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Stochastic linear models are fitted to hydrologic data for two main reasons: to enable forecasts of the data one or more time periods ahead and to enable the generation of sequences of synthetic data. These techniques are of considerable importance to the design and operation of water resource systems. Short sequences of data lead to uncertainties in the estimation of model parameters and to doubts about the appropriateness of particular time series models. A premium is placed on models that are economical in terms of the number of parameters required. One such family of models is multiplicative seasonal autoregressive integrated moving average (Arima) models that have been described by G. E. P. Box and G. M. Jenkins. In this paper we illustrate the process of identifying the particular member of the family that fits logarithms of monthly flows, estimating the parameters, and checking the fit. The seasonal Arima model accounts for the seasonal variability in the monthly means but not the seasonal variability of the monthly standard deviations: for this reason its value is limited. The forecasting of flows one or more months ahead is described with an example.

A single trait factor, Calculation Speed, and five state factors, General Energy Level, Fatigue, Excitation, Rhythm and Volitional Tention, are extracted from a (30×30) correlation matrix obtained from a 30-minute performance of simple additions of two single-digit numbers in the Uchida-Kraepelin Psychodiagnostic Test. Factor scores are estimated by the multiple regression method and thirty error factors are calculated by substituting them into the formula of the factor analytic model. The distributions of common and error factor scores are completely normal with the zero mean and the unit variance in each. Discrepancies between the observed and synthetic data are found to be very small. A computer program in Fortran language is also prepared as an objective evaluation of the Uchida-Kraepelin Test.

An investigation is made of the application of the autocorrelation matrix method to the analysis of moveout. A number of multichannel Wiener filters are computed to pass/reject correlated events. Experiments are performed on a variety of synthetic data in order to ascertain how such filters will perform in a given situation. It is shown that very low levels of signal can be successfully recovered and a very close match must exist between the moveouts present in the data and those specified for the particular filter design. The “average” or second order autocorrelation matrix is developed, and a multichannel filter scan of a column from this matrix is incorporated in the former method. Experiments are performed on synthetic data for different noise conditions in order to ascertain the ability of these methods to detect low‐power correlated events. It is shown that such detection can be successful, provided the random noise is not too severe. Results from the analyses of real data records are consistent with those of the synthetic case.

Several writers have proposed the use of multichannel filters for the elimination of coherent noise on seismic records. One filter of this type which can be constructed is a multichannel Wiener filter which has a multichannel input and a single channel output. In this form, it is applicable to data collected for vertical or horizontal common‐depth‐point stack processing. The choice of desired output characteristics for this Wiener filter is flexible and, for example, can be tuned to correspond to multichannel deconvolution. The results of the application of filters of this type to field and synthetic data, in general, show little if any advantage over single‐channel deconvolution. This failure appears to be connected with the low cross coherence of both noise and reflection signal on field‐recorded, common‐depth‐point traces.

When a vessel's track consists of a sequence of intersecting segments, discrepancies in the values of the phenomena under survey arise at the intersections. A statistical method is described for reducing these discrepancies by adjustment of the track. A computer program implementing this method has been written and has been tested with synthetic data.

The standard method for determining the relative transition probabilities of vibrational-rotational bands from experimental data is examined. It is shown that the relative intensities in the high overtone sequences are likely to be similar to those for a linear dipole moment, and that it is impossible to predict either the dipole moment or the intensities in sequences with lower Δv values from them. When this is taken into account the method is found to be satisfactory in its application to synthetic data derived from a model moment. Procedures for determining the signs of ratios of transition integrals based on experimental data are noted. The method is applied to OH using the experimental data of Krassovsky et al. and Garvin et al. on the sequences Δv = 2 to Δv = 7, though the lack of measurements on the Δv = 1 sequence gives rise to ambiguity in the dipole moment and in the transition probabilities for this sequence. Further experimental work is needed. An attempt is made to calculate the rates of population of the various vibrational levels and the effect on these of the ambiguity is demonstrated.

This paper presents a model for synthesising daily average streamflow data that is suitable for most rivers in Great Britain. The method is based on a linear interpolation of the logorithms of 5-day average flows. The 5-day average flows are produced using N.T. Kottegoda's statistical model (Thesis, Univ. of Birmingham, 1970). The 5-day model preserves the long-term statistical characteristics of the daily data, while the short-term characteristics such as hydrograph shape are imposed by the interpolation method. A stochastic error term is superimposed on the interpolated daily flows. This term represents the non-deterministic component of the daily time series. The analysis of the observed error terms represents an important part of this paper. The riverflow in the Severn at Bewdley is used to demonstrate both the analysis of actual data and the generation of synthetic data. The technique is then applied to data from two other rivers with widely differing characteristics to demonstrate the range of the method.

A method is described for analyzing ligand-binding data for proteins. The method has the advantages of objectivity and minimal computer requirements. Its use is illustrated for synthetic data and for real data. The method is shown to be particularly advantageous in the case of co-operative binding.

A procedure for synthesizing 500-mb. height and temperature data over oceanic areas using only sea level data as an input is developed. The procedure is tested by comparing the synthetic data to observed 500-mb. data at 4YN and 4YP and by comparing 500-mb. synthetic data charts to 500-mb. charts based on observed data. The North Pacific is chosen for the latter test. The 500-mb. height variations given by synthetic data compare favorably with the observed height variations at 4YN and 4YP. While certain errors are apparent in the synthetic 500-mb. chart it does show many of the features of the chart produced from observed data.

The description of biomechanical heredity processes leads to the study of non-linear Volterra integral equations such as that given by Eq. 3. It is shown that under convenient assumptions regarding the nature of the memory of the system approximate equations of evolution of the process can be given in terms of a system of difference-differential equations subject to initial conditions. Alternative formulations are discussed, mainly emphasizing numerical aspects. The identification problem is briefly discussed, and some numerical examples, obtained using synthetic data, are presented to exhibit the application and the principal ideas of the method.

The study describes an algorithm and specifies conditions for obtaining data by sedimentation equilibrium to study nonideal associating systems. The algorithm can analyze systems of up to four macromolecular components, and it returns values of their concentrations plus a single nonideality coefficient B. By appropriate mathematical manipulations the algorithm is made to solve by least squares for linear coefficients, rather than attempting the more cumbersome iterative fitting of nonlinear regression. Synthetic data with realistic errors, both random and systematic, were used to test the reliability of the answers obtained and to define the optimum experimental conditions which can yield data capable of analysis. Equations are derived which permit the simultaneous solution of data from two ultracentrifuge cells and thus extend the concentration range studied. Extension of the concentration range is essential for all systems with more than two macromolecular components. Synthetic systems of widely varying association constants, number of species, and degree of nonideality were analyzed successfully. The dissociation constants varied from 1.0 x 10-3 m to 3.74 x 10-6 m; the second virial coefficient was varied from 0 to 3.0 x 10-5 mole dl g-2. Statistical tests for constructing confidence regions are used in determining the reliability of the parameters obtained.

A study was made of the possibility of representing the hourly rainfall process observed at a point by a simple model using a digital computer. The purpose of the modeling procedure is to produce synthetic hourly rainfall data for the Stanford watershed model to make possible long records of synthetic streamflows. The model represents the hourly rainfall process by a sixth-order Markov chain and is shown to be a suitable procedure. Dry periods between storms are found to be longer than dry periods that occur in nature. When used in the Stanford watershed model, the synthetic data produce hydrologic characteristics for a test watershed that are in good agreement with those derived from historic data.

This article compares, via synthetic data analysis, the performance of five different methods for scaling averaged dissimilarities data under conditions involving individual differences in “perception.” All methods perform well when no “degradation” of the (simulated) ratings is entailed. When the data are transformed to zero-one values—a procedure sometimes followed in applied studies—all procedures perform poorly compared to the no-degradation case. Implications of these results for scaling applications involving group solutions are discussed.

Correlation between Snyder's basin constants Ct and 640 Cp and basin physical characteristics of drainage area, total length, average slope length to the center of area, and elongation is established using synthetic data. Actual data are used to compensate for discrepancies introduced through initial use of synthetic data. Effects of nonuniform rainfall distribution on the basin runoff hydrograph is represented by a unit graph, selected from three computed for each basin. Computation formulas are presented to define each of the three basin unit graphs. A dimensionless unit graph is introduced for a more accurate definition of the rising and falling links of each unit graph.

The more recent developments in common‐depth‐point techniques to attenuate multiple reflections have resulted in an exploration capability comparable to the development of the seismic reflection method. The combination of new concepts in digital seismic data processing with CDP techniques is creating unforeseen exploration horizons with vastly improved seismic data. Major improvements in multiple reflection and reverberation attenuation are now attainable with appropriate CDP geometry and special CDP stacking procedures. Further major improvements are clearly evident in the very near future with the use of multichannel digital filtering‐stacking techniques and the application of deconvolution as the first step in seismic data processing. CDP techniques are briefly reviewed and evaluated with real and experimental data. Synthetic data are used to illustrate that all seismic reflection data should be deconvolved as the first processing step.

Stratigraphic data constitute an open system of numbers that can be converted to a closed system by computing percentages and ratios. Percentage and ratio maps play an important part in facies studies, and it is appropriate to consider the changes that occur in maps when the data are closed. In two- and three-component systems the statistical correlations that arise from closing the data can be exactly predicted, but the restraints are less rigid for systems with four or more components. The behavior of two- and three-component systems is illustrated with synthetic data, followed by an example of a four-component stratigraphic unit from the Upper Permian of western Kansas and eastern Colorado. Analysis of the map data before the maps are actually drawn shows that some general predictions can be made regarding the particular maps that may give most information within the context of a specific problem. This theme is developed with the Permian example, and leads to a tentative flow chart of map selection based on computation of summary statistics, of all linear correlations, and of trend surfaces and their deviations.

Simple structure and other common principles of factor rotation do not in general provide strong grounds for attributing explanatory significance to the factors which they select. In contrast, it is shown that an extension of Cattell's principle of rotation to proportional profiles (PP) offers a basis for determining explanatory factors for three‐way or higher‐order multimode data. Conceptual models are developed for two basic patterns of multimode data variation, system‐ and object‐variation, and PP analysis is found to apply in the system‐variation case. The generalized PP model provides the basis for a program called PARAFAC (for parallel factors). This program is demonstrated to give unique “correct” solutions with oblique nonsimple structure and even nonlinear factor structures. A series of tests, conducted with synthetic data of known factor composition, provides data on the minimal necessary conditions of uniqueness and the properties of the analysis procedures when these minimal conditions are not fulfilled. PARAFAC has been applied to several different sets of real data with meaningful results (as reported elsewhere in this meeting).

This paper discusses the design and application of numerical pass filters. Pass filtering is a general term encompassing low-pass, high-pass, band-pass, and band-reject (notch) filtering. These filters are zero phase, and they can provide roll-offs of almost any desired amount. Truncation effects are reduced with Fejer weights. Relatively short time-domain operators are obtained through a combined use of parabolic rounding, linear attenuation, and Fejer weighting. These operators decay with an approximate 7/2 power law, as compared with the 2/2 decay of the sin x/x function. The emphasis in this paper is on applications rather than theory. Examples illustrate data processing with pass filters. Examples of digital-alias filters are given, Applications include synthetic data as well as actual field examples. The applications relate to exploration seismology; however, these filters are quite general, applying equally well to other geophysical, geological, and scientific problems.