Set Of Simultaneous Equations Research Articles

Amazon Wildfires and Respiratory Health: Impacts during the Forest Fire Season from 2009 to 2019.

The Brazilian Amazon, a vital tropical region, faces escalating threats from human activities, agriculture, and climate change. This study aims to assess the relationship between forest fire occurrences, meteorological factors, and hospitalizations due to respiratory diseases in the Legal Amazon region from 2009 to 2019. Employing simultaneous equation models with official data, we examined the association between deforestation-induced fires and respiratory health issues. Over the studied period, the Legal Amazon region recorded a staggering 1,438,322 wildfires, with 1,218,606 (85%) occurring during August-December, known as the forest fire season. During the forest fire season, a substantial portion (566,707) of the total 1,532,228 hospital admissions for respiratory diseases were recorded in individuals aged 0-14 years and 60 years and above. A model consisting of two sets of simultaneous equations was constructed. This model illustrates the seasonal fluctuations in meteorological conditions driving human activities associated with increased forest fires. It also represents how air quality variations impact the occurrence of respiratory diseases during forest fires. This modeling approach unveiled that drier conditions, elevated temperatures, and reduced precipitation exacerbate fire incidents, impacting hospital admissions for respiratory diseases at a rate as high as 22 hospital admissions per 1000 forest fire events during the forest fire season in the Legal Amazon, 2009-2019. This research highlights the urgent need for environmental and health policies to mitigate the effects of Amazon rainforest wildfires, stressing the interplay of deforestation, climate change, and human-induced fires on respiratory health.

THERMALLY-OPTICALLY-THERMALLY STIMULATED LUMINESCENCE

A three-stage energy band model was studied. The model consists of electrons thermally stimulated from the ground state to the first excited state, after which they were optically stimulated into the second excited state and they were finally stimulated thermally into the conduction band. A set of simultaneous differential equations was generated from the models and three assumed conditions were applied to this model, which they were solved analytically and analytical expressions were obtained. The same set of simultaneous equations were solved numerically using ode 15s MATLAB solver. When considering first-order peaks, the kinetic parameters obtained were found to be in good agreement with the analytical expressions. But when considering non first-order peaks, the kinetic parameters obtained numerically were not in good agreement with the analytical expressions and explanations had been given. Second-order peaks could not be obtained despite careful selection of the kinetic parameters because the traps were quickly saturated and the quasi- equilibrium conditions assumed could no longer be satisfied. The stability of the excited TA-OSL signals produced by the model was also studied. The real stability of the excited TA-OSL signals produced by this model was found to be about 46 million years.

A comparison between multivariate linear model and maximum likelihood estimation for the prediction of elemental composition of coal using proximate analysis

The elemental composition of coal is essential for analysing the overall process of energy conversion systems. Simultaneous information regarding the elemental composition of coal fed into a boiler is of increasing interest for plant operation. In this study, methods of estimating the elemental composition of coal using proximate analysis and a higher heating value were developed to meet requirements in boiler operation. A novel method was developed to formulate the multivariate linear model (MLM) for predicting the elemental composition of coal by solving a set of simultaneous equations with extensive correlations between coal components and an elemental composition constraint. The maximum likelihood estimation (MLE) approach for predicting carbon, hydrogen, oxygen, nitrogen, and sulphur contents was also developed based on a series of extensive correlations among coal components. The maximum likelihood estimator employs more correlations and considers the performance of prediction residuals for each correlation, offering a better prediction accuracy than the MLM, which is based on fewer correlations and neglects prediction residuals. A total of 743 data points was used to derive the MLM and MLE models, which were validated and verified by another set of data that included the same variety of coal types. The proposed methods can estimate the complete elemental composition of coal (C, H, O, N, and S) with acceptable prediction accuracy for engineering purposes. Another important finding is that average absolute error corresponding to the measured values of nitrogen is only 14.14%, although the predicted nitrogen contents did not follow the trends of the measured values.

Performance Analysis of Simulated Annealing and Genetic Algorithm on systems of linear equations

Problem solving and modelling in traditional substitution methods at large scale for systems using sets of simultaneous equations is time consuming. For such large scale global-optimization problem, Simulated Annealing (SA) algorithm and Genetic Algorithm (GA) as meta-heuristics for random search technique perform faster. Therefore, this study applies the SA to solve the problem of linear equations and evaluates its performances against Genetic Algorithms (GAs), a population-based search meta-heuristic, which are widely used in Travelling Salesman problems (TSP), Noise reduction and many more. This paper presents comparison between performances of the SA and GA for solving real time scientific problems. The significance of this paper is to solve the certain real time systems with a set of simultaneous linear equations containing different unknown variable samples those were simulated in Matlab using two algorithms-SA and GA. In all of the experiments, the generated random initial solution sets and the random population of solution sets were used in the SA and GA respectively. The comparison and performances of the SA and GA were evaluated for the optimization to take place for providing sets of solutions on certain systems. The SA algorithm is superior to GA on the basis of experimentation done on the sets of simultaneous equations, with a lower fitness function evaluation count in MATLAB simulation. Since, complex non-linear systems of equations have not been the primary focus of this research, in future, performances of SA and GA using such equations will be addressed. Even though GA maintained a relatively lower number of average generations than SA, SA still managed to outperform GA with a reasonably lower fitness function evaluation count. Although SA sometimes converges slowly, still it is efficient for solving problems of simultaneous equations in this case. In terms of computational complexity, SA was far more superior to GAs.

Global Intercomparison of Hyper-Resolution ECOSTRESS Coastal Sea Surface Temperature Measurements from the Space Station with VIIRS-N20

The ECOSTRESS multi-channel thermal radiometer on the Space Station has an unprecedented spatial resolution of 70 m and a return time of hours to 5 days. It resolves details of oceanographic features not detectable in imagery from MODIS or VIIRS, and has open-ocean coverage, unlike Landsat. We calibrated two years of ECOSTRESS sea surface temperature observations with L2 data from VIIRS-N20 (2019–2020) worldwide but especially focused on important upwelling systems currently undergoing climate change forcing. Unlike operational SST products from VIIRS-N20, the ECOSTRESS surface temperature algorithm does not use a regression approach to determine temperature, but solves a set of simultaneous equations based on first principles for both surface temperature and emissivity. We compared ECOSTRESS ocean temperatures to well-calibrated clear sky satellite measurements from VIIRS-N20. Data comparisons were constrained to those within 90 min of one another using co-located clear sky VIIRS and ECOSTRESS pixels. ECOSTRESS ocean temperatures have a consistent 1.01 °C negative bias relative to VIIRS-N20, although deviation in brightness temperatures within the 10.49 and 12.01 µm bands were much smaller. As an alternative, we compared the performance of NOAA, NASA, and U.S. Navy operational split-window SST regression algorithms taking into consideration the statistical limitations imposed by intrinsic SST spatial autocorrelation and applying corrections on brightness temperatures. We conclude that standard bias-correction methods using already validated and well-known algorithms can be applied to ECOSTRESS SST data, yielding highly accurate products of ultra-high spatial resolution for studies of biological and physical oceanography in a time when these are needed to properly evaluate regional and even local impacts of climate change.

Statistical Analysis of Nigeria’s Price Sector: An Econometric Approach

This study analyzed Nigeria’s price sector using a formulated model for the price sector of the Nigeria economy. A set of simultaneous equations were used to reflect the implicit gross domestic product deflators for each of the sectors of the Nigeria economy and was found to be over identified under the order condition for identification. The model was estimated by ordinary least square method and two stage least square methods. All the variables have expected signs and as indicated by the F –statistic, the overall performance of the entire regression is significant. The high measure of R2 and Ṝ2, in each case indicates that the explanatory variables included in the equation jointly account for the entire variation. The small RMSE also indicates that the equations have good fit. Durbin –Watson statistics shows that there is no positive first order autocorrelation. The small value of the Theil’s inequality indicates that the equation has good predictive performance. The researcher therefore recommends that government should employ the model so as to be able to monitor price of each of the sectors of the economy and put proper mechanism in place to control those sectors that affect the overall price sector of the economy.

Preliminary design of a three-element zoom system based on a variable-focal-power lens

In this paper, we present the evaluation of a predesign of a zoom system based on a variable-focal-power lens by paraxial and aberration analyses only. This zoom system involves less mechanics and has a more compact design than those using a fixed-power lens. We derive corresponding formulas between the focal power of each lens and the magnification of the zoom system, and the bendings of the variable-focal-power lens elements are determined by introducing primary spherical aberrations. The calculation of the paraxial properties of zoom systems involves finding the solution of sets of simultaneous equations. Examples of the procedure are shown by applying it to previously designed three-element zoom systems. The results of this study are valuable in the design of future zoom systems.

A Computational Approach to Solve a System of Transcendental Equations with Multi-Functions and Multi-Variables

A system of transcendental equations (SoTE) is a set of simultaneous equations containing at least a transcendental function. Solutions involving transcendental equations are often problematic, particularly in the form of a system of equations. This challenge has limited the number of equations, with inter-related multi-functions and multi-variables, often included in the mathematical modelling of physical systems during problem formulation. Here, we presented detailed steps for using a code-based modelling approach for solving SoTEs that may be encountered in science and engineering problems. A SoTE comprising six functions, including Sine-Gordon wave functions, was used to illustrate the steps. Parametric studies were performed to visualize how a change in the variables affected the superposition of the waves as the independent variable varies from x1 = 1:0.0005:100 to x1 = 1:5:100. The application of the proposed approach in modelling and simulation of photovoltaic and thermophotovoltaic systems were also highlighted. Overall, solutions to SoTEs present new opportunities for including more functions and variables in numerical models of systems, which will ultimately lead to a more robust representation of physical systems.

On the Calculation of the Relative Amounts of Endmember Constituents For Garnet

ABSTRACT It is commonly accepted that the calculation of the proportions of endmember constituents in garnet is dependent on the particular sequence of calculating the amounts of the endmembers, and this belief has been used to justify avoiding the use of endmembers in the definition of a mineral species. Calculating the amounts of endmember constituents to represent a specific mineral formula involves the solution of a set of simultaneous equations, and hence the idea that the solution is dependent on the order in which the amounts of endmember constituent are determined conflicts with the meaning of the term “simultaneous equations”. Here I examine the data on which these conclusions are based and show that these sequence-dependent results arise because of the use of non-stoichiometric formulae that are not electroneutral. If a garnet formula is adjusted slightly such that it exactly fits the general formula of a garnet, [8]X3[6]Y2[4]Z3O12, and is electroneutral, the simultaneous equations relating its chemical formula to a set of endmember constituents have a single unique solution. Thus, the argument that has been used to justify avoiding the use of endmembers in the definition of a mineral species is specious.

HOW TO ESTIMATE BLACK WATTLE ABOVEGROUND BIOMASS FROM HETEROSCEDASTIC DATA?

This study developed a system of equations for predicting total aboveground and component biomass in black wattle trees. A total of 140 black wattle trees at age 10 years were measured regarding their diameter at 1.30 m height above the ground (d), total tree height (h), basic wood density (branches and stem), and biomass (stem, crown, and aboveground). We evaluated the performance of linear and nonlinear allometric models by comparing the statistics of R2adj., RRMSE%, and BIC. Nonlinear models performed better when predicting crown biomass (using only d as an independent variable), and stem and aboveground biomass (using d and h as independent variables). Adding basic density did not significantly improve biomass modeling. The residuals had non-homogeneous variance; thus, the fitted equations were weighted, with weights derived from a function containing the same independent variables of the fitted biomass function. Subsequently, we used a simultaneous set of equations to ensure that the sum of each component's estimated biomass values was equal to the total biomass values. Simultaneous fitting improved the performance of the equations by guaranteeing the components' additivity, and weighted regression allowed to stabilize error variance, ensuring the homoscedasticity of the residuals.

Farmscape Composition and Livelihood Sustainability in Deforested Landscapes of Colombian Amazonia

In this article, we operationalized a sustainability framing based on the Sustainable Rural Livelihood Resources Framework (SLF), which consists of five capitals—human, physical, social, financial, and natural. We proposed a sustainability index (SI) for two landscapes dominated by two agricultural systems: cattle ranching and small-scale family agriculture. Farm variables within each capital were analyzed using confirmatory factor analysis. Key variables were identified and index values were calculated for each capital. These were combined through a set of simultaneous equations to estimate farm-specific capitals and SI from the observed farm variables. Principal component and cluster analyses were used to group the farms according to their index scores and to further compare their characteristics. Furthermore, with the purpose of comparing the index scoring with an independent metric, a landscape indicator, which comes from a continuous forest, was calculated. From the results, the capitals that contributed to a higher SI score the most were financial and physical. As cattle ranching was associated with higher economic returns and infrastructure investments, this livelihood was identified as the most sustainable. Yet, cattle ranching has been a deforestation driver in the region. These results are attributed to the current conceptual framework design, which gives greater weight to material and economic variables; therefore, it generates a weak sustainability measure. Although the framework allowed us to identify land-use alternatives that could improve SI scores (i.e., silvopastoral systems), corrections to the proposed framework and methodological approach will need to include additional environmental benefits currently unaccounted for. Farmers that use their farms for conservation purposes should be recognized and compensated. An improved environmentally focused SI operational framework could help to endorse and promote sustainable livelihoods and to generate a strong sustainability measure.

Developing Capacity Reduction Factors for Curbside Bus Stops Under Heterogeneous Traffic Conditions

Traffic characteristics of a roadway are significantly influenced by bus stops, and it becomes more chaotic when a separate bus bay is not provided and the bus has to be stopped in the curb lane. The present study analyzes the effect of such bus stops on the midblock capacity of urban arterials under heterogeneous traffic conditions. Field data have been collected at six sections of six-lane divided urban arterial roads; out of that, three sections have not marked with bus stop and referred to as base sections, whereas the other three have designated bus stops and denoted as friction sections. A set of simultaneous equations are established using speed and flow data to estimate the midblock capacity of urban roads and found to be 6000 PCU/h for one direction of flow. Further, a set of simultaneous equations are developed between the speed of a vehicle type and the densities of all categories of vehicles with densities of stopped buses in their individual terms. These equations are solved for any number of stopping buses at a fixed proportion of vehicles in the traffic stream for different traffic volume levels and thus get the capacity curves. It has been found that the capacity of urban arterials is reduced by 6.6–28.3% when the frequency of stopping buses is as low as 20 bus/h to as high as 120 bus/h, respectively. The study outcomes are useful for designing bus stops and at the same time estimating the capacity of such sections in order to reduce the congestion on the urban roadway network.

Research of optimal fast-acting systems of automatic control on laboratory bench

Purpose. This paper presents a laboratory bench for research of optimal and quasi-optimal automatic control system in respect of its operation speed. Laboratory bench consists of thermal unit and software and hardware suite which includes VIPA System 200 V and HMI/SCADA logic controller and Zenon Supervisor 7.0 system. Thermal unit is described by differential equation of second order pursuing the control channel “amperage in electric heating unit power converter – air temperature inside thermal unit.” Differential equation coefficients depend on screen position and centrifugal blower rotation frequency. Methodology. Reported the methodology of synthesis and results of calculation of optimal relay hypothesis for thermal unit control through the chosen channel. Were demonstrated the results of experiment in transition of thermal unit from various initial states to final states. Was shown the possibility of implementing optimal control system in respect of its operation speed in real time scale by means of software development by including algorithms for transcendence set of simultaneous equations into it or by means of development of predicative model of thermal unit. Results. Were specified conditions for application of quasi-optimal relay control hypothesis in respect of its operation speed. Reported the methodology of synthesis and results of calculation of the length of the first control interval depending on the predetermined value of readjustment. Were demonstrated the results of modeling of quasi-optimal automatic control system in respect of its operation speed in Simulink of Matlab app. Was established functional relation of the length of the first control interval depending on the predetermined value of readjustment for implementing of quasi-optimal automatic control system in real time scale with application of basic operational units of programmed logic controller.

Linear Model of Information System of Industrial Processes in Crop Production

the timeliness of information received during the implementation of any production process, as well as its qualitative composition, significantly affect the efficiency of this process. This is particularly important for agricultural production, as field operations are highly time-sensitive, both in terms of start-up and duration. The real possibility of a significant increase in the efficiency of production processes in crop production appeared at the current level of information technology and the spread of digitalization of various fields of activity. On the basis of existing methodological approaches to the system representation of objects of reality, we have developed an information system of the production process in crop production. The elements of the system are the amount of knowledge of specialists in the cultivation of crops, information about the composition of cultivation technologies, information about the objects processed in the implementation of production processes, recorded in any form replenished special knowledge, knowledge about the goals achieved in the implementation of production processes. Elements of the system are interconnected, developing in a certain way and, at the same time, limiting each other. The formalization of the relationships between elements of the information system allowed us to obtain ten equations with fifteen unknowns. To solve this system of equations, a set of simultaneous linear equations is presented. Five sets of simultaneous equations can be defined, depending on which of the relationships of the elements of the information system of the type fi will be broken. The obtained solutions will allow to quantify the volume of elements of the information system, which makes it very promising to use information technology management of production processes in crop production.

An analytical algorithm for reasonable central tower stiffness in the three-tower suspension bridge with unequal-length main spans

Three-tower suspension bridges with unequal-length main spans more easily adapt to different terrain features and therefore, have broader application prospects. However, due to the unique “central tower effect” of the three-tower suspension bridge, it is required that the lateral stiffness of the central tower in the longitudinal direction of the bridge girder should be neither too large nor too small. To calculate the reasonable range for the central tower stiffness in the three-tower suspension bridge with unequal-length main spans, this study proposes an analytical algorithm based on the segmental catenary theory. Firstly, hanger tensile forces under the joint action of dead and live loads are calculated. Next, the governing equations for the main cable shape of each span are constructed for the following conditions: closure of elevation difference, closure of span length, moment balance of splay saddle, and conservation of unstrained length of the main cable. The solutions of the derived set of simultaneous equations are obtained for (i) deflection-to-span ratio limit of the stiffening girder and (ii) anti-slip control between the main cable and saddle conditions, which yield the lower and upper limits of reasonable stiffness of the central tower, respectively. This study discusses a three-tower suspension bridge spanned as 248 m + 1060 m + 1360 m + 380 m. The calculation is performed for the two cases of the live load application: (i) to the longer main span and (ii) to the shorter main span. The results obtained proved the feasibility and effectiveness of the proposed algorithm. The following findings are reported: The upper and lower limits of reasonable central tower stiffness are derived from the above two cases, respectively. Therefore, it is not sufficient to consider only the former case when calculating the central tower stiffness of the three-tower suspension bridge with equal-length main spans. The dead-to-live load ratio and friction coefficient between the main cable and saddle also strongly influence the central tower stiffness: their increase can expand the reasonable range of the latter but if they are too small, no such optimization can be provided.

Multivariate modeling for retained protein and lipid.

Energy efficiencies and maintenance parameters have been traditionally estimated with a linear regression model that treated metabolizable energy intake as the dependent variable and protein and lipid depositions as the independent variables. Several studies have described the statistical issues associated with this approach, such as the reverse role of dependent and independent variables and a potential multicollinearity issue due to the high correlation between protein and lipid depositions. Biased regression techniques have been proposed to minimize the harmful effects of multicollinearity on the estimates of energy efficiencies. These approaches, however, only partially addressed the issues described for the linear regression approach. A first multivariate approach was developed by L. J. Koong in the 1970s, who estimated the energy parameters using a set of simultaneous equations. This multivariate approach has been considerably extended in the past two decades with the complete characterization of model’s biological interpretation under different feeding conditions, the simultaneous estimation of maintenance requirements, the extension of the model to a mixed-effects framework, and the implementation of a Bayesian framework for model fitting. The multivariate approach has been successfully applied to model energy deposition and partitioning by mice, pigs, salmon, and rainbow trout. However, multivariate models are, in general, harder to fit than linear regression models due to 1) larger number of parameters, 2) issues with parameter identifiability, and 3) overall lack of algorithm convergence. Therefore, with the recent availability of easy to use and efficient computer packages for model fitting, the use of a Bayesian framework seems to be an attractive approach for fitting multivariate models describing protein and lipid deposition.

Optimization and kinetics modeling of okara isoflavones extraction using subcritical water

In this study, the soybean milk and tofu byproduct okara was subjected to subcritical water extraction with the intention of recovering isoflavones with minimal degradation. Response Surface Methodology (RSM) of the extraction variables indicated that optimized conditions would be T = 146.23 °C, P = 3.98 MPa, and α = 20 mg (solid)/mL (extractant). Mathematical models for the conversion and degradation of isoflavones were solved as a set of simultaneous equations leading to rate constants and time-dependent concentration profiles for each genistein- and daidzein-based compound. These kinetic analyses suggested that an optimum extraction time, under RSM-optimized conditions, would be 213.5 ± 8.7 min. The results of our study suggest that okara byproducts could be valorized efficiently, as a source of bioactive isoflavone aglycones, using subcritical water. The mathematical models and optimized extraction conditions that we established in this study could be employed, as process control-optimized variables, in the exploitation of okara, specifically in the isolation of genistein and daidzein.

How the Economics Profession Overlooked Keynes’s IS-LM (LP) model in chapter 21 of the General Theory in the 20th and 21st centuries

J M Keynes’s brief and detailed summaries of his original IS-LM(LP) models in Chapters 15 and 21, respectively, of the General Theory were overlooked by the economics profession in the 20th and 21st centuries due to three main reasons. The first reason was the claims made by the Keynesian fundamentalists,Joan Robinson and GL S Shackle, that Keynes’s supposed definition of uncertainty as complete and total ignorance of the future precluded the use of formal mathematical and statistical models of the macro economy. Therefore, any IS-LM model in the General Theory was an impossibility. The second reason was based on the supposed claim ,made by Backhouse,Young, Dimand, and Rubin, that Keynes was an adherent of the Marshallian view of formal mathematical exposition, which supposedly emphasized presenting only verbal chains of exposition that were based on mathematical analysis, which was then removed from the published work. Therefore, although Keynes created earlier versions of IS-LM to work with in 1933,1934, and 1935, Keynes decided to remove his formal IS-LM model by scattering the three components of his IS-LM model throughout the General Theory in 1936. Keynes substituted a strictly verbal, literary, prose analysis in its place. The third reason was based on the claim, made by Hawtrey, Robertson, Robinson, Viner, Hansen and Ahiakpor, that Keynes’s liquidity preference analysis was restricted only to chapter 13 of the General Theory alone ,where the crucial analysis was based on the equation specified on page 168 of the General Theory, M=L(r),so that the demand and supply for money is identical to the demand and supply for liquidity. Keynes’s chapter 15 definition of liquidity preference on p.199 is ignored. Adherents of this view likewise overlooked Keynes’s analysis on pages 207-209 of chapter 15 and pages 298-306 of chapter 21 of the General Theory. All three of these reasons are badly flawed because they all ignore Keynes’s analysis on pages 199, 207-209 of Chapter 15 and pages 298-306 of chapter 21 of the General Theory. Once these pages are studied carefully, it becomes clear that Keynes incorporated an improved IS-LM(LP) model in the General Theory that was superior to the earlier 1933,1934,and 1935 versions he had presented in his student lectures and the mid 1934 draft copy of the General Theory. Keynes makes it crystal clear in his discussion on pages 298-299 that he was presenting a mathematical model that had the following characteristics : • One can calculate a quantitative answer using it • The model is composed of three elements a), b), and c). a) is the consumption function, C, with both the mpc and investment multiplier specified. b) is the Investment function, I, which is a downward sloping function of the rate of interest, r. c) is the Liquidity Preference equation from page 199 of chapter 15, which is a function of both of Y and r, where Y=C+I. • The three elements provide an analysis that is valuable in introducing order and method to the enquiry • The three elements define a set of simultaneous equations • The equations provide a determinate answer or result. • This result is an equilibrium position • There is no uncertainty /expectations in Keynes’s chapter 21 model of IS-LM(LP) Keynes perfectly described his IS-LM(LP) model in the General Theory on pp.298-299. It is a major improvement over his earlier 1933, 1934, and 1935 versions. The Hicksian IS-LM version, like Harrod’s version, of Keynes’s General Theory, which was published in Econometrica,1937, is an inferior version of Keynes’s IS-LM model in chapter 21 of the General Theory because the Hicksian version has no supporting, micro, D-Z analysis that incorporates expectations and uncertainty, as presented by Keynes in chapters 20 and 21 of the General Theory.

Electromagnetic Interference Caused by Parasitic Electric-line Current on a Digital Module in a Closed Cabinet

This paper conducts an electromagnetic analysis of the closed cabinet used for instrumentation and control in nuclear power plants for the case of a parasitic electric-line current that is undesirably generated onto an interior digital module. Based on the Helmholtz equation in conjunction with both the separation of variables and superposition principle, we employed a mode-matching method to obtain the analytic solution to an electromagnetic interference problem. Using modal coefficients from a set of simultaneous equations by enforcing boundary conditions, we computationally derive electromagnetic-field distributions in terms of various geometric parameters. The results provide us with information useful for estimating the level of influence of the undesired current source on the adjacent digital modules.

Implementation of Nuttall’s vector-sensing target motion analysis algorithm

Direction cosine measurements, obtained from an array of vector sensors, are processed and used to estimate the location and motion of a moving source. Nuttall derived this unpublished TMA algorithm in 2006, which recently was reviewed, implemented, and evaluated using newly collected experimental data. Both stationary and moving sources are examined. It is assumed that the array element positions are known; however, at any given time instant, the measured direction cosines are presumed noisy. Originally, Nuttall’s formulation only utilized the three orthogonal components of acoustic particle velocity at each array location, however adding acoustic pressure is straight-forward and resolves bearing ambiguity. For a non-moving source, the TMA processor (linear in source position variables) generates a 3-by-3 set of simultaneous equations, with solution vector for the source position (xs, ys, zs). Range from any array element to the source can then be estimated. For a moving source, a 6-by-6 set of simultaneous equations is obtained; the solution vector is now the initial source position (xs, ys, zs) and the source velocity components (a, b, c). It is presumed that the source velocity is constant over the observation interval.