Background on Orthogonal Functions and Covariance

2006 ◽  
Author(s):  
Huug van den Dool
Keyword(s):  
Orthogonal Functions

The purpose of this chapter is to present some basic mathematics and statistics that will be used heavily in subsequent chapters. The organization of the material and the emphasis on some important details peculiar to the geophysical discipline should help the reader.

Reconstruction of Objects from their Projections Using Orthogonal Functions

1973 ◽  
Vol 31 ◽  
pp. 268-269
Author(s):  
Elrnar Zeitler
Keyword(s):  
Unit Area ◽  
Three Dimensional ◽  
One Dimension ◽  
Spatial Density ◽  
Dimensional Representation ◽  
Orthogonal Functions ◽  
Object A ◽  
Plane Normal ◽  
Dimensional Object ◽  
Spatial Density Distribution

Considering any finite three-dimensional object, a “projection” is here defined as a two-dimensional representation of the object's mass per unit area on a plane normal to a given projection axis, here taken as they-axis. Since the object can be seen as being built from parallel, thin slices, the relation between object structure and its projection can be reduced by one dimension. It is assumed that an electron microscope equipped with a tilting stage records the projectionWhere the object has a spatial density distribution p(r,ϕ) within a limiting radius taken to be unity, and the stage is tilted by an angle 9 with respect to the x-axis of the recording plane.

Rubber Composition–Properties Relationships during Tire Numerical Simulation and Design Optimization

2017 ◽  
Vol 45 (1) ◽  
pp. 71-84 ◽  
Author(s):  
Alexey Mazin ◽  
Alexander Kapustin ◽  
Mikhail Soloviev ◽  
Alexander Karanets
Keyword(s):  
Numerical Simulation ◽  
Design Optimization ◽  
Strain Tensor ◽  
General Purpose ◽  
Orthogonal Functions ◽  
Element Analysis ◽  
Time Investment ◽  
Footprint Analysis ◽  
Composition Properties ◽  
Nonlinear Elastic

ABSTRACT Numerical simulation based on finite element analysis is now widely used during the design optimization of tires, thereby drastically reducing the time investment in the design process and improving tire performance because it is obtained from the optimized solution. Rubber material models that are used in numerical calculations of stress–strain distributions are nonlinear and may include several parameters. The relations of these parameters with rubber formulations are usually unknown, so the designer has no information on whether the optimal set of parameters is reachable by the rubber technological possibilities. The aim of this work was to develop such relations. The most common approach to derive the equation of the state of rubber is based on the expansion of the strain energy in a series of invariants of the strain tensor. Here, we show that this approach has several drawbacks, one of which is problems that arise when trying to build on its basis the quantitative relations between the rubber composition and its properties. An alternative is to use a series expansion in orthogonal functions, thereby ensuring the linear independence of the coefficients of elasticity in evaluation of the experimental data and the possibility of constructing continuous maps of “the composition to the property.” In the case of orthogonal Legendre polynomials, the technique for constructing such maps is considered, and a set of empirical functions is proposed to adequately describe the dependence of the parameters of nonlinear elastic properties of general-purpose rubbers on the content of the main ingredients. The calculated sets of parameters were used in numerical tire simulations including static loading, footprint analysis, braking/acceleration, and cornering and also in design optimization procedures.

Empirical Methods in Short-Term Climate Prediction

2006 ◽  
Author(s):  
Huug van den Dool
Keyword(s):  
Climate Prediction ◽  
Empirical Orthogonal Functions ◽  
Data Sets ◽  
Seasonal Forecasts ◽  
Short Term ◽  
Orthogonal Functions ◽  
Ocean Atmosphere Interaction ◽  
Global Coverage ◽  
Atmosphere Interaction ◽  
Global Data

This clear and accessible text describes the methods underlying short-term climate prediction at time scales of 2 weeks to a year. Although a difficult range to forecast accurately, there have been several important advances in the last ten years, most notably in understanding ocean-atmosphere interaction (El Nino for example), the release of global coverage data sets, and in prediction methods themselves. With an emphasis on the empirical approach, the text covers in detail empirical wave propagation, teleconnections, empirical orthogonal functions, and constructed analogue. It also provides a detailed description of nearly all methods used operationally in long-lead seasonal forecasts, with new examples and illustrations. The challenges of making a real time forecast are discussed, including protocol, format, and perceptions about users. Based where possible on global data sets, illustrations are not limited to the Northern Hemisphere, but include several examples from the Southern Hemisphere.

scholarly journals Kernel-based learning of orthogonal functions

2020 ◽  
Vol 53 (2) ◽  
pp. 2305-2310
Author(s):  
Anna Scampicchio ◽  
Gianluigi Pillonetto ◽  
Mauro Bisiacco
Keyword(s):  
Orthogonal Functions

scholarly journals Spatio-Temporal Patterns of Mass Changes in Himalayan Glaciated Region from EOF Analyses of GRACE Data

2021 ◽  
Vol 13 (2) ◽  
pp. 265
Author(s):  
Harika Munagapati ◽  
Virendra M. Tiwari
Keyword(s):  
Temperature Anomaly ◽  
Mass Gain ◽  
Snow Accumulation ◽  
Empirical Orthogonal Functions ◽  
Total Water ◽  
Orthogonal Functions ◽  
Grace Data ◽  
Grace Satellite ◽  
Spatio Temporal ◽  
Gravity Recovery

The nature of hydrological seasonality over the Himalayan Glaciated Region (HGR) is complex due to varied precipitation patterns. The present study attempts to exemplify the spatio-temporal variation of hydrological mass over the HGR using time-variable gravity from the Gravity Recovery and Climate Experiment (GRACE) satellite for the period of 2002–2016 on seasonal and interannual timescales. The mass signal derived from GRACE data is decomposed using empirical orthogonal functions (EOFs), allowing us to identify the three broad divisions of HGR, i.e., western, central, and eastern, based on the seasonal mass gain or loss that corresponds to prevailing climatic changes. Further, causative relationships between climatic variables and the EOF decomposed signals are explored using the Granger causality algorithm. It appears that a causal relationship exists between total precipitation and total water storage from GRACE. EOF modes also indicate certain regional anomalies such as the Karakoram mass gain, which represents ongoing snow accumulation. Our causality result suggests that the excessive snowfall in 2005–2008 has initiated this mass gain. However, as our results indicate, despite the dampening of snowfall rates after 2008, mass has been steadily increasing in the Karakorum, which is attributed to the flattening of the temperature anomaly curve and subsequent lower melting after 2008.

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