SAMPLING AND MINIMUM PHASE FROM BOTH A CONTINUOUS AND DISCRETE POINT OF VIEW*

Examples show that the sampling operation–i.e., the change from the continuous time domain to the discrete time domain–does not necessarily preserve the minimum-phase property. Further examples can be constructed to show that the resampling operation on the discrete time domain does not necessarily preserve the minimum-phase property. Finally it can be shown that the minimum-phase property can be either created or destroyed by sampling or resampling.     

MIGRATION IN THE PRESENCE OF NOISE*

It is shown that the so-called Kirchhoff-summation operator is of a very wide-band nature and even contains an evanescent part. As a consequence, discretization may cause serious aliasing errors, particularly for small extrapolation steps. It is proposed to use in all practical cases band-limited versions of the summation operator, the spatial cut-off frequency being determined by the spatial Fourier spectrum of the coherent noise.     

Correspondence analysis applied to volcanic aerosols

A statistical analysis of volcanic aerosol based on 1,083 variables including concentration, grain-size and site variabilities establishes correlations between sites and among chemical elements.Variation curves of element concentration show parallel behaviour. High coefficients of correlation between anions and cations confirm associations between elements.     

An experimental assessment of the added mass of some plates vibrating in water

A reservoir of water is contained by a concrete valley block, a ferrocement wall and a steel plate. Both wall and plate contain an array of pressure transducer sockets (Figures 1 and 2). Using the M.A.M.A.¹ equipment pure modes of vibration are excited. Frequency and mode shape are measured with the reservoir empty. When the reservoir is full hydrodynamic pressure is also measured. These hydrodynamic pressures are compared with Chopra's² two-dimensional, series solution, which includes compressibility of water, and with two- and three-dimensional finite element solutions of Laplace's equation, which do not include compressibility. Chopra's solution is unsatisfactory for modes which contain a vertical node line. The best agreement between experimental and theoretical hydrodynamic pressure is obtained when the latter is obtained from three-dimensional solutions of Laplace's equations, indicating that compressibility does not play a significant rǒle. This conclusion is supported by agreement between experimental frequencies (reservoir full) and those calculated using added mass obtained from the Laplace solution. Similar conclusions were reached from tests on a floating steel plate, suspended in the surface of the reservoir by a soft spring. Here, dynamic pressure measurements were not made, reliance being placed on agreement between calculated and measured frequencies and mode shapes.     

A synthetic model for daily streamflow

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.     

Subsurface electric fields of a grounded cable of finite length for both frequency and time domain

Summary The subsurface electric fields of a current-carrying cable are examined for both steady-state and transient situations. Closed form expressions are obtained for two of the electric field components, and a numerical integration is used to obtain the third component. Waveforms for a stepfunction current excitation are displayed. The results have possible application to pulsed downlink communication in mine rescue operations.     

Phosphorus accumulation rates in metalliferous sediments on the East Pacific Rise

Based on phosphorus, iron and manganese analyses in 16 cores (5 dated) from the crest and flanks of the East Pacific Rise and the Bauer Deep we estimate that phosphorus is being deposited about 20 times faster in metalliferous sediments near the rise crest than in adjacent flank deposits, and about 40 times faster on the crest than in the Bauer Deep. Almost all of the phosphorus on the rise crest is contained in poorly crystallized hydrothermal iron oxyhydroxides, supporting Berner's (1973) proposal of phosphate sorption by these phases. The phosphate is probably derived from seawater, but some hydrothermal contribution cannot be excluded at this time. Flux estimates indicate that metalliferous sedimentation could remove 15–40% of the pre-agricultural river input of dissolved phosphate.     

Input-output models

Summary Elementary mass balance and export models are explored in relation to eutrophication as caused by phosphorus and nitrogen. New criteria for phosphorus loading are given in relation to the ratio ‘mean depth-water fill-in-time’ . The results suggest that lakes having long water renewal times are much more sensitive to phosphorus loading than would appear from mean depth only. Further, from comparison of the relative residence time of nitrogen and phosphorus, it is deduced that—with increasing eutrophication—the nitrogen metabolism is speeded up beyond the point of simple proportionality which would explain the transition from phosphorus to nitrogen limitation in highly eutrophied lakes. It is further suggested that the principles derived from eutrophication in regard to the metabolism of phosphorus and nitrogen in lakes are applicable also to other environmental compartments and stress factors.

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Zusammenfassung Elementare Massenbilanzen und Exportmodelle werden bezüglich ihrer Bedeutung für die Eutrophierung von Seen mit Phosphor und Stickstoff untersucht. Als neues Kriterium für die Phosphorbelastung wird das Verh?ltnis ?mittlere Wassertiefe-Zeit der Wassererneuerung? eingeführt. Die Beobachtungen lassen vermuten, dass Seen mit langen Wassererneuerungsperioden viel empfindlicher auf die Phosphorbelastung ansprechen, als aus ihrer mittleren Tiefe geschlossen werden k?nnte. Aus dem Vergleich der mittleren Aufenthaltszeiten von Stickstoff und Phosphor ergibt sich, dass mit zunehmender Eutrophie der Stickstoffumsatz weit st?rker beschleunigt wird, als der einfachen Proportionalit?t zu seiner Konzentrationszunahme entsprechen würde. Dies erkl?rt die Verschiebung von Phosphor zu Stickstoff als limitierendem Faktor in hocheutrophierten Seen. Es wird vermutet, dass die Prinzipien, welche bezüglich der Eutrophierung aus dem Umsatz von Phosphor und Stickstoff in Seen abgeleitet werden, auch für andere Lebensr?ume und andere Belastungsfaktoren angewendet werden k?nnen.

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Résumé Un équilibre massive élémentaire et des modèles de sortie ont été explorés en relation avec l’eutrophisation provoquée par le phosphore et l’azote. De nouveaux critères ont été décrits en relation avec le taux ?profondeur moyenne?—durée de remplissage d’eau? . Les résultats suggèrent que les lacs dont la durée de renouvellement de l’eau est longue, sont beaucoup plus sensibles aux charges de phosphore, que ne laisserait appara?tre le critère seul de la profondeur moyenne. De plus, en comparant les durées relatives de séjour de l’azote et du phosphore, on déduit que—avec la progression de l’eutrophisation—le métabolisme de l’azote est accéléré au delà du point de simple proportionalité qui expliquerait la transition du phosphore à l’azote en tant que nutriment limitant dans des lacs fortement eutrophes. En outre, il est suggéré que les principes, dérivant de l’eutrophisation des lacs en relation avec le métabolisme du phosphore et de l’azote, sont également applicables à d’autres compartiments et facteurs de containte de l’environnement.

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Paper given at the Conference on Chemical-Ecological Considerations for Defining the Goals of Water Pollution Control, Kastanienbaum, Switzerland, April 19–21, 1972.     

Helium isotopes in the earth's interior and in the atmosphere: A degassing model of the earth

A first-order degassing model was applied to describe the evolution of helium content and isotope composition in the earth and in the atmosphere. The main events described by the model are: (1) the earth-trapped primordial rare gases at the moment of its accretion; (2) later, the solid earth lost primordial and radiogenic rare gases, and (3) they were accumulated in the atmosphere; (4) in addition,³He was formed in the atmosphere due to cosmic irradiation, accretion from solar wind, etc.; (5)³He and⁴He dissipated into space at different loss rates.Study of this model confirms the concept that some of primordial helium is retained in the interior of the earth; terrestrial helium (³He/⁴He～ 2 × 10^?5) was most probably formed as a mixture of primordial (³He/⁴He= 3 × 10^?4) and radiogenic (³He/⁴He～ 3 × 10^?8) helium. For achondritic concentrations of heavy radioactive elements (U= 2.25 × 10^?8g/g) the calculated⁴He flux from the earth is equal to 5.7 × 10⁶ at cm^?2 sec^?1. The corresponding³He flux is about 114 at cm^?2 sec^?1. In discussing the aeronomic problem of helium it is necessary to take into account that the earth is the main source of the light helium isotope.     

A generalized continuum method for dynamic analysis of asymmetric tall buildings

The paper presents a continuum method for dynamic analysis of asymmetric tall buildings with uniform cross-section in which the horizontal stiffness is provided by shear walls and columns of arbitrary shape and layout, coupled by horizontal beams. The equations of motions are formulated in variational terms, including axial strain energy. Numerical solutions, obtained by using finite time differences and infinite polynomials, are presented for the response of a twenty-storey building with six shear walls to an impact load and earthquake accelerations. It is shown that omission of the axial deformations results in a substantially distorted pattern of behaviour, some of its effects being:

• 1 Overestimation of the bending stiffness of the coupled shear walls, with corresponding changes in their stiffness ratios.
• 2 Underestimation of the periods of the principal modes, with a corresponding change in the dynamic response.
• 3 Distortion of the magnitude, form, time of onset and coupling of the maximum displacements.
• 4 Pronounced change in the shear force and moment diagrams for the shear walls, the beams and the building as a whole.