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Normal Mode and Surface Wave Observations 1.04 Theory and Observations: Normal Mode and Surface Wave Observations G Laske, Scripps Institution of Oceanography, University of California San Diego, La Jolla, CA, USA R Widmer-Schnidrig, Black Forest Observatory, Wolfach, Germany ã 2015 Elsevier B.V. All rights reserved. 1.04.1 Introduction 117 1.04.1.1 Free Oscillations 117 1.04.1.2 Surface Waves 118 1.04.1.3 What We See in Seismograms: The Basics 119 1.04.2 Free Oscillations 120 1.04.2.1 Modes of a Spherically Symmetric Earth 120 1.04.2.2 Modes of a Heterogeneous Earth 122 1.04.2.2.1 Mode splitting 122 1.04.2.2.2 Mode coupling 126 1.04.2.3 Measuring Mode Observables 128 1.04.2.3.1 Multiplet stripping and degenerate mode frequencies 130 1.04.2.3.2 Singlet stripping and receiver stripping 132 1.04.2.3.3 Retrieving the splitting matrix with the matrix autoregressive technique 133 1.04.2.3.4 Retrieving the splitting matrix with iterative spectral fitting 133 1.04.2.3.5 Observed mode coupling 135 1.04.2.4 Example of a Mode Application: Inner-Core Rotation 136 1.04.2.5 Example of a Mode Application: Earth’s Hum 139 1.04.3 Surface Waves 141 1.04.3.1 Standing Waves and Traveling Waves 141 1.04.3.2 The Measurement of Fundamental Mode Dispersion 144 1.04.3.2.1 Group velocity 148 1.04.3.2.2 The study of ambient noise 149 1.04.3.2.3 Phase velocity 149 1.04.3.2.4 Time-variable filtering 151 1.04.3.3 Other Surface Wave Observables 152 1.04.3.4 Higher Modes 154 1.04.3.4.1 Higher-mode dispersion and waveform modeling 154 1.04.3.4.2 Love waves and overtones 157 1.04.3.5 Surface Waves and Structure at Depth 157 1.04.3.6 The Validity of the Great-Circle Approximation and Data Accuracy 160 1.04.4 Concluding Remarks 160 Acknowledgments 161 References 161 1.04.1 Introduction Valdivia, Chile, earthquake. It was not until 1975 – after the analysis of digitized records of the deep, magnitude 8.0 31 July 1.04.1.1 Free Oscillations 1970 Colombia earthquake – that enough mode measure- The destructive MW ¼9.0, 11 March 2011 Tohoku, Japan, ments were available to construct the first one-dimensional earthquake most certainly raised recent awareness of extremely (1-D), normal-mode derived Earth that could withstand large earthquakes. And also remembering the MW ¼9.1 decades of testing as a reference Earth model (REM) (1066A 26 December 2004 Sumatra–Andaman earthquake and the by Gilbert and Dziewonski, 1975). Not much later, great pro- MW ¼8.8 27 February 2010 Maule, Chile, earthquake, one gress was achieved to facilitate the collection of high-quality can easily forget that a seismologist has to wait many years mode data by installing the global digital seismic network IDA for a great, preferably deep earthquake to occur in order to (International Deployment of Accelerometers), a network of observe Earth’s free oscillations. After all, the last earthquake in LaCoste–Romberg gravimeters that were specifically designed this league of ‘mega earthquakes’ occurred 40 years before the to observe Earth’s free oscillations (Agnew et al., 1976, 1986). Sumatra–Andaman event: the ‘Good Friday’ 27 March 1964 In the meantime, permanent stations of several other global Alaska earthquake. It is therefore not surprising that, compared seismic networks have been upgraded with very broadband to other seismic studies, free oscillation studies started rela- seismic sensors – typically Wielandt–Streckeisen STS-1 vault tively late last century, after the great MW ¼9.5 22 May 1960 seismometers or Geotech–Teledyne KS-54000 borehole Treatise on Geophysics, Second Edition http://dx.doi.org/10.1016/B978-0-444-53802-4.00003-8 117 118 Theory and Observations: Normal Mode and Surface Wave Observations seismometers – and digital recording units. This includes early oscillations and surface waves, and Masters et al. (1982) dis- networks that were designed to monitor global seismicity and covered harmonic degree 2 variations in the transition zone the comprehensive test ban treaty (CTBT), such as the high-gain that are associated with subducting slabs. The first widely used long-period (HGLP) network (first digitally recording network; 3-D models of Earth’s upper mantle, M84A and M84C Savino et al., 1972), the seismic research observatory (SRO; (Woodhouse and Dziewonski, 1984), were obtained using Peterson et al., 1980), the World-Wide Standardized Seismo- mode theory. Still today, careful analysis of high-precision graph Network (WWSSN; Oliver and Murphy, 1971), and the mode measurements provides crucial clues to answer some of China Digital Seismograph Network (CDSN; Peterson and the most fundamental geodynamical questions that remain Tilgner, 1985). Many of these stations as well as the upgraded elusive to other seismic techniques. For example, the analysis IDA network are now part of the US Global Seismic Network of Masters and Gubbins (2003) provided updated estimates of (GSN) operated by IRIS (Incorporated Research Institutions for the density jump across the IC boundary, which is relevant to Seismology), but other global networks exist such as the French the discussion of the growth rate and the age of the IC. They GEOSCOPE (Romanowicz et al., 1984) and the German GEO- also argue against a significant overall excess density in the FON (Hanka and Kind, 1994). All these and more operate lowermost mantle that was proposed by Kellogg et al. (1999) under the umbrella of the international Federation of Digital for locally varying hot abyssal layers for which seismic evidence Seismograph Networks (FDSN; Romanowicz and Dziewonski, was presented by Ishii and Tromp (1999). Earth’s density 1986). Roughly 25 years into very broadband seismic networks, structure and the solidity of the IC are best constrained by a normal mode seismologist now can enjoy a more than tenfold mode data. Similarly, modes help determine Earth’s internal increase in high-quality vertical-component long-period seismic anelastic and anisotropic structure. There are many more records for a given earthquake. But even in the late 1990s – a few aspects where mode data can help out to understand how years after the deep magnitude 8.2 09 June 1994 Bolivia earth- our planet works. One example is the differential rotation of quake provided spectacular spectra – great effort was invested in the IC. Evidence for this was first observed using body-wave digitizing the legendary 1970 Colombia records. Before the data and was initially reported to be between 1 and 3 per year 26 December 2004 Sumatra–Andaman earthquake, there have (Song and Richards, 1996; Su et al., 1996) but hotly debated been a handful of other great earthquakes since ‘Bolivia.’ But (e.g., Souriau, 1998). As subsequent studies accumulated, this even the great 23 June 2001 Arequipa, Peru, earthquake, the number decreased dramatically and is currently estimated at largest digitally recorded earthquake until then, did not excite 1/10 of the initial rate. Mode observations in particular (Laske the relatively few normal mode observers whose interest lies in and Masters, 1999, 2003; Sharrock and Woodhouse, 1998) unraveling the deep secrets of the inner core (IC). ‘Peru’ was provided the conclusive constraints to correct the rotation simply not deep (or great) enough to make some of the modes estimates downward. ring that the observers are interested in. On the other hand, since In the first part of this chapter, the reader gets acquainted normal modes involve the vibration of the whole planet, mode with the jargon used in normal mode seismology, some of observations at a single station readily reveal a wealth of infor- which requires to summarize the theoretical background that mation about Earth structure that no other seismic techniques is described in Chapter 1.02. We then introduce some of the can provide. Modes are intrinsic low-pass filters of Earth struc- most commonly used measurement techniques that we have ture. It is relatively easy to collect unbiased estimates of mode been involved in. Mode analysis involves more than simply observables that constrain the spherical average of Earth as well reading the peak frequencies and amplitudes from a spectrum. as long-wavelength perturbations to it. It is therefore not sur- In fact, in most cases, such an approach leads to biased esti- prising that in the current efforts to remove the more than three- mates. One also has to bear in mind that the most basic decade-old ‘preliminary’ from PREM (preliminary reference analysis techniques treat modes as being isolated from their Earth model, Dziewonski and Anderson, 1981)–thecurrently neighbors in which case only Earth structure of even-degree still most widely accepted REM of the spherical average – a symmetry can be retrieved. Earth’s rotation and lateral varia- suitable mode data set for an updated model exists (a ‘best- tions cause modes to couple with each other. This complicates estimate’ data set can be obtained from the REM website), mode analysis but also facilitates the assessment of odd-degree while we still struggle to obtain an unbiased body-wave data set. structure. This is briefly described. Normal mode seismology has facilitated other great achievements that we can proudly look back to. For example, 1.04.1.2 Surface Waves the analysis of modes provided the ultimate proof that the IC is solid (Dziewonski and Gilbert, 1971). Normal mode studies Surface waves can be understood as a superposition of free were at the forefront to assess Earth’s attenuation (Smith, oscillations. It is therefore not surprising that many long- 1972) and to retrieve earthquake moment tensors (Gilbert period surface wave seismologists analyzed normal modes and Dziewonski, 1975), which has been continued in the at some time in their career.
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