travel time curves

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Travel times ¶

A seismic wave travelling through an isotropic homogeneous medium will propagate at a constant velocity. Therefore, the time \(t\) required for a seismic wave to travel from source to receiver in a homogeneous earth layer with velocity \(v\) is simply given by the formula

where \(d\) is the distance travelled in the layer. In a seismic survey we measure source to receiver travel times and use those data to estimate the properties of the subsurface. Basic seismic interpretation methods assume that the earth is composed of a series of uniform layers and attempt to compute the thicknesses, velocities, and sometimes dips of each layer. We will discuss specific techniques for computing layer thicknesses and velocities in the reflection and refraction survey sections. However, we will introduce the concept of travel time computations and how they relate to geometry here, using the example of a two layered earth.

Consider a layer of thickness h and velocity \(v_1\) overlying a uniform halfspace of velocity \(v_2\) . A source is detonated at time \(t=0\) . We are interested in the waves and arrival times of those waves at a receiver which is located at a distance \(x\) from the source at position \(D\) in the figure below. There are three principal waves that will travel through the earth and arrive at position D. i) direct waves, ii) reflected waves, and iii) critically refracted waves.

The travel time curves for these ray paths are shown below, where the horizontal axis represents distance from the source along the flat surface of the earth, \(x_{crit}\) is called the critical distance, and \(x_{cross}\) the crossover distance. The critical distance is the closest surface point to the source at which the refracted ray can be observed. The crossover distance is the surface point at which the direct and refracted rays arrive at the same time. At offsets from the source greater than the crossover distance the refracted ray will be the first signal to arrive from the source. You can explore these concepts interactively using the Science seismic refraction app . Instructions for using the app can be found here . The following video illustrates the propagation of headwaves and how they relate to the crossover distance

Travel times of visible arrivals are related to the distance between source and receiver ( \(x\) ), thickness of the layer ( \(h\) ) and the wave velocities in the upper layer and basement ( \(v_1\) and \(v_2\) ). Let us denote the arrival times at point \(x\) for the direct, reflected and refracted waves as \(t_{dir}\) , \(t_{refl}\) \(t_{refr}\) respectively. These times are given by the following formulas

Note that the formulas for the direct and refracted waves are linear in \(x\) but that the reflected arrival time formula is not.

Before moving on, let’s look at an example of how travel times show up in the field. The horizontal axis of the following plot shows offset from a seismic receiver. Each line plots the displacement vs time curve of a geophone at a given offset. The plot clearly shows a set of events moving linearly in time from one geophone to the next.

We will now discuss the computation of traveltimes in more detail. It is important to note here that computing traveltimes for an arbitrary, heterogeneous earth is a complex problem well beyond the scope of this course. However, much insight can be gained by assuming that the subsurface consists of a series of homogeneous layers with horizontal or possibly dipping interfaces.

Refracted ray in a two layered-earth ¶

We need to identify specific ray paths and their associated travel times. Consider an earth composed of a uniform layer with velocity \(v_1\) and thickness \(z\) overlying a medium with velocity \(v_2\) . Let \(\theta\) be the critical angle and x denoted the distance between the source at \(A\) and a receiver at \(D\) . Let \(x_c\) denote the critical distance.

From elementary geometry the following relationships hold:

The travel time is the cumulative time for the wave to traverse the path \(ABCD\) . This is \(t=t_{AB}+t_{BC}+t_{CD}\) .

Generally time = distance / velocity, so we can write \(t_{AB} = L/v_1 = (z/cos\theta) / v1\) , (using \(L\) from just above).

Also, we can note that \(t_{AB} = t_{CD}\) and the distance \(BC\) is \(x-x_c\) . So we can now state that \(t=2t_{AB}+t_{BC}\) , or

It is convenient to rearrange this slightly differently. Using the definition for critical angle \(\sin\theta=v_1/v_2\) , we can make the “velocity triangle”, so expressions for the angle arise directly from simple trigonometry:

Use these two relations for \(\cos\) and \(\tan\) in the expression for t above to obtain a useful set of relations.

This simple relation says that the travel time curve is a straight line which has a slope of \(1/v_2\) and an intercept of \(t_i\) . This intercept time is the time where the refraction line extends to intercept the \(y\) -axis –above the source position–. This is not a real “time” - it is derived from the graph.

The velocities of the seismic layers and the layer thickness are obtained in the following manner.

Plot the times of first arrivals on an time-offset plot (“offset” is distance between source and geophone).

The direct arrivals are observed to lie along a straight line joining the origin. The slope of this line is \(1/v_1\) , giving the velocity of the upper layer.

The refracted arrivals appear as a straight line with smaller slope equal to \(1/v_2\) , giving the velocity of the lower layer.

For the refracted wave, this intercept time is

We therefore can obtain all three useful parameters about the earth, \((v_1, z, v_2)\) .

There is another useful point that is observable from the first arrival travel-time plot. We can often discern the crossover distance. Since this is the location where the direct wave and the refracted wave arrive at the same time, we can write

This can be used as a consistency check, or it can be used to compute one of the variables given values for two others.

Refracted ray in a three layered-earth ¶

The extension to more layers is in principle straight forward. Snell’s law holds for waves at all interfaces, so for a multi-layered medium

For a three layer case, the algebra is slightly more involved compared to a two layer example because we need to compute the times due to the ray path segments in the two top layers. Consider the diagrams below:

Using arguments that are entirely analagous to the two layer case (above) the travel time for the wave refracted at the top of layer three is given by

All quantities are defined in the diagrams, and the angles are

Note that \(\theta_2\) is a critical angle while \(\theta_1\) is not. You can prove the relation for \(\theta_1\) yourself by using Snell’s law at the two interfaces, and recalling that the angle of the ray coming from point \(B\) is the same as the angle arriving at point \(C\) . The straight line that corresponds to an individual refractor provides a velocity (from its slope) and a thickness (from the intercept). Thus the information on the above travel-time plot allows us to recover all three velocities and the thickness of both layers.

The travel time curves for multi layers are obtained from obvious extension of the above formulation.

Reflected rays - single layer ¶

Consider the situation to the right, in which there is a source \(S\) and a set of receivers on the surface of the earth. The earth is a single uniform layer overlying a uniform halfspace. A reflection from the interface will occur if there is a change in the acoustic impedance at the boundary.

Let \(x\) denote the “offset” or distance from the source to the receiver. The time taken for the seismic energy to travel from the source to the receiver is given by

\[t = \frac{(x^2 + 4z^2)^\frac{1}{2}}{v}\]

This is the equation of a hyperbola. In seismic reflection (as in radar) we plot time on the negative vertical axis, and so the seismic section (without the source wavelet) would look like.

Two way travel time:

Normal Moveout:

In the above diagram \(t_0\) is the 2-way vertical travel time. It is the minimum time at which a reflection will be recorded. The additional time taken for a signal to reach a receiver at offset \(x\) is called the “Normal Moveout” time, \(\Delta t\) . This value is required for every trace in the common depth point data set in order to shift echoes up so they align for stacking. How is it obtained? First let us find a way of determining velocity and \(t_0\) .

For this simple earth structure the velocity and layer thickness can readily be obtained from the hyperbola. Squaring both sides yields,

This is the equation of a straight line when \(t^2\) is plotted against \(x^2\) . Now, to find \(\Delta T\) , we must rearrange this hyperbolic equation relating \(t_0\) , \(x\) , the \(Tx\) – \(Rx\) offset, \(t\) at \(x\) , or \(t(x)\) , and the ground’s velocity, \(v\) .

Apply binomial expansion to get

Now, since normal moveout is \(\Delta T = t_x - t_0\)

The algebra has only one complicated step–a binomial expansion must be applied to obtain a simple relation without square roots etc.

The approximation is valid so long as the source-receiver separation (or offset) is “small” which means much less than the vertical depth to the reflecting layer (i.e. \(x << vt_0\) ). The result is a simple expression for normal moveout.

Each echo can be shifted up to align with the \(t_0\) position, so long as the trace position, \(x\) , the vertical incident travel time, \(t_0\) , and the velocity are known. Velocity can be estimated using the slope of the \(t^2\) – \(x^2\) plot, or with several other methods, which we will discuss in pages following.

Travel Time Curves for Multiple Layers ¶

If there are additional layers then the seismic energy at each interface is refracted according to Snell’s Law. The energy no longer travels in a straight line and hence the travel times are affected. It is observed that for small offsets, the travel time curve is still approximately hyperbolic, but the velocity, which controls the shape of the curve, is an “average” velocity determined from the velocities of all the layers above the reflector. The velocity is called the RMS (Root Mean Square) velocity, \(v_{rms}\) .

The complex travel path of a reflected ray through a multilayered ground. (b) The time–distance curve for reflected rays following the above type of path. Note that the divergence from the hyperbolic travel-time curve for a homogeneous overburden of velocity Vrms increases with offset.

As outlined in the figure above, the reflection curve for small offsets is still like a hyperbola, but the associated velocity is \(v_{rms}\) , not a true interval velocity.

For each hyperbola:

By fitting hyperbolas to each reflection event one can obtain \(t_n,v_n^{rms}\) for n = 1, 2, … The interval velocity and layer thickness of each layer can be found using the formulae below:

These formulae for the interval velocity and thickness of the \(n^{th}\) layer are directly obtainable from the definition of \(v_n^{rms}\) given above. The RMS velocity for the \(n^{th}\) layer is given by:

where \(v_i\) is the velocity of the \(i^{th}\) layer, and \(\tau_i\) is the one-way travel time through the \(i^{th}\) layer.

• Traveltime curves how they are created
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Travel-time Curves: How they are created

• Closed Captions (.srt)
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• GIF Travel time short (Excerpt from animation) Your browser does not support the video tag.
• GIF Travel time Curve (Excerpt from animation) Your browser does not support the video tag.

Where do travel-time graphs come from?

A travel time curve is a graph of the time that it takes for seismic waves to travel from the epicenter of an earthquake to seismograph stations at varying distances away. The velocity of seismic waves through different materials yield information about Earth's deep interior. IRIS' travel times graphic for the 1994 Northridge, CA earthquake (described in No.5. Exploring the Earth Using Seismology) is animated to show how travel times are determined. Seismic waves "bounce" the buildings to merely illustrate arrival times and wave behavior, not to depict reality. The resultant seismograms show that stations around the world record somewhat predictable arrival times.

CLOSED CAPTIONING: A .srt file is included with the download. Use an appropriate media player to utilize captioning.

• Time vs. distance graph of P and S seismic waves
• Seismic waves recorded at different points are a function of distance from the earthquake
• Seismic velocities within the earth have been calculated from these data

Related Animations

Seismic waves travel through the earth to a single seismic station. Scale and movement of the seismic station are greatly exaggerated to depict the relative motion recorded by the seismogram as P, S, and surface waves arrive.

We use exaggerated motion of a building (seismic station) to show how the ground moves during an earthquake, and why it is important to measure seismic waves using 3 components: vertical, N-S, and E-W. Before showing an actual distant earthquake, we break down the three axes of movement to clarify the 3 seismograms. 

A cow and a tree in this narrated cartoon for fun and to emphasize that seismic waves traveling away from an earthquake occur everywhere, not just at seismic stations A, B, C, and D. A person would feel a large earthquake only at station A near the epicenter. Stations B, C, D, and the cow are too far from the earthquake to feel the seismic waves though sensitive equipment records their arrival.

A gridded sphere is used to show a single station recording five equidistant earthquakes.

A gridded sphere is used to show: 1) the seismic stations don't need to be lined up longitudinally to create travel-time curves,       as they appear in the first animation, and 2) a single station records widely separated earthquakes that plot on the travel-time curves.

The shadow zone is the area of the earth from angular distances of 104 to 140 degrees from a given earthquake that does not receive any direct P waves. The different phases show how the initial P wave changes when encountering boundaries in the Earth.

The shadow zone results from S waves being stopped entirely by the liquid core. Three different S-wave phases show how the initial S wave is stopped (damped), or how it changes when encountering boundaries in the Earth. 

Related Fact-Sheets

Earthquakes create seismic waves that travel through the Earth. By analyzing these seismic waves, seismologists can explore the Earth's deep interior. This fact sheet uses data from the 1994 magnitude 6.9 earthquake near Northridge, California to illustrate both this process and Earth's interior structure.

NOTE: Out of Stock; self-printing only.

Related Software-Web-Apps

jAmaSeis is a free, java-based program that allows users to obtain and display seismic data in real-time from either a local instrument or from remote stations.

Seismic Waves is a browser-based tool to visualize the propagation of seismic waves from historic earthquakes through Earth’s interior and around its surface. Easy-to-use controls speed-up, slow-down, or reverse the wave propagation. By carefully examining these seismic wave fronts and their propagation, the Seismic Waves tool illustrates how earthquakes can provide evidence that allows us to infer Earth’s interior structure.

Related Videos

Video lecture on wave propagation and speeds of three fundamental kinds of seismic waves.

A video demonstration of how a slinky can be a good model for illustrating P & S seismic waves movement.

Related Lessons

Learning occurs as students work first in small groups and then as a whole class to compare predicted seismic wave travel times, generated by students from a scaled Earth model, to observed seismic data from a recent earthquakes. This activity uses models, real data and emphasizes the process of science.

To understand plate tectonic processes and hazards, and to better understand where future earthquakes are likely to occur, it is important to locate earthquakes as they occur. In this activity students use three-component seismic data from recent earthquakes to locate a global earthquake.

In this multi-step lab, students explore the concepts of seismic wave propagation through materials with different mechanical properties, and examine seismic evidence from a recent earthquake to infer Earth’s internal structure and composition.  This lab is designed to be done with an instructor present to answer questions and guide students to conclusions

Related Posters

Seismic waves from earthquakes ricochet throughout Earth's interior and are recorded at geophysical observatories around the world. The paths of some of those seismic waves and the ground motion that they caused are used by seismologists to illuminate Earth's deep interior.

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• Travel time curves described

Travel Time Curves Described

How can we calculate the distance to an epicenter?

Related animations.

A travel time curve is a graph of the time that it takes for seismic waves to travel from the epicenter of an earthquake to the hundreds of seismograph stations around the world. The arrival times of P, S, and surface waves are shown to be predictable. This animates an IRIS poster linked with the animation.

A cow and a tree in this narrated cartoon for fun and to emphasize that seismic waves traveling away from an earthquake occur everywhere, not just at seismic stations A, B, C, and D. A person would feel a large earthquake only at station A near the epicenter. Stations B, C, D, and the cow are too far from the earthquake to feel the seismic waves though sensitive equipment records their arrival.

We use exaggerated motion of a building (seismic station) to show how the ground moves during an earthquake, and why it is important to measure seismic waves using 3 components: vertical, N-S, and E-W. Before showing an actual distant earthquake, we break down the three axes of movement to clarify the 3 seismograms. 

Seismic waves travel through the earth to a single seismic station. Scale and movement of the seismic station are greatly exaggerated to depict the relative motion recorded by the seismogram as P, S, and surface waves arrive.

A gridded sphere is used to show: 1) the seismic stations don't need to be lined up longitudinally to create travel-time curves,       as they appear in the first animation, and 2) a single station records widely separated earthquakes that plot on the travel-time curves.

The Earth has 3 main layers based on chemical composition: crust, mantle, and core. Other layers are defined by physical characteristics due to pressure and temperature changes. This animation tells how the layers were discovered, what the layers are, and a bit about how the crust differs from the tectonic (lithospheric) plates, a distinction confused by many.

Related Lessons

To understand plate tectonic processes and hazards, and to better understand where future earthquakes are likely to occur, it is important to locate earthquakes as they occur. In this activity students use three-component seismic data from recent earthquakes to locate a global earthquake.

We encourage the reuse and dissemination of the material on this site as long as attribution is retained. To this end the material on this site, unless otherwise noted, is offered under Creative Commons Attribution ( CC BY 4.0 ) license

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Earthquake Travel Times

High resolution graph in  PDF format  (1MB)

Travel Time Curves

Table of p and s-p versus distance.

P and S-P travel times as a function of source distance for an earthquake 33 km deep. The Time of the first arriving P phase is given, along with the time difference between the S and P phases. The latter time is known as the S minus P time.

This table is based on the iasp91 model of Kennett and Engdahl (1991) and was generated with the program ARTIM written by R. Buland.

Kennett, B. L. N. and E. R. Engdahl (1991). Travel times for global earthquake location and phase identification, Geophys. J. Int., v 105, p 429-465.

• API Overview
• obspy.taup - Ray theoretical travel times and paths
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obspy.taup - Ray theoretical travel times and paths 

The ObsPy Development Team ( devs @ obspy . org )

GNU Lesser General Public License, Version 3 ( https://www.gnu.org/copyleft/lesser.html )

This package started out as port of the Java TauP Toolkit by [Crotwell1999] so please look there for more details about the algorithms used and further information. It can be used to calculate theoretical arrival times for arbitrary seismic phases in a 1D spherically symmetric background model. Furthermore it can output ray paths for all phases and derive pierce points of rays with model discontinuities.

Basic Usage 

Let’s start by initializing a TauPyModel instance. Models can be initialized by specifying the name of a model provided by ObsPy.

ObsPy currently ships with the following 1D velocity models:

1066a , see [GilbertDziewonski1975]

1066b , see [GilbertDziewonski1975]

ak135 , see [KennetEngdahlBuland1995]

ak135f , see [KennetEngdahlBuland1995] , [MontagnerKennett1995] , and http://rses.anu.edu.au/seismology/ak135/ak135f.html (not supported)

ak135f_no_mud , ak135f with ak135 used above the 120-km discontinuity; see the SPECFEM3D_GLOBE manual at https://specfem3d-globe.readthedocs.io/en/latest/

herrin , see [Herrin1968]

iasp91 , see [KennetEngdahl1991]

jb , see [JeffreysBullen1940]

prem , see [Dziewonski1981]

pwdk , see [WeberDavis1990]

sp6 , see [MorelliDziewonski1993]

Custom built models can be initialized by specifying an absolute path to a model in ObsPy’s .npz model format instead of just a model name. Model initialization is a fairly expensive operation so make sure to do it only if necessary. See below for information on how to build a .npz model file.

Travel Times 

The models’ main method is the get_travel_times() method; as the name suggests it returns travel times for the chosen phases, distance, source depth, and model. By default it returns arrivals for a number of phases.

If you know which phases you are interested in, you can also specify them directly which speeds up the calculation as unnecessary phases are not calculated. Please note that it is possible to construct any phases that adhere to the naming scheme which is detailed later.

Each arrival is represented by an Arrival object which can be queried for various attributes.

Ray Paths 

To also calculate the paths travelled by the rays to the receiver, use the get_ray_paths() method.

The result is a NumPy record array containing ray parameter, time, distance and depth to use however you see fit.

Pierce Points 

If you only need the pierce points of ray paths with model discontinuities, use the get_pierce_points() method which results in pierce points being stored as a record array on the arrival object.

If ray paths have been calculated, they can be plotted using the Arrivals.plot_rays() method:

( Source code , png )

Plotting will only show the requested phases:

Additionally, Cartesian coordinates may be used instead of a polar grid:

Travel times for these ray paths can be plotted using the Arrivals.plot_times() method:

Alternatively, convenience wrapper functions plot the arrival times and the ray paths for a range of epicentral distances.

The travel times wrapper function is plot_travel_times() , creating the figure and axes first is optional to have control over e.g. figure size or subplot setup:

The ray path plot wrapper function is plot_ray_paths() . Again, creating the figure and axes first is optional to have control over e.g. figure size or subplot setup (note that a polar axes has to be set up when aiming to do a plot with plot_type='spherical' and a normal matplotlib axes when aiming to do a plot with plot_type='cartesian' . An error will be raised when mixing the two options):

More examples of plotting may be found in the ObsPy tutorial .

Phase naming in obspy.taup 

This section is a modified copy from the Java TauP Toolkit documentation so all credit goes to the authors of that.

A major feature of obspy.taup is the implementation of a phase name parser that allows the user to define essentially arbitrary phases through a planet. Thus, obspy.taup is extremely flexible in this respect since it is not limited to a pre-defined set of phases. Phase names are not hard-coded into the software, rather the names are interpreted and the appropriate propagation path and resulting times are constructed at run time. Designing a phase-naming convention that is general enough to support arbitrary phases and easy to understand is an essential and somewhat challenging step. The rules that we have developed are described here. Most of the phases resulting from these conventions should be familiar to seismologists, e.g. pP , PP , PcS , PKiKP , etc. However, the uniqueness required for parsing results in some new names for other familiar phases.

In traditional “whole-Earth” seismology, there are 3 major interfaces: the free surface, the core-mantle boundary, and the inner-outer core boundary. Phases interacting with the core-mantle boundary and the inner core boundary are easy to describe because the symbol for the wave type changes at the boundary (i.e., the symbol P changes to K within the outer core even though the wave type is the same). Phase multiples for these interfaces and the free surface are also easy to describe because the symbols describe a unique path. The challenge begins with the description of interactions with interfaces within the crust and upper mantle. We have introduced two new symbols to existing nomenclature to provide unique descriptions of potential paths. Phase names are constructed from a sequence of symbols and numbers (with no spaces) that either describe the wave type, the interaction a wave makes with an interface, or the depth to an interface involved in an interaction.

P - compressional wave, upgoing or downgoing; in the crust or mantle, p is a strictly upgoing P -wave in the crust or mantle

S - shear wave, upgoing or downgoing, in the crust or mantle

s - strictly upgoing S -wave in the crust or mantle

K - compressional wave in the outer core

I - compressional wave in the inner core

J - shear wave in the inner core

m - interaction with the Moho

g appended to P or S - ray turning in the crust

n appended to P or S - head wave along the Moho

c - topside reflection off the core mantle boundary

i - topside reflection off the inner core outer core boundary

ˆ - underside reflection, used primarily for crustal and mantle interfaces

v - topside reflection, used primarily for crustal and mantle interfaces

diff appended to P or S - diffracted wave along the core mantle boundary; appended to K - diffracted wave along the inner-core outer-core boundary

kmps appended to a velocity - horizontal phase velocity (see 10 below)

ed appended to P or S - an exclusively downgoing path, for a receiver below the source (see 3 below)

The characters p and s always represent up-going legs. An example is the source to surface leg of the phase pP from a source at depth. P and S can be turning waves, but always indicate downgoing waves leaving the source when they are the first symbol in a phase name. Thus, to get near-source, direct P -wave arrival times, you need to specify two phases p and P or use the “ ttimes compatibility phases” described below. However, P may represent a upgoing leg in certain cases. For instance, PcP is allowed since the direction of the phase is unambiguously determined by the symbol c , but would be named Pcp by a purist using our nomenclature.

With the ability to have sources at depth, there is a need to specify the difference between a wave that is exclusively downgoing to the receiver from one that turns and is upgoing at the receiver. The suffix ed can be appended to indicate exclusively downgoing. So for a source at 10 km depth and a receiver at 20 km depth at 0 degree distance P does not have an arrival but Ped does.

Numbers, except velocities for kmps phases (see 10 below), represent depths at which interactions take place. For example, P410s represents a P -to- S conversion at a discontinuity at 410km depth. Since the S -leg is given by a lower-case symbol and no reflection indicator is included, this represents a P -wave converting to an S -wave when it hits the interface from below. The numbers given need not be the actual depth; the closest depth corresponding to a discontinuity in the model will be used. For example, if the time for P410s is requested in a model where the discontinuity was really located at 406.7 kilometers depth, the time returned would actually be for P406.7s . The code would note that this had been done. Obviously, care should be taken to ensure that there are no other discontinuities closer than the one of interest, but this approach allows generic interface names like “410” and “660” to be used without knowing the exact depth in a given model.

If a number appears between two phase legs, e.g. S410P , it represents a transmitted phase conversion, not a reflection. Thus, S410P would be a transmitted conversion from S to P at 410km depth. Whether the conversion occurs on the down-going side or up-going side is determined by the upper or lower case of the following leg. For instance, the phase S410P propagates down as an S , converts at the 410 to a P , continues down, turns as a P -wave, and propagates back across the 410 and to the surface. S410p on the other hand, propagates down as a S through the 410, turns as an S -wave, hits the 410 from the bottom, converts to a p and then goes up to the surface. In these cases, the case of the phase symbol ( P vs. p ) is critical because the direction of propagation (upgoing or downgoing) is not unambiguously defined elsewhere in the phase name. The importance is clear when you consider a source depth below 410 compared to above 410. For a source depth greater than 410 km, S410P technically cannot exist while S410p maintains the same path (a receiver side conversion) as it does for a source depth above the 410. The first letter can be lower case to indicate a conversion from an up-going ray, e.g., p410S is a depth phase from a source at greater than 410 kilometers depth that phase converts at the 410 discontinuity. It is strictly upgoing over its entire path, and hence could also be labeled p410s . p410S is often used to mean a reflection in the literature, but there are too many possible interactions for the phase parser to allow this. If the underside reflection is desired, use the pˆ410S notation from rule 7.

Due to the two previous rules, P410P and S410S are over specified, but still legal. They are almost equivalent to P and S , respectively, but restrict the path to phases transmitted through (turning below) the 410. This notation is useful to limit arrivals to just those that turn deeper than a discontinuity (thus avoiding travel time curve triplications), even though they have no real interaction with it.

The characters ˆ and v are new symbols introduced here to represent bottom-side and top-side reflections, respectively. They are followed by a number to represent the approximate depth of the reflection or a letter for standard discontinuities, m , c or i . Reflections from discontinuities besides the core-mantle boundary, c , or inner-core outer-core boundary, i , must use the ˆ and v notation. For instance, in the TauP convention, pˆ410S is used to describe a near-source underside reflection. Underside reflections, except at the surface ( PP , sS , etc.), core-mantle boundary ( PKKP , SKKKS , etc.), or outer-core-inner-core boundary ( PKIIKP , SKJJKS , SKIIKS , etc.), must be specified with the ˆ notation. For example, Pˆ410P and PˆmP would both be underside reflections from the 410km discontinuity and the Moho, respectively. The phase PmP , the traditional name for a top-side reflection from the Moho discontinuity, must change names under our new convention. The new name is PvmP or Pvmp while PmP just describes a P -wave that turns beneath the Moho. The reason why the Moho must be handled differently from the core-mantle boundary is that traditional nomenclature did not introduce a phase symbol change at the Moho. Thus, while PcP makes sense since a P -wave in the core would be labeled K , PmP could have several meanings. The m symbol just allows the user to describe phases interaction with the Moho without knowing its exact depth. In all other respects, the ˆ - v nomenclature is maintained.

Currently, ˆ and v for non-standard discontinuities are allowed only in the crust and mantle. Thus there are no reflections off non-standard discontinuities within the core, (reflections such as PKKP , PKiKP and PKIIKP are still fine). There is no reason in principle to restrict reflections off discontinuities in the core, but until there is interest expressed, these phases will not be added. Also, a naming convention would have to be created since “ p is to P ” is not the same as “ i is to I ”.

Currently there is no support for PKPab , PKPbc , or PKPdf phase names. They lead to increased algorithmic complexity that at this point seems unwarranted. Currently, in regions where triplications develop, the triplicated phase will have multiple arrivals at a given distance. So, PKPab and PKPbc are both labeled just PKP while PKPdf is called PKIKP .

The symbol kmps is used to get the travel time for a specific horizontal phase velocity. For example, 2kmps represents a horizontal phase velocity of 2 kilometers per second. While the calculations for these are trivial, it is convenient to have them available to estimate surface wave travel times or to define windows of interest for given paths.

As a convenience, a ttimes phase name compatibility mode is available. So ttp gives you the phase list corresponding to P in ttimes . Similarly there are tts , ttp+ , tts+ , ttbasic and ttall .

Building custom models 

Custom models can be built from .tvel and .nd files using the build_taup_model() function.

Classes & Functions 

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Paradox-Free Time Travel Is Theoretically Possible, Researchers Say

Matthew S. Schwartz

A dog dressed as Marty McFly from Back to the Future attends the Tompkins Square Halloween Dog Parade in 2015. New research says time travel might be possible without the problems McFly encountered. Timothy A. Clary/AFP via Getty Images hide caption

A dog dressed as Marty McFly from Back to the Future attends the Tompkins Square Halloween Dog Parade in 2015. New research says time travel might be possible without the problems McFly encountered.

"The past is obdurate," Stephen King wrote in his book about a man who goes back in time to prevent the Kennedy assassination. "It doesn't want to be changed."

Turns out, King might have been on to something.

Countless science fiction tales have explored the paradox of what would happen if you went back in time and did something in the past that endangered the future. Perhaps one of the most famous pop culture examples is in Back to the Future , when Marty McFly goes back in time and accidentally stops his parents from meeting, putting his own existence in jeopardy.

But maybe McFly wasn't in much danger after all. According a new paper from researchers at the University of Queensland, even if time travel were possible, the paradox couldn't actually exist.

Researchers ran the numbers and determined that even if you made a change in the past, the timeline would essentially self-correct, ensuring that whatever happened to send you back in time would still happen.

"Say you traveled in time in an attempt to stop COVID-19's patient zero from being exposed to the virus," University of Queensland scientist Fabio Costa told the university's news service .

"However, if you stopped that individual from becoming infected, that would eliminate the motivation for you to go back and stop the pandemic in the first place," said Costa, who co-authored the paper with honors undergraduate student Germain Tobar.

"This is a paradox — an inconsistency that often leads people to think that time travel cannot occur in our universe."

A variation is known as the "grandfather paradox" — in which a time traveler kills their own grandfather, in the process preventing the time traveler's birth.

The logical paradox has given researchers a headache, in part because according to Einstein's theory of general relativity, "closed timelike curves" are possible, theoretically allowing an observer to travel back in time and interact with their past self — potentially endangering their own existence.

But these researchers say that such a paradox wouldn't necessarily exist, because events would adjust themselves.

Take the coronavirus patient zero example. "You might try and stop patient zero from becoming infected, but in doing so, you would catch the virus and become patient zero, or someone else would," Tobar told the university's news service.

In other words, a time traveler could make changes, but the original outcome would still find a way to happen — maybe not the same way it happened in the first timeline but close enough so that the time traveler would still exist and would still be motivated to go back in time.

"No matter what you did, the salient events would just recalibrate around you," Tobar said.

The paper, "Reversible dynamics with closed time-like curves and freedom of choice," was published last week in the peer-reviewed journal Classical and Quantum Gravity . The findings seem consistent with another time travel study published this summer in the peer-reviewed journal Physical Review Letters. That study found that changes made in the past won't drastically alter the future.

Bestselling science fiction author Blake Crouch, who has written extensively about time travel, said the new study seems to support what certain time travel tropes have posited all along.

"The universe is deterministic and attempts to alter Past Event X are destined to be the forces which bring Past Event X into being," Crouch told NPR via email. "So the future can affect the past. Or maybe time is just an illusion. But I guess it's cool that the math checks out."

• time travel
• grandfather paradox

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A traveler's guide to Novosibirsk, the unofficial capital of Siberia

Trans-Siberian heritage

Residents of Novosibirsk love trains and are proud of the fact that their city played a significant role in the history of the grand Trans-Siberian railway, which spans the breadth of Russia. The railway is such a part of Novosibirsk identity that it is even depicted on the city’s emblem, along with the bridge that crosses the Ob river and two Siberian sables standing on their hind legs.  

In the city, there are as many as five monuments to trains, and an open-air locomotive museum is located in the vicinity of the train station Seyatel’. The museum has more than 100 steam locomotives, diesel locomotives and carriages, reflecting the history of rail transportation in Russia from pre-revolutionary times to the present day. Wondering around the stationary trains and comparing your height with the diameter of the gigantic iron wheels of the first steam locomotives is all very well, but why not climb inside the carriages and see how the nobility once traveled across Russia in pre-revolutionary times? These tours will however need to be booked in advance. The museum opens from 11:00 until 17:00 every day except Mondays. 

Novosibirsk spans both sides of the river Ob. In the early twentieth century, the border of two different timezones passed right through the city which led to a strange situation- morning on the east bank started one hour earlier than on the west bank! The two-kilometer covered metro bridge that crosses the river is considered the longest in the world. Due to the fluctuations in temperature across the year (on average +30 °C to -30 °C), during the summer the metro bridge expands, and in the winter it contracts by half a meter. To counter these effects, the bridge’s supports are equipped with special rollers that allow it to move.   

The cultural center of Siberia

The repertoire of the theatre can be viewed on its official website . The theatre season runs from September to July, and comprises mainly classical performances, like the ballet “The Nutcracker” by Tchaikovsky, Borodin’s opera “Prince Igor” and Verdi’s “La Traviata”.  

The large Siberian sea and ligers

Weekends are best spent at the Novosibirsk zoo . The zoo is known for breeding big cats, although surrounded by controversy, hosts a successful crossing of a tiger and lion, which of course would not otherwise breed in wildlife. Ligers, or exotic cubs of an African lion and Bengal tigress, feel quite comfortable in the Siberian climate and even produce offspring. The zoo is open to visitors year-round, seven days a week, and even has its own free mobile app, Zoo Nsk .

Every year at the beginning of January, the festival of snow culture takes place bringing together artists from across Russia and around the world to participate in a snow sculpting competition. The tradition started in 2000 inspired by the snow festival in Sapporo, Novosibirsk’s twin-city.

Siberian Silicon Valley

Despite the fact that Akademgorodok was built half a century ago in the middle of the uninhabited Siberian taiga, architecturally it was ahead of its time. No trees were destroyed for its construction, and houses were built right in the middle of the forest. A man walking through the woods would seemingly stumble upon these structures. At that time, no one had built anything similar in the world and ecovillages only became fashionable much later.

For residents of the Novosibirsk Akademgorodok is a different world. When you step out the bus or car, you are immediately on one of the hiking paths through the forest, between the scientific buildings and clubs. On a walk through Akademgorodok, it is possible to unexpectedly encounter art-like objects handmade by residents of the city which have been erected as monuments and some monuments fixed up by city authorities. For example, the monument to the laboratory mice, which knits a strand of DNA on to some needles, can be found in the square alongside the Institute of Cytology and Genetics. In Akademgorodok there are many cafes and restaurants, in which it is possible to rest after a long walk. Grab a coffee and go to eat at Traveler’s Coffee , or eat lunch at the grille and bar People’s or Clover .

Winters in the Akademgorodok are slightly colder than in the city, so wrap up. Spring and summer are usually wetter, so waterproof boots are recommended. In the summer the Ob sea provides respite from the heat, so do not forget your swimsuit to go for a dip.

Memento Mori

Among the exhibits of the museum is one dedicated to world funeral culture — hearses, memorial jewellery from the hair of the deceased, samples from a specific photo-genre of  "post mortem", a collection of funeral wear from the Victorian era, deathmasks, statues and monuments. There’s also an impressive collection of coffins. One of them, resembling a fish, was manufactured on a special visit to Novosibirsk by a designer coffin-maker from Africa, Eric Adjetey Anang, who specializes in the production of unusual coffins.

Surprisingly, the crematorium itself does not look at all gloomy in appearance and definitely does not look like infernal scenes from movies, or like crematoriums of other cities that gravitate towards gloomy temple aesthetics. The Novosibirsk crematorium is decorated in “cheerful” orange tones and is surrounded by a park with a children’s playground nearby. A visit to the museum then leaves you with mixed feelings. 

Novosibirsk underground

Tourists from all over the world go down into the Moscow metro to take a ride and a few selfies in the most famous underground museum. The Novosibirsk metro is also quite a museum in itself — it has 13 stations, the most beautiful of which is Gagarinskaya, Sibirskaya and Rechnoy Vokzal.

The ultramodern Gagarinskaya station is like a real cosmos underground. Its technologically themed design includes marble walls with metallic elements, dark blue backlighting and portraits of Yuri Gagarin. The Sibirskaya station looks like an underground treasure trove, decorated by Altai masters craftsmen with mosaics of precious Siberian stones. The Rechnoy Vokzal station is framed with ten glowing stained glass windows depicting the largest cities of Siberia, including Novosibirsk itself, Omsk, Barnaul and others. The platform resembles a big ship sailing on the Ob, from which ancient Siberian cities are visible through its windows.  

How to get there

The easiest way to get to Novosibirsk is by plane with Aeroflot or Novosibirsk airline S7 with one-way tickets from Moscow costing from 200-250 USD. If you decide to take from the train from Moscow, you’ll have to travel approximately a third of the Trans-Siberian Railway. That’s 3,300 kilometers over almost a three-day journey. 

Where to stay

There are many great hotels in Novosibirsk. Amongst the best include a four-star Doubletree hotel by Hilton , which is located near Lenin Square (per room from $200). After renovations and repairs, the congress-hotel Novosibirsk has improved (per room from $100) and is located across from the train station. Less expensive but of a similar standard is the four-star River Park hotel near Rechnoy Vokzal metro station, which costs $80 per night.

All rights reserved by Rossiyskaya Gazeta.

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