Search

This feasibility study evaluates an approach for prediction of sandstone plug porosity and permeability based on low-angle illumination imaging, the Angle Measure Technique (AMT) and chemometric multivariate calibration/validation. The AMT approach transforms 2-D texture images of drill core plug ends into 1-D ‘complexity spectra’ in which inherent porosity- and permeability-correlated features are subsequently extracted and subjected to multivariate calibration modelling. A training data set was selected because of its wide-spanning porosity and permeability ranges allowing evaluation of realistic prediction performance for typical North Sea/Scandinavian sandstone oil/gas reservoir rocks. This first study makes use of sand stone plugs from a single drill core from the Danish underground. Contingent upon proper test set validation (deliberately not deleting a few small, potential outliers), prediction performance assessment were for porosity [%] slope: 0.86; RMSEP: 2.2%; R2 = 0.90 and for permeability [mDarcy]: slope: 0.91; RMSEP: 458 mDarcy; R2 = 0.87, which translates into RMSEPrel of 12% and 19% respectively. These results pertain to a typical, well-spanning training data set (18 sandstone plugs); it is therefore concluded that the AMT approach to poro-perm prediction from images is feasible, but further, extended calibrations must be based on a more comprehensive training data sets covering the full geological regime of reservoir sandstones. We discuss possible application potentials and limitations of this approach.

Image Analysis combined with multivariate regression on Angle Measure Technique (AMT) transformed imagery and the Theory of Sampling (TOS) is here presented as a comprehensive for Image Analysis Sampling (IAS), which takes all aspects of sampling representativity into account — especially 2-dimensional image versus 3-dimensioanl bulk compositions issues. Every IAS application has to be based on optimized image acquisition parameters: camera and illumination type, illumination angle, sample thickness as well as image post-processing, which are all examined here in order to delineate the general requirements for optimal prediction models for particle size distribution of natural and industrial bulk solid aggregates. We present a complete optimization study in order to show its intrinsic problem-dependent nature. This optimization allowed an original 60-sample data set to be compressed to an essential 22 natural coastal sands array with equally varying composition ranges — which was subjected to IAS in order to characterize the specific particle size distribution curves. In addition to D50 (50%-tile), six other size classes were successfully predicted, while extreme size classes (extreme low or high particle sizes) showed a too narrow training data set span, illustrating a critical grain size contrast which will always bracket successful models of particulate matter being imaged for grain size characterization. All classes with a satisfactory (representative) calibration interval span can be quantitatively predicted due to the powerful scale-dependency of the central AMT feature extraction combined with PLS multivariate calibration. The present application to natural sand aggregate size distributions forms a vehicle to illustrate the full potential of image analysis in general, IAS in particular, also for technological/industrial manufacturing on-line product and process monitoring applications, or quality control purposes, with similar grain-size prediction objectives. There is a significant carrying-over potential to parallel industrial scenarios.

Didactic data sets representing a range of real-world processes are used to illustrate “how to do” representative process sampling and process characterisation. The selected process data lead to diverse variogram expressions with different systematics (no range vs. important ranges; trends and/or periodicity; different nugget effects and process variations ranging from less than one lag to full variogram lag). Variogram data analysis leads to a fundamental decomposition into 0-D sampling vs. 1-D process variances, based on the three principal variogram parameters: range, sill and nugget effect. The influence on the variogram from significant trends and outliers in the original data series receive special attention, due to their critical adverse effects. We highlight problem-dependent interpretation of variographic analysis a.o. the problem-dependent background for periodicities and trends. All presented cases of variography either solved the initial problems or served to understand the reasons and causes behind the specific process structures revealed in the variograms. Process Analytical Technologies (PAT) are not complete without process TOS.

An in-depth heterogeneity analysis of wheat lots from varying field experiments with respect to protein concentration was conducted in order to quantify and compare both sampling and analytical errors as defined by the Theory of Sampling (TOS). Thirty wheat samples of forty-two seeds were extracted from three different wheat lots. Half of these were extracted using a non-optimal spoon (grab sampling) and the other half were extracted using a riffle splitter. Ten additional samples of forty-two seeds were extracted using a riffle splitter from ten different wheat lots. The protein content of every single-seed was determined by Near-Infrared Transmission (NIT) spectroscopy based on a multivariate calibration to Kjeldahl with Partial Least Squares Regression (PLS-R). The effect of orientation and number of replicate measurements of the individual seeds in the NIT beam was investigated for minimizing the analytical error as well the resulting time requirements. Remarkably, the best prediction model with respect to Root Mean Squared Error of Cross Validation (RMSECV) was obtained by only recording three replicate NIT spectra of the seeds in only one specific orientation. The variance of the Global Estimation Error (GEE) of both the riffle splitter and the spoon sampling processes was estimated as well as its components, the Fundamental Sampling Error (FSE), the Grouping and Segregation Error (GSE), the Incorrect Sampling Errors (ISE) and the Total Analytical Error (TAE). The bias induced by non-probabilistic spoon extraction was also estimated. The GEE variance of the spoon extractions was seventy percent higher than that of the riffle split samples. The sampling variances of FSE, GSE and ISE were all of the same order of magnitude, each approximately ten times higher than the TAE variance. The squared bias of the spoon sampling was approximately twice the magnitude of the sampling variances and thus contributed significantly to the representativity score. Spoon sampling representativity was three times the size of that for the riffle splitter. Order of magnitude estimates of the Constitutional Heterogeneity (CH L ) as well as the Distributional Heterogeneity (DH L ) for a forty-two seed riffle split sample was derived. In this investigation the fundamental concepts of TOS have been investigated and estimates of all sampling and analytical errors have been presented for a specific zero-dimensional composite material. The ability of TOS for quantifying and evaluating the various error contributions to the overall estimation of the protein concentration was confirmed.

Extensive experimentation and evaluation of alternative sampling methods of highly heterogenous bioslurries are carried out in the context of the theory of sampling (TOS) in order to delineate optimal, representative sampling procedures in PAT (process analytical technologies). The analytical methods investigated are at-line NIR and image analysis (IA) for monitoring of an industrial bioenergy anaerobic digestion processes (AD), subject to stringent economic bracketing: PAT solutions have to be both practical and inexpensive in the bioenergy sector where cost efficiency of instrumentation and monitoring systems is an absolute must. Focus is on development of minimum expenditure methods for sufficient characterization of the very heterogeneous types of biomass feedstock as used in industrial scale, continuously stirred tank reactors (CSTR). Product and process characterization necessarily involves a chemometric multivariate calibration predictor (PLS regression). The general goal is development of appropriate sampling/PAT facilities for at-line/on-line process monitoring in typically low-tech bioenergy, agro-industrial sectors. Experimental laboratory reactor evaluations, based on biomass feedstock and digested products from a full-scale biogas plant, were initially run in batch mode for a week, followed by fed-batch addition of maize silage, introducing a systematic increase in total solids allowing properly spanning multivariate calibration models. Measurements on 55 laboratory reactor samples taken during a complete 14-day fermentation cycle included three key process parameters: total solids (TS), volatile solids (VS) and chemical oxygen demand (COD), representing difficult-to-sample analytes. NIR spectroscopy and image analysis, including the angle measure technique transform (AMT), were evaluated for characterization of different feedstocks as well as continuously extracted process samples with respect to selected chemistry and dry matter characteristics. Optimized sampling on four different scale-levels allowed acceptable PLS prediction models for TS and VS for both NIR and image analysis compared to chemical reference analysis, while it was not possible to predict the COD levels satisfactorily due to large uncertainties in mandatory reference measurement protocols. This feasibility study is promising for NIR as at-line prediction of TS and VS content as well as other AD parameters, which can be measured on the same sample types, while image analysis is currently too complex and expensive for these industry sectors. The findings in the present bioslurries study have a considerable generalization potential: all PAT approaches are critically dependent on representative reference calibration sampling, which has to be fully compliant with the theory of sampling (TOS).

The Theory of Sampling (TOS) provides a description of all errors involved in sampling of heterogeneous materials as well as all necessary tools for their evaluation, elimination and/or minimization. This tutorial elaborates on—and illustrates—selected central aspects of TOS. The theoretical aspects are illustrated with many practical examples of TOS at work in typical scenarios, presented to yield a general overview. TOS provides a full scientific definition of the concept of sampling correctness , an attribute of the sampling process that must never be compromised. For this purpose the Fundamental Sampling Principle (FSP) also receives special attention. TOS provides the first complete scientific definition of sampling representativeness . Only correct (unbiased) mass reduction will ensure representative sampling. It is essential to induct scientific and technological professions in the TOS regime in order to secure the necessary reliability of: samples (which must be representative, from the primary sampling onwards), analysis (which will not mean anything outside the miniscule analytical volume without representativity ruling all mass reductions involved, also in the laboratory) and data analysis (“data” do not exist in isolation of their provenance). The Total Sampling Error (TSE) is by far the dominating contribution to all analytical endeavours, often 100+ times larger than the Total Analytical Error (TAE).We present a summarizing set of only seven Sampling Unit Operations (SUOs) that fully cover all practical aspects of sampling and provides a handy “toolbox” for samplers, engineers, laboratory and scientific personnel.

Optimization of an industrial image analytical application which uses the Angle Measure Technique (AMT) as an integrated part of estimation of the quality parameter dust level for encapsulated enzyme granules has been carried out. The principal AMT results from this study can be generalized to all similar types of segregating particulate matter. The first part of the study addresses pixel-sampling strategies within one image prior to application of the Angle Measure Technique, where the unfolded image is regarded as a one-dimensional object. Optimal parameters for the AMT algorithm is found to be systematic or stratified sampling of 500+ initiating points, unfolded in a novel spiral- or snake-wise fashion from the fluorescent image; this new unfolding procedure eliminates certain algorithmic “wrap around” artifacts which have not been sufficiently in need of rectification until now. The second part is directed towards delineating optimal image analytical sampling of a series of such images, i.e., optimization of image process sampling. The minimum number of images needed to uphold a sufficiently good online estimate of the QC parameter in question is found to be corresponding to a relatively high sampling rate of 15–20%, related to an intrinsic measurement error level of a mandated analytical reference method. Increasing the number of image increments included in the image analytical sample primarily affects the accuracy of the estimates, whereas precision is much less affected, attesting to a satisfactory stability of the image analytical method developed, which is now under industrial implementation.

We here present a comprehensive survey of current mass reduction principles and hardware available in the current market. We conduct a rigorous comparison study of the performance of 17 field and/or laboratory instruments or methods which are quantitatively characterized (and ranked) for accuracy (bias), reproducibility (precision), material loss (external as well as internal loss), user-dependency, operation time, and ease of cleaning. Graphical comparison of these quantitative results allow a complete overview of the relative strengths and weaknesses of riffle splitters, various rotational dividers, the Boerner Divider, the “spoon method”, alternate/fractional shoveling and grab sampling. Only devices based on riffle splitting principles (static or rotational) passes the ultimate representativity test (with minor, but significant relative differences). Grab sampling, the overwhelmingly most often used mass reduction method, performs appallingly—its use must be discontinued (with the singular exception for completely homogenized fine powders). Only proper mass reduction (i.e. carried out in complete compliance with all appropriate design principles, maintenance and cleaning rules) can always be representative in the full Theory of Sampling (TOS) sense. This survey also allows empirical verification of the merits of the famous “Gy's formula” for order-of-magnitude estimation of the Fundamental Sampling Error (FSE).

The conditions inside the vessels of the living human organism are different from those in the external environment outside the human body. Sampling and handling procedures must ensure that these in vivo conditions are preserved as well as possible even though invasive techniques must be used. However, such procedures will trigger biological processes, which are continued during the sample handling, altering the state and even the concentration of analytes. Thus routine blood collection will inevitably result in activation of the coagulation system, with the potential for highly significant increment preparation errors (IPE) if not compensated properly. We discuss the difficulties encountered when dealing with the in vivo–ex vivo interface being established when sampling blood for analysis. Physical sampling and the subsequent preparatory handling of blood increments from human individuals are shown to be critically dependent upon standardised procedures in order to obtain representative samples. In particular, we here focus on pre-analytical conditions affecting the coagulation system in the human body. In this context, blood sampling IPE may severely influence the outcome of the determination of many sensitive analytes. The samples collected are used for analytical determination of numerous biochemical quantities, and each of these may in fact require specific collection or handling procedures, but international guidelines are only available for the most common biochemical quantities. However, even the recommendations given in these guidelines are not necessarily evidence-based and may not suffice for all quantities to be determined. There is thus a need for further research and standardisation, which per force must include theory of sampling (TOS). These problems have prompted us to establish a new interdisciplinary collaboration between our two disciplines in order to elucidate this interesting sampling terra incognito.

This Special Issue is dedicated to one man's distinguished achievements in science and technology: 50 years of Pierre Gy's “Theory of Sampling” (TOS); the First World Conference on Sampling and Blending (WCSB1) was dedicated to this same purpose. This tribute covers a summary of Gy's professional career with a natural focus on TOS and its relationships to fundamental as well as applied sciences and technology, and why it is timely and fitting to present Pierre Gy with the tribute of WCSB1.

Measurement is the process of determining a correct value of a measurand; definition of the measurand is decisive for an expedient choice of a measurement method. The result of a measurement refers in principle only to the amount of substance actually contributing to the analytical signal. An appropriate definition of the measurand, however, must include a specification of the entire system, for which the result of the measurement is required to apply, i.e., which it is supposed to represent. The concept of representativity therefore assumes importance for the metrological quality of a measurement, and the process with which to ascertain representativity is proper sampling. All systems from which a sample is to be selected are inherently heterogeneous (this is only a matter of the scale of observation), and it is therefore necessary to take into account a contribution from this characteristic, when expressing the uncertainty of the reported value of the measurand. Examples are presented, where methods from metrology are combined with the Theory of Sampling to evaluate the influence of sampling on the uncertainty of results obtained by the analysis of biological materials.

This paper gives an overview of multi-way methods in image analysis, termed N-way image analysis. Both weak and strong multi-way methods are applied in order to decompose and characterize image data, and obtain insight into their abilities to capture and model the interpretable data structure. Multivariate Image Analysis (MIA) is a typical example based on weak multi-way methods like unfold-PCA/PLS. Strong multi-way methods such as PARAFAC, Tucker3, N-PLS are also introduced and applied to image analysis in this work. Which method to use is problem-dependent. Through macroscopic satellite images, virtual fluorescence images and microscopic functional property image examples, the performance of each alternative method is presented, as well as comparisons between weak and strong multi-way models. It is demonstrated that efficient handling of multiple images requires a clear a priori overview of the relationship between problem formulation and data array configuration. Appropriate preprocessing techniques, such as 2-D FFT and Wavelet transform, may also be needed in order to transform and configure some special types of image data to forms specifically suited for multi-way modeling. Application I shows the possibility for application of strong multi-way methods on multispectral images, otherwise conventionally analyzed by MIA. By contrast, application II attempts to investigate the feasibility of applying MIA models on typical three-way data, normally handled by the strong multi-way methods and provides a new perspective of dealing with fluorescence spectra as images. In application III, attempts have been made to predict rheological parameters from microscopic cheese images by multi-way methods. The present didactic exposition allows to draw some tentative first conclusions as to the dominant relationships between strong and weak multi-way data decompositions, their pros and cons and their relative merits.

This study explores how to combine variographic spatial characterization with multivariate data analysis, by showing how Principal Component Analysis (PCA) can be applied to unconventional types of data matrices, X variogram . This is here performed on a specific data set from an agricultural field in western Jutland, Denmark, but the data analytical approach is generic . In order to characterize the heterogeneity of a typical sandy soil, a variographic experiment along a 1-D profile is performed on 38 different minerogenic variables (geochemical elements). While the variogram is defined for one variable only, it is shown how PCA is able to characterize a multitude of variograms simultaneously, facilitating subject-matter interpretation of the fingerprints of the process(es) responsible for the spatial heterogeneity encountered. PCA scores and loadings contain information pertaining to the specific matrix type consisting of variograms, X variogram . Together with a companion paper, a complete approach for characterizing scale-varying spatial heterogeneity is presented with a view of developing sampling procedures for managing the intrinsic variability in natural soil and in similar systems (e.g. environmental characterization and monitoring, pollution in time and space, applied geochemistry, medical geology). Sampling in all of these contexts is shown to be much more than a ‘materials handling’ issue, by force involving the Theory of Sampling, TOS. The PCA (X variogram ) approach can be applied for tuning in of sampling procedures and 1-D and 2-D sampling plans in soil, environmental substrates, pollution and medical geology studies with a carrying-over potential to many other application fields with similar heterogeneity management needs.

We present an introduction to Multivariate Image Regression (MIR) with a selection of illustrative application studies. Generalisation from two-way multivariate calibration to the three-way regimen leads to—at least—three alternative image regression cases depending on the nature of the available Y-data: IPLS-Ydiscrim; IPLS-Ygrid; IPLS-Ytotal. A systematic image regression typology is briefly introduced. We here present the core of the principles of applied MIR. Two major MIR application studies are worked through, a food mass product industrial inspection study (IPLS-Ydiscrim) and a food product (fruit) storage stability image analytical monitoring (IPLS-Ygrid). These exemplifications are presented as archetypes, representing a much wider range of potential industrial/technological application areas. Based on simple three-channel imagery (in order to simulate many industrial systems), they nevertheless represent all higher-dimensional multivariate image cases as well, since the pertinent MIR principles and software are invariant w.r.t. any number of channels/variables employed. The present paper represents one major element of our work towards establishing a complete, stand-alone facility for Multivariate Image Regression (MIR); the second paper in this series deals with the development, implementation and extensive exemplifications of a complementary cross-validation facility.

This paper presents a novel way to predict the bulk powder characteristics from unilaterally illuminated powders using Angle Measure Technique (AMT) image analysis in combination with multivariate calibration. It is demonstrated that the AMT transform can account for the complexity of images in the scale domain and be used as a strong preprocessing facility for multivariate regression modeling. The concept of multivariate AMT regression has been proposed for this purpose. A wide variety of types of powders was collected in order to study the reliability, reproducibility and representativity of the methods. PLS models have been established to quantitatively predict key physical and behavioral powder properties such as particle size, density, minimum fluidization velocity, wall friction angle, and angle of repose. Finally, a first attempt at prediction of mixing component fractions in powder mixtures has also been implemented, which can be used for on-line monitoring of many types of mixing process if fast digital imaging is available.

A new approach for powder characterization based on image analysis, “Angle Measure Technique” (AMT) and multivariate data modeling is presented. AMT is designed to describe signal complexity as a function of geometrical scale from local to global. In this application, powder images are first unfolded to produce 1-D measurement series, which AMT, subsequently, transforms into multivariate scale characterizations. This new compound approach is able to extract and characterize powder features, such as particle size(s), shape(s), smoothness, coarseness, graininess, mixing homogeneity, as well as to classify and discriminate between different powders and even predict bulk behavioral properties. Experimental work reported here involves digital imaging of several tens of different types of powders, using a problem-dependent, low-angle, asymmetric illumination. The unilateral illumination setup brings about a significant simplification of traditional image analysis in powder studies, which is usually orientated towards characterizing all individual particles before aggregating this information. The present new technique achieves the same objectives by a simple and direct imaging, followed by AMT chemometric analysis. Principal Component Analysis (PCA) on AMT spectra derived from this type of imagery is used here to illustrate the power of this new technique, specifically to discriminate between powder types.

Acoustic chemometrics is presented as an interdisciplinary approach, covering diverse fields including applied engineering, electronics, signal analysis and chemometrics. The objectives for acoustic chemometrics are manifold, but quantitative analysis (chemical/physical) and process monitoring plays a major role as does physical characterisation of products, machinery—and process states. Potential applications in many industry sectors abound. In one particular sense, acoustic chemometrics is simple: obtaining problem-dependent `acoustic signals' (by relevant technical means), which—followed by some form of pertinent signal analysis—are subjected to chemometric data analysis. In this context it is often the power of multivariate calibration that comes to the fore. Here we give one major exemplar of the use of applied acoustic chemometrics—non-invasive monitoring of pneumatic gas/particle transportation processes. Many of our acoustic chemometrics forays so far have specifically been oriented towards `listening only', exclusively relying on passive sensors: we are deliberately only interested in utilising whatever complex acoustic signals may be discerned from `noisy' processes/products—because we find this approach the most challenging e.g., in contrast to ultrasound approaches, in which one also is in control of an appropriate acoustic input impulse or signal with which to excite, or probe, the system under investigation (`active listening', `active sensors', transducer technology). While many of our first generation applications thus for the most part have had the character of strictly empirical calibrations in which the detailed physical/chemical signal–response relation need not per force be known, we have also started parallel work of a more basic research nature, aimed at elucidating the fundamental mechanisms behind the satisfactory first achievements of acoustic chemometrics.

The nearest neighbor coordinate (NNC) representation of molecules provides a way of comparing the spatial structures between molecules. A method is described that transforms distance matrices into a comparable coordinate system. NNC is invariant to rotations and translations in Euclidean space. This representation may be useful in quantitative structure—activity relationships for comparison of molecular structures. NNC is also a general tool for problems where objects described by coordinates are to be compared.

Esbensen, K.H., Geladi, P.L. and Grahn, H.F., 1992. Strategies for multivariate image regression. Chemometrics and Inteligent Laboratory Systems, 14: 357–374. We present multivariate image regression (MIR) as a set of typically problem-dependent strategies for image decomposition guided by the nature of the Y variable and/or training data set delineation in the ( X , Y ) image domains. Regression techniques common in chemometrics may be applied also to the image regimen (in this paper we treat mainly two-dimensional images). We present applications of both IMPCR and IMPLS-DISCRIM in an effort to delineate the various possibilities for image regression. IMPCR builds directly on our earlier bilinear multivariate image analysis projection approach, while IMPLS-DISCRIM is trained on scene space binary classification masking with subsequent off-screen partial least squares analysis; the results are back-projected as images in the original scene space. Regression may either be carried out for modelling purposes and/or for subsequent prediction purposes. In the image domain this duality is accompanied by several optional training data set delineations in the scene space and/or in the spectral domain. We try to cover as complete a survey as possible of typical, representative regression problem types. We illustrate some of these MIR strategies with an MR-imaging example as well as a simple didactic MIR calibration from analytical chemistry.

An acoustic chemometric Process Analytical Technologies (PAT) feasibility study of fluidized bed granulation of a fertilizer product (urea) is applied to a semi-industrial pilot plant (SIPP) in a final evaluation before implementation for full-scale monitoring of the industrial production process. Conventional process monitoring and control is sub-optimal due to slow and labour-intensive laboratory analysis: particle size and liquid urea-concentration (correlated with water content) which are typically delayed by up to some 2 + h of analysis time. It is critical to be able to detect process transients and unwanted upsets, e.g. critical conditions in the granulator which may lead to uncontrolled shutdown situations, necessitating days of labour-intensive work to clean the granulator. The acoustic chemometrics approach goes directly into a real-time, on-line domain. In this study we focus on: 1. Optimal localization of acoustic sensors 2. Testing of a new sensor type (high-temperature microphone) in a semi-industrial granulator 3. Assessment of the feasibility to predict (by PLS-regression): • bed movement (airflow trough granulator) • liquid feed concentration (urea) • reflux of fine material to granulator 4. Monitoring and visualization of critical trajectories– early warnings –in an operator-friendly fashion. Results show that both process state and product quality can be monitored to a satisfactory degree, e.g. detecting unwanted lump-formation already at an incipient stage as well as bottom plate clogging. Such early warning allows process operators to change relevant process parameters (fluidization or atomization airflow, bed temperature, feed flux) to control product quality or to prevent critical shutdown situations. Successful validation of this type of PLS-prediction model signifies that acoustic chemometrics is now maturing into a proven on-line technology in process analytical chemometrics and PAT domains.

The partial least squares regression (PLSR) algorithms of Wold and Martens provide alternative, powerful tools for handling ill-conditioned multivariate regression data, see e.g. Martens and Nees (1989) and Hoskuldson (1996) for overviews of the fundamental multivariate calibration concept, in which are presented the above two slightly different versions of the PLSR method. Both algorithms assume collinear regressor variables generated by underlying latent variables, and although they are different with respect to the score and loading matrices involved, they give identical predictors. There is certainly no practical need for yet another algorithm that provide the same predictor as the well-known algorithms of Wold and Martens. However, these algorithms, as presented in the chemometrical literature, make no use of the dynamic systems theory available for readers from the control community, and the main aim of the present paper is therefore to provide a predictor derivation at first especially for this category of readers. This actually also results in a simplified version of the Martens prediction algorithm making use of the so-called loading weight vectors only, which would appear to be of general interest (see Esbensen (2000) for a discussion of loadings and loading weights). The score vectors, which are also essential elements of chemometrics, are easily computed once a preliminary predictor is found. The new algorithm has a lot in common with an algorithm developed by Helland (1988), and a similar algorithm by Di Ruscio (2000). Again, the reason behind the present modifications is mainly didactic. The key step in the simplification is the use of an explicit latent variables model, which facilitates an early introduction of the Helland predictor form using only loading weight vectors. The present exposition, we believe, constitute a novel, easy, and complete introduction to the prediction aspect of multivariate calibration.

Esbensen, K.H. and Martens, H, 1987. Predicting oil-well permeability and porosity from wire-line petrophysical logs—a feasibility study using partial least squares regression. Chemometrics and Intelligent Laboratory Systems, 2: 221–232. A method for predicting permeability, porosity and other petrophysical core parameters directly from wire-line logs in oil wells is reported, based on multivariate calibration using principal component analysis and partial least squares (PLS) regression. The method is applied to data from a single North Sea oil well over a depth interval of 125 m. Samples with log parameters indicating abnormal lithologies or with excessive missing data were rejected as “outliers”. Calibration was carried out by relating a set of seventeen non-selective, in situ wire-line petrophysical log variables from various depths in the well to petrophysical laboratory determinations subsequently performed on drill core samples from the corresponding depths. Calibration modelling was effected by treating the samples both as a single lithology class (reservoir sand) and as high and low permeability classes (coarse and fine sands). Various linearity transformations of the input data were tested; (permeability)13 was found to be optimal. The predicted permeability and porosity closely covary down the entire well with the core control data. Approximately 50% of their total data variance was correctly reproduced; the same stratigraphic horizons were identified. The residual variance is due to measurement error, shift discrepancies and model errors (non-linearities and data subgroupings). The PLS model was found to be statistically stable by two different validation methods (leverage correction and cross-validation). This study demonstrates in situ wall-rock parameter prediction from core calibration data in one well, and simultaneously simulates inter-well prediction for other non-cored wells. PLS prognosis for oil/gas wells appears feasible at the present precision level of log and laboratory measurements and uncertainty in core-depth assignment.

Multivariate data analysis of 2-dimensional object arrays is explained and its application to images is demonstrated. The possibility of creating soft models in latent variables is a great aid in this process. The pattern classification, cognition and recognition processes as used in the SIMCA method can be translated to latent variables of the multivariate images. A small example of a satellite image is presented. The method aspires to make maximal use of the possibilities of standard image analysis equipment and to combine the results obtained with more traditional image analysis techniques. This article concentrates mainly on theoretical aspects.

Bilinear decomposition (soft modelling using principal component analysis) of multivariate imagery results in: score and loading plots, score images, classification projections and residual images in the scene space. Feature space score plots are used as a starting point for pixel class delineations, followed by iterative scene space evaluation. This is a reversal of traditional image processing practice, which selects training samples in the scene space. The present feature space class definitions can be shown to have certain optimality characteristics with respect to traditional scene space delineations. After problem-dependent relevant pixel class delineations have been obtained, one can compute corresponding local class PC-models that serve as an alternative basis for problem-dependent classification and sequential segmentation. Multivariate image analysis (MIA) allows interactive exploration and classification of most types of technical multivariate imagery. We present a general strategy for multivariate image analysis, illustrated by a remote sensing showcase.

Principal Component Analysis (PCA) is a multivariate exploratory analysis method, useful to separate systematic variation from noise. It allows to define a space of reduced dimensions that preserve ...

Esbensen, K., Lindqvist, L., Lundholm, I, Nisca, D. and Wold, S., 1987. Multivariate modelling of geochemical and geophysical exploration data. Chemometrics and Intelligent Laboratory Systems , 2: 161–175. We describe an integrated data-analytical strategy for overburden geochemical prospecting, based on principal component analysis (SIMCA), partial least-squares regression (PLS) and geostatistics (kriging). The strategy extracts and integrates relevant prospecting information existing in multivariate data from several geological sample media. Data-analytical modelling with subsequent geographical interpolation and plotting compresses the relevant geological information into a manageable number of maps. The procedure defines a multivariate divide between the background and anomalies and combines single variables into geologically useful components. Classification of all survey samples or new data with respect to background models emphasizes anomalies in their geographical context, giving quantitative indications of target areas for follow-up exploration activities. To further improve a model of an area, the strategy also optionally involves integration of geochemistry with petrophysical and other geophysical and geological data. The emphasis is on characterizing the underlying bedrock into areas with similar chemical and petrophysical properties, resulting in a digital geological map of the subsurface local geology. This output is designed for further image analysis etc. A glacial-overburden prospecting campaign from Northern Sweden illustrates the various elements of the present approach.