About

About PySeisTuned^2.0
Seismic tuning basics
A note about wavelets
The seismic tuning wedge
Spectral decomposition as a special case

About PySeisTuned^2.0

PySeisTuned^2.0 is a tool for calculating seismic tuning wedges. Input the velocity and density properties of the wedge layers, specify the wavelet parameters, and a synthetic tuning wedge and tuning curve are generated.

The initial release, PySeisTuned^1.0, was inspired by Agile Scientific's "X Lines of Code" Jupyter Notebook series and was written as a standalone GUI using Python 3 and PyQT5. In this iteration, a user needs to create a virtual conda environment using the provided environment.yml file and then launch the app from a command line prompt. While this works, it probably isn't the most user-friendly deployment. PySeisTuned^1.0 is open source and licensed under GPL-3.0 for compatibility with PyQT5 licensing.

Enter PySeisTuned^2.0, which is now a web app! The PyQT5 GUI is gone and now PySeisTuned^2.0 runs using Python 3, Flask, HTML5, Bootstrap, & JavaScript. PySeisTuned^2.0 is open source and licensed under the Apache-2.0 license. My primary goal in re-writing PySeisTuned^2.0 as a web app is to make the tool as easy as possible to use. Web apps have changed the way we interact with software and simplifies end-user consumption. To me, this means removing the environmental setup steps necessary to run PySeisTuned^1.0 and also removing the need to launch from a terminal. In fact, unless you want to clone the repository locally to customize PySeisTuned^2.0, there is no longer any "installation" steps required to get running!

Flask is a powerful and easy to use Python-based web app framework that connects the Python code in the background with the HTML5 user-interface. Styling is made simple by utilizing Bootstrap pre-built CSS style sheets. Additionally, some custom JavaScript is required to enable several interactive features. One more enhancement is that all plotting is now done using Bokeh instead of Matplotlib. Using Bokeh enables embedding feature-rich, interactive plots that are more appropriate in a web app.

Contributions, suggestions, and bug reports are all welcome! I hope you enjoy using PySeisTuned^2.0 and please let me know if there are ways I can make it even better:

Follow the links below to view the respective project repositories:

You can also send an email directly using the Contact link down below in the footer.

Citing PySeisTuned^2.0

If you use PySeisTuned^2.0 as part of a publication, please include a citation in the references section:

Dowdell, B.L., 2020, PySeisTuned2.0, https://www.pyseistuned.com/, accessed MM DD, YYYY.

Let us know if your work is published and we will include a link to it!

Seismic Tuning Theory Basics

return to top

Seismic Tuning as a concept is a basic geophysical tenet and is crucial in understanding the seismic response. The seismic tuning phenomenon arises from constructive interference of reflected waves at layer interfaces as the bed thickness between layers begins to thin. As bed thickness decreases, the reflection from the top interface and the bottom interface begin to align and constructively interfere. At a certain point, the constructive interference between the two layers will reach a maximum and this is known as the Seismic Tuning Thickness and is the layer thickness below which the top and base reflections are no longer indistinguishable from one another.

Widess (1973) published the seminal work on the topic titled "How thin is a thin bed?", which sought to dispel common misconceptions about thin bed resolvability. In the paper, Widess considers two identical wavelets offset by a time delay. The two wavelets effectively represent the reflections from the top and base of a layer while the time delay effectively represents the bed thickness in Two-Way Time (TWT). Several very important definitions arise from Widess's paper:

Onset of Tuning Thickness: $\lambda/2$
Tuning Thickness: $\lambda/4$
Limit of resolution: $\lambda/8$

where $\lambda$ is the velocity-dependent wavelength of the layer's dominant frequency. Recall the relationship between velocity, frequency, and wavelength:

$$v = f \times \lambda$$

where $v$ is velocity in (m or ft)/sec, $f$ is Frequency in Hz (or cycles/sec), and $\lambda$ is wavelength in (m or ft). This equation commonly gets rewritten as:

$$\lambda = {v\over f}$$

because we typically have an idea about what the velocity and frequency of our zone of interest is and instead want to know wavelength. Having estimated the wavelength, we can then estimate the tuning parameters using Widess's definitions above. For example, the tuning thickness is given by:

$$\lambda_{tuning} = {v\over f}/4$$

We can also define the tuning parameters in terms of TWT thickness. If we take the inverse of frequency, we instead get the TWT period, $T$ in seconds. The equation above can now be written as:

$$\lambda = v\times T/2$$

with the period $T$ being divided by two to convert to One-Way Time (OWT). Converting the period to OWT is essential as otherwise the depth thickness we measure would be twice the actual bed thickness. This means that when the Widess tuning parameters are expressed in terms of TWT, the divisor becomes half the wavelength divisor so that when the TWT tuning parameter is converted to OWT, the overall relationship remains the same:

Onset of Tuning Thickness: $\lambda/2 = T$
Tuning Thickness: $\lambda/4 = T/2$
Limit of resolution: $\lambda/8 = T/4$

such that the tuning parameter thicknesses are given by:

\begin{align} &\lambda_{onset} = v\times T/2 \\ &\lambda_{tuning} = v\times (T/2)/2 \\ &\lambda_{limit} = v\times (T/4)/2 \end{align}

The advantage to expressing the various tuning parameters in TWT versus depth units is that doing so only requires knowledge of the layer's dominant frequency and as such, TWT tuning thickness is always constant for a given frequency. In other words, if the frequency of a particular reflection pair is 25 Hz, then the TWT tuning thickness is simply:

\begin{align} T_{tuning} & = 1/f/2 \\ & = 1/25\;Hz/2 \\ & = 20\;ms\;TWT\;, \end{align}

and if we know that the velocity of the layer, either from well log control or a seismic velocity model, is 3,000 m/s, then the tuning thickness in meters is:

\begin{align} \lambda_{tuning} & = v\times (T/2)/2 \\ & = 3000\;m/s\times 20\;ms\;TWT/2 \\ & = 3000\;m/s\times 0.020\;s\;TWT/2 \\ & = 30\;m \end{align}

A note about wavelets

return to top

Much has been written and discussed about wavelets, reflecting their central importance to understanding the seismic response. In theory, a seismic wavelet is a stationary filter which is convolved with the earth's reflectivity series, producing reflections that are recorded as seismic data. One of the most commonly used wavelets in seismic forward modelling is the Ricker wavelet , named after Norman Ricker.

The Ricker wavelet amplitude response as a function of time is defined as:

$$A(t) = (1 - 2\pi^2f_{peak}^2t^2)e^{-\pi^2f_{peak}^2t^2}$$

where $f_{peak}$ is the peak frequency used to define the Ricker wavelet, $t$ is the time duration of the wavelet, $\pi$ is the constant Pi, and $e$ is the constant Euler's number. The Ricker wavelet is a zero-phase wavelet meaning that the energy is centered about $t=0$ and is mathematically the second derivative of the Gaussian function. An important distinction of the Ricker wavelet is that its amplitude spectra is asymmetric which means that $f_{peak}$ is not equal to $f_{central}$. This is important because for a wavelet with $f_{peak} = 25\;Hz$, one might expect the period of the wavelet to be $T = 1/f = 40\;ms$. However, for a Ricker wavelet, the period is actually defined as:

$$T = \frac{\sqrt{6}}{\pi f_{peak}}$$

This means that for a Ricker wavelet defined with $f_{peak} = 25\;Hz$, the period is actually 31.2 ms instead of 40 ms, and the apparent frequency is actually $f_{apparent} = 1 / 0.031188\;sec = 32.06\;Hz$. For the Ricker wavelet, it turns out that $f_{apparent}$ is equal to $f_{central}$. If we flip the equation of the Ricker wavelet period, we can show the relation between $f_{central}$ and $f_{peak}$ to be:

$$f_{central} = \frac{\pi}{\sqrt{6}}f_{peak}$$

As it relates to PySeisTuned^2.0, I have taken the assumption that when the user specifies the frequency for a Ricker wavelet, that is in fact the frequency with which the wavelet period should correspond. Therefore, in defining the Ricker wavelet, I assume that $f_{central}$ is specified and apply the $\frac{\pi}{\sqrt{6}}$ correction to pass the corresponding $f_{peak}$ to the Ricker wavelet function definition. In doing so, the resulting period can be used directly for estimating seismic tuning parameters for the frequency of interest. This means that specifying a Ricker wavelet with $f_{central} = 25\;Hz$ will result in a tuning thickness of $T_{tuning} = 1/25\;Hz/2 = 20\;ms\;TWT$ as expected, as opposed to a tuning thickness of $T_{tuning} = 1/32.06\;Hz/2 = 15.59\;m$s when instead $f_{peak} = 25\;Hz$.

The user also has the option of defining an Ormsby wavelet in constructing the synthetic tuning wedge model. I personally do not think that there is much value in using the Ormsby wavelet for this purpose, but I included it as an option. My reason for this preference is that seismic tuning loses its meaning if the wavelet frequency spectrum becomes very broad. Ormsby wavelets tend to be very ringy, especially if the high-pass to high-cut slope is steep, making the Ormsby wavelet a poor candidate for narrow pass-band filters, which is another reason I do not think they are ideal for tuning wedge models. Nonetheless, the option is there.

Seismic Tuning Wedge Model

return to top

Now that the foundational concepts behind seismic tuning are laid out, we can now understand the utility of the Seismic Tuning Wedge Model. The seismic tuning wedge model is a simple three-layer forward model in which the top and base layers are typically the same and the middle wedge layer simulates the layer of interest. In other words, the top and base layers usually represent the encasing shale and the wedge layer represents a sandstone. The wedge layer is typically given a maximum thickness well above tuning at one end of the model and thins out to zero thickness at the other. The thicknesses of the top and base layers are not important as we only care about the reflections generated at the wedge top and base interfaces.

For a very simple wedge model, it is only necessary to know the P-velocity, $V_p$, and bulk density, $\rho$, of the three layers. Generally, and as implemented in PySeisTuned^2.0, the properties of the top and base layers are the same. More detailed forward models can be built to understand reflectivity from differing top and base properties, but if the goal is simply to understand seismic resolution of a sand body, then in my opinion it is best to keep the top and base properties the same. The underlying assumption is that the encasing shales do not have drastically varying properties, but that is not always the case.

Once $V_p$ - $\rho$ pairs are assigned at each sample, the Acoustic Impedance, $AI = V_p\times\rho$, (also sometimes denoted as $Z_p$) of each layer is calculated. In a simple wedge model, $AI$ does not vary laterally or vertically within a given layer. In order to have a seismic reflection generated at a layer boundary, there must be a contrast in $AI$ across the interface. The reflectivity of a layer interface at normal incidence is called the Reflection Coefficient:

$$RC = \frac{AI_2 - AI_1}{AI_2 + AI_1}$$

If there is no contrast in $AI$ at the interface of two layers, no seismic reflection is generated ($RC = 0$) and the layer below the interface is deemed to be acoustically invisible, assuming that the lower layer is sufficiently thick to be seismically resolvable otherwise. This assumption is why half-space models, which consider the layers above and below an interface to be infinitely thick, are only useful for understanding reflectivity at an interface. Wedge models allow us to understand how thickness impacts reflectivity, as well!

Calculating the $RC$ of the wedge model results in an impulse response at the top and base of the wedge and zero values everywhere else. Once the reflection coefficients are calculated, the model can be convolved with a wavelet to produce a wedge model with a synthetic trace for each thickness step. For example, here is the $AI$, $RC$, and synthetic trace for a blocky model with the following properties:

Top & Base Layer: $V_p = 3,000\;m/s\;,\rho = 2.5\;g/cm^3$
Wedge Layer: $V_p = 2,700\;m/s\;,\rho = 2.3\;g/cm^3$
Ricker Wavelet: $F_{central} = 32\;Hz\;,duration = 0.100\;sec\;,dt = 0.001\;sec$

This figure illustrates the relationship between Acoustic Impedance, $AI$, Reflection Coefficient, $RC$, and the synthetic seismic trace. This simple blocky model is extracted from a wedge model at the point where the wedge thickness is 80 ms TWT. The $AI$ of the middle layer decreases relative to the layers above and below. This results in a negative $RC$ at the top interface of the layer and a positive $RC$ at the bottom interface. When a $RC$ series is convolved with a wavelet, the result is a seismic trace.

This raises an important point about polarity conventions. PySeisTuned^2.0 adopts SEG Normal Polarity, meaning that a decrease in $AI$ results in a negative reflection coefficient, or a trough in the seismic response. Additionally, a decrease in impedance is represented by red while an increase is represented by blue. While this seems simple, understanding the polarity convention of a seismic dataset is fundamental to successful interpretation.

We can now look at a wedge model which extends the blocky impedance model above. Using the properties for the layers as defined above but varying thickness of the middle wedge layer from 0 to 100 ms TWT results creates the following wedge model:

blocky wedge model — Wedge model showing blocky acoustic impedance for every fifth trace. The layer impedances are rendered in transparent color.

Next, the $RC$ at the top and base wedge interfaces are calculated and then convolved with the same Ricker wavelet with $F_{central} = 32\;Hz$ to create the synthetic seismic wedge model:

Cool! So we have a synthetic wedge model, but now what? What does this tell us and why do we care about what all we just went through? If we extract the amplitude across the top of the wedge, we can create a plot called the Tuning Curve, shown below:

From the tuning curve we can quickly make observations about how layer thickness impacts relfection amplitude. First, when the wedge is much thicker than tuning thickness, the amplitude at the top interface is unaffected. Once the wedge thins to a thickness of $\lambda/2$ or $T$, we see that the tuning curve begins to turn upwards as the reflections from the top and base of the wedge begin to constructively interfere. Continuing to thin the wedge results in stronger constructive interference until thinned to a thickness of $\lambda/4$ or $T/2$ at which point constructive interference reaches a maximum. This is the Tuning thickness and is the thinnest a layer can be and still seismically resolve a layer's top and base. Thinning the wedge beyond tuning will then result in destructive interference between the top and base wedge reflections and the tuning curve decreases in amplitude rapidly. If a layer is thinner than the tuning thickness, the top and base reflections can no longer be fully resolved and the layer thickness cannot be directly estimated from seismic mapping. More on this in a moment ...

Let's take another look at the interpolated density and wiggle trace plot of the synthetic wedge model:

On this version of the synthetic wedge model I've added the true wedge top and base surfaces in thick black lines and the apparent wedge top and base surfaces in dashed grey lines. As shown in the above figure of true versus apparent wedge thickness, the true and apparent wedge top and base surfaces are the same until the onset of tuning thickness. At this wedge thickness, the apparent top and base surfaces diverge from the true top and base surfaces and the wedge thickness is actually apparently thinner than true thickness. At tuning thickness, the true and apparent surfaces cross and for wedge thickness below tuning thickness, the apparent top and base surfaces are now thicker than the true wedge thickness. If the layer we are mapping is below the tuning thickness and we pick a trough-peak pair for the top and base surfaces, our interpreted thickness will overestimate the true layer thickness.

So far I have only discussed the relation between layer thickness and reflection amplitude and how it relates to seismic resolution. However, let's turn this discussion on its head and talk about amplitude as a function of thickness. It should be very obvious by now (I hope!) that thickness variations impact amplitude around tuning thickness. This means that if our layer of interest is close to tuning thickness, any amplitude attribute extractions we make will have a tuning component embedded in it. In other words, when beds are thin, amplitudes cannot be trusted! If we look at the portion of the tuning curve where thickness is well above tuning, the amplitude response is the true amplitude response. At and approaching tuning thickness, amplitude is over-estimated as a result of constructive interference. If the layer becomes very thin, the amplitude rapidly drops off and is much less than the true amplitude response. This leads us to the topic of Detuning which refers to correcting amplitude maps for tuning thickness phenomenon. Detuning is a bit beyond the scope of this discussion so I would instead refer you to the works of Patrick Connolly.

Seismic Tuning & Spectral Decomposition

return to top

Spectral decomposition in many ways is a very special case of seismic tuning. While there are many different ways to implement spectral decomposition, the basic idea is to decompose a seismic trace into narrow frequency bands. The Law of Superposition allows us to additively sum narrow frequency band representations of a seismic trace to reconstruct the full bandwidth trace. Consider for a moment that in performing spectral decomposition of a trace, we create iso-frequency representations of that trace from 5 to 60 Hz in 5 Hz increments. If we then compare each of those iso-frequency traces to one another, we will see that for a given layer thickness, one of the iso-frequencies will have peak magnitude compared to the others. If that were, for example, 25 Hz, then we would say that the event "tunes up" at 25 Hz.

Panel showing iso-frequency gather response for layers of varying TWT (ms) thickness. As layers increase in thickness, the peak magnitude of the iso-frequency gather shifts towards lower frequencies. For each layer, the tuning frequency is notated on the gather by a black '+'. The layer thicknesses are 10, 20, 30, 40, 50, & 100 ms TWT with a 100 ms TWT layer in between, as illustrated in the impedance model. The reflectivity series shows the reflection coefficients at the top and base of each layer. Each trace in the iso-frequency gather represents the convolution of the reflectivity series with a narrow-band filter which in this instance is a Ricker wavelet, displayed as wiggle trace and interpolated density.

Tying this back to the seismic tuning wedge model, we know that peak amplitude occurs at the tuning thickness. As the frequency used to construct the tuning wedge varies, the tuning thickness varies, as well, but peak amplitude is always at the tuning thickness. If we compare the amplitude responses of a given layer at different frequencies, we see that one frequency will have peak amplitude compared to all the others. This frequency is the tuning frequency and can be used for estimating the layer's thickness. The figure above shows an impedance model where each subsequent layer is thicker. We can convolve the reflectivity series derived from this model with a suite of Ricker wavelets to create an iso-frequency gather. What we observe is that as layer thickness changes, the maximum energy changes as a function of frequency. Thinner layers have maximum energy at higher frequencies while thicker layers have maximum energy at lower frequencies. We now know from the tuning wedge model that the maximum amplitude at a layer interface occurs at the tuning thickness of the wedge. So, for a set layer thickness, the iso-frequency trace with maximum magnitude for that layer is at tuning (hence "tuning frequency"), and we can then use the equations above to relate tuning frequency to period and estimate the layer's thickness. Spectral Decomposition is really cool!

return to top

About

About PySeisTuned2.0

Citing PySeisTuned2.0

Seismic Tuning Theory Basics

A note about wavelets

Seismic Tuning Wedge Model

Seismic Tuning & Spectral Decomposition

About PySeisTuned^2.0

Citing PySeisTuned^2.0