application of the considered approaches to a number of examples.
The generalized wave equation and generalized sine-Gordon equations are known to be natural multidimensional differential geometric generalizations of the classical two-dimensional versions. In this paper, a system of linear differential equations is associated with these equations, and it is shown how the direct and inverse problems can be solved for appropriately decaying data on suitable lines. An initial-boundary value problem is solved for these equations.
Collision-free trajectory planning for robotic manipulators is investigated. The task of the manipulator is to move its end-effector from one point to another point in an environment with polyhedral obstacles. An on-line algorithm is developed based on finding the required joint angles of the manipulator, according to goals with different priorities. The highest priority is to avoid collisions, the second priority is to plan the shortest path for the end effector, and the lowest priority is to minimize the joint velocity for smooth motion. The pseudo-inverse of the Jacobian matrix is applied for inverse kinematics. When a possible collision is detected, a constrained inverse kinematic problem is solved such that the collision is avoided. This algorithm can also be applied to a time-variant environment.