# x lines of Python: AVO plot

Amplitude vs offset (or, more properly, angle) analysis is a core component of quantitative interpretation. The AVO method is based on the fact that the reflectivity of a geological interface does not depend only on the acoustic rock properties (velocity and density) on both sides of the interface, but also on the angle of the incident ray. Happily, this angular reflectivity encodes elastic rock property information. Long story short: AVO is awesome.

As you may know, I'm a big fan of forward modeling — predicting the seismic response of an earth model. So let's model the response the interface between a very simple model of only two rock layers. And we'll do it in only a few lines of Python. The workflow is straightforward:

1. Define the properties of a model shale; this will be the upper layer.
2. Define a model sandstone with brine in its pores; this will be the lower layer.
3. Define a gas-saturated sand for comparison with the wet sand.
4. Define a range of angles to calculate the response at.
5. Calculate the brine sand's response at the interface, given the rock properties and the angle range.
6. For comparison, calculate the gas sand's response with the same parameters.
7. Plot the brine case.
8. Plot the gas case.
9. Add a legend to the plot.

That's it — nine lines! Here's the result:

Once we have rock properties, the key bit is in the middle:

    θ = range(0, 31)
shuey = bruges.reflection.shuey2(vp0, vs0, ρ0, vp1, vs1, ρ1, θ)

shuey2 is one of the many functions in bruges — here it provides the two-term Shuey approximation, but it contains lots of other useful equations. Virtually everything else in our AVO plotting routine is just accounting and plotting.

As in all these posts, you can follow along with the code in the Jupyter Notebook. You can view this on GitHub, or run it yourself in the increasingly flaky MyBinder (which is down at the time of writing... I'm working on an alternative).

What would you like to see in x lines of Python? Requests welcome!

### Matt Hall

Matt is a geoscientist in Nova Scotia, Canada. Founder of Agile Scientific, co-founder of The HUB South Shore. Matt is into geology, geophysics, and machine learning.

# White magic: calibrating seismic attributes

This post is part of a series on seismic attributes; the previous posts were...

Last time, I hinted that there might be a often-overlooked step in attribute analysis:

Calibration is a gaping void in many published workflows. How can we move past "that red blob looks like a point bar so I drew a line around it in PowerPoint" to "there's a 70% chance of finding reservoir quality sand at that location"?

Why is this step such a 'gaping void'? A few reasons:

• It's fun playing with attributes, and you can make hundreds without a second thought. Some of them look pretty interesting, geological even. "That looks geological" is, however, not an attribute calibration technique. You have to prove it.
• Nobody will be around when we find out the answer. There's a good chance that well will never be drilled, but when it is, you'll be on a different project, in a different company, or have left the industry altogether and be running a kayak rental business in Belize.
• The bar is rather low. The fact that many published examples of attribute analysis include no proof at all, just a lot of maps with convincing-looking polygons on them, and claims of 'better reservoir quality over here'.

This is getting discouraging. Let's look at an example. Now, it's hard to present this without seeming over-critical, but I know these gentlemen can handle it, and this was only a magazine article, so we needn't make too much of it. But it illustrates the sort of thing I'm talking about, so here goes.

Quoting from Chopra & Marfurt (AAPG Explorer, April 2014), edited slightly for brevity:

While coherence shows the edges of the channel, it gives little indication of the heterogeneity or uniformity of the channel fill. Notice the clear definition of this channel on the [texture attribute — homogeneity].
We interpret [the] low homogeneity feature [...] to be a point bar in the middle of the incised valley (green arrow). This internal architecture was not delineated by coherence.

A nice story, making two claims:

1. The attribute incompletely represents the internal architecture of the channel.
2. The labeled feature on the texture attribute is a point bar.

I know explorers have to be optimists, and geoscience is all about interpretation, but as scientists we must be skeptical optimists. Claims like this are nice hypotheses, but you have to take the cue: go off and prove them. Remember confirmation bias, and Feynman's words:

The first principle is that you must not fool yourself — and you are the easiest person to fool.

### The twin powers

Making geological predictions with seismic attribute analysis requires two related workflows:

1. Forward modeling — the best way to tune your intuition is to make a cartoonish model of the earth (2D, isotropic, homogeneous lithologies) and perform a simplified seismic experiment on it (convolutional, primaries only, noise-free). Then you can compare attribute behaviour to the known model.
2. Calibration — you are looking for an explicit, quantitative relationship between a physical property you care about (porosity, lithology, fluid type, or whatever) and a seismic attribute. A common way to show this is with a cross-plot of the seismic amplitude against the physical property.

When these foundations are not there, we can be sure that one or more bad things will happen:

• The relationship produces a lot of type I errors (false positives).
• It produces a lot of type II error (false negatives).
• It works at some wells and not at others.
• You can't reproduce it with a forward model.
• You can't explain it with physics.

As the industry shrivels and questions — as usual — the need for science and scientists, we have to become more stringent, more skeptical, and more rigorous. Doing anything else feeds the confirmation bias of the non-scientific continent. Because it says, loud and clear: geoscience is black magic.

The image is part of the figure from Chopra, S and K Marfurt (2014). Extracting information from texture attributes. AAPG Explorer, April 2014. It is copyright of the Authors and AAPG.

### Matt Hall

Matt is a geoscientist in Nova Scotia, Canada. Founder of Agile Scientific, co-founder of The HUB South Shore. Matt is into geology, geophysics, and machine learning.

# An attribute analysis primer

A question on Stack Exchange the other day reminded me of the black magic feeling I used to have about attribute analysis. It was all very meta: statistics of combinations of attributes, with shifted windows and crazy colourbars. I realized I haven't written much about the subject, despite the fact that many of us spend a lot of time trying to make sense of attributes.

### Time slices, horizon slices, and windows

One of the first questions a new attribute-analyser has is, "Where should the window be?" Like most things in geoscience: it depends. There are lots of ways of doing it, so think about what you're after...

• Timeslice. Often the most basic top-down view is a timeslice, because they are so easy to make. This is often where attribute analysis begins, but since timeslices cut across stratigraphy, not usually where it ends.
• Horizon. If you're interested in the properties of a strong reflector, such as a hard, karsted unconformity, maybe you just want the instantaneous attribute from the horizon itself.
• Zone. If the horizon was hard to interpret, or is known to be a gradual facies transition, you may want to gather statistics from a zone around it. Or perhaps you couldn't interpret the thing you really wanted, but only that nice strong reflection right above it... maybe you can bootstrap yourself from there.
• Interval. If you're interested in a stratigraphic interval, you can bookend it with existing horizons, perhaps with a constant shift on one or both of them.
• Proportional. If seismic geomorphology is your game, then you might get the most reasonable inter-horizon slices from proportionally slicing in between stratigraphic surface. Most volume interpretation software supports this.

There are some caveats to simply choosing the stratigraphic interval you are after. Beware of choosing an interval that strong reflectors come into and out of. They may have an unduly large effect on most statistics, and could look 'geological'. And if you're after spectral attributes, do remember that the Fourier transform needs time! The only way to get good frequency resolution is to provide long windows: a 100 ms window gives you frequency information every 10 Hz.

### Extraction depends on sample interpolation

When you extract an attribute, say amplitude, from a trace, it's easy to forget that the software has to do some approximation to give you an answer. This is because seismic traces are not continuous curves, but discrete series, with samples typically every 1, 2, or 4 milliseconds. Asking for the amplitude at some arbitrary time, like the point at which a horizon crosses a trace, means the software has to interpolate between samples somehow. Different software do this in different ways (linear, spline, polynomial, etc), and the methods give quite different results in some parts of the trace. Here are some samples interpolated with a spline (black curve) and linearly (blue). The nearest sample gives the 'no interpolation' result.

As well as deciding how to handle non-sampled parts of the trace, we have to decide how to represent attributes operating over many samples. In a future post, we'll give some guidance for using statistics to extract information about the entire window. What options are available and how do we choose? Do we take the average? The maximum? Something else?

### There's a lot more to come!

As I wrote this post, I realized that this is a massive subject. Here are some aspects I have not covered today:

• Calibration is a gaping void in many published workflows. How can we move past "that red blob looks like a point bar so I drew a line around it in PowerPoint" to "there's a 70% chance of finding reservoir quality sand at that location"?
• This article was about  single-trace attributes at single instants or over static windows. Multi-trace and volume attributes, like semblance, curvature, and spectral decomposition, need a post of their own.
• There are a million attributes (though only a few that count, just ask Art Barnes) so choosing which ones to use can be a challenge. Criteria range from what software licenses you have to what is physically reasonable.
• Because there are a million attributes, the art of combining attributes with statistical methods like principal component analysis or multi-linear regression needs a look. This gets into seismic inversion.