Number Of Empirical Orthogonal Functions Research Articles

Thermographic Data Analysis for Defect Detection by Imposing Spatial Connectivity and Sparsity Constraints in Principal Component Thermography

Data analysis methods have been extensively used in active thermography for defect identification. Among them, principal component thermography (PCT) is popular for dimensionality reduction and feature extraction. PCT summarizes the thermal images with a small number of empirical orthogonal functions that better reflect the information of defects. However, PCT does not induce sparsity, which limits the interpretation of PCT results. Recently, sparse PCT (SPCT) has been proposed to provide more interpretable analysis results. However, SPCT does not consider the spatial connectivity between pixels, omitting the fact that a defective region is usually spatially connected. In this article, a novel thermographic data analysis method is proposed to overcome the shortcomings of the existing methods. The proposed method imposes both spatial connectivity and sparsity constraints in PCT. Finally, one case study on an ancient marquetry sample and another on a carbon fiber-reinforced polymer composite illustrate the feasibility of the proposed method.

A Bayesian hierarchical model for climate change detection and attribution

AbstractRegression‐based detection and attribution methods continue to take a central role in the study of climate change and its causes. Here we propose a novel Bayesian hierarchical approach to this problem, which allows us to address several open methodological questions. Specifically, we take into account the uncertainties in the true temperature change due to imperfect measurements, the uncertainty in the true climate signal under different forcing scenarios due to the availability of only a small number of climate model simulations, and the uncertainty associated with estimating the climate variability covariance matrix, including the truncation of the number of empirical orthogonal functions (EOFs) in this covariance matrix. We apply Bayesian model averaging to assign optimal probabilistic weights to different possible truncations and incorporate all uncertainties into the inference on the regression coefficients. We provide an efficient implementation of our method in a software package and illustrate its use with a realistic application.

Fully probabilistic seismic source inversion – Part 1: Efficient parameterisation

Abstract. Seismic source inversion is a non-linear problem in seismology where not just the earthquake parameters themselves but also estimates of their uncertainties are of great practical importance. Probabilistic source inversion (Bayesian inference) is very adapted to this challenge, provided that the parameter space can be chosen small enough to make Bayesian sampling computationally feasible. We propose a framework for PRobabilistic Inference of Seismic source Mechanisms (PRISM) that parameterises and samples earthquake depth, moment tensor, and source time function efficiently by using information from previous non-Bayesian inversions. The source time function is expressed as a weighted sum of a small number of empirical orthogonal functions, which were derived from a catalogue of >1000 source time functions (STFs) by a principal component analysis. We use a likelihood model based on the cross-correlation misfit between observed and predicted waveforms. The resulting ensemble of solutions provides full uncertainty and covariance information for the source parameters, and permits propagating these source uncertainties into travel time estimates used for seismic tomography. The computational effort is such that routine, global estimation of earthquake mechanisms and source time functions from teleseismic broadband waveforms is feasible.

Reconstruction of sea surface temperature by means of DINEOF: a case study during the fishing season in the Bay of Biscay

The Spanish surface fishery operates mainly during the summer season in the waters of the Bay of Biscay. Sea surface temperature (SST) data recovered from satellite images are being used to improve the operational efficiency of fishing vessels (e.g. reduce search time and increase catch rate) and to improve the understanding of the variations in catch distribution and rate needed to properly manage fisheries. The images used for retrieval of SST often present gaps due to the existence of clouds or satellite malfunction periods. The data gaps can totally or partially affect the area of interest. Within this study, an application of a technique for the reconstruction of missing data called DINEOF (data interpolating empirical orthogonal functions) is analysed, with the aim of testing its applicability in operational SST retrieval during summer months. In this case study, the Bay of Biscay is used as the target area. Three months of SST Moderate Resolution Imaging Spectroradiometer (MODIS) images, ranging from 1 May 2006 to 31 July 2006, were used. The main objective of this work is to test the overall performance of this technique, under potential operational use for the support of the fleet during the summer fishing season. The study is designed to analyse the sensitivity of the results of this technique to several details of the methodology used in the reconstruction of SST, such as the number of empirical orthogonal functions (EOFs) retained, the handling of the seasonal cycle or the length (number of images) of the SST database used. The results are tested against independent SST data from International Comprehensive Ocean–Atmosphere Data Set (ICOADS) ship reports and standing buoys and estimations of the error of the reconstructed SST fields are given. Conclusions show that over this area three months of data are enough for efficient SST reconstruction, which yields four EOFs as the optimal number needed for this case study. An extended EOF experiment with SST and SST with a lag of one day was carried out to analyse whether the autocorrelation of the SST data allows better performance in the SST reconstruction, although the experiment did not improve the results. The validation studies show that the reconstructed SSTs can be trusted, even when the amount of missing data is very high. The mean absolute deviation maps show that the error is greatest near to the coast and mainly in the upwelling areas close to the French and north-western Spanish coasts.

The Gulf Stream and Atlantic sea-surface temperatures in AD1790-1825

We present gridded sea-surface temperatures (SSTs) for the Atlantic basin (45°S–60°N) as averages over the period AD1790–1825, based on early-instrumental SST data. The original measurements were compiled by Major James Rennell and made by numerous British naval vessels on behalf of the British Admiralty. We describe the digitization of this dataset and the reconstruction of spatially coherent, averaged conditions for the boreal cold (November-March) and warm (May–September) season using a reduced space optimal interpolation (RSOI) technique, in which the data is projected on a limited number of empirical orthogonal functions. This approach is validated on modern data that are sampled in a similar way as the early-instrumental data. The reconstruction for the November–March period shows a large area with anomalously high temperatures from the point where the Gulf Stream separates from the coast until ca. 20°W. A tongue of anomalous cool water is found at the eastern side of the North Atlantic basin, along the coast of Europe and northern Africa. In the northeastern South Atlantic, anomalously high temperatures are found, while temperatures in the southwestern South Atlantic are anomalously cool. For the March–September season, anomalous temperatures in the South Atlantic are similar, but stronger, compared with those in the boreal cold season. Over the North Atlantic, there is not much similarity between the current SST reconstructions and those published in the late 1950s. Copyright © 2009 Royal Meteorological Society

Application of a SEEK filter to a 1D biogeochemical model of the Ligurian Sea: Twin experiments and real in-situ data assimilation

The Singular Evolutive Extended Kalman (SEEK) filter has been implemented to assimilate in-situ data in a 1D coupled physical-ecosystem model of the Ligurian Sea. The biogeochemical model describes the partly decoupled nitrogen and carbon cycles of the pelagic food web. The GHER hydrodynamic model (1D version) is used to represent the physical forcings. The data assimilation scheme (SEEK filter) parameterizes the error statistics by means of a set of empirical orthogonal functions (EOFs). Twin experiments are first performed with the aim to choose the suitable experimental protocol (observation and estimation vectors, number of EOFs, frequency of the assimilation,…) and to assess the SEEK filter performances. This protocol is then applied to perform real data assimilation experiments using the DYFAMED data base. By assimilating phytoplankton observations, the method has allowed to improve not only the representation of the phytoplankton community, but also of other variables such as zooplankton and bacteria that evolve with model dynamics and that are not corrected by the data assimilation scheme. The validation of the assimilation method and the improvement of model results are studied by means of suitable error measurements.

Spectral bidirectional reflectance of Antarctic snow: Measurements and parameterization

The bidirectional reflectance distribution function (BRDF) of snow was measured from a 32‐m tower at Dome C, at latitude 75°S on the East Antarctic Plateau. These measurements were made at 96 solar zenith angles between 51° and 87° and cover wavelengths 350–2400 nm, with 3‐ to 30‐nm resolution, over the full range of viewing geometry. The BRDF at 900 nm had previously been measured at the South Pole; the Dome C measurement at that wavelength is similar. At both locations the natural roughness of the snow surface causes the anisotropy of the BRDF to be less than that of flat snow. The inherent BRDF of the snow is nearly constant in the high‐albedo part of the spectrum (350–900 nm), but the angular distribution of reflected radiance becomes more isotropic at the shorter wavelengths because of atmospheric Rayleigh scattering. Parameterizations were developed for the anisotropic reflectance factor using a small number of empirical orthogonal functions. Because the reflectance is more anisotropic at wavelengths at which ice is more absorptive, albedo rather than wavelength is used as a predictor in the near infrared. The parameterizations cover nearly all viewing angles and are applicable to the high parts of the Antarctic Plateau that have small surface roughness and, at viewing zenith angles less than 55°, elsewhere on the plateau, where larger surface roughness affects the BRDF at larger viewing angles. The root‐mean‐squared error of the parameterized reflectances is between 2% and 4% at wavelengths less than 1400 nm and between 5% and 8% at longer wavelengths.

Reduced‐rank Kalman filters applied to an idealized model of the wind‐driven ocean circulation

The goal of this study is to evaluate a specific type of reduced‐rank Kalman filter for application to realistic ocean models. Data assimilation experiments were performed using an idealized nonlinear model of the wind‐driven ocean circulation. Separate configurations of the model were employed that exhibit either a quasi‐periodic behavior on the decadal timescale or a statistically stationary behavior accompanied by a high level of mesoscale eddy activity. The model consists of about 104 prognostic variables with observations of the model state taken at 30 locations concentrated in the region of highest variability near the western boundary. The assimilation scheme is an approximation to the extended Kalman filter in which the error covariances and corrections to the forecasts are only calculated in a reduced‐dimension subspace spanned by a small number of empirical orthogonal functions. The filter was implemented using both temporally evolving and stationary error covariances. Additionally for the quasi‐periodic model configuration, the model state space was partitioned according to three distinct flow regimes exhibited by the model and the asymptotically stationary error covariances calculated for each regime. The importance of specifying appropriate model and observation error covariances and the difficulties related to using stationary basis functions are also discussed. The performance of these reduced‐rank approaches is compared with the more traditional approach of using stationary error covariances with a simple prescribed functional form. The results show that the reduced‐rank Kalman filter is able to reduce the error in all assimilation experiments and consistently performs better than using simple prescribed error covariances for the quasi‐periodic case. Also, the performance of the filter with stationary error covariances is surprisingly similar to the much more computationally expensive filter with flow‐dependent covariances. The forecast errors from the assimilation experiments are examined with respect to their consistency with the error statistics calculated by the filter and their projections onto the resolved and unresolved subspaces.

Multivariate Empirical Orthogonal Function analysis of the upper thermocline structure of the Mediterranean Sea from observations and model simulations

Abstract. Multivariate vertical Empirical Orthogonal Functions (EOF) are calculated for the entire Mediterranean Sea both from observations and model simulations, in order to find the optimal number of vertical modes to represent the upper thermocline vertical structure. For the first time, we show that the large-scale Mediterranean thermohaline vertical structure can be represented by a limited number of vertical multivariate EOFs, and that the "optimal set" can be selected on the basis of general principles. In particular, the EOFs are calculated for the combined temperature and salinity statistics, dividing the Mediterranean Sea into 9 regions and grouping the data seasonally. The criterion used to establish whether a reduced set of EOFs is optimal is based on the analysis of the root mean square residual error between the original data and the profiles reconstructed by the reduced set of EOFs. It was found that the number of EOFs needed to capture the variability contained in the original data changes with geographical region and seasons. In particular, winter data require a smaller number of modes (4–8, depending on the region) than the other seasons (8–9 in summer). Moreover, western Mediterranean regions require more modes than the eastern Mediterranean ones, but this result may depend on the data scarcity in the latter regions. The EOFs computed from the in situ data set are compared to those calculated using data obtained from a model simulation. The main results of this exercise are that the two groups of modes are not strictly comparable but their ability to reproduce observations is the same. Thus, they may be thought of as equivalent sets of basis functions, upon which to project the thermohaline variability of the basin. Key words. Oceanography: general (water masses) – Oceanography: physical (hydrography; instruments and techniques)

The Retrieval of Cloud Structure Maps in the Equatorial Region of Jupiter Using a Principal Component Analysis of Galileo/NIMS Data

The cloud structure of the jovian atmosphere at pressures less than 2 bars has previously been estimated using near-infrared observations such as those by both the Solid State Imager (SSI) and Near-Infrared Mapping Spectrometer (NIMS) instruments on board the Galileo spacecraft. Unfortunately, complete near-infrared spectra, such as those measured by NIMS, take a long time to be analyzed with multiple-scattering radiative transfer models and thus it has until now been rather difficult to use these data to produce wide-area cloud maps. In this paper we show how principal component analysis may be employed to isolate a small number of empirical orthogonal functions (EOFs) from spectra of Jupiter made by Galileo/NIMS. These EOFs may be used to represent the variance of real NIMS spectra to a high degree of accuracy and with good noise and “drop-out” discrimination. Because of this, a small set of representative spectra may then be calculated using these EOFs and input into a retrieval model that generates a table of fitted cloud profiles for each case. This approach avoids the long times required for analyzing a large number of spectra with full multiple-scattering radiative transfer models and allows us to represent the variability of the 3000 spectra contained in the observations of the North Equatorial Belt (NEB) used in this study with only 75 representative spectra. The cloud structures fitted to these representative spectra were interpolated for the spectra found at individual locations in the measured NIMS data set to produce maps of cloud opacity and mean particle size. We find that the dominant opacity variation, anticorrelated with 5-μm brightness, exists in the 1–2 bar pressure range. The distribution of the cloud at 0.72 bars is mapped and found to be more zonally diffuse than the lower clouds. We find at least one 2000-km-sized deep convective cloud in the NEB vertically extending to all the pressure levels that can be sensed by NIMS.

Sea level from temperature profiles in the tropical Pacific Ocean, 1975–1982

We investigate how well the 1975–1992 sea level interannual variability in the tropical Pacific is captured by dynamic height from temperature profiles. For each temperature profile, a surface dynamic height relative to 300 m is estimated, assuming a constant temperature‐salinity relationship. After multiplication by a latitudinally varying factor and the removal of a seasonal cycle, the dynamic height deviations fit the tide gauge sea level variability to within the sampling errors, except at a few sites near the equator west of the date line, where surface salinity variability is large. The dynamic height data are assimilated into a wind‐forced linear numerical model of the sea level in the tropical Pacific, applying a Kalman filter in a space of reduced dimension. A limited number of empirical orthogonal functions of the unfiltered run (1975–1992) define the reduced space, into which the Kalman Filter covariance evolution calculation is done [Cane et al., 1996]. Experiments indicate that results are better with 32 functions than with a smaller number but are not improved by retaining more functions. The resulting analyzed fields of sea level are compared to withheld dynamic height estimates from moorings, sea level data from tide gauges, and sea level analyses made with the same Kalman filter formalism applied to tide gauge measurements. The comparisons to observations suggest that the temperature profiles were usually sufficient to constrain the monthly analyzed fields to be close to the observed sea level with errors typically less than 3 cm near the equator. The comparison to tide gauge sea level reveals that this analysis is often more accurate than the analysis of tide gauge sea level data with which it shares many characteristics. Near the equator west of the date line, salinity variations are large and their neglect in estimating dynamic height has a negative impact on the analysis. The analyzed signal is underestimated in the southwest Pacific and at more than 20° off the equator. The reanalysis of the temperature data done with a primitive equation model at the National Meteorological Center (NMC) [Ji et al., 1995; Enfield and Harris, 1995] does not share this problem. On the other hand, NMC reanalysis (RA4) departs more from the observations elsewhere, although more data were included than in our analysis.

Empirical orthogonal function representations of stochastic sound-speed ensembles

The representation of stochastic ensembles of sound-speed profiles by convenient and efficient means is examined. The procedure is based on using discrete empirical orthogonal functions (EOFs) with random coefficients. The eventual objective is to use the EOF representations in random wave propagation models, for which their efficiency is expected to reduce significantly the computational expense. To illustrate the procedure, two specific model ensembles are considered. The first consists of deep-water (Munk) sound-speed profiles, and the second has shallow-water profiles obtained from temperature profiles generated by a portion of the generalized digital environmental model (GDEM). After a statistical distribution of stochastic environmental input parameters is assumed, EOFs are calculated for ensembles of up to 10 000 profile samples. Relations between the statistics of the input parameters and the sound speed profiles that are obtained from EOF representations are investigated. In each case examined, the number of EOFs needed to account for virtually all of the sample variance is at most four. Additional cases are considered using Continental Shelf data by assuming measured profiles represent ensemble samples. [Work supported by ONR.]