Methods of Experimen taI PhysicsVOLUME 24 GEOPHYSICSPART 6

Fleld Measurements

METHODS OF EXPERIMfNTAL PHYSICSRobert Celotta and Judah Levine, Editors-in-Chief

Founding Editors

L MARTON . C. MARTON

Volume 24

GeophysicsPART B Field Measurements

Edited by

Charles G. SammkDepartment of Geological Sciences University of Southern Cafifornia Los Angeles, California

Thomas L. HenyeyDepartment of Geologlcai Sciences University of Southern California Los Angeles, Californla

ACADEMIC PRESS, INC.Hercourt Brace Jovanovich, Publishers

Orlando San Dlego New York Austln Boston London Sydney Tokyo Toronto

COPYRIGHT @ 1987 BY ACADEMIC PRESS. INC. ALL RIGHTS RESERVED. NO PART OFTHIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS. ELECTRONIC OR MECHANICAL. INCLUDING PHOTOCOPY. RECORDING. OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM. WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS, INC.Orlando. Florida 32887

ACADEMIC PRESS INC.

Unired Kingdom Edition published by(LONDON) 24-28 Oval Road, London NWI 7DX

LTD

Library of Congress Cataloging in Publication Data (Revised for volume 24, Part B) Geophysics. (Methods of experimental physics; v. 24) Includes indexes. Contents: pt. A. Laboratory measurements - pt. B. Field measurements. 1 . Geophysics. I . Sammis, Charles G. 11. Henyey, Thomas L. (Thomas Louis), Date I l l . Title. IV. Series. QE501.G48 1987 551 86-17439 ISBN 0-12-475967-X (v. B : alk. paper)

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PRINTED IN THE UNITED STATES OF AMERICA8 1 8 8 8 9 9 0 9 8 1 6 5 4 3 2 1

CONTENTSPREFACE .

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10. Seismic Instrumentation TA-LIANG TENG1. 2. 3. 4. 5. 6. 7. 8.

Introduction . . . . . . . Historical Development . . . . Nature of Seismic Ground Motions . Basic Types of Seismic Sensors . . Damping Devices and Transducers . Pendulum-Galvanometer Interaction Central Recording and Networking . Recent Developments in Seismographs References . . . . . . . .

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1 2 11 14 25 32 35 50 73

11. Marine Acoustic Techniques F . N . SPIESS1 2 3 4 5

. Echo Sounders . . . . . . . Bottom-Imaging Sonars . . . . . Acoustics for Position Determination . A System Example . . . . .

. Introduction: The Ocean Acoustic Environment. . . References . . . . . . . . . . . . .

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77 87 97 110 120 123

12. Surface Measurement of the Earths Gravity Field JAMES . WHITCOMB H1. Introduction . 2 . Instrumentation 3 . Applications . References . .

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13. Satellite Measurement of the Earths Gravity Field WILLIAM . KAULA M1 2 3 4

. Introduction and HistoricalSummary .References .

. Geodetic Satellites . . . . . . . . . . . . . Satellite Orbit Dynamics . . . . . . . . . . . . DataAnalysis . . . . . . . . . . . . . .

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163 164 170 178 186

14 Experimental Methods in Continental Heat Flow

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DAVID BLACKWELL ROBERT . SPAFFORD D AND E

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. Introduction . . . . . . . Temperature . . . . . . . Thermal Conductivity . . . . Heatproduction . . . . . . Heat Flow Calculation . . . . Miscellaneous Techniques . .

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189 192 206 213 213 217 219 221

15. Measurement of Oceanic Heat Flow R. P VON HERZEN

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. . 3. Thermal Conductivity 4. Heat Flow . . . .

1 Introduction . . . 2 Temperature Gradients

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16. Electrical Methods in Geophysical Prospecting STANLEY . WARD H1 Introduction . . . . . . . 2 Elementary ElectromagneticTheory

3 Electrical Properties of Earth Materials.Surveys

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4 Basic Principles of Resistivity and Induced Polarization

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5 MagnetotelluricMethod . . . . . . 6 Controlled-SourceElectromagneticMethods

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17. Measurement of In Situ Stress BEZALEL HAIMSON C

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. Surface Measurements . . . . . . . . . . . . Near-Surface Measurements . . . . . . . . . . . Deep Measurements . . . . . . . . . . . .. State of Stress in the Earths Crust . . . . . . . . . Future Research . . . . . . . . . . . . .References .

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377 379 383 393

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18. Continuous Measurement of Crustal Deformation DUNCAN CARR AGNEW

. Measurement . . . . . 2 . Quantities to be Measured . 3 . General Design Features . . 4 . Tiltmeters: Particular Designs 5 . Strainmeters . . . . .

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1 Aims and Problems of Continous Deformation

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19. Geophysical Well Logging JAYTITTMAN1 Introduction . . . . . . . . . . . . . . 2. Geological and Petrophysical Interpretation of Logging

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Measurements .

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PREFACEGeophysics uses many of the methods and techniques of physics to study the solid earth and planets. Geophysical data are collected both in the laboratory and in the field. Volume 24, Part A of this two-volume set discusses the laboratory techniques, and Volume 24, Part B discusses the field techniques used in geophysics. Field measurements provide the basic data set against which all geophysical models must be tested. Geophysical field methods are directed principally toward investigation of the earths interior-that portion of the earth that is inaccessible to geologic observation. The methodology involved in field research generally falls into one of three categories. In the first category, instruments are strategically deployed to passively detect the natural waves and potential fields of the earth. Properties of these fields are governed by the physics of the earths interior-its composition, rheology, and structure. Data from these measurements can then be inverted to recover or constrain the physics. Examples of these methods include the detection of wavefields generated by earthquakes and the measurement of gravitational and geomagnetic fields. In the second category, wave or potential fields are artificially generated at the earths surface and the resulting reflected or secondary fields are measured using carefully designed arrays of surface instruments. Examples include reflection seismology and electromagnetic methods. Finally, the third category involves the direct measurement of i situ properties of the earth, n such as the state of crustal stress or the physical properties of rock into which deep borings have been made. Seismology is the most powerful method used for exploring the earths interior. In Chapter 1 ,T. Teng discusses instrumentation used to detect and 0 record seismic wavefields. Instrumental techniques used to exploit the broad seismic bandwidth are described. In a related discussion in Chapter 11, F. Spiess reviews acoustic techniques used in the marine environment to characterize the sea floor. The emphasis is on high-resolution signal generators or transducers operating at frequencies above 1 kHz. We have not included a discussion of the seismic reflection method as carried out on land or at sea. This method uses signals generated in the 1 to 100-Hz bandwidth to map geologic structures in the continental and oceanic crust, and has been highly refined by the petroleum exploration industry. Excellent discussions of the experimental techniques and operational methods involved can be found in publications of the Society of Exploration Geophysics, and in texts such as Exploration Seismology, Volumes 1 and 2, by R. E. Sheriff and L. P. Geldhart.ix

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Knowledge of the distribution of mass in the earth, particularly in the outer heterogeneous layers, is fundamental to understanding the structure of the lithosphere and the geometry of convection currents that drive the lithospheric plates. In Chapters 12 and 13, J. Whitcomb and W. Kaula describe techniques for measuring the earths gravity field from the surface and from satellites, respectively. Surface measurements provide high resolution but spotty coverage of the earth as a whole. Satellites, on the other hand, obtain a synoptic view of the gravity field, uncovering areas that are otherwise inaccessible. Estimates of temperatures in the earth, particularly for the deep interior, vary widely. Much of the imprecision in our knowledge stems from an incomplete understanding of the distribution of heat sources. The nature of convection in the mantle is critically dependent on the internal temperature distribution. Surface heat flow measurements provide the only direct estimates of internal temperatures, particularly within the continental and oceanic lithospheres. Chapter 14, by D. Blackwell and R. Spafford, and Chapter 15, by R. Von Herzen, review methods for making heat flow measurements on the continents and in the oceans, respectively. Oceanic heat flow measurements have played a fundamental role in the development of the theory of sea floor spreading. Chapter 16, by S. Ward, reviews electrical methods in geophysical exploration. These methods, which include resistivity, induced-polarization, and magnetotelluric and electromagnetictechniques, have been used extensively for groundwater exploration, and by the mining and geothermal industries. Some, most notably magnetotelluric methods, are now being applied to deep exploration of the crust and mantle. We have not included a discussion of the magnetic method of exploration, a passive measurement technique based on perturbations of the earths largely dipolar magnetic field by rock masses displaying remanent and induced magnetization. Magnetic measurements are also important in paleomagnetism (see Chapter 9 in Volume 24, Part A) and in descriptions of spatial and temporal characteristics of the earths dipole and non-dipole fields. Discussions of magnetic methods can be found in a variety of texts, such as Interpretation Theory in Applied Geophysics, by F. S. Grant and G. F. West. The in situ state of stress is directly related to tectonic processes in the lithosphere-specificaIly earthquakes and the mechanics of faulting. In Chapter 17, B. Haimson reviews the methods for measuring stress in the crust. Perhaps the most important advance in in situ stress determinationin the past several years has been the refinement and general use of the borehole hydraulic fracturing technique. Tectonic stresses applied by one plate on another across lithospheric plate boundaries, by convection-induced basal tractions or by gravitational body forces, result in nonuniform strain fields with the plates. Faults represent important singularities in these

PREFACE

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strain fields. Chapter 18, by D. Agnew, summarizes the methods for continuous measurement of crustal deformation, many of which have been refined over the past 10 years in response to a national program in earthquake hazard reduction and fault zone monitoring. More than any other kind of instrument, strainmeters require the ultimate in long-term stability. Finally, Chapter 19, by J . Tittman, reviews the methods used in geophysical well logging. Because of the importance of these methods to hydrocarbon exploration and production, many specialized instruments and techniques, some proprietary, have been developed and refined by the petroleum industry. However, as more deep borings become available to the general scientific community, well logging methods will find their way into the arsenal of basic geophysical research tools for the characterization of rock physics. This area of geophysical instrumentation has the potential for significant scientific and economic profit.

THOMAS HENYEY L. CHARLES SAMMIS 0.

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10. SEISMIC INSTRUMENTATION

Ta-Liang TengDepartment of Geological Sciences University of Southern California Los Angeles, California 90089-0741

1. IntroductionElastic radiation emanating from a seismic source propagates as waves traveling over the surface and through the interior of the earth. An instrument registering seismic waves is a seismograph, and a record so registered is a seismogram. From seismograms comes our knowledge of the global distribution of earthquakes, of the internal structure of the earth, and of the nature of the seismic source process. In modern seismological observations, a number of seismographs form a network, which is the basic scientific tool (analogous to major telescopes in astronomy) that provides a continuing data base fundamental to the science of seismology. The quantities to be measured by seismological instruments are the time history of displacements and their derivatives (velocities, accelerations, and strains) at the surface, which give the boundary values from which the earth's internal constitution as well as the seismic source structures are deduced. The measurement of these seismic boundary values is itself a science called seismometry. It is the purpose of this chapter to give a comprehensive account of the seismic instrumentation that forms the backbone of seismometry. One of the characteristics of seismometry is that measurements are made over an enormous magnitude range (for displacements from lo-'' to 10" m and accelerations from lo-' to 10"g) and a broad frequency band to lo2 Hz). In some applications, a frequency range down to DC or up to lo4 Hz is required. Another salient feature of seismometry is that at the same time that measurements are being made, both the observer and the object are subject to disturbing ground motions. The measurement sought is the motion history of this observed object with respect to an inertial frame which does not exist on the earth at the time of the passage of seismic waves. Therefore, much of the effort in the development of seismometry has been devoted to establishing an adequate pseudostationary point for an inertial reference1METHODS OF EXPERIMENTAL PHYSICS Vol. 24, Part B Copyright 0 1987 by Academic Press, Inc. All rights of reproduction in any form reserved.

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frame against which measurements may be made to a sufficient degree of accuracy. There are basically three devices that can be used for measuring seismic ground motions: (1) the pendulum sensor, which makes use of an inertial mass loosely coupled to the sensor housing ; (2) the strainmeter, which measures the difference between displacements at two distinct points ; and (3) the pressure sensor. A possible fourth device can be developed, at least in principle, using a gyroscope suspended in a balanced, frictionless gimbal system. Conservation of angular momentum of such a system offers the potential for measuring the rigid body rotation on top of displacement. Such devices are in common use in inertial navigation. If their state-of-the-art sensitivity and stability can be improved, they could potentially be adapted to seismological applications and could provide more complete descriptions of the earths ground motions. Although the sensor is the most crucial element, a complete seismic monitoring system today consists of five parts : the sensor, the signalconditioning device, the recorder, the timing device, and the seismic data telemetry link. To provide better areal coverage with an accurate common time base and to improve operational economy, seismic stations are interconnected through telemetry Iinks to form regional networks. Depending on the type of application, a multitude of designs for all these five parts are in current use. Detailed discussions of the designs and applications will be given in the following sections.

2. Historical DevelopmentSeismology is a young science. Yet the design and construction of earthquake-detecting devices can be traced to as early as A.D. 132, when a Chinese astronomer and mathematician, Chang Hang, invented a machine that would register the occurrence of an earthquake and give the approximate direction of the wave approach. It is quite an ingenious device, making use of mass inertia as the triggering mechanism. Once triggered, the instrument would cause the release of a copper ball held in the mouth of one of eight dragon heads arranged in eight equal azimuthal directions. The released ball would drop into the open mouth of one of eight waiting toads below, thus also giving the direction of wave approach. In the early 18th century, a French device consisting of a bowl of mercury with small holes drilled on the rim would indicate an earthquake and its direction by mercury overflowing through the holes during the passage of earthquake waves. The first European pendulum-based measuring devices that recorded the strengths of earthquakes came into use in the mid-1700s. A suspended mass, such as a

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simple pendulum, remains momentarily stationary as the earth quakes. A track record can be obtained on a thin sand layer or on smoked glass that gives the relative motion between the pendulum and the earth. These devices, called seismoscopes, have since been reinvented many times in different parts of the world. An obvious improvement is to have the instrument make a mark on a rotating drum to record the time of earthquake occurrence. An instrument that writes the earthquake motion as a time function is called a seismograph; it was formally developed in about 1880 in Japan by a group of British workers-Gray, Milne, and Ewing. A record giving the time function of the earths ground motions is called a seismogram. It is the study of seismograms that has established seismology as a branch of quantitative physical science. Since the first introduction of seismographs, the progress in seismic instrumentation has by and large followed the advances of technology. With the establishment of new and better seismic stations, the progress of observational seismology has been punctuated by milestone findings. Early examples include the 1889 first identification of a distant earthquake on a seismogram (an instrument in Potsdam, Germany, recorded an earthquake in Japan), the 1909 discovery of the base of the continental crust, now known as the Moho, the 1913 determination of the depth of the earths core, and the 1922 discovery of deep-focus earthquakes. The rate of new findings accelerates as more and better instruments are being deployed. T o follow the developmental history of seismic instrumentation, one must also take note of the worldwide deployment programs of seismograph networks-a special feature that makes seismology a truly international science. A seismograph system generally consists of four parts: the sensor, the signal-conditioning device, the clock, and the recorder. If a number of seismic stations are interconnected for central recording with a common time base, a fifth part-the telemetry links-will also be an integral component of the overall seismic network. Of the five parts that generally form a seismic network, the sensor is a component unique in seismic instrumentation. We will devote more discussion to its design principles and its evolution with time. Development of the other four components, which are common to many other instrument systems, followed closely the progress of technology. Taking the timing device, for instance, at the turn of the century, when the seismograph first came into common use, it was difficult to keep the daily drift rate of the mechanical clock of a seismograph system to within seconds. Synchronization of clocks among distance seismic stations was an impossible task. This difficulty was not alleviated until the 1940s, when electric clocks were incorporated in seismograph systems together with standard time radio receivers that allowed synchronization with the worldwide standard time broadcasting

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service that was then just beginning. This improvement permitted timekeeping in a seismograph station to be within a fraction of a second. As clocks with quartz oscillators (in a constant-temperature oven) came into common use in the 1960s, satellite broadcasting of standard time commenced in the late 1970s, and continuous synchronization became practical with modern electronic devices, the timing problem was essentially solved in seismograph operation with millisecond accuracy. Early seismographs made use of a rotating drum as a recorder. A sheet of smoked paper was wrapped around the drum. As the drum rotated, it also translated slowly, allowing a very fine helicoidal line to be scratched on the smoked paper by a pen attached to the inertial mass-the pendulum. Timing minute marks appeared as square pulses on the signal trace. A whole days record would register on one sheet of paper with a time scale of 30-60 mm to the minute. These smoked-paper recorders later evolved into other types that make use of an ink pen on ordinary paper or a light beam on photosensitive paper. The light beam device was able to increase the instrument magnification and remove the trace curvature on the recorded waveforms because of the short arm length of the recording pen. As technology progressed, the use of analog tape recorders together with multipen chart recorder playback greatly increased the recording capacity. In normal seismic network operation, the highest frequency t o be recorded is about 25-30 Hz. It was then possible to multiplex many signal channels (usually eight) on each recording tape track. A 1-inch-wide tape with 14 recording tracks, for example, has the capacity to record 112 seismic signal channels. This large capacity plus the advant of signal conditioning and telemetry electronics brought about in early 1960s the current mode of seismic network operation that combines central recording with data telemetry. These analog tape units require one long tape (7200 ft) per day continuously running at a low speed (15/16 ips). Normally, 99% of the tape content gives non-earthquake-related signals or background noise. Therefore, as digital computers come to common use, many seismic networks are converted to digital recordings. With a signal-discriminating device (usually software in the computer CPU, or a bank of microprocessors, one for each input signal channel) it is possible to record only seismic events, thus greatly compressing the tape contents. The digital tape of seismic events lends itself conveniently to downstream computer data processing pertinent to the seismic events. Although computerbased seismic recording is the current state of the art, drum recorders are still in common use, if sometimes only as a supplement. The visible helicoidal paper records, after all, provide a good means for quick diagnosis of network performance as well as an up-to-the-minute summary on seismicity. The signal-conditioning device was not part of the system when the seismography was first introduced, for then the differential motions between

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the pendulum and the recorder were magnified mechanically through level arm arrangements. The device became desirable in the 1900s, when electromagnetic transducers were introduced that converted penduludmotions into electric signals. These signals would have to be amplified and filtered before they could be applied to recorders. With the progress of electronics, the signal-conditioning device became increasingly complex and advanced to meet various applications. Up to the 1950s, a good deal of work was devoted to the design of stable long-period galvanometers that could be coupled with the electromagnetic seismographs. As low-noise, high-impedance preamplifiers became available, galvanometers were replaced by electronic amplifiers and filters, which not only eliminated the need for photographic recording as required by the galvanometer, but also gave high amplification and broadband performance. At the same time, high magnification permitted the seismometer to become much smaller. The low power consumption of these new signal-conditioning devices permitted their operation on DC power, and this has brought about unattended telemetered seismic stations. Telemetry is the latest entry of the seismograph system. The need for large areal coverage prompted a rapid increase in the number of seismic stations. For economical operation, telecommunication technology was first applied to seismic monitoring in the early 1960s. Through either telephone or radio transmission links, signals from seismometers, after preliminary conditioning, were used to modulate bands of carrier frequencies, then multiplexed and transmitted through the telemetry link. At the recording end (usually the observatory), the received signals were demultiplexed and further conditioned before being applied to recording devices. Today, a telemetered seismic network may consist of as many as several hundred seismic stations, with all signals centrally recorded on a common time base. Since field stations do not depend on AC power and are unattended, with infrequent maintenance visits, it is much easier to site stations in remote and strategic locations with a quiet background, resulting in networks of high sensitivity and good coverage. Lately, satellite telemetry is being tested on a small scale. The successful application plus a cost reduction in satellite transmission will lead to the establishment of a global digital seismic network. This would extend the current telemetered seismic networks from a regional to a global scale, significantly increase the network aperture. Finally, we will give a historical account of the development of the part unique to a seismograph-the pendulum. A simple pendulum has a natural period proportional to the square root of the pendulum length divided by gravitational acceleration. To sense short-period (e.g., with a pendulum natural period To = 1 sec) ground motions, a pendulum length of 25 cm will suffice. However, for picking up long-period surface waves (e.g., 6 = 20 sec), it is entirely impractical to erect a 100-m-high pendulum. This

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severe limitation prompted the search for a pendulum that could be operated at a long period with stability. The limitation was overcome with a garden gate-type horizontal pendulum, whose natural period was extended by tilting the hinge axis at a small angle 8 from the vertical, so that only a small fraction of the gravitational restoring force (g sin 8)was used. As 6 is made small, the natural period becomes long. A second design was the inverted pendulum shaped like a freestanding top-an unstable system. The inverted pendulum was stabilized by supporting springs attached to the instrument frame. With appropriate adjustments, the natural period of the inverted pendulum could be lengthened to seven to eight times that of simple pendulum of equivalent length. Besides the period problem, the sensitivity of these pendulum sensors was also a problem, because at the turn of the century instrument magnification could be achieved only by lengthening the mechanical level arm that held the recording pen. To increase the instrumental magnification, a larger pendulum mass was required to overcome the friction of the pen at the end of the long level arm. At the turn of the century, the German scientist Wiechert was building larger pendulums, with masses from 100 to 1000 kg. The heaviest one, which was built in 1906, weighed 17 tons and realized a maximum magnification of 2 x lo3. In the same year the Russian scientist Galitzin developed an electromagnetic seismograph, which employed a coil as the pendulum mass surrounded by a permanent magnet affixed to the instrument frame. The electric signal generated by the relative motion between the magnet and the coil was used to drive a galvanometer. Optical registration of the galvanometer deflection provided the needed instrumental magnification. The problem of instrumental magnifications was basically solved, and Galitzin further showed that a pendulum mass of 7 kg was adequate. However, the total system performance became much more complex due t o the coupling of the transducer circuit with the galvanometer circuit. While the garden-gate-type pendulum answered the problem of a longperiod horizontal sensor, the problem of constructing a long-period vertical sensor was not solved until LaCoste introduced the zero-length-spring seismograph in 1934. This spring required a special method of winding that imparted a residual compressional stress in its natural state and would have reduced the spring to zero length if it could collapse on itself. Theoretically, this spring could achieve an infinite period when operated in a vertical vibration mode; it has been used typically in vertical seismometers that have a natural period of about 30 sec. With progress in the development of seismometers, the size of the sensors has become smaller. This has made practical the installation of sensors inside small-diameter (4-6 in.) boreholes. Downhole installation of seismic sensors has significantly reduced background noise and increased sensitivity. The

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small sensor size also made possible portable seismometers, ocean bottom seismometers, and seismometers deployed on the moon. The recent development of force-balanced accelerometers (FBAs) has resulted in smaller sensors with broader response bandwidths. Velocity and displacement signals can be obtained by simply integrating the FBA output. Rapid developments in digital electronic technology and the universal availability of standard time broadcasting have made possible the miniaturization of yesterdays seismological observatory into a briefcase-size package. The widespread availability of data telemetry and particularly, in the near future, satellite telemetry can potentially link together a seismic array of continental dimensions. Such large-aperture seismic arrays will be operated as inverted telescopes pointing into and imaging the earths interior. A new era of exploring the detailed properties of the earths interior is soon to begin. Parallel to instrumental developments are deployments of seismic networks. Although the first seismic network with some degree of worldwide distribution was established by John Milne in 1896, a network of real global distribution was not deployed until the early 1960s as part of the U.S. national effort to improve capability in detecting and identifying underground nuclear explosions. Known as the World-Wide Standardized Seismograph Network (WWSSN), it consists of stations which each have three-component longand short-period seismographs with uniform calibration, time synchronization, and means of data archiving and dissemination. The WWSSN today comprises 110 stations operating in 54 countries (Fig. l), and it has provided fundamental data for seismological research in the past two decades not only in the United States, but around the world. In 1973 the United States began the development and global deployment of 13 Seismic Research Observatories (SROs) that combined the new borehole seismometer with an advanced analog and digital recording system. The availability of high-quality digital data produced by the SROs has opened up exciting new directions and opportunities for seismological research utilizing the newly acquired digital computing power. The success of the SRO prompted the development in the late 1970s of a digital recorder that could be attached to thus upgraded existing WWSSN systems. Seventeen such recorders are being installed at WWSSN stations (termed DWWSSN). Moreover, an early version of five high-gain long-period (HGLP) seismographs installed in the late 1960s has been modified by using more advanced computer-controlled digital recording. These five stations are called Abbreviated Seismic Research Observatories (ASROs). Today, SRO, ASRO, and DWWSSN stations together form the Global Digital Seismic Network (GDSN) shown in Fig. 2, which provides the bulk of digital data for seismological research. Another modern development is the installation of special seismographs to record ultra-long-period surface waves and the free oscillation of the earth out t o a period approaching DC.

m

1 FIG. 1. Map showing the distribution of WWSSN stations.

10Y

0 INSILLLED

LOYC-PERIOD (HGL?) SlhlIOYf ILLWID)

FIG. Map showing the distribution of GDSN stations. 2.

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There are over 20 ultra-long-period observatories around the world. Instruments consist of strainmeters (quartz tube type or carbon fiber type) and gravimetric accelerometers. A detailed discussion of these ultra-longperiod instruments can be found in a recent review paper (Agnew, 1986). To improve the capability for detecting weak signals from nuclear explosions in the midst of microseisms, seismograph arrays have been installed by a number of countries. A notable example is the large-aperture seismic array (LASA) installed near Billings, Montana, in the mid-1960s by the Department of Defense. LASA consists of 525 linked seismometers grouped in 21 subarrays. In each subarray 25 seismometers are arranged in a hexagonal geometry. A schematic diagram is shown in Fig. 3. The array covers an area 200 km in diameter and is similar in operation to a radio telescope array except that it points downward. The advantages of array operation include timing synchronization, microseism suppression, and identification of the direction of energy approach. However, the array operation is very expensive, especially for data processing and archiving. During the past 20 years there have been about 20 arrays in operation ;today only about 5 large seismic arrays are still active.N

FIG. Schematic diagram of the LASA array. [Reprinted with permission from Aki, K . , and 3. Richardson, P. (1980). Methods o f Quantitative Seismology. W. H. Freeman and Company, San Francisco, California. Copyright 0 1980.1

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11

To monitor regional seismic activity for earthquake hazard studies, a large number of telemetered short-period microearthquake networks have been constructed. This activity began in the early 1960s, when data telemetry (both radio and telephone) became economically feasible. These networks are typically composed of single (vertical) component short-period band (1 -20 Hz) sensors with their outputs telemetered to a central recording facility, where timing is supplied. Some large networks may consist of several hundred stations covering areas of hundreds of thousands of square kilometres ; the smaller networks may have only a dozen stations over an area of 100 km. High gain is of primary concern for these networks rather than high fidelity of reproduction of actual ground motions. The principal objective of these networks is to locate the hypocenters and determine the magnitudes and the fault plane solutions. In California alone, there are probably more than 500 such stations. To avoid excessive cultural noise in areas where stations must be set up to provide uniform coverage, downhole installations are not uncommon. For coastal areas, ocean bottom installations sometime become desirable to pinpoint offshore events. Finally, there is another type of seismic station for the recording of nearfield strong ground accelerations. These instruments are usually referred to as strong motion accelerographs ;they are inactive until triggered into motion by a preset level of ground acceleration (usually 1070 of a g. These accelero) graphs can also be linked to form an array, but most of them are operated independently with internal relative timing. The data output is invaluable for engineers engaging in earthquake-resistant designs.

3. Nature of Seismic Ground MotionsA seismic disturbance (earthquake or explosion) excites elastic waves, which propagate in the form of body waves and surface waves. For a very large earthquake, these propagating waves interfere with each other to form standing waves known as the earths free oscillations. From the analysis of these waves, we have derived information which constitutes our present basic understanding of the earths interior as well as the nature of the seismic source. Observations of seismic ground motions include the waveforms of particle acceleration, velocity, displacement, and strain (or the spatial derivative of the displacement field). A special feature of seismic observations is the signals broad range in both amplitude and frequency. These broad ranges typically span six orders of magnitude. This large variation not only is a result of the great difference in energy content between earthquakes of different magnitudes, but also strongly depends on the epicentral distance. The most prominent signals recorded by a standard long-period seismograph

12

TA-LIANG TENG

for a distant shallow earthquake are surface waves with a period of about 20 sec. Shorter-period surface waves tend to suffer from scattering due to shallow heterogeneities, and longer-period surface waves more easily lose energy into the asthenosphere. Nevertheless, surface waves give rise to signals of much larger amplitude in a seismogram than do body waves, mainly because body waves suffer stronger amplitude drop-off due to spreading of the wavefront. A small (Ms 3) earthquake will show a surface-wave amplitude of 100 mp (10-5 cm) at an epicentral distance A = 20" and 10 m p at A = SO", whereas the largest earthquake will show an amplitude of several centimeters at A = 20" and several millimeters at A = 80". Only to record surface waves over these distance and magnitude ranges requires an instrumental dynamic range of at least 120 dB. For observation very close t o the epicenter, an additional 40dB in instrumental dynamic range would be necessary for an adequate recording of the displacement field, which can reach a maximum amplitude of several meters in close-in range for very large earthquakes. Body waves usually have shorter periods ;their wave amplitude drops off faster with distance due to both a stronger wavefront spreading factor and a heavier attenuation effect, which increases rapidly with the wave frequency. At a large distance, the displacement amplitude of body waves can be one to two orders of magnitude smaller than that of surface waves. In the absence of background noises, an ideal seismic instrument for recording all waveforms over the entire distance range would require a dynamic range of nearly 200 dB. This stringent requirement imposes great difficulty on various aspects of the seismic instrument design, from the sensor, the amplifier, and the telemetry electronics to the recording device. In terms of acceleration, the corresponding required sensitivity is about lo-'' g , and the corresponding strain is about This resolution is very difficult to achieve, and the always present seismic background noise prevents the detection of these small signals. In fact, the ambient seismic noise level on the earth's surface is generally several orders of magnitude higher than this minimum resolution and is both frequency- and site-dependent. Sources of the seismic noise include meteorological effects, ocean wave motions, industrial activities, and traffic. The effect of the ocean wave motions is particularly pronounced and persistent. An analysis of the noise power spectral density shows a generalized picture such as that in Fig. 4 with two noise peaks at about 0.07 and 0.14 Hz. The latter is much stronger, and both can be traced to an origin related to ocean wave motions. The level of the ambient noise is also site-dependent ; it can differ by almost two orders of magnitude between a quiet and a noisy site. The main noise peak at 0.14 Hz cm for a noisy site and has an equivalent displacement of about cm for a quiet site. This lessens somewhat the above stringent requirement on the dynamic range. It would therefore be sufficient for the instrument to

-

FIG.4. Generalized power spectrum density of seismic background noises (after Melton, 1976). Earth noise is plotted as squared acceleration per millihertz. The Queen Creek spectrum is shown with two branches, the lower branch (thin line) representing the noise after substraction of the measured instrumental noise included in the upper branch. Elsewhere along this curve the instrumental noise is a negligible portion of the energy represented. The Camp Elliott curve represents data from a Southern California site about 20 km from the West Coast. Thermal acceleration energy for several assumed seismometers is shown by the horizontal solid lines, and a proposed frequency plot of an earthquake with a surface wave magnitude of 3 at 60" epicentral distance is included to show the relation between earthquake and noise energy.

14

TA-LIANG TENG

resolve a ground displacement of cm at that frequency. However, the Ievel of ambient noise tends to drop away from these two noise peaks. An ambient noise minimum occurs at about 30-40 sec. A nominal resolution of lo-' cm for long period is desirable for this period band. For the short-period band that is commonly dominated by microearthquake spectral energy over the frequency range from 1 t o 30 Hz, a desirable resolution is about cm. From low8to lo2cm, the amplitude of seismic signals still covers 10 orders of magnitudes or 200 dB in dynamic range. But for different applications of earthquake recording, it is general practice to design different instruments to cover different bands. To avoid the noise peaks at 0.07 and 0.14 Hz, seismic instruments are typically divided into two classes : long-period instruments with their amplitude response peaking at 30 sec, and short-period instruments with their amplitude response peaking at about 10 Hz. Each class of instrument is generally operated at the maximum gain permitted by the ambient noise of the site, and a system dynamic range of about 120 dB is quite adequate. Even this reduced 120-dB dynamic range has been achievable only with the advent of digital computers. For the recording of very long-period earthquake motions such as the free oscillations of the earth as a consequence of large earthquakes, ultra-long-period instruments are designed. They require a strain sensitivity of about lo-" or a corresponding 10-9g in acceleration. However, the amplitude of the earth's free oscillations typically spans four to five orders of magnitude, and a recording device of 80 to 100 dB is adequate.

4. Basic Types o Seismic Sensors fA distinct feature of the measurement of seismic ground motions is the absence of a stationary reference frame on which the sensing device can be placed. In other words, the observer moves with the object during an earthquake, which makes careful measurement of the trajectory of motion of the object difficult. An effective and practical method for overcoming this difficulty is t o make use of an inertial mass that is loosely coupled t o the moving frame (the earth) through certain pendulum arrangements. Therefore, a discussion of seismic sensors is basically an analysis of the motion of a damped pendulum. Since seismic ground motions are a vector field, this analysis should involve the equation of motion of both vertical and horizontal pendulums. Besides pendulum seismometers, other sensors have also been developed that measure the derivatives of the displacement field such as strain and pressure. Strainmeters, or strain seismometers, measure the spatial derivatives of the displacement. Quartz tube strainmeters, carbon fiber strainmeters, and various tiltmeters all belong to this category, which

10. SEISMIC INSTRUMENTATION

15

makes no use of the inertial property of a pendulum. Their response to very low frequency (down to DC) motions makes them very useful in the study of the earths free oscillations. Another category consists of hydrophones, which measure the pressure field (in fluid) induced by seismic ground motions. Their sensing elements are typically piezoelectric pressure transducers. Principal applications of hydrophones are for seismic exploration work in water-covered areas. Some downhole seismometers use hydrophones as sensors to relax the requirement of locking the downhole package against the hole. There is one last method that, at least theoretically, can be used to measure the vector field of seismic ground motions. Based on the conservation of angular momentum, it is possible to use a gyroscope as a stationary point against which the rigid-body rotation of a point on the earth can be measured. Gyroscope-based systems are available for navigation applications as well as for downhole orientation in drilling. The state-of-the-art sensitivity and stability are not yet good enough for measurement of seismic ground motions. As mentioned before, a pendulum can provide an inertial mass for the needed stationary reference frame that is loosely coupled to the earth. Ground motions are thus measured with respect to this simulated reference frame. To adequately record seismic ground motions, an important and continuing effort has been the design and operation of stable long-period pendulums with minimum cross-axis coupling and parasitic vibrations. Since the ground motions are a vector field, both vertical and horizontal pendulums are used. A brief recapitulation of the analysis of pendulum motion is given here together with a discussion on period-lengthening measures commonly employed for stable long-period pendulums.4.1. Vertical Pendulums

The simplest way to create a stationary reference frame for vertical motions is a vertical spring with a suspended mass M (Fig. 5a). Let LO the be length of the unstressed spring and L the length of the spring under stress. Then the equilibrium condition requiresMg = k(L - Lo)

where g is the gravitational acceleration and k the spring constant. The equation of motion for the rectilinear vertical motion of the mass M isM d 2 y / d t 2= - k [ ( L

+ y ) - Lo] + Mg = - [Mg/(L - Lo)]y

(1)

or

16Earth

TA-LIANG TENG

Earth

A

0

Mg(a 1

(b)

FIG. 5 . (a) Simple vertical pendulum consisting of a mass and a spring. (b) Improved suspension for period lengthening and cross-axis coupling reduction.

where o1is the angular frequency of the simple harmonic motion0 1

= 2 7l = 7f

J g / ( L - Lo)

(3)

Thus the pendulum period is=

~ z J ( L Lo)/g -

(4)

Clearly, with constant g the pendulum period depends on L , which physically has a practical limit. A natural period longer than 1 sec is difficult to realize. Also, this simple suspension lends itself easily to a nonlinear effect caused by cross-axis coupling ; that is, horizontal ground motions can contaminate the rectilinear motions in the vertical direction. A method of lengthening the pendulum period and reducing the degree of cross-axis coupling can be achieved by a new suspension shown in Fig. 5b. Here the equilibrium condition requires :MRog = ka(L - LO)(5)

where ROis the distance from the center of gravity of the pendulum t o the turning axis, and a is the connecting point of the spring and boom to the turning axis. With a small angular motion 8, the equation of motion of this system isj l d28/dt2 5:

- ka[(L + ad) - Lo] + MRog cos 8+ ae) - Lo] + MRog

= -ka[(L

(6)

where jl is the moment of inertia of A4 about 0.

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SEISMIC INSTRUMENTATION

17

Putting Eq. (5) into Eq. (6), we getj1 d28/dt2

+ ka28 = 0

(7)

or

Thus the angular frequency of the pendulum is0 1

=

2nf1 =

a

(9)

and the corresponding period is

7 = 27rdjZ2 iNeglecting the mass of the boom, j~ = MR;, givingT I

=

2

n

m = 2 n 4 L - Lo)/gJRo/a

(1 1)

4, Comparing Eq. (1 1) with Eq. ( ) one finds that this new suspension can realize a lengthening of the pendulum period by a factor of (Ro/a). For RO = 4a, the original period can be lengthened by a factor of 2 . However, mechanically, Ro cannot be much larger than a, which also imposes a practical limit to this approach. An improved approach is to place the point B, which joins the spring with the boom, below the line connecting the mass M and the pivot 0 as shown in Fig. 6 . For this system the equilibrium condition isMRog = kr(L - lo)

(12)

Here, again, Lo is the length of the unstressed spring, L the length of the spring as the mass Mis at the equilibrium condition, and r the distance from the pivot point 0 to the spring. A small deflection of the boom by an angle t is described by the equation of motion: 9jl

d28/dt2 = - MRog cos 8

+ k(L -

LO)^'

(13)

The primed values correspond to those for the disturbed state. Combining Eqs. (12) and (13) gives

(14) Setting up rectangular coordinates centered at 0, the point of spring suspension A is (x, y ) ; the point B is (XI, y l ) , which reduces to (a, 0) at the equilibrium position. Expanding r and r in Taylor series for small 8 and keeping terms up to 02, we havej~ d 2 8 / d t 2

j1 d26/dt2= - k(L - L O ) ~ C6 S k(L - LO)^' O

+

+ kax[l - L o / L + Loay2/xL3]6 + (tka2yLo(a- X)[X(X - a) + y 2 ] / L 5 ) 8= 0 2

(15)

18

TA-LIANG TENG

Y

X

FIG.6. Further improvement on suspension for a linear vertical pendulum.

The term with 8 in Eq. (15) causes a departure of the pendulum from the simple harmonic motion. Since O2 is always positive, during the oscillatory motion, the mass will spend more time below the equilibrium position than above it. However, the O2 term will vanish if any one of the following three conditions exists. (1) x = a. This reduces the pendulum suspension to the one shown in Fig. 5b. Equation (15) collapses back to Eq. (7) and the pendulum period given in Eq. (10) results. Thus the elimination of the e2term by setting x = a does not bring about further improvement in period lengthening. (2) LO = 0. This is the condition of a spring of zero initial length. When a spring is stretched, the applied load f is proportional to the elongation ( L - LO),where L is the actual length and LOis known as the initial length. Theoretically, LO = 0 implies that this spring will collapse to zero length if the size of the coils vanishes such that a residual tension can no longer be supported by the coils. For LO = 0, Eq. (15) reduces toj l d2B/dt2

+ kaxe = 0=

(16)

or d28/dt2+ with0 1

0

(17)

= 2Xfi =

10. SEISMIC INSTRUMENTATION

19

and

TI

=

2ndjzZ

Here, as x = 0, that is, placing the supporting point A on they axis (Fig. 6), the pendulum period 2i approaches infinity. At equilibrium, the angle AOB = n / 2 and the equilibrium condition gives

MRog = kLr = kayyielding

(20)

y = MRog/kaPutting Eq. (21) into Eq. (19), we have

(21)

TI

= 2 rx 7-

=

2nd10tan(cu

+ p)/g

(22)

where 10 is the equivalent pendulum length of the physical pendulum and angles CY and pare shown in Fig. 7. It is interesting to note that by maintaining the condition (21), one can adjust the pendulum period by changing the position o f x .Figure 8 illustrates a number of possible spring suspensions that result in varying degrees of pendulum period lengthening possibilities as dictated by the geometry. (3) x(x - a)2 + y2 = 0. For fixed Q , this is an equation of a circle centered at ( a / 2 , 0 )with radius a / 2 . This condition gives the locus of the spring suspension point A, which must fall on the above circle. The geometry is shown in Fig. 9. This, again, reduces Eq. (15) to Eq. (16) and leads to theY

FIG. 7. Angular geometry of suspension for a linear vertical pendulum.

20

TA-LIANG TENG

FIG. 8 . Various spring suspensions resulting in varying degrees of pendulum period lengthening for a linear vertical pendulum.Y

0

FIG,9. Suspension that gives a linear vertical pendulum only at the equilibrium condition.

same result as for the case of a zero-initial-length spring. However, this result can be achieved only when the pendulum is at the equilibrium position; the effect of the O2 term will again appear during dynamic vibrations. Comparing the above three conditions, the case LO = 0 (zero-initial-length spring) is the only one that can both remove the nonharmonic oscillation and

10.

SEISMIC INSTRUMENTATION

21

effectively extend the period of the pendulum. Since the sensitivity of a pendulum to acceleration at low frequency is proportional to the square of the pendulum period, finding a stable long-period vertical pendulum was the most important problem in instrumental seismology for many years. This problem was not solved until LaCoste and Romberg (1942) invented the zero-initial-length spring in 1935. LaCostes spring can theoretically achieve an infinite period,* but in actual applications, long-term stability can be achieved only at a period up to about 30 sec. The zero-initial-length spring must be wound with a twist applied to the wire as it is coiled; this plus the stringent requirement on the material properties makes its production nontrivial. Press et a/. (1948) were the first to introduce the zero-initial-length spring in the long-period vertical siesmometer design, which later became the backbone of the World-Wide Standardized Seismograph Network (WWSSN). A common problem associated with the vertical pendulum is the temperature dependence of the spring constant k, which, in turn, causes the pendulum to drift from its equilibrium position. If 4 is the pendulum drift angle, the drift period T ( 4 )can be approximated byTI(+)= Tldcos 4 - B sin 4

where B = 4n2/0/(7i2g). Thus, an upward drift of the pendulum (4 > 0) tends to reduce the pendulum period, and a downward drift increases it. Therefore, it is common practice to choose alloys with low temperature coefficients and control the temperature inside the instrument housing in order to stabilize the long-period vertical pendulum. In the area of period lengthening, much recent research has been done using electronic feedback and compensation circuits. These measures are generally referred to as the force-balanced approach and have achieved varying degrees of success ;some have realized a pendulum period up to several hundred seconds and a sensor response close to DC. A discussion of the force-balanced sensors will be given in a later section.4.2. Horizontal Pendulums

The simplest horizontal pendulum consists of a mass M suspended by a string from a point 0 on the frame (Fig. 10a). The pendulum can couple to the frame through a leaf spring (Fig. lob). Again, if the moment of inertia of the pendulum about the point 0 isjl and the distance between 0 and the* In practice, this is unattainable because of inexact spring length and hinge positions, finite restoring force of the hinge, temperature dependence of k, variation of gravity, and other disturbances. The maximum period maintained in a routine observation was 80 sec, by Francis Lehner of the Seismological Laboratory of the California Institute of Technology.

22

TA-LIANG TENG

F I G . 10. (a) Simple horizontal pendulum. (b) Simple horizontal pendulum with a leaf spring hinge suspension. (c) Equivalent pendulum length of a physical pendulum.

1=

Earth

Earth

center of gravity of the pendulum is R o , a small deflection B of the pendulum will excite a simple harmonic motion described byj , d2B/dt2= - M R o g sin B

(24)

For a small deflection, sin 8 = 0 , we haved28/dt2 + ( M R o g / j , ) 8= 0

(25)(26)

This leads to the same equation as Eq. ( 2 ) , givingw1 = 2nf, =

andT, =-2 n

2nJl/g

(27)

where10= jl/MRo

is the equivalent pendulum length as described in Fig. 1Oc. It can be shown that 10 > R o . For a spring-coupled pendulum (Fig. lob), the right-hand side of Eq. (24) has an additional term - CB to account for the elastic restoring force, with C ( > O ) being the equivalent spring constant. The natural frequency for the spring-coupled pendulum is01

=

27rfi = Jg/Io

+ C/jl

(28)

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SEISMIC INSTRUMENTATION

23

0'

I

i

I

( C )

FIG.1 1 . (a) Garden gate suspension of a horizontal pendulum. (b) Modification of the garden gate suspension resulting in tensional hinge points. ( c ) Period lengthening of a garden gate suspension.

Therefore, a spring-coupled pendulum will result with an increased natural frequency or reduced pendulum period. From Eq. (27), one finds that a 20-sec pendulum period requires a suspension length of almost 100 m. Thus, a simple pendulum cannot be a practical device for long-period horizontal seismic sensors. However, a garden gate-type suspension (Fig. 1la) can provide stable long-period pendulums for a horizontal seismometer. The pendulum make use of a fraction of the restoring force (Mg sin i), where i is the angle between the support axis 0-0' and the vertical. For a small deflection 8, sin 8 = 9. We have the equation of motionj , d28/dt2 (MRog sin i)B = 0

+

(29)

The natural frequency of the pendulum isWI

=

2nf

=

J ~ M R ~i/j, = & G Z & sin

(30)

24

TA-LIANG TENG

or the pendulum periodT,=

2nJ10/(gsin i)

TI becomes large for small i, and approaches infinity as i approaches zero. Extending the support axis 0-0, will intersect at a point 0with a vertical it line passing through the center of the pendulum mass M(Fig. 1lc). One finds that the equivalent pendulum length 16 is

I6

= lo/sin

i.

As i becomes small, 16 becomes long. For i = 1 minute, I6 is about 3600 times longer than l o , which corresponds to a 60-fold pendulum period lengthening. A minor modification in suspension (Fig. l l b ) results in tension at both supporting points 0 and 0,thus reducing the friction at pivots such as may exist in the case shown in Fig. l l a . This type of long-period horizontal pendulum is commonly used in present-day seismometers, especially those of the WWSSN. The maximum stable period for a standard instrument is about 30 sec. A longer operating period is again limited by the nonnegligible temperature coefficients of materials used in the pendulum construction, as well as the constancy to which the small tilt angle i can be held. It may be noted in passing that inverted pendulums were used in designing horizontal seismometers that are still being operated in a small number of older seismic stations around the world. The mass M is supported by a leaf spring (Fig. 12a) or by a rod and coupled to the pivot 0 through a leaf spring (Fig. 12b). Since the system is unstable, two springs are needed to balance it.

Eorth(a)

Eorth

(b)

FIG. 12. (a) Inverted pendulum. (b) Inverted pendulum with a leaf spring hinge support.

10.

SEISMIC INSTRUMENTATION

25

If the spring constants of both springs are ko, then for a small deflection 8, sin B = 8 and the equation of motion isj l d2B/dt2 = (MRog - 2ka2)8

where a is the length between 0 and A. The pendulum period is

& = 2 d l o / g ( D - 1)where D = 2ka2/MRog. For D 5 1, the inverted pendulum loses its stability. For D -+ 1+, TI 03 ; this can never be achieved, however. In actual applications, the inverted pendulum is far less stable even though it can realize a period seven to eight times longer than that of a simple pendulum.-+

5. Damping Devices and TransducersTo provide meaningful ground motion data, the pendulum seismometer must give an output bearing a definite relationship to the input disturbance. For an undamped pendulum, it will oscillate freely and indefinitely when excited. For impulsive earthquake wave arrivals, the output signal will be an envelope of varying amplitude containing primarily oscillations at the free period of the pendulum, thus completely obscuring the real information and rendering the recording useless. To suppress the undesirable oscillation, damping is introduced. Either viscous damping or electromagnetic damping can be used. The latter is much more effective and easily adjustable, and it is not subject to the undesirable temperature dependence of a viscous fluid. Therefore, electromagnetic damping is in common use in almost all seismometer designs. A damping force is proportional to the angular velocity of the pendulum and in a direction opposite to the pendulum motion. Damping of instruments with velocity transducers is almost always accomplished by energy loss in the resistive elements of the output circuit. This may be a galvanometer circuit, the input resistance of an amplifier, or simply a resistive shunt.5.1. Motion of a Damped Pendulum

Let B I be a damping factor such that the damping moment is given by - B I 8. Incorporating this damping term in Eq. (8) or Eq. (17) gives

Bwhere~ E = I

+Z

E , ~

+ o:e = o

(32)

B l / j l and c1 is called the damping coefficient. Solutions of

26

TA-LIANG TENG

Eq. (32) with initial conditions

8 = B0

and

8 = O0

.

.

at t = 0

can take one of three forms:(1)with0 1

> El:

tan 4 = Bovl/(B0

+ E~ e0)

This is a damped oscillation with period T = 2n/v1 = Z/(1 - D?). As i in Eq. (10) or Eq. (19), Z = 2n/01 is the natural period of the undamped pendulum, and D 1= E I / W ~is commonly referred to as the damping constant. Since 01 > e ~DI < 1 gives the case of underdamped oscillation with , TI> Tl . The behavior of the pendulum is shown in Fig. 13. The damping ratio u is defined as = I ykl/lyk+ll eTel/vl = eTDI/(1--Df)l = (2) o1= E1(or D1 = 1):

8

=

[(I

+ mlt)e0 + &t]e-lf.

This marks the inception of a nonperiodic motion, and DI = 1 is referred to as the critical damping constant. (3) 01 < E1(or DI > 1):

8 = e-I(Cl sinh v1 twhere71 = ( E f

+ C2 cosh vl t )c2= eo

-

01)

2 1/2

c1= (eo + E1eo)/vlThis also gives a nonperiodic motion.

FIG. 13. Motions of a damped pendulum.

10.

SEISMIC INSTRUMENTATION

27

It therefore can be concluded that (a) as D I= 0, the pendulum performs an undamped oscillation, (b) as D I< 1, the pendulum performs an underdamped oscillations, (c) as D I = 1, the pendulum gives a critically damped nonperiodic motion, and (d) D I > 1, an overdamped oscillation results.5.2. Forced Oscillation of a Damped Pendulum

Assuming that the motion is restricted to the x direction, we denote the motion of the earth in the inertial reference frame as u(t). Then in view of Eq. (32), the equation of forced oscillation is where [ = lo0 for length. Equation (33) shows that a linear combination of [ ( t )and its time derivatives can reproduce the acceleration ii of the earths motion. If the earths motion has a characteristic frequency u, then(1) For u ol, the first term on the left-hand side of Eq. (33) dominates, and ( becomes nearly equal to - ii. Thus the damped pendulum essentially records the earths displacement. (2) For w 4 U I ,the last term on the left-hand side of Eq. (33) dominates, : and u [ approximates - ii. Thus the damped pendulum essentially records the earths acceleration.

g + 2E4 + u:[ = -ii small motion, and AI is the effective pendulum

(33)

The response of a damped pendulum to an arbitrary forcing function u ( f ) can be obtained by directly solving Eq. (33). However, the frequency response X ( w ) ,or the system response, of a damped pendulum can easily be obtained by considering a sinusoidal input u ( t ) = exp( - i d ) ; then the response [ ( t ) = X(u) exp( - i u t ) with

X(U) ( - u2)/(u22i.m = +X(W) = IX(0)Iexp[i4(w)l

-0 ) :

If we define the amplitude response IX(u)I and the phase delay $(u) by then

IX(U)( =and

UZ/[U2 -

Uh2 + 4 E 2u2 I 1/2

4(0)

= - tan-[2Eo/(u2 - a:)] n +

Figure 14 gives the amplitude response and phase delay over a range of damping constant D 1= 8/01 and frequency band. For u 0 1 , IX(u)I 4 1 and 4(u) + n ;the pendulum records the ground displacement correctly but with a reversed sign. For u 4 0 1 , there is no phase delay, but the pendulum

*

28

TA-LIANG TENG

0

1w,/w

2

3

-n

1

I

0

Iw1lw

2

FIG. 14. Amplitude response IX(w)l and phase delay +(w) of a pendulum seismometer. [Reprinted with permission from Aki, K., and Richards, P. (1980). Methods of Quantitative Seismology. W . H. Freeman and Company, San Francisco, California. Copyright 0 1980.1

has little or no sensitivity to ground displacement with a period much longer than the pendulum period. However, the pendulum becomes a good accelerometer. The response [(t) for an arbitrary ground displacement u ( t ) can be obtained by convolution :

[ ( t )=

:l

u(t - t)x(r)dr

where x ( t ) is the inverse Fourier transform of X(o).

5.3. Types of TransducersEarly seismometers are arranged for direct registration of the relative motion between the pendulum mass and the frame by either mechanical or

10.

SEISMIC INSTRUMENTATION

29

optical means. In the former method, a system of levers magnifies the motion of a very heavy mass and applies it to a pen or stylus in contact with the recording surface. In order to develop enough force to overcome the stylus friction and lever inertia, the pendulum mass may weigh several tons. Very few of these mechanical magnifying, direct-writing instruments are now in use. For optical magnifying systems, a mirror is usually coupled to the suspended system, and a light beam focused through the mirror onto a sheet of photosensitive paper wrapped over a revolving drum exposes a data trace which becomes visible when the sheet is processed. Since no work is done in using the light beam, the suspended mass may be much smaller. Modern seismometers make use of transducers that convert the pendulum motion into an electrical signal which can conveniently be manipulated (amplified, filtered, etc.) and applied to various recording media. By and large, there are two types of transducers : electromotive and parametric. The former include electromagnetic transducers, piezomagnetic transducers, and piezoelectric transducers. All of them can convert the mechanical motion directly into electromotive force. Parametric transducers instead make use of the effect of mechanical motion on an element of an external circuit which indirectly modifies its current. Capacitive transducers, inductive transducers, and resistive transducers belong to this category. Electromagnetic transducers are most commonly used in seismometer design ; they are called velocity transducers as their output is directly proportional to the relative velocity between the pendulum mass and the frame. Displacement or acceleration signals can be simply obtained by integrating or differentiating circuits, respectively. Piezoelectric or piezomagnetic transducers have their output proportional to the acceleration and therefore are acceleration transducers. Since the piezo effect makes no use of the inertia, these transducers can be made very small and are particularly useful for high-frequency work. They are also made into the sensing elements of hydrophones. However, their sensitivity is relatively low. Capacitive transducers make use of the hyperbolic relationship of the gap between plates and the resulting capacitance. As the gap is small, the relationship between the displacement gap and the capacitance approaches linearity and gives rise to highly sensitive displacement transducers. These transducers are commonly used for long-period work. Various modifications of circuits are used in designs for ultra-long-period strain and tilt measurements.5.4. Electromagnetic Transducer and Resistive Damping

The relative motion of a pendulum and the seismometer frame is most commonly measured by the electromagnetic transducer. Therefore the observed output voltage is a measure of the velocity. It can be either a moving

30

TA-LIANG TENG

Eanh

FIG. 15. Schematic system configuration of an electromagnetic transducer. [Reprinted with permission from Aki, K . , and Richards, P. (1980). Methods of Quantitative Seismology. W . H . Freeman and Company, San Francisco, California. Copyright 0 1980.

coil system in which a coil is attached to the pendulum mass and moves through a magnetic field, or a moving magnet system in which the pendulum mass is made into a magnet moving through a fixed coil. To generate sufficient output, the coil is made of a long conductor that has a finite resistance Ro. An adjustable shunt resistor R is commonly connected across the output terminals for damping. Following Aki and Richards (1980), Fig. 15 shows a schematic system configuration. Let 1 be the length of the core conductor within the magnetic field of flux density B , and assume that the directions of coil movement and magnetic field are mutually perpendicular. Then the force F necessary to drive the coil motion is

F = IlBwhere l i s the current generated in thecoil. The mechanical power produced is

10.

SEISMIC INSTRUMENTATION

31

The power must dissipate through the resistive elements of the circuit R + Ro; this gives

VZ = ZIBt

or

V = IBt

Writing G for IB, we find that V = G t and F = GI. Therefore G is called the electromotive constant and has units of volts per centimeter per second. We can solve for Z and F :Z = GO; and

'

+dF-2)2p' - 21 (- 1)' ( I h-r 7

(

)

(

- 2p' +2P

h = k , g'O; p' = p , q' = q for p s N 2 ;

hzk-q',

p' = 1 - p , q' = - q for p > 1/2

For small eccentricity e, useful forms are (Allan, 1967):1

GlP(-1)

+ l)e/2 = (- I + 4p + l)e/2= (31 - 4p

However, Colombo (1983) suggests that these are inadequate at very high degree I > 100. Again, FFT techniques are most efficient at these high degrees (Goad, 1987).

3.3. Forced PerturbationsPrecessions of the perigee and node at rates on the order of one revolution per 3 months can be taken as observed. These arise from the term lmpq = 2010 in Eq. (27),used in Eq. ( 5 ) with Eq. (15) for the Poisson brackets [a, el, [a, and [a, (K39): I], I]d a / d t = 3nNzo(1

- 5 cos2I)(ae/a)2C2~/4(1 e2)2 -

(30) Higher even zonal harmonics (I even, m = 0) also make a contribution a factor of smaller to these secular motions. Sinusoidal oscillations of about 3-month period arise from zonal harmonics of odd degree due to the motion of the perigee with respect to latitude. These effects are generated

dWdt = 3nN20(cosZ)(a,/a)zC2~/2(1 e2)2 -

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WILLIAM M. KAULA

chiefly by terms in Eq. (27) with I - 2p = - q = f 1 . The most prominent effects generated by tesseral harmonics, m # 0, have rates of about m cycles per day arising from terms with I - 2p + q = 0 in Eq. (27). To calculate the amplitudes of these perturbations, normally it suffices to take (I, e, Z constant and n, w , A secularly changing on the right of the 4 equations of motion, Eq. (5). For example, the rate of change of the inclination obtained by using Eq. (27) in Eq. (5) is(dI1dt)lmpq

=

i - 2p)[I, 01 + mIZ, a l 1 R l m p q W

(31)

The assumption of secular change then leads to an integral with respect to time, using Eq. (15), of (K40):AZ/mpq

= ((1 - ~P)[Z, 1 0

+ m[Z,o l l R / m p q / V / / m p q = [(I - 2p) cos I - m ] ( ( ~ J a ) ' A , m f i m p G / p q x e~p(iy/lm,,)n/V//mpq(~(l e2)1'2 sin I -

(32) (33)

where the rate in the denominator istj/mpq

= ( I - 2p)ci,

+ ( I - 2p + q)ni + m(hz - 4)

Perturbations by all the harmonics of all the Kepler elements can be calculated in a similar manner (K40). In cases of very small eccentricity, numerical difficulties can be avoided by using in place of the elements e, o the variablesP = ecosoQ = esino

(34)

Substitution of AP = Ae cos o - e A o sin o and so forth leads to forms in which terms that have divisors going to zero have zero numerators. As a consequence of the linear perturbations described above, a particular term in the spherical harmonic expansion of the gravity field can be considered to generate a spectrum of variations in a nearly circular satellite orbit which have frequencies that are different combinations of the four rates &, 0, 6 : one generated by the central term of the earth's gravity, two by the h, oblateness, and one by the earth's rotation. For a term with I even, there will be an m cyclejday term corresponding t o p = 112, plus I terms with rates of about I - 2p cycles per orbit. Terms dependent on q = f 1 will exist, but have high rates. For a term with I odd, there will be no m cyclelday term for q = 0, but there can be significant terms of q = i;that is, skew symmetry 1 in the field will force an eccentricity. Since the effects of interest are of order or less, in an analytic calculation of the orbit it is necessary to take into account the nonlinear perturbations by the oblateness CZOsince it is of order , In the decade 1957-1967, an extraordinary number of papers were published on this

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problem. The options which fostered this abundance included : (1) definition of intermediary : there are several refinements to the secularly precessing Kepler ellipse taken empirically here ; (2) independent variable :longitude of intermediary and so forth, rather than time; (3) choice of instantaneous elements: Kepler, Delaunay, Hill, and so forth; (4) means of carrying out nonlinear interactions : simple Taylor expansion, Von Zeipel transformation, Lie transformation, numerical iteration. Probably the most effective theories are those which use an intermediary entailing a linear C20 departure from the simple Kepler ellipse, employ Hill elements or P,Q [Eq. (34)] as the final variables (to avoid difficulty with zero eccentricity), and perform the nonlinear interactions by applying Lie transformations to the Poisson brackets (Aksnes, 1970; Kinoshita, 1977). The principal defect of these algorithms for gravity field determination is thought to be interaction between C 0 and tesseral harmonics (Lambeck and Coleman, 1983). 23.4. Resonance

The procedure outlined above can break down in cases where Glmpq = 0. In general, for any orbit with v revolutions per day, there will exist harmonics generating perturbations with rates less than a half-cycle per day, arising from m = v, 2v, 3v, ... .Since v is necessarily less than 17 [a good enough rule is 17(ae/~)~~] 1 1 m,these normally will be rather small terms and the and effects can be treated as small divisor terms rather than true resonances, since i///mpq goes through full cycles, 0 to 27r. Some systematic attempts have been made to select sets of satellites of 16, 15, 14, 13, ... cycles per day and analyze them forterms of m = 16,32, 15,30, ...order, but they havenot beenmajor contributions to the overall determination of the field. On the other hand, in comprehensive analyses of the field, higher-degreeterms for the orders m corresponding to low rates for satellites in the set used will often be included. One case of true resonance which does often occur arises fromgeostationary satellites placed in orbit for the purpose of communication, meteorology, or other surveillance. Since these have d u e of 6.6 from the rule of thumb given above, they are sensitive mainly to the I, m = 2,2 harmonic, corresponding to an ellipticity of the equator. The most useful quantity based on observations is an acceleration along track arising from changes in the energy: that is, in the semimajor axis. From Eqs. (5), (lo), (15), and (27) (KSl),

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WILLIAM M. KAULA

A pendulumlike solution of Eq. (39, resulting in a libration about 1 - A22 = d 2 with a period proportionate to 1/Jj12can be made (K50-53), but information about the gravity field is actually obtained by comparing observed accelerations to Eq. (35) extended to include higher-degree terms (Wagner, 1983).3.5. Miscellaneous Effects

The most comprehensive analyses (Lerch et al. , 1974; Gaposchkin, 1980) take into account (1) atmospheric drag, using an idealized atmospheric model (Jacchia, 1965);(2) radiation pressure from the sun ;(3) lunar and solar direct attractions; and (4)oceanic and solid planet tides. All these effects cause smaller accelerations on a close geodetic satellite than the variations of the gravitational field. However, they build up to larger displacements because they are nonconservative or long-periodic. Hence it is necessary to take them into account in classical satellite geodesy, in which the distribution of observations is nonuniform.3.6. Numerical Integration

The major determinations of the gravitational field now all utilize numerical integration of the orbits and the partial derivatives with respect to parameters, since the termination of the effort at the Smithsonian Astrophysical Observatory in 1981.Examples of numerical integration systems are the Jet Propulsion Laboratory (JPL) Development Ephemeris (Devine, 1967) and the NASA/GSFC GEODYN (Martin et al., 1972). In the GEODYN system, position is obtained by a second-order Cowell predictor-corrector scheme, while velocity is obtained by a first-order Adams-Moulton predictor-corrector. These programs are quite reliable but lavish in use of computer time.

4. Data Analysis4.1. General Considerations

In classical satellite geodesy-category 1 of Section I-fairly elaborate corrections of the observations are needed because of the variations in direction and rate of the earths rotation, atmospheric refraction, and aberration (Kaula, 1966a, K82-K86; Lerch et a/., 1974). Like the miscellaneous perturbations of the orbit, they are significant mainly because of the nonuniform distribution of observations. This nonuniformity affects the determination of the gravity field because the spectrum of effects given in Eq. (33) is generated by the same rotation that affects the observability by

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179

a particular tracking station. A satellite is observable when its earth-referred latitude and longitude $ and 1 are close to those of the tracking station. But from the spherical triangle formed by the orbit, the equator, and the meridian for 1,

o

+ f = sin-(sin $/sin I)

(36) Hence any one station will see the satellite only near two values of the angle vfmpq corresponding to a northward pass and the other to a southward :one pass. For this reason, determinations of the gravity field from systems of category 1-ground station to satellite tracking-must be large-scale leastsquares computations, solving simultaneously for station locations and orbital constants of integration with the spherical harmonic coefficients of the gravity field. It is a different matter, however, with the newer systems, where the measurement is made on board the spacecraft and hence is not so limited by intervisibility constraints. Radar altimetry is by far the simplest and most direct method, despite the nonidentity of the mean sea level with the geoid, since the measurement is linearly proportional to the quantity of interest :the potential. Although there is some art in the removal of orbital error by use of track crossings and so forth, altimetry comes closest of all systemsto direct mapping. Satellite-to-satelliterange rate, necessary to get the gravity to comparable resolution over the land, is more complicated and indirect because the range rate is affected by both the angular rate (which is only energy dependent) and the position (which is both energy and angular momentum dependent). Hence spherical harmonics of the same order rn and parity (l odd or even) have overlapping spectra, entailing simultaneous solution for each such string. Therefore alternative means of representation are being tried (Kaula, 1983).4.2. Observation Equation Formation

CI - e = I - cot-1( i dtan~csc 4 - sec2I )

To use an observation to correct orbit-affecting parameters, the effect of the parameters on the observation must be computable, as well as the associated partial derivative. Thus in Eq. (1) for range rate, the station location and rotation rate of the earth must be taken into account (Kaula, 1966a, K63):

r = [Rxqq - Rxuu][Rxqq - (tJRxu/tJ19)ue]/r

(37)

where u is the earth-fixed rectangular coordinate of the tracking station and R,, is R3(- 6). In differentiating Eq. (37) (or the analogous expression for

180

WILLIAM M. KAULA

range or direction), the principal cautions which must be exercised concern the facts that (1) q and q depend on the eccentricity e through the eccentric anomaly E as well as directly, and ( 2 ) the instantaneous angular elements a, w , M depend on the constants of integration a0 , e o ,10through their effects on the precession rates, Eq. (30). Radar altimetry does not require such complicated differencing. However, it is instructive to derive the dependence of an altitude on a spherical harmonic through the radial coordinate as well as the geoid height. Using r = a(1 - e cos E), from Eq. (8), and allowing for the dependence through a, e, and M givesAhm =A m n

C fimp(ae/a)[2(1 - 2p)/@imp0 + Gip(-i)/eCi/(-i)mpoP

- Glpl/e@(i+ i)mpo] exp(iy/rmpo)- UePIm exp(imA)

(38)

Satellite-to-satellite range rate turns out to be a good deal more complicated because of the dependence of the range rate on the sum of two radial displacements as well as the difference between the two angular rates. For two satellites of angular separation 6M, and ( I - 2 p ) 6M 4 1,starting from the cosine law for the range,

R = (r? + rf - 2rlt-z cos 6M)one obtains (Kaula, 1983):ARrmp = A 1 m n ~ a ( ~ e / ~ ) F l m , 6 ~ [ 2 G 2p i+l 1 / 2 ) / 3 ( 1 + l ) m p 0 /p ( -

(39)

- 2G/p(-i)(I - 2p - 1/2)/@(/-1)mp0 - 3(1- 2 ~ ) ~ / @ / m p o l i exp(iy/rmpo)(40)

4.3. Techniques of Solution

4.3.1. Classical Satellite Geodesy. As mentioned, the overlapping spectra of different gravity coefficients together with the nonuniformity of tracking distribution and the stroboscopic effect, Eq. (36), necessitate the use of several satellites of varied specifications in solutions from classical ground-to-satellite data. The most elaborate solution (Lerch et al. , 1979, 1982) has 7954 unknowns : 592 spherical harmonic coefficients of the gravity field, 3 coordinates each for 146 tracking stations; 6 constants of integration each for 699 orbital arcs; and, for 2 . 5 years of Lageos, 3 components of earth orientation for 5-day intervals. But since each of the 1,300,000 observation equations has only one set of six constants of integration (plus, for Lageos, three earth orientation parameters), the normal equations for each arc can be partitioned so that only the parameters common with other arcs must be carried over for combination with other arcs. This partitioned normals

13.

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181

technique (Kaula, 1966a, K105) thus reduces the maximum normal matrix which must be inverted to that corresponding to the parameters common to all arcs. Since most of the 146 tracking sites are associated with only a subset of the orbital arcs, a second partitioning can be performed to remove them from the final solution. These procedures make it most convenient computationally to retain separately the normal equations for the gravitational harmonic coefficients and station coordinates. Another technique employed to reduce instability of the inversion is to use zero apriori estimates of the coefficients with standard deviations according to a rule such as lO-/I2. The inverse square of this quality is added to the diagonal of the normal equations before the final inversion. In these massive solutions, the errors of the observations are necessarily assumed to be uncorrelated. This assumption is obviously untrue, not so much because of systematic errors in the instrumentation as because the force model is unavoidably incomplete and the observations are nonuniformly distributed. Consequently, the determination of the weight factors which must be applied to the normal equations from different satellites plus the a priori weight described in the previous paragraph is a rather empirical matter. Figure 3 is a global map of the most recent solution by Lerch et al. (1982). 4.3.2. Altimetry. The great quantity of data generated by the radar altimetry necessitates further improvisation. The normal procedure is t o determine the orbits by using gravity fields from previous solutions, calculate the radial coordinates of the observations from these observations, and then obtain the geoid height by subtraction of the measured altitude (plus reference ellipsoid and tidal effects). This procedure gives good determinations of the short-wavelength (less than 2000 km) variations in the geoid, which then are affected mainly by ocean dynamics and instrumental effects. However, there remain longer-wavelength warps in the geoid of a few meters arising from errors in the orbits, which, in turn, are caused mainly by errors in the gravity field. For a map of the geoid over an ocean, the long wavelengths are largely removed by requiring consistency of the geoid heights at track crossings. In effect, the error varies smoothly enough that it can be replaced by a linear trend. The most detailed global compilation of oceanic geoid heights is probably that by Rapp (1982) ;see Fig. 4 for an example. Another global map at smaller scale is by Marsh and Martin (1982). Since then, the emphasis has been on exploiting the full resolution of the Seasat altimetry-about 20 kmfor regional imagery, using digital processing techniques developed for other remote-sensing data. Haxby et al. (1983) subtract out a 12th degree gravity field from the geoid, apply a two-dimensional fast Fourier transform, and use a flat earth approximation to obtain gravity anomalies which are represented on a multicolored map with slant illumination. Sandwell (1984)

FIG.3. Geoid heights referred to a mean ellipsoid (flattening 1/298.257), calculated from spherical harmonic coefficients to degree 36 based on a combination of satellite and surface data: solution GEM 10B of Lerch et al. (1981).

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183

LONGITUDE

FIG. 4. Detailed altimetry geoid; heights are in meters referred to a mean earth ellipsoid (flattening V298.257). From orbital radii determined at NASA Goddard Space Flight Center (Marsh and Williamson, 1980) minus Seasat altitudes and tidal heights (Schwiderski and Szeto, 1979), with crossover adjustments at Ohio State University (Rowlands, 1981). One of 53 maps in Rapp (1982).

takes along-track differences of altitudes and then uses an autocorrelation function to interpolate among passes. The resulting map of geoid height slopes highlights submarine features such as seamounts and fracture zones remarkably; see Fig. 5 . 4.3.3. Combination of Classical Data and Altimetry. To improve the long-wavelength features of the gravity field from radar altimetry, ideally observation equations based on Eq. (39) would be added. These would then be used to yield improved orbit constants of integration, as well as spherical harmonic coefficients of the gravity field. In practice, this use has not yet been made of the altimetry. Instead area means are formed, either of the geoid heights themselves (Gaposchkin, 1980; Lerch et al., 1981 ;Rapp, 1981) or of the gravity anomalies calculated therefrom (Reigber et al. , 1982). These area means are then subjected to harmonic analysis to obtain estimates of the spherical harmonic coefficients to combine with est