Generalized Inverse Approach Research Articles

Predicting plant available water holding capacity of soils from crop yield

Quantification of variations in plant available water holding capacity (PAWC) of soils helps to improve yield forecast and inform spatially variable management practices in dryland agriculture systems. We developed and tested a general inverse approach to estimate PAWC from crop yield. The APSIM model was used to simulate wheat yield on synthetic soils with contrasting PAWC and climates. The simulated results were used to develop an empirical model to relate simulated yield to PAWC. The empirical model was inversely used to predict PAWC from observed crop yield. Potential prediction ability was quantified using independently simulated wheat yield on actual soils. The actual ability was assessed with measured wheat yields and PAWC. The approach had higher accuracy for sites with high rainfall or dominant summer rainfall. It could potentially provide acceptable PAWC predictions across contrasting climate regions (prediction error < 37 mm, 33.5%). The prediction error using crop yield against measured PAWC was <25 mm (26.5%). Our results demonstrate that soil PAWC can be reliably predicted from crop yield. This approach provides an alternative way to predict PAWC rather than directly measuring it via soil sampling, with profound implications for reducing labour and costs.

The Classical Braess Paradox Problem Revisited: A Generalized Inverse Method on Non-Unique Path Flow Cases

The classical Braess paradox problem refers to a user-equilibrium assignment model which all started with Braess’s (Unternehmensforschung 12; 258–268, 1968) demonstrated example network. Some variants of Braess paradox and related theories were subsequently developed to detect this paradoxical phenomenon on a general network. In this paper, the authors are devoted to the classical Braess paradox problem involving situations whenever considering new links to be added to a network. Historical literature told us that existing theories for this problem were limited to networks which admit unique path flow solution. A generalized inverse approach is suggested to solve this problem without the assumption of unique path flow solution in this study. The change of equilibrium cost after link additions is derived as a generalized inverse formulation of which solution possesses the non-uniqueness and flow conservation over all perturbed paths. Based on this generalized inverse formulation of the change of equilibrium cost, the authors show that there exists at least one of the O/D pairs, connected by new added routes, such that Braess paradox doesn’t (does) occur if the proposed test matrix is positive (negative) semi-definite. The derivations extend existing theories towards the situations when multiple routes are arbitrarily generated after link additions. These new theories deliver prior information to foresee Braess paradox taking place on a class of transportation networks which is more general than before and never reached by existing studies on the indicated classical Braess paradox problem.

Image Reconstruction in Optical Interferometry

This tutorial paper describes the problem of image reconstruction from interferometric data with a particular focus on the specific problems encountered at optical (visible/IR) wavelengths. The challenging issues in image reconstruction from interferometric data are introduced in the context of a general inverse problem approach. This framework is then used to describe state-of-the-art image reconstruction algorithms specifically developed for optical interferometry.

Design of unknown input functional observers for nonlinear systems with application to fault diagnosis

This paper considers the design of low-order unknown input functional observers for robust fault detection and isolation of a class of nonlinear Lipschitz systems subject to unknown inputs. The proposed functional observers can be used to generate residual signals to detect and isolate actuator faults. By using the generalized inverse approach, the effect of the unknown inputs can be decoupled completely from the residual signals. Conditions for the existence and stability of reduced-order unknown input functional observer are derived. A design procedure for the generation of residual signals to detect and isolate actuator faults is presented using the proposed unknown-input observer-based approach. A numerical example is given to illustrate the proposed fault diagnosis scheme in nonlinear systems subject to unknown inputs.

Comparative study on mode-identification algorithms using a phased-array system in a rectangular duct

To identify multiple acoustic duct modes, conventional beam-forming, CLEAN as well as L2 (i.e. pseudo-inverse) and L1 generalized-inverse beam-forming are applied to phased-array pressure data. A tone signal of a prescribed mode or broadband signal is generated upstream of a curved rectangular duct, and acoustic fields formed in both upstream and downstream stations of the test section are measured with identical wall-mounted microphone arrays. Sound-power distributions of several horizontal and vertical modes including upstream- and downstream-propagating waves can be identified with phased-array techniques, and the results are compared among the four approaches. The comparisons using synthetic data demonstrate that the L2 generalized-inverse algorithm can sufficiently suppress undesirable noise levels and detect amplitude distributions accurately in over-determined cases (i.e. the number of microphones is more than the number of cut-on modes) with minimum computational cost. As the number of cut-on modes exceeds the number of microphones (i.e. under-determined problems), the L1 algorithm is necessary to retain better accuracy. The comparison using test data acquired in the curved duct test rig (CDTR) at NASA Langley Research Center suggests that the L1/L2 generalized-inverse approach as well as CLEAN can improve the dynamic range of the detected mode by as much as 10dB relative to conventional beam-forming even with mean flow of M=0.5.

Single-hole imaging using borehole gravity gradiometry

We study the feasibility of imaging density variations away from boreholes using gravity gradient measurements in a single borehole. The objective is to develop numerical interpretation tools and to understand the utility and limitations of borehole gravity gradiometry in anticipation of the development of such instruments for the exploration and production of minerals, oil, and gas. We develop an analytic solution for directly locating discrete off-hole mass anomalies and apply a generalized inverse approach to image anomalous density distribution. In the analytic approach, we solve for the distance and direction angles to a point source using the multiple components of the gravity gradient tensor from a set of observations in a single borehole; and then apply this approach to the scenarios of prismatic sources. In the generalized inverse approach, we invert multiple gradient tensor components to construct 3D density contrast distributions around the borehole. As few as two independent gravity gradient components in a single borehole can be used to locate and image isolated anomalous density bodies.

Assimilation of Ship-Mounted ADCP Data for Barotropic Tides: Application to the Ross Sea

Abstract The application of a generalized inverse approach for assimilating vessel-mounted acoustic Doppler current profiler (VM-ADCP) data into numerical solutions of barotropic tides is described. The derived estimates of tidal currents can be used to detide the VM-ADCP data and expose underlying mean circulation. The methodology is illustrated with data assimilation models of tidal currents in the Ross Sea. The prior solution, obtained by solving the nonlinear shallow-water equations by time stepping with a linear bottom friction parameterization and elevation of open boundary conditions obtained from a circum-Antarctic tide model, provides reasonably good fit to most available moored current meter data. Two inverse solutions were obtained: one assimilating moored current meter records and the other assimilating three cruises of VM-ADCP data. Fitting either the mooring time series or the VM-ADCP records leads to only small changes relative to the prior solution currents, except over the shelf break where short length scale, energetic diurnal topographic vorticity waves are present. It is shown that the dynamics embedded in the representer functions provides reasonable tidal corrections even with no prior information about forcing at open boundaries.

Resolution, stability and efficiency of resistivity tomography estimated from a generalized inverse approach

SUMMARY Inconsistencies between an object and its image delivered by tomographical methods are inevitable. Loss of information occurs during the survey through incomplete and inaccurate data sampling and may also be introduced during the inverse procedure by smoothness constraints inadequate to the resolving power of the experimental setup. A quantitative appraisal of image quality (spatial resolution and image noise) is therefore not only required for successful interpretation of images but can be used together with measures of efficiency of the experimental design to optimize survey and inverse procedures. This paper introduces a low-contrast inversion scheme for electrical resistivity tomography that supports the reconstructed image with estimates of model resolution, model covariance and data importance. The algorithm uses a truncated pseudo-inverse and a line search approach to determine the maximum number of degrees of freedom necessary to fit the data to a prescribed target misfit. Though computationally expensive, the virtue of the method is that it reduces subjectivity by avoiding any empirically motivated model smoothness constraints. The method can be incorporated into a full non-linear inversion scheme for which a posteriori quality estimates can be calculated. In a numerical 2-D example the algorithm yielded reasonable agreement between object and image even for moderate resistivity contrasts of 10:100:1000. On the other hand, the resolving power of an exemplary four-electrode data set containing classical dipole–dipole and non-conventional configurations was shown to be severely affected by data inaccuracy. Insight into the resolving power as a function of space and data accuracy can be used as a guideline to designing optimized data sets, smoothness constraints and model parametrization.

Estimation of Finite and Closed Population Size: An Inverse Sampling Approach with Unequal Recapture Probability

AbstractBased on capture‐mark‐recapture sampling methods the problem of estimating unknown population size was considered. The sampling started with the assumption that at the beginning of the experiment all the individuals were unmarked, and the unmarked individuals caught in each sample will be marked and returned to the original population before the next sample is drawn. It is also assumed that the population is closed by birth, death, emigration and immigration. Using a general inverse sampling approach, the unknown population size N is estimated by a maximum likelihood estimator (MLE), and a simple form for approximate MLE is obtained. The probability function for S (the minimum number of samples required to be drawn to have L (L ≥ 1) samples, each of which contains at least one marked individual) and the form for E[S] are also obtained. In addition, corrections and improvements of some previous works in this field are given.

Fast adaptive RLS algorithms: a generalized inverse approach and analysis

A generalized inverse approach is used to derive two fast adaptive recursive least squares (RLS) algorithms and an exact and stable initialization algorithm for the prewindowed signal case. The partitioning techniques are directly applied to the signal matrix rather than the covariance matrix. The simulations show that the method has better numerical properties than existing fast algorithms, especially when the Kalman gain is concerned. Interesting relations among the variables in the fast RLS algorithms are provided. These relations can be used to derive other variations of fast RLS algorithms. They can also be used to develop new rescue schemes. Finally, a numerical analysis illustrates some pitfalls of the fast RLS algorithms.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A full invariant under dynamic output feedback: a generalized inverse approach

A novel full invariant of linear time-invariant multivariable systems under causal dynamic output feedback is derived. It makes use of a generalized inverse approach.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A general inverse approach for 2-D edge waves

Edge waves, which are of interest in many applications of wave motion since such waves carry energy over greater distances than the analogous body waves, are found for example in oceanography as waves trapped along continents bounded by oceans with various bottom topographies. A method is presented by which the topography associated with a given edge wave is determined. This is the inverse of the usual treatment which specifies topography and seeks edge wave solutions and is simpler mathematically. Some new solutions are obtained using this approach.

The magnetic field of Jupiter: A generalized inverse approach

The estimation of planetary magnetic fields from observations of the magnetic field gathered along a spacecraft flyby trajectory is examined with the aid of generalized inverse techniques, with application to the internal magnetic field of Jupiter. Model non‐uniqueness resulting from the limited spatial extent of the observations and noise on the data is explored and quantitative estimates of the model parameter resolution are found. The presence of a substantial magnetic field of external origin due to the currents flowing in the Jovian magnetodisc is found to be an important source of error in estimates of the internal Jovian field, and new models explicitly incorporating these currents are proposed. New internal field models are derived using the vector helium magnetometer observations and the high field fluxgate observations of Pioneer 11, and knowledge of the external current system gained from the Pioneer 10 and Voyagers 1 and 2 encounters.

Some remarks on the undersized sample problem in econometrics

This paper discusses three approaches to the problem of undersized samples: the estimable function, the generalized inverse and the principal component method. It turns out that the k-class estimation methods amount to ordinary least squares if use is made of estimable function or generalized inverses. The generalized inverse approach is a special case of the estimable function approach and of the principal component solution.

The Mahalanobis classifier with the generalized inverse approach for automated analysis of imagery texture data

Multicollinearity exists in texture measures, implying that the dispersion matrices are nearly singular. The generalized inverse was employed to solve this problem, yielding an 8% improvement in classification accuracy.

Targets, instruments and generalized inverses

The purpose of this paper is to demonstrate the usefulness of generalized inverses for problems associated with macro economic policy. Previous research on the selection of targets and instruments has specified the problem so that both the values of the instruments and the values realized for the targets are a joint outcome of the constrained optimization of an arbitrarily defined welfare function. Two theorems are developed which suggest that: (1) if the number of instruments exceeds the number of targets, a welfare function need not be defined. It is possible to discriminate between policies with a minimum norm generalized inverse which is based on the values of the instruments: (2) given the number of targets exceeds that of instruments the generalized inverse approach reduces to a special case of Theil's certainty equivalence method. Klein's six equation model from ‘Economic Fluctuations in the United States’ is used to illustrate these approaches.