TIMING Analysis of the Multiple Passages of the Pressure Wave Generated by the 2022 Hunga Tonga-Hunga Ha’apai and Comparison with the 1883 Krakatoa Pressure Wave

Abstract

The Hunga Tonga-Hunga Ha’apai (HTHH) eruption of 15 January 2022 was an exceptional event by the period, magnitude, and duration of propagation of the atmospheric waves it generated, which circled the Earth multiple times. This event, taking into account the magnitude of the atmospheric pressure waves, is comparable only to the Krakatoa eruption of 1883. To compare both eruptive sequences, a method similar to the analysis of the timing of the arrival of multiple phases at barometric stations, as reported in (Strachey, R., Stokes G.G., Scott, R.H. (1888). On the air waves and sound caused by the Krakatoa eruption of August 1883, in “The eruption of Krakatoa and subsequent phenomena,” Symons, G. J. (ed.). Report of the Krakatoa Committee of the Royal Society (Trübner and Co., London)) for Krakatoa, was used. Since the HTHH volcanic event gave rise to the only volcanic pressure wave known to have circled within the Earth’s atmosphere multiple times in the last 139 years, it is of interest to perform similar timing statistics on the multiple passages of the waves at stations that recorded them. A review of the Krakatoa analysis and a comparison with the HTHH are presented, with possible implications on the physical parameters affecting its speed of propagation. Changes in the global state of the atmosphere during the interval between the two events may also explain some of the differences observed.

1 Introduction

Even though several volcanic eruptions in the past 150 years generated notable barometric disturbances, the Hunga Tonga-Hunga Ha’apai (HTHH)¹ eruption is comparable only to the Krakatoa² eruption of 1883 by the magnitude of the atmospheric pressure waves that it generated (Matoza et al., 2022). For instance, the very energetic Mt Pinatubo eruption of 1991, and the 1956 eruption of Bezymianny, among the most recent and notable volcanic eruptions (Yokoo et al., 2006) did not produce pressure waves of the same period and magnitude as the Krakatoa or HTHH eruptions.

The 15 January 2022 volcanic eruption of the HTHH was an exceptional event and was instrumentally well documented in this era where numerous satellite and ground-based sensors allow immediate or quasi-immediate observation and data sharing (Matoza et al., 2022; Vergoz et al., 2022). The satellite coverage was sufficient to not only observe the spread of the ash cloud, but also of a large atmospheric non-dispersive pressure wave, known as a Lamb wave (Lamb, 1895), which modified the temperature sufficiently on its passage that it could be observed on infrared satellite images (Otsuka, 2022). While HTHH was known to be active for a number of years, it followed a period of intensified volcanic activity that started in December 2021, including the formation of ash clouds affecting air transport and leading to warnings by the Volcanic Ash Advisory Centres (VAACs) of Wellington and Sidney (Global Volcanism Program, 2022a; b). Such atmospheric ash plume activity had already been noticed in an earlier eruptive episode during the period of December 2014 to January 2015 (Global Volcanism Program, 2015). The paroxysmal explosion of 15 January 2022, by its magnitude, and especially the generation of an unusually large low frequency pressure wave, attracted the attention of scientists with a variety of specializations (volcanologists, seismologists, acousticians, atmospheric scientists, specialists of the ionosphere, and of lightning), as is evidenced by the many publications available less than two years after the event.

The event is of interest to volcanologists, with a volcanic explosive index (VEI) of at least 5 (Global Volcanism Program, 2023b; Poli & Shapiro, 2022), the largest since the 1991 Pinatubo eruption with an index of 6.0 (Global Volcanism Program, 1991). The mass of material ejected, 9.5 km³ (NIWA, 2022) and the composition of the gasses is specific to each individual volcanic eruption, and in the case of the HTHH, water vapor was dominant, increasing the stratospheric concentration by several orders of magnitude locally within the cloud, and by more than 5% globally (Vömel et al., 2022.) Unlike what was observed after the Pinatubo eruption, the quantity of SO₂ (0.4–0.5 Tg) into the stratosphere seemed to have been comparably small and inconsequential in terms of an atmospheric cooling effect (Carn et al., 2022). Table 1 shows the basic parameters of the volcanic sources for Krakatoa and HTHH.

Table 1 Basic parameters of the volcanic sources for Krakatoa and HTHH

A review of the data presented in Strachey et al. (1888), and of the importance of the Lamb wave observations made at the time of the Krakatoa eruption to the understanding of the physics of pressure oscillations in the atmosphere is first presented in this paper. This includes the effects of temperature variations with latitude and of the global wind patterns. Then a method is presented that seeks to obtain comparable multiple pass timing data on the HTHH event of 2022. Finally, a discussion compares the results obtained from the HTHH eruption with the Krakatoa eruption.

2 The Krakatoa Atmospheric Lamb Wave. A Major Inspiration for Pioneering Earth Scientists and Physicists Studying the Oscillations of the Atmosphere

The major eruption of Krakatoa in August 1883 is an event that had worldwide effects. It was one of the first natural events with mediatic coverage and one that many people around the world could observe the effect first hand. At the end of the nineteenth century, most people still had better connection with nature and the direct observation of the skies, and one of the effects of the eruption was to bring unusual optical effects to sunsets worldwide (Russell & Archibald, 1888). Indeed, the extensive scientific report published by the Royal Society (London) on the eruption and its effects, chaired (Symons, 1888) dedicates 312 pages to the optical phenomena and only 31 pages to the air waves, and 62 pages to the sea waves observations. Another effect which touched on peoples’ lives was the planetary cooling effect due to ash and SO₂ gas injected into the atmosphere (Rampino & Self, 1982; Schaller et al., 2009), although it appears to have been a loss milder than the 1815 Tambora eruption.

The end of the nineteenth century is an era of fast development of scientific instruments. Probably the most widespread atmospheric pressure instruments at the time were mercury barometers and most meteorological observatory recorded the barometric pressure only a few times a day. There were however barographs in existence, and they provided the first continuous measurements of the atmospheric pressure. Figure 1 is an assembly into a single figure of the records published in Scott (1883) in multiple figures. It should be noted that at the time of his publication (i.e. in December 1883) only European stations (at the exception of Toronto, in Canada) were assembled. He did notice the apparent east to west propagation of the passage of the first pulse and the opposite direction of propagation of the second pulse but stopped at further interpretation until extra-European recordings were received. Four European records (Pawlowsk, Vienna, Lisbon, and Serra da Estrella) existed but had not yet been engraved and were not included in the figures shown by Scott (1883). In a note about Scott’s publication, Strachey (1884) clearly formulates the hypothesis that the record may be attributable to a disturbance originating at Krakatoa, expanding over the Earth, reaching the stations a first time, reconcentrating at the antipode, and reaching the stations a second time (see diagram on Fig. 2). Many authors (e.g. Rockwood, 1885) attribute credit to that publication for the discovery that a large pressure wave had originated at Krakatoa and circled the Earth several times. The observation of the phenomenon on extra-European stations was confirmed later in Part II authored by Strachey, Scott, and Stokes (of Navier–Stokes equations fame) in Strachey et al. (1888). Furthermore, it seems that at that time, the records from the missing European stations had been engraved and snippets of the waveforms showing the different passages of the air wave at fifty stations are presented in the report. It is unfortunate that only snippets are presented of the waveforms and not the entire traces. Efforts are under way to assemble a data set (Agnew, 2023) that would recover these data, digitize them, and make them available to researchers for further investigations.

Figure 1

Section showing the barograms from stations assembled from the separate plates published in Scott (1883). Four days of waveforms are displayed

Figure 2

Graphical form of the combined data for Tables VI, VII, and VIII of Strachey et al. (1888) for complete 360° passages at each barograph. Note the general trend of increasing travel time differences with distance from the volcano. Krakatoa’s latitude being close to the equator, the horizontal axis is also generally indicative of the increasing latitudes of the stations and therefore of the portion of the paths crossing higher latitudes. The stations are displayed in order of increasing distance from Krakatoa. Only a few names are placed on the x-axis and the numbers refer to station rows in Table 2 in Appendix B, which also lists the latitude and longitudes of the stations and their distances to Krakatoa

Strachey et al. (1888) published tables of the times of arrival at the available barograph stations of as many passages of the Lamb wave as possible, as well as the travel time differences between the passages of the waves propagating in the same direction. This latter data is reproduced in Fig. 2. Like Rayleigh waves’ successive passages at individual stations, the nomenclature Lamb1 for the minor arc, and Lamb2 for the major arc are used. Subsequent identified passages follow up to Lamb7. The short forms 1, 2, 3, 4, 5, 6, and 7 are used for the sake of conciseness, and the travel time between 3 and 1 is noted as T3-T1.

The observations of the Krakatoa Lamb wave marked a milestone in the understanding of propagation of pressure waves in the atmosphere. Early authors, such as Laplace, in chapter V of the fourth book of “Traité de Mécanique Céleste” Laplace (1799), entitled “Des oscillations de l’atmosphère” analysed the influence of the tides on the atmosphere and concluded that their propagation would be analogous to gravity waves (tides) in an ocean with an equivalent depth (roughly 9 km by his estimation) whose value depends on the temperature of the atmosphere. He did not consider any other forces acting on the atmosphere besides the gravity exerted by the Sun and Moon. Much closer to our time, this is also the model in the numerical simulation computed by Amores et al. (2022). Their numerical model designed for propagation of gravity waves (tsunamis) in the ocean assume an analogy with an equivalent ocean. The equations of propagation can indeed be equated provided that the depth of an equivalent ocean is computed as H = γRT/Mg where T is the air temperature, γ the specific heat ratio for air, R the universal gas constant, M the specific molecular mass of air, and g the gravitational constant. Amores et al. (2022) computed the propagation of the HTHH Lamb wave using a time-averaged (1979 to 2022) temperature model, with variations confined to the troposphere, and a constant temperature stratosphere. Using a temperature of 293 K, the equivalent depth would be about 12 km, slightly larger than Laplace’s estimate. Their computed waveforms agree generally with the general features of the observed waveforms for HTHH but do show that the arrival times at some stations deviates from the simulation, especially for the third passage of the wave at a station. From this it can be inferred that the average temperature model is either not sufficiently accurate to explain the data or that other physical parameters need to be considered. Perhaps the temperature model needs to reflect the actual state of the tropospheric temperatures during the few days of the Lamb wave propagation. Other parameters, such as atmospheric winds, or topography, may also influence the propagation.

The extensive works of Lamb (1882, 1895, 1911, 1917) on wave propagation in various media make him one of the pioneers of seismology and atmospheric acoustic, along with Rayleigh (1890). Lamb waves usually refer to elastic waves in plates (Lamb, 1917), however he also introduced what came to be known as atmospheric Lamb waves in art. 272 of his book “Hydrodynamics” (Lamb, 1895). Lamb’s work is purely theoretical, and he did not attempt to apply his work to the Krakatoa observations. A few years later, Taylor (1929) sought to explain the observed semi-diurnal period of barometric pressure. The observations of the semi-diurnal components are much larger than expected from either the tides or the thermal effect of the sunshine variations on the atmosphere. This could possibly be explained by a resonance effect inherent to the atmosphere. However, he used the observed Krakatoa Lamb wave velocity to discount the possibility that a free period of oscillation of the atmosphere is close enough to 12 h to account for a resonance effect.

In Strachey et al. (1888), the authors explain the observations of travel time differences by the temperatures and winds encountered. The observation that the paths having a westerly direction, against the direction of rotation of the Earth are slower than the paths having an easterly direction is made (see excerpt one in Appendix A). An allusion to the effect of the temperature differences due to latitude differences in the paths is also made (see excerpt two in Appendix A).

Taylor (1929) used an approach based on a graphical approximation of the Lamb wave wavefront. This wavefront diagram is reproduced from Strachey et al. (1888) on Fig. 3. It shows an interpolation and interpretation of the first passage of the wavefront as it is about to travel back to Krakatoa for the first time. The three-lobe pattern of the 36 h isochrone on the (b) diagram of Fig. 3 was interpreted by Taylor (1929) as a possibility that global wind patterns alone could explain the timing anomalies of the wavefront. It is worth noting however that this pattern does not seem to persist at the second passage back to Krakatoa (see diagram (d) of Fig. 3), and the subsequent passages of the Lamb wave at the stations do not seem to support this interpretation.

Figure 3

Graphical interpretation of the Lamb wave wavefront based on tables of travel time. Panels a and b show the wavefronts of the first passage back to Krakatoa from the antipodes. Panels c and d show the second passage back to Krakatoa from the antipodes. This is reproduced from Strachey et al. (1888), and b is used in Taylor (1929) to argue that average wind patterns alone could explain the deformation of the wavefront of the Lamb wave

Pekeris (1939) used a few of the Krakatoa barograms, including the Montsouris (Paris) recording to search for evidence of pure gravity waves in addition to the large Lamb wave. In the later era of atmospheric nuclear tests, many authors referred to the Krakatoa wave as an example of large Lamb waves like the ones triggered by atmospheric nuclear explosions (e. g. Harkrider & Press, 1967). Gabrielson (2010) presents an excellent summary of the evolution in understanding of the Krakatoa atmospheric waves observations.

3 Influence of the Latitude-Dependent Temperatures and Dominant Winds on the Propagation Speed of the Lamb wave

3.1 Influence of the Temperature Differences Due to Latitude

Acoustic velocity is dependent on the temperature of the medium. The relationship between these two quantities can be used to estimate the portion of the travel time variation observed that can be attributed to the temperature difference encountered along the path of propagation of the Lamb wave. Assuming a simple model for the temperature with latitude-only dependence, the amount of polar, temperate, or tropical air encountered along each great circle path is a function of the inclination of the path with respect to the axis of rotation (angle γ). Appendix C details the definition of ϕ = cos(γ) and the derivation of its value from the latitudes and longitudes of the volcanic sources and the stations. Figure 15 illustrates the relationship between the great circle path hosting the volcano and station and this parameter. The global temperature model chosen in this study is inspired from the model shown in Fig. 4 for surface temperatures. It is symmetrical with respect to the Northern and Southern hemispheres and is derived from Feulner et al., 2013. Considering now the great circle with an inclination γ, and the length of the path s along this great circle, the delay encountered along the path due to the temperature variation needs to be evaluated as a function of $\phi$, and this is done in Appendix C, resulting in the relationship between T and $\phi :$

$$T={T}_{0}+\Delta T\left(\phi \right)$$

(1)

where ${T}_{0}=-2\pi R \frac{\sqrt{{K}_{0}}}{5}\left( 1-\sqrt{1+\frac{45}{{K}_{0}}}\right)$, and $\Delta T \left(\phi \right)=2\pi R \frac{\sqrt{{K}_{0}}}{5} \left[\left( \sqrt{1+\left|\phi \right| \frac{45}{{K}_{0}}} \right)-1\right]$,

Figure 4

Adapted from Feulner et al. (2013). For order of magnitude estimates, the surface temperature curve used in this study is symmetrical with respect to the equator and varies only with latitude: Θ = − 20 + 45* cos(λ), where θ is the temperature in °C, and λ is the latitude in radian. The curve chosen for this study is shown in thick and transparent light purple. CRU stands for Climatic Research Unit (Jones et al., 1999). The data presented is the average over the 1961–90 period

When using values ${K}_{0}=298$ K, and R = 6378 km for the radius of the Earth at the equator, and the approximation $v =20 \sqrt{\theta }$, where v is the acoustic velocity, and $\theta$ the temperature, the travel time in hours around the equator is calculated to be 32.24 h, simply dividing the circumference by the velocity. There are several assumptions in this calculation. One is that the Lamb wave velocity is approximately equal to the sound speed in the medium, which is known to be true (e.g. Harkrider & Press, 1967), and the other is that the average temperature of the troposphere is the same as the surface temperature, which is not correct, the temperature decreasing rapidly with altitude in the troposphere (Gerber et al., 2012). Compared with the observed values both for Krakatoa and HTHH, this travel time estimate is too short, indicating an overestimation of the temperature. This is expected as the Lamb wave samples the colder and slower air in altitude. The empirically observed value is on average around 35 h. Estimating the temperature value at the equator corresponding to 35 h yields about 250 K, or − 23 °C. This value is compatible with what is known about the temperature profile in the troposphere, which may vary rapidly, for example between about 10 °C at the surface and − 50 °C at the tropopause for northern mid-latitudes in January (Gerber et al., 2012).

3.2 Influence of the Winds

Taylor (1929) showed that the timing of the Lamb wave at its first passage back to the volcano could be explained in large parts by a latitude-dependent wind pattern of the form:

$$v (\lambda )= {V}_{0} \left[1+a\text{cos}\left(3\lambda \right)+ b\text{cos}\left(\lambda \right)\right]$$

where a and b are dimensionless constants, λ is the latitude, and ${V}_{0}$ is a constant velocity vector with a value corresponding to the velocity of low frequency waves (310 m/s) representing the wind along the meridian (positive pointing East). This is of course an oversimplification of the wind systems in the troposphere, as it is purely latitude dependent and steady state. Note that the sum over latitudes does not vanish, implying an overall westerly component. A simpler model than Taylor’s is proposed, and it is assumed that the velocity v is of a form accommodating the general westerly circulation at temperate latitudes:

$$v = {V}_{0} \left[1+c\text{sin}\left(2\lambda \right)\right]$$

In doing so, the trade winds are considered to have a smaller impact but could be added in a more refined study.

Assuming the Lamb wave is slowed down by a factor of $v cos\left(\gamma \right)$, similarly to the derivation for the latitude temperature variations, the expected delay along the great circle parametrized by γ can be estimated based on this form of the v function. Appendix D derives the expression for the delay around the great circle as a function of γ. If the expressions for the delays due to the latitude-dependent temperature and the latitude-dependent wind patterns are combined, the travel time on the great circle parameter by φ = cos(γ) is of the form:

$$T\left(\phi \right)={T}_{P}+{C}_{T} \left[{\left(1+ \frac{45}{{K}_{0}}\left|\phi \right|\right)}^\frac{1}{2}-{\left(1+ \frac{45}{{K}_{0}}\right)}^\frac{1}{2}\right]+{C}_{w}\phi \left[1-\frac{1}{ 1+0.024 \left|\phi \right|\sqrt{1-{\phi }^{2}}}\right]$$

(2)

where T_P, C_T, and C_W are coefficients that can be adjusted from the Lamb wave observations made for the HTHH and Krakatoa events.

4 Analysis of the HTHH Data

To compare the observations made at the time of the Krakatoa eruption with the HTHH eruption, it is first essential to estimate the time of as many passages of the Lamb wave as possible at barometric and infrasound stations. Strachey et al. (1888) mentioned the challenging aspect of picking times on the barograph data and the authors resorted to picking on a specific feature of each wave train, the first trough. The method used in this work is explained in the next paragraph.

4.1 Method of Picking on Selected Stations

Waveforms from thirty Global Seismological Network (GSN) stations (Scripps Institution of Oceanography, 1986), eleven openly available International Monitoring System (IMS) stations (Various Institutions, 2022), and five additional IMS stations were collected. Since the frequency responses of the LDO and LDI (‘L’ stands for long period, ‘D’ is the code for pressure data, ‘I’ means the instrument is inside and ‘O’ outside) of the GSN stations differ from the BDF (‘B’ is for broadband, ‘D’ is the code for pressure data, and ‘F’ is for infrasound) channels at the IMS stations and at some of the GSN stations, the set of LDO and LDI channels was first analysed, since they most closely resemble the historical barograph data from Krakatoa by their frequency range. IMS stations were then analysed in a frequency range chosen from a comparison of BDF and LDI channels on two IMS stations co-located with GSN stations. Table 3 in Appendix B lists the coordinates of the GSN and IMS stations and Fig. 14 is a map showing their location and a few values of parameter ϕ on the azimuthal equidistant projection. Table 4 in Appendix B lists the channels used for each station. Station IS57 is co-located with GSN station PFO, and this pair as well as the IS50-ASCN pair was used to assess the best bandwidth to filter IMS and IRIS data such that the data from these two different networks could be compared. Figure 5 shows a comparison of the waveforms for channel BDF at site H1 of the IS57 station, and the PFO LDI channel. The frequency band [0.1–0.5] mHz seems appropriate to compare the two types of data.

Figure 5

Comparison of waveforms for IS57 (BDF channel) and PFO (LDI channel) in various frequency bands. Note the unambiguous arrivals for the four passages in the frequency band [0.0001–0.0005] Hz. Note also that the time difference between passages 3 and 1 is clearly less than 36 h. In contrast, it is clearly larger for the difference between passages 4 and 2

Figure 6 shows the filtered (0.0001 to 0.4 Hz) waveforms from the 30 GSN stations on which time picking is done on each of the first four passages. They are ordered by increasing distance from the HTHH volcano and the time frame includes data from the 15 to 18 January 2022. From Fig. 6, the first and second passages of the atmospheric Lamb wave at all stations is quite clear. The eye also picks up similar patterns for the third and fourth passages, but looking at an individual station, it can be challenging to pick the time of these two passages.

Figure 6

GSN stations waveforms filtered in [0.0001–0.4] Hz band and shown in order of increasing distance (in degrees) from HTHH

To help with the picking, the following method was used on the data filtered in the [0.0001–0.4] Hz frequency band:

• 1. The first passage wave train was aligned for all thirty stations and the picking done on the waveforms as consistently as possible from station to station. In the case of the first passage, the arrivals are quite clear on all stations and thanks to the relatively broad frequency range, there is limited risk of cycle-skipping the alignment.

• 2. The third passage waveforms were then aligned to maximize by eye the waveform similarity between the stations, and manual picks made on these aligned waveforms. Figure 7 shows two 8-h sections of waveforms around the first and third passage. The time is aligned on the first passage picks for both sections.

• 3. The second passage wave train was aligned for all thirty stations and the picking done on the waveforms as consistently as possible from station to station. This alignment was slightly more challenging than for the first passage, but picking could be done for most stations with a high degree of confidence.

• 4. Like for the picking of the third passage, the fourth passage waveforms were then aligned to maximize by eye the similarity between the stations. This proved much more challenging than for the third passage, as illustrated by Fig. 8.

Figure 7

Two 8-h sections of waveforms around the first and third passages. Waveforms are aligned on the first passage picks for both sections. The 0 and 36-h times after the first passage are shown with a thin black line. Stations are ordered top to bottom in increasing distance from the source. Note the very clear onset of the first passage for most stations. As expected, the third passage is not as clear, but picks were made at all stations except ASCN

Figure 8

Two 8-h sections of waveforms around the second and fourth passages. Waveforms are aligned on the second passage picks for both sections. The 0 and 36-h times after the second passage are shown with a thin black line. Stations are ordered top to bottom in increasing distance from the source. Note that the second passage signal is still strong but not as clear as the first passage. The fourth passage is more difficult to pick that the third, the signal to noise ratio having decreased substantially, and is not recognizable at all stations

Table 4 in Appendix B details the values obtained for the picks using the method detailed above.

Pressure data from the BDF channels of the IMS stations are analyzed separately in the frequency band [0.0001–0.0005] Hz which was shown to yield results like the LDI channel at co-located stations PFO and IS57. The time picking of the four passages is done in a manner similar to the GSN stations and the results are listed in Table 5 of Appendix B. Figure 9 shows the filtered waveforms with the picks indicated by maroon circles for the second passage, green for the third, and yellow for the fourth.

Figure 9

Waveforms for the 14 IMS stations that are not co-located with a GSN station. The waveforms are aligned on the first passage picks and the absolute time is different for each station. The times picked are listed in Table 5 of Appendix B. Not all stations have identifiable picks for all passages

4.2 Estimation of Parameters for the Latitude-Dependent Temperature and Wind Models

The picks on GSN and IMS stations as a function of parameter f, the cosine of the inclination of the great circle joining HTHH to the station were compared to the models developed in Sect. 3. It was shown that the time variation as a function of f is of the form: $\Delta T \left(\phi \right)=2\pi R \frac{\sqrt{{K}_{0}}}{5} \left[\left( \sqrt{1+\left|\phi \right| \frac{45}{{K}_{0}}} \right)-1\right]$. Since $\left[\left|\phi \right| \frac{45}{{K}_{0}}\right] \ll 1$, an approximation can be used for ∆T $\Delta T \left(\phi \right)=2\pi R\frac{4.5}{\sqrt{{K}_{0}}} \left|\phi \right|$. Assuming the functional form ${T= T}_{P}+{\Delta T\left(\phi \right)={T}_{P}+ C}_{T}\left|\phi \right|$, parameters T_P and C_T can be estimated from the HTHH data using the least-squares method. This results in a value of 36.10 h for T_P and − 0.594 h for C_T.

Panel (a) in Fig. 10 shows the time differences between the third and first passage (T3-T1) and the fourth and second passage (T4-T2) for the GSN picks shown in Table 4 of Appendix B. The error bars are uniformly chosen to be 0.25 h from the experience gained from manually picking the times of arrival of the different Lamb wave passages. The dashed line shows the best fit model (T_P and C_T listed above) for a latitude-dependent temperature variation. Most points on the negative ∅ side are above the dashed line and most points on the positive side are below the dashed line, as would be expected from the influence of the mid-latitude westerlies. In the next step, an estimate of the influence of these winds is made by estimating parameter C_W from Eq. (2) and estimating coefficient C_w. Since $\left[0.024 \left|\phi \right|\sqrt{1-{\phi }^{2}}\right]\ll 1$, an approximation can be used for $W\left(\phi \right)=\phi \left[1-\frac{1}{ 1+0.024 \left|\phi \right|\sqrt{1-{\phi }^{2}}}\right] \cong 0.024 \phi \left|\phi \right| \sqrt{1-{\phi }^{2}}$. It is found that a value of 57.2 for C_W minimizes the residual between the data points and the latitude-dependent model, when care is taken to avoid instabilities near small values of ϕ by damping the least-squares estimate by 1%. The solid line on panel (a) of Fig. 10 shows the model when both the latitude-dependent temperature and the winds are considered.

Figure 10

a Delays (T3-T1) and (T4-T2) as a function of parameter ϕ for the 30 GSN stations recording the HTHH Lamb wave considered in this study. The dashed line is the best fit latitude-dependent temperature model, the solid line is with the addition of the effect of the dominant westerlies. b Delays from the 14 IMS stations are added to the data set and the corresponding best fitting latitude-dependent temperature model and temperature and winds model are updated based on these additional data points. GSN stations are shown as open circles to avoid overcharging the plot

Panel (b) of Fig. 10 shows the results of adding the IMS picks. A very similar value for the parameters of the latitude-dependent temperature model is obtained when adding the IMS data: 36.6 h for T_P and − 1.5 h for C_T. Parameter C_w is more affected by the addition of the IMS picks, since a lot of the values of ∅ for these stations are near zero. Using the same damping for the computation, the value of C_w is now 63.6. A possible interpretation is that the influence of the temperature differences due to latitude creates time differences of about 0.45 h between purely polar and purely equatorial paths, while the wind’s influence in the temperate zone occasion maximum differences of about 1 h. It is clear from Fig. 10 that the delay experienced by the Lamb wave is increased for negative values of ϕ and reduced for positive values. This can be explained by a model of westerly winds at temperate latitudes. Figure 11 further illustrates the influence of the westerlies. It is showing the time delay differences at each station between the great circle path with an easterly component of propagation and the great circle path with a westerly component of propagation. Most stations (27 out of 33) are on the positive side of this difference, meaning that the Lamb wave on average experienced westerly headwinds. The six stations on the negative side are either on a near-polar trajectory (BFO, BORG, IS27, SUR) which are not influenced by longitudinal winds, or an equatorial trajectory subject to easterly trade winds (IS52 and JTS).

Figure 11

Delays (T3-T1) and (T4-T2) as a function of parameter ϕ for the historical Krakatoa data set. The black dashed line is the best fit latitude-dependent temperature model, the solid black line is with the addition of the effect of the dominant westerlies. The dashed grey line is the temperature model and the solid grey line the combined model for the HTHH data set

5 Comparison with the Krakatoa Historical Data

Most of the Krakatoa historical data was used to constrain a latitude-dependent temperature model and a combined temperature plus wind model. Several data points are left out for the purpose of fitting the model to the data: South Georgia for the (T3-T1) and the North American (including Havana) stations for (T4-T2). The result is shown in Fig. 11. The dashed line shows the temperature model. The difference between the equatorial value and the polar value is 1.99 h (parameter C_T), larger than the value for the combined GSN plus IMS station, which was 1.5 h. The value for parameter C_W is 48.3, a value smaller than the values obtained when using GSN-only (57.2) and GSN + IMS (63.6) data points. The largest difference between the models derived from the HTHH data points and the historical data points is therefore the latitude-dependent temperature part of the model. The Krakatoa data suggests a delay spanning between about 35 h and 37 h from equatorial to polar paths whereas the HTHH data suggests a smaller variation between about 35.5 h and 36 h. This may be due in large part to the sparse sampling of the historical data set, with very few data points for small ∅ values. The North America stations are the exception, and consistently measure large delays in the neighbourhood of 37–38 h for (T3-T1) and very small delays slightly bigger than 32 h for (T4-T2). These two sets of measurements are incompatible with a latitude-dependent temperature variation as well as any wind system parallel to the equator. They could be explained by the encounter of winds with very large north–south components that would delay the 1, 3 and all odd passages and help the 2, 4 and all even passages. This could for instance be southerlies in the western Atlantic Ocean.

To better illustrate the difference between eastward and westward propagation, Fig. 12 plots the time difference between an eastward and westward propagation. In order to do this, the difference plotted is T(3-1)– T(4-2) when ∅ is negative, as in the case if the European stations, and T(4-2)– T(3-1) when ∅ is positive, where both are available. Several observations can be made from this display which emphasizes the difference between eastward and westward propagation of the Lamb wave. For the Krakatoa stations, in maroon on Fig. 12:

• The large group of European stations, towards which the Lamb wave propagates eastward for odd passages show a positive difference (0.5 h to 2.5 h), meaning that this eastward propagation is slower than the westward propagation for the even passages at these stations. This can be explained by the dominant westerlies against the direction of propagation on the odd passages and with the propagation for the even passages. These paths also cross the intertropical trade wind areas, but their effect seem to be smaller than the temperate zone westerlies.

• The Indian station Bombay has a ϕ value close to the European stations as they are located on great circles from Krakatoa very close to the Krakatoa-Europe great circles. Not surprising, they show similar differences between eastward and westward propagations.

• The two southern hemisphere tropical stations in Africa and the Indian Ocean, Luanda (Angola) and Mauritius (Indian Ocean) both show a negative difference. This can be easily explained by the dominant trade winds which would have been opposite to the propagation on the odd passages and in the direction of propagation on the even passages.

• The group of Australian and New-Zealand stations show a positive difference. The odd passages in this group of stations are propagating mostly eastward, aided by the dominant westerlies, and the even passages are slower, contrary to the European stations. The difference is however smaller in absolute value (less than 1 h) than for the European group, perhaps due to the encounter of easterly trade winds on a larger proportion of their paths.

• The group of North American stations, which are not displayed in Fig. 12 and whose T4-T2 are not displayed in Fig. 11 have a low value of ϕ, indicating nearly polar paths. They should therefore be insensitive to the east–west wind patterns considered in this work. The difference between the odd and even paths is strikingly high and could not possibly be attributed to any phenomenon that would be generally constant on latitude lines, such as temperature differences, trade winds, or temperate zones westerlies.

• The South Georgia value is a clear outlier. This may possibly be attributable to erroneous picks on the second, third, and fourth passage, as illustrated on Fig. 13.

Figure 12

Difference in time delay between Lamb waves travelling in the easterly direction minus travelling in the westerly direction at the same station. This is computed as (T3-T1)–(T4-T2) for stations on the negative side of the ϕ axis (for T3-T1) and (T4-T2)–(T3-T1) for the ones of the positive side (for T3-T1). Error bars are 0.5 h. Purple symbols are for the HTHH stations and maroon for the Krakatoa stations. North American stations and South Georgia for Krakatoa are not displayed. Note that a large majority of stations (37 out of 48) indicate westerly headwinds. Some stations with near-polar paths are less affected by winds with equatorial components (BORG, IS27, SUR, BFO), and plot on the negative side of the abscissa. Stations on near-equatorial paths (SACV, US51, IS52, Mauritius, Loanda) also plot on the negative side, perhaps because they are affected by easterly trade winds

Figure 13

This figure is reproduced from Plates VIII and IX in Strachey et al. (1888). The waveforms are snippets around passages one to four (left to right) at the Melbourne and South Georgia barographs. from Plate VIII the picks shown by black arrows have been added from the information on Plate IX. The difficulty in deciding on a pick for some of the barographs for the later passages is clearly illustrated here. Should the red arrows be used instead, for instance, the South Georgia station would not be an outlier in Fig. 11. Having a complete continuous trace would be of great advantage to decide on a proper pick and confirm or infirm he picks from the original report

In conclusion of the re-analysis of the Krakatoa data tabulated in Strachey et al. (1888), and of the HTHH data picks, a lot of the delays between multiple passages of the Lamb wave can be explained by a simple latitude-dependent model of temperature and general wind patterns, but not surprisingly, it would take a more complex model including the actual meteorological observations on the days when the propagation occurred to possibly explain the departures from a simple latitude-dependent model.

6 Discussion

Comparing two sets of measurements so far in time from each other is challenging for several reasons. Perhaps the most significant is the difference in the instrumentation used. While the values of pressure recorded by a nineteenth century barograph are likely to be accurate, as they were usually regularly checked against the readings of a barometer, the timing of the measurements may suffer from inaccuracies in several ways. Clocks were not as accurate as in 2022, and the tracing may not always have been of high quality. Table 1 in Strachey et al. (1888) details the type of instrument, comments on the quality of the time scale and the tracing of each recording received from the station operators. In many instances, they mention that the time scale is very contracted, making a time reading difficult. At least ten types of instruments are mentioned in the table, making this historical data set more heterogeneous than the modern data set. All these reasons imply that perhaps the standard error of 15 min for the Krakatoa picks used in our study is optimistic. As many records are now missing from the set that was available to Strachey et al. (1888), there is little choice but to trust the picks that were made at that time.

The picking itself is discussed at quite some length in Strachey et al. (1888). It is difficult to pick the onset of the Lamb wave due to its very low frequency and emergent character and the nineteen century scientists decided to pick the time of the first sharp negative peak of the waveform, as it was recognisable in many of the records. This is also the approach taken in this work for the HTHH records. The first and second passages, are easily identifiable on all stations and while their features in the frequency broad band that was used do vary from one passage to the next, the picking was done on the onset of a sharp large negative peak. This was quite identifiable for the first and second passages on all stations. It was still identifiable on the third passage for most stations, but on fewer stations for the fourth passage. There is always some amount of subjectivity in picking times on waveforms, and the conclusions reached in this paper depend on the interpretation of the waveforms in Figs. 7 and 8.

The stations recording the HTHH Lamb wave sampled the Earth more evenly than those recording the Krakatoa wave, which were concentrated in only a few regions, the majority in Europe and North America with a few exceptions. Two of these exceptions are Luanda (Angola) and Mauritius (Indian Ocean) with a fast delay (respectively 35.0 and 34.6 h for T3-T1) as expected for a path close to the equator. The measurements at these two stations agree well with the observations made in 2022 for HTHH, where stations SACV (34.9 h), IS52 (35.5 h), and IS51 (35.7 h) have very similar delays for T3-T1. Near-equatorial paths are consistently faster for the two sets of observations. This is to be expected since the wave travels in warm regions implying a faster speed. Furthermore, the winds are generally weaker in the equatorial area implying that the difference between the T4-T2 and T3-T1 delays are not very large, which is observed especially for HTHH. The Krakatoa observations at Luanda and Mauritius did differ by about 1 h, with T4-T2 being slower, which implies an easterly headwind for both these stations. Similar observations can be made for HTHH at SACV, where T4-T2 is 34.7, 0.2 h faster than T3-T1, also compatible with an encounter with easterly headwinds.

Stations sampling higher latitudes present more differences between odd and even passages. This is very clear for the North American stations in the case of Krakatoa, where the slowest paths are encountered for T3-T1, with differences of up to 5.7 h with T4-T2 for the Toronto station. Given that the polar paths sample the same latitude regions for both odd and even passages, the difference can be attributed to the effect of the winds in the troposphere. In the case of the North American stations, it would mean that the odd passages have encountered more head winds than the even passages and that these headwinds had a strong North–South component.

One of the initial motivations of this work was to potentially detect the effect of a global increase in tropospheric temperature between 1883 and 2022, affecting the speed of propagation of the Lamb wave. An order of magnitude estimation of the expected change in the delay between passages of the wave at a single station for a temperature increase of 1.5 degrees is detailed in Appendix E. A value of 0.1 h is derived from this estimation. This is well below the variances observed between different paths within both events, and to be able to detect this low level of decrease in the delays, one would need to perform a more detailed and elaborate modelling involving the measured tropospheric temperatures and wind strength and directions. Amores et al. (2022) performed such a numerical simulation using an average temperature model. They showed good general agreement with the arrival times, however differences with the observations may be explainable when using a more complete numerical simulation considering the actual meteorological observations for 15–18 January 2022, and adding wind effects which their model ignores. Sepúlveda et al. (2023) assess that these winds played the most important role in affecting the propagation and waveshapes. In addition, they consider topography as a third source of perturbation, although of more limited effect.

One important difference between the Krakatoa and HTHH eruptions is that they happened on different locations and at different times of the year (boreal summer for Krakatoa and boreal winter for HTHH). The location difference is only important because the sampling for Krakatoa is limited to a few limited regions that were instrumented at the time. For waves that sample the whole Earth, if the instrumentation is dense enough, the difference in location should not matter. The pressure wave between consecutive odd or even passages would sample the same great circle paths. The difference in seasons however is of more significance as the global wind patterns do vary between summer and winter, and it has been argued in this study, and others (Sepúlveda et al., 2023; Taylor, 1929) that wind patterns affect the propagation of the Lamb wave.

7 Conclusions

A comparison between the time delays between different passages of the Lamb waves triggered by the Krakatoa and HTHH eruptions lead to the conclusion that it is affected by both the temperature and wind patterns in the troposphere. Even though it is clear from examining the barograph recordings for the HTHH that the wave propagated multiple times around the Earth (see Fig. 6), one of the most challenging aspects of this comparison is to identify its passage at each individual station. Some general observations can be drawn which are common to both events:

• The near-equatorial paths (|ϕ| between 0.8 and 1) show remarkable stability between the odd and even passage delays and between the 1883 event and the 2022 event.

• Both events show that on average the dominant westerlies affect the paths with a large portion in the intermediate latitudes (values of |ϕ| between 0.2 and 0.8)

• The polar paths have large discrepancies between odd and even passages, for both events, probably due to winds with north–south components encountered on these paths.

Lamb waves are very low frequency and generated only very occasionally by volcanic events, large bolides, and large atmospheric nuclear explosions. Short of deconvolving BDF channels for the instrument response (including a high-pass filtering), as was done in Vergoz et al. (2022), GSN LDI channels perform well compared to the BDF channels used at IMS infrasound stations. Since the IMS was designed to detect all nuclear explosions, including large ones that would trigger Lamb waves, BDF channels are optimised for the IMS passband between 0.02 and 4 Hz. Based on these observations, adding one LDI channel per infrasound array could be a low-cost addition to the IMS infrasound network. These additional instruments would also help in determining the size of large natural event.

Appendices

Appendix A

1.1 Two Excerpts from Strachey et al. (1888) Relevant to Attributing the Delays Observed to Global Wind Patterns and Latitude-Dependent Temperatures

1.1.1 Excerpt One

“The difference of the velocities of the waves that travelled with and against the direction of the earth’s rotation amounts to about four-tenth of a degree, or 28 English miles per hour, and it may probably be accounted for by the circumstance that winds along the paths of this portion of the wave would, on the whole, have been westerly, which would have caused an increase of velocity in the wave moving with the earth’s rotation, and an equal diminution in that moving in the opposite direction, so that the observed difference of 28 miles could be produced by an average westerly current of 14 miles per hour, which is not unlikely.”

1.1.2 Excerpt Two

“There is some appearance of a greater retardation of the wave passing in a direction opposed to the earth’s rotation over the northern European stations as compared with those in the south of Europe, which may possibly be due to the lower temperature of the more northern part of the zone traversed.”

Appendix B

2.1 Station Coordinates, Distances to Sources and Pick Times

See Tables 2, 3, 4, 5 and Fig. 14 here.

Table 2 Station coordinates and distances from Krakatoa (from Strachey et al., 1888)

Table 3 Station coordinates, distances, and ∅ parameter

Table 4 GSN Stations and channels, time of picks for four Lamb wave passages, and time intervals between odd and even passages

Table 5 IMS Stations and channels, time of picks for four Lamb wave passages, and time intervals between odd and even passages

Figure 14

Location of the GSN (purple lines) and IMS (maroon line) stations used in this study on an azimuthal equidistant projection map centred on the HTHH event

Appendix C

3.1 Definition of Directional Parameters Used in the Analysis of the Krakatoa and HTHH Data

See Fig. 15 here.

The speed of propagation of an atmospheric wave is affected in the first order (among other environmental parameters) by two important parameters: the temperature of the atmosphere, and the angle the propagation makes with the wind direction. Temperature is sensitive to the latitude, and the first parameter computed from the configuration of the stations and volcano locations is the latitude sampling of the great circle along which the pressure wave propagates. The lower the temperature, the slower the propagation. Thus, it would be expected that both the latitudes and the season at which the event occurred would influence the propagation time between two consecutive passages at the same station. The wind direction encountered is strongly dependent on the amount of East–West direction of the propagation. The propagation will be slowed down by head winds and aided by tail winds. This paragraph describes a very basic approach (method) to estimate the effect of two parameters indicative of the latitude of the propagation and the direction of propagation compared to the East–West direction.

3.2 Parameter ∅

If K is the unit vector pointing to Krakatoa in the coordinate system and A the vector pointing to the station, C is the cross product of K × A, and γ the angle between ζ and C. The cosine of γ, ϕ, is the main parameter in the analysis conducted in this work.

Using the coordinate system (X, Y, Z) of the unit vectors at the centre of the Earth pointing to the (lat 0$^\circ$, lon 0$^\circ$) point, the (lat 0$^\circ$, lon 90$^\circ$) point, and the (lat 90$^\circ$, lon 0$^\circ )$, X = $\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]$, Y = $\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]$, and Z = $\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$. Point K given by (lat μ, lon υ) can be written in that coordinate system as: $\left[\begin{array}{c}\text{cos}\left(\nu \right)\text{cos}\left(\mu \right)\\ \text{sin}\left(\nu \right)\text{cos}\left(\mu \right)\\ \text{sin}\left(\mu \right)\end{array}\right]$, and A (lat α, lon β), as$\left[\begin{array}{c}\text{cos}\left(\beta \right)\text{cos}\left(\alpha \right)\\ \text{sin}\left(\beta \right)\text{cos}\left(\alpha \right)\\ \text{sin}\left(\alpha \right)\end{array}\right]$.

If C is the vector cross product of K and A (C = K × A), the normalized dot product of C with the Z vector is the cosine of the angle between the polar direction and the normal to the great circle joining the two points on the sphere:

$\text{cos }\left(\upgamma \right)=\frac{\text{C}.\text{Z}}{\Vert \text{C}\Vert },$ where Z = $\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right].$ This can be used as a substitute for the proportion of east–west versus north–south propagation along the great circle. This cosine is expressed as:

$$\upphi =\text{ cos }(\upgamma ) = \left[\text{cos}\left(\nu \right)\text{cos}\left(\mu \right)\text{sin}\left(\beta \right)\text{cos}\left(\alpha \right)-\text{cos}\left(\beta \right)\text{cos}\left(\alpha \right)\text{sin}\left(\nu \right)\text{cos}\left(\mu \right)\right]/\Vert \text{C}\Vert$$

Figure 15

Illustrating sketch of the spherical geometry simplification used to derive parameter ∅. The notations used are the same as in the text. ∅ is an indicator of the inclination of the great circle with respect to the polar axis. A small value for ∅ indicates that the path is nearly equatorial, and a large value that it is close to polar. A positive value for ∅ means that the shorter great circle path from Krakatoa to the station is towards the East, a negative value that it is towards the West

Appendix D

4.1 Time Delay Due to Latitude-Dependent Temperature Around a Great Circle Parametrized by its Inclination Angle (γ)

See Eqs. 3and4 here.

Using the square root relationship between acoustic velocity and temperature in the range of temperatures appropriate for the troposphere (Kinsler et al., 2000), $v =20 \sqrt{\theta }$, and the relationship derived from fitting the CRU (Feulner et al., 2013; Jones et al., 1999): θ = K₀ −  + 45 cos(λ) is the absolute temperature in °K, K₀ = 273 °K, corresponding to 0 °C, v is in m/s, and λ is the latitude in degrees. Along an elementary portion, ds, of the great circle parametrized by γ, the difference in travel times is:

$dT=\frac{R}{20} \left[\frac{1}{\sqrt{{K}_{0}+45*\text{cos}\left(\lambda +d\lambda \right)}}-\frac{1}{\sqrt{{K}_{0}+45*\text{cos}\left(\lambda \right)}}\right]$, where R is the radius of the Earth, and ds = 2πRdλ.

$$dT=\frac{2\pi R}{20}\left(-\frac{45}{2}\right)\left[\frac{\text{sin}\left(\lambda \right)d\lambda }{\sqrt{{K}_{0}+45*\text{cos}\left(\lambda \right)}}\right]$$

Integrating along the whole great circle with inclination γ leads to four times the value between 0 and π/2 with inclination γ and substituting λ with x = cos(λ):

$$T\left(\gamma \right)= -4\frac{2\pi R}{20}\frac{45}{2}\frac{1}{\sqrt{{K}_{0}}}{\int }_{0}^{\frac{\pi }{2}}\frac{\text{sin}\left(\lambda \right)d\lambda }{\sqrt{1+\frac{45}{{K}_{0}}\text{cos}\left(\lambda \right)}}=-8\frac{R\sqrt{{K}_{0}}}{40}{\left[{\left(1+\frac{45}{{K}_{0}}x\right)}^\frac{1}{2}\right]}_{1}^{\text{cos}\left(\gamma \right)}$$

$$T\left(\Phi \right)=- 2\pi R\frac{\sqrt{{K}_{0}}}{5} \left( \sqrt{1+\Phi \frac{45}{{K}_{0}}}- \sqrt{1+\frac{45}{{K}_{0}}}\right)$$

(3)

The value for ϕ = 0 is:

$${T}_{0}=-2\pi R \frac{\sqrt{{K}_{0}}}{5}\left( 1-\sqrt{1+\frac{45}{{K}_{0}}}\right),$$

The difference in travel time between the great circle parametrized by φ and the equatorial travel time (φ = 1) is:

$$\Delta T \left(\Phi \right)=2\pi R \frac{\sqrt{{K}_{0}}}{5} \left[\left( \sqrt{1+\Phi \frac{45}{{K}_{0}}} \right)-1\right]$$

(4)

Note that Taylor (1929) used the assumption that the temperature varies like (1− cos (2*λ)), however a cosine law for the temperature has a good fit to the modern observations of Feulner et al. (2013), and equation (4) allows a direct comparison with the data.

Appendix E

5.1 Delay Due to the Global Pattern of Westerlies Around a Great Circle Parametrized by its Inclination Angle (γ)

Starting with the elementary segment of path ds along the great circle, an estimation of the delay on ds is:

$$dT =\text{cos}\left(\gamma \right)\left[\frac{1}{v\left(\lambda +d\lambda \right)} - \frac{1}{v\left(\lambda \right)}\right]$$

$$dT=\frac{cos\left(\gamma \right)}{{V}_{0}}\left[\frac{-dv/d\lambda }{{v}^{2}\left(\lambda \right)} \right]d\lambda$$

$$dT=\frac{cos\left(\gamma \right)}{{V}_{0}}\left[\frac{2c\text{cos}\left(\lambda \right)}{{\left(1+c\text{sin}\left(2\lambda \right)\right)}^{2}} \right]d\lambda$$

where v is assumed to be of the form $v = V_{0} \left[ {1 + c\sin \left( {2\lambda } \right)} \right].$ Integrating over the great circle with circumference R, and considering the symmetries:

$$\Delta T\left(\gamma \right)= \frac{2\pi R cos\left(\gamma \right)}{{V}_{0}} {\int }_{0}^{\gamma }\left[\frac{2c\text{cos}\left(\lambda \right)}{{\left(1+c\text{sin}\left(2\lambda \right)\right)}^{2}} \right]d\lambda =- \frac{2\pi R cos\left(\gamma \right)}{{V}_{0}} {\left[\frac{1}{1+c\text{sin}\left(2\lambda \right)} \right]}_{0}^{\gamma }$$

where R is the circumference of the Earth. Setting x = cos(λ) and expanding sin(2λ), the following function of γ is obtained:

$$\Delta T\left(\gamma \right)=\frac{2\pi R }{{V}_{0}} cos\left(\gamma \right)\left[1 - \frac{1}{ 1+c\text{sin}\left(2\gamma \right)}\right] = \frac{2\pi R }{{V}_{0}} cos\left(\gamma \right)\left[1 - \frac{1}{ 1+\left.2c cos\left(\gamma \right.\right)\text{sin}\left(\gamma \right)}\right]$$

Considering the sign of ϕ, the delay is negative when the wind is in the direction of the propagation, and positive otherwise. The magnitude of the delay is the same for positive or negative ϕ:

$$\Delta T\left(\phi \right)=\frac{2\pi R }{{V}_{0}} \phi \left[1-\frac{1}{ 1+2c \left|\phi \right|\sqrt{1-{\phi }^{2}}}\right]$$

Appendix F

6.1 Estimate of a Temperature Difference Explaining a Velocity Difference

Assuming that the velocity is proportional to the square root of the absolute temperature, and assuming that the difference in propagation time between Krakatoa and HTHH for the time differences between the third (III) and first (I) passage of the Lamb wave is entirely due to a difference in temperature, we obtain the following relationships, where v is the average velocity, and q the average air temperature:

$$\frac{\Delta v}{v} = \frac{1}{2}\frac{\Delta \theta }{\theta }$$

If the travel time between III and I is called T, and the distance between the two passages X:

$${\text{T}} = \frac{{\text{X}}}{{\text{v}}}$$

$$\Delta T = - X\frac{\Delta v}{{v^{2} }} = - T\frac{\Delta v}{v} = - \frac{1}{2}\frac{\Delta \theta }{\theta }$$

$$\frac{\Delta \theta }{\theta } = - 2\frac{\Delta T}{T}$$

For a value of the global difference in atmospheric temperature of 1.5 °C expected between 1883 and 2022 and using 36 h as a nominal difference, the delay change estimate is, at a temperature of 273 K:

$$\Delta T = - 36*\frac{1.5}{{273}} = 0.1{\text{ hour}}$$