# x lines of Python: synthetic wedge model

Welcome to a new blog series! Like the A to Z and the Great Geophysicists, I expect it will be sporadic and unpredictable, but I know you enjoys life's little nonlinearities as much as I.

The idea with this one — x lines of Python — is to share small geoscience workflows in x lines or fewer. I'm not sure about the value of x, but I think 10 seems reasonable for most tasks. If x > 10 then the task may have been too big... If x < 5 then it was probably too small.

Python developer Raymond Hettinger says that each line of code should be equivalent to a sentence... so let's say that that's the measure of what's OK to put in a single line.

### Synthetic wedge model

To kick things off, follow this link to a live Jupyter Notebook environment showing how you can make a simple synthetic three-rock wedge model in only 9 lines of code.

The sentences represented by the code that made the data in these images are:

1. Set up the size of the model.
2. Make the slanty bit, with 1's in the wedge and 2's in the base.
3. Add the top of the model as 0; these numbers will turn into rocks.
4. Define the velocity and density of rocks 0 to 2.
5. Distribute those properties through the model.
6. Calculate the acoustic impedance everywhere.
7. Calculate the reflection coefficients in the model.
8. Make a Ricker wavelet.
9. Convolve the wavelet with the reflection coefficients.

All of the notebooks we share in this series will be hosted on mybinder.org. I'm excited about this because it means you can run and edit them live, without installing anything at all. Give it a go right now.

You can see them on GitHub too, and fork or clone them from there. Note that if you look at the notebook for this post on GitHub, you'll be able to view it, but not change or run code unless you get everything running on your own machine. (To do that, you can more or less follow the instructions in my User Guide to the TLE tutorials).

Please do take this notion of x as 'par' as a challenge. If you'd like to try to shoot under par, please do — and share your efforts. Code golf is a fun way to learn better coding habits. (And maybe some bad ones.) There is a good chance I will shoot some bogies on this course.

We will certainly take requests too — what tasks would you like to see in x lines of Python?

### 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.

# A focus on building

We've got some big plans for modelr.io, our online forward modeling tool. They're so big, we're hiring! An exhilarating step for a small company. If you are handy with the JavaScript, or know someone who is, scroll down to read all about it!

Here are some of the cool things in Modelr's roadmap:

Interactive 1D models – to support fluid substitution, we need to handle physical properties of pore fluids as well as rocks. Our prototype (right) supports arbitrary layers, but eventually we'd like to allow uploading well logs too.

Exporting models – imagine creating an earth model of your would-be prospect, and sending it around to your asset team to strengthen it's prognosis. Modelr solves the forward problem, PickThis solves the inverse. We need to link them up. We also need SEG-Y export, so you can see your model next to your real data.

Models from sketches – Want to do a quick sketch of a geologic setting, and see what it would look like under the lens of seismic? At the hackathon last month, Matteo Niccoli and friends showed a path to this dream — sketch a picture, take a photo, and upload it to the the app with your phone (right).

3D models Want to visualize how seismic amplitudes vary according to bed thickness? Build a 2D wedge model and you can analyze a tuning curve. Now, want to explore the same wedge spanning a range of physical properties? That's a job for a 3D wedge model.

Seismic attributes – Seismic discontinuity attributes, like continuity, or curvature can be ineffective when viewed in cross-section; they're really meant to be shown in time-slices. There is a vast library of attributes and co-rendering technologies we want to provide.

If you get excited about building simple tools on the web for difficult tasks under the ground, we'd love to talk to you. We have an open position for a full-time web developer to help us carry this project forward. Check out the job posting.

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# The race for useful offsets

We've been working on a 3D acquisition lately. One of the factors influencing the layout pattern of sources and receivers in a seismic survey is the range of useful offsets over the depth interval of interest. If you've got more than target depth, you'll have more than one range of useful offsets. For shallow targets this range is limited to small offsets, due to direct waves and first breaks. For deeper targets, the range is limited at far offsets by energy losses due to geometric spreading, moveout stretch, and system noise.

In seismic surveying, one must choose a spacing interval between geophones along a receiver line. If phones are spaced close together, we can collect plenty of samples in a small area. If the spacing is far apart, the sample density goes down, but we can collect data over a bigger area. So there is a trade-off and we want to maximize both; high sample density covering the largest possible area.

### What are useful offsets?

It isn't immediately intuitive why illuminating shallow targets can be troublesome, but with land seismic surveying in particular, first breaks and near surface refractions clobber shallow reflecting events. In the CMP domain, these are linear signals, used for determining statics, and are discarded by muting them out before migration. Reflections that arrive later than the direct wave and first refractions don't get muted out. But if these reflections arrive later than the air blast noise or ground roll noise — pervasive at near offsets — they get caught up in noise too. This region of the gather isn't muted like the top mute, otherwise you'd remove the data at near offsets. Instead, the gathers are attacked with algorithms to eliminate the noise. The extent of each hyperbola that passes through to migration is what we call the range of useful offsets.

The deepest reflections have plenty of useful offsets. However if we wanted to do adequate imaging somewhere between the first two reflections, for instance, then we need to make sure that we record redundant ray paths over this smaller range as well. We sometimes call this aperture; the shallow reflection is restricted in the number of offsets that it can be illuminated with, the deeper reflections can tolerate an aperture that is more open. In this image, I'm modelling the case of 60 geophones spanning 3000 metres, spaced evenly at 100 metres apart. This layout suggests merely 4 or 5 ray paths will hit the uppermost reflection, the shortest ray paths at small offsets. Also, there is usually no geophone directly on top of the source location to record a vertical ray path at zero offset. The deepest reflections however, should have plenty of fold, as long as NMO stretch, geometric spreading, and noise levels were good.

The problem with determining the range of useful offsets by way of a model is, not only does it require a velocity profile, which is easily attained from a sonic log, VSP, or velocity analysis, but it also requires an estimation of the the speed, intensity, and duration of the near surface events to be muted. Parameters that depend largely on the nature of the source and the local ground conditions, which vary from place to place, and from one season to the next.

In a future post, we'll apply this notion of useful offsets to build a pattern for shooting a 3D.

Click here for details on how I created this figure using the IPython Notebook. If you don't have it, IPython is easy to install. The easiest way is to install all of scientific Python, or use Canopy or Anaconda.

This post was inspired in part by Norm Cooper's 2004 article, A world of reality: Designing 3D seismic programs for signal, noise, and prestack time-migration. The Leading Edge23 (10), 1007-1014.DOI: 10.1190/1.1813357

Update on 2014-12-17 13:04 by Matt Hall
Don't miss the next installment — Laying out a seismic survey — with more IPython goodness!

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# Cross sections into seismic sections

We've added to the core functionality of modelr. Instead of creating an arbitrarily shaped wedge (which is plenty useful in its own right), users can now create a synthetic seismogram out of any geology they can think of, or extract from their data.

### Turn a geologic-section into an earth model

We implemented a color picker within an image processing scheme, so that each unique colour gets mapped to an editable rock type. Users can create and manage their own rock property catalog, and save models as templates to share and re-use. You can use as many or as few colours as you like, and you'll never run out of rocks.

To give an example, let's use the stratigraphic diagram that Bruce Hart used in making synthetic seismic forward models in his recent Whither seismic stratigraphy article. There are 7 unique colours, so we can generate an earth model by assigning a rock to each of the colours in the image.

If you can imagine it, you can draw it. If you can draw it, you can model it.

### Modeling as an interactive experience

We've exposed parameters in the interface and so you can interact with the multidimensional seismic data space. Why is this important? Well, modeling shouldn't be a one-shot deal. It's an iterative process. A feedback cycle where you turn knobs, pull levers, and learn about the behaviour of a physical system; in this case it is the interplay between geologic units and seismic waves.

A model isn't just a single image, but a swath of possibilities teased out by varying a multitude of inputs. With modelr, the seismic experiment can be manipulated, so that the gamut of geologic variability can be explored. That process is how we train our ability to see geology in seismic.

Hart's paper doesn't specifically mention the rock properties used, so it's difficult to match amplitudes, but you can see here how modelr stands up next to Hart's images for high (75 Hz) and low (25 Hz) frequency Ricker wavelets.

There are some cosmetic differences too... I've used fewer wiggle traces to make it easier to see the seismic waveforms. And I think Bruce forgot the blue strata on his 25 Hz model. But I like this display, with the earth model in the background, and the wiggle traces on top — geology and seismic blended in the same graphical space, as they are in the real world, albeit briefly.

Subscribe to the email list to stay in the loop with modelr news, or sign-up at modelr.io and get started today.

This will add you to the email list for the modeling tool. We never share user details with anyone. You can unsubscribe any time.

Seismic models: Hart, BS (2013). Whither seismic stratigraphy? Interpretation, volume 1 (1). The image is copyright of SEG and AAPG.

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# Getting started with Modelr

Let's take a closer look at modelr.io, our new modeling tool. Just like real seismic experiments, there are four components:

• Make a framework. Define the geometries of rock layers.
• Make an earth. Assign a set of rock properties to each layer.
• Make a kernel. Define the seismic survey.
• Make a plot. Set the output parameters.

Modelr takes care of the physics of wave propagation and reflection, so you don't have to stick with normal incidence acoustic impedance models if you don't want to. You can explore the full range of possibilities.

### 3 ways to slice a wedge

To the uninitiated, the classic 3-layer wedge model may seem ridiculously trivial. Surely the earth looks more complicated than that! But we can leverage such geometric simplicity to systematically study how seismic waveforms change across spatial and non-spatial dimensions.

Spatial domain. In cross-section (right), a seismic wedge model lets you analyse the resolving power of a given wavelet. In this display the onset of tuning is marked by the vertical red line, and the thickness at which maximum tuning occurs is shown in blue. Reflection profiles can be shown for any incidence angle, or range of incidence angles (offset stack).

Amplitude versus angle (AVA) domain. Maybe you are working on a seismic inversion problem so you might want to see what a CDP angle gather looks like above and below tuning thickness. Will a tuned AVA response change your quantitative analysis? This 3-layer model looks like a two-layer AVA gather except our original wavelet looks like it has undergone a 90 degree phase rotation. Looks can be deceiving.

Amplitude versus frequency domain. If you are trying to design a seismic source for your next survey, and you want to ensure you've got sufficient bandwidth to resolve a thin bed, you can compute a frequency gather — right, bottom — and explore a swath of wavelets with regard to critical thickness in your prospect. The tuning frequency (blue) and resolving frequency (red) are revealed in this domain as well.

Wedges are tools for seismic waveform classification. We aren't just interested in digitizing peaks and troughs, but the subtle interplay of amplitude tuning, and apparent phase rotation variations across the range of angles and bandwidths in the seismic experiment. We need to know what we can expect from the data, from our supposed geology.

In a nutshell, all seismic models are about illustrating the band-limited nature of seismic data on specific geologic scenarios. They help us calibrate our intuition when bandwidth causes ambiguity in interpretation. Which is nearly all of the time.

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### Evan Bianco

On Tuesday we announced our new web app, modelr.io. Why are we so excited about it?

• We love the idea that subsurface software can cost dollars, not 1000's of dollars.
• We love the idea of subsurface software being online, not on the desktop.
• We love the idea that subsurface software can be open source. Here's our code!
• We love the idea of subsurface software that doesn't need a manual to master.
• We love the idea of subsurface software that runs on a tablet or a phone.
• We see software as an important way to share knowledge and connect people.

OK, that's enough reasons. There are more. Those are the main ones.

The point is: we love these ideas. And we hope that you, dear reader, at least like some of them a bit. Because we really want to keep developing modelr. We think it can be awesome. Imagine 3D earth models, imagine full waveform modeling, imagine gravity and magnetic models. We get very excited when we think about all the possiblities. There's no better way to calibrate your seismic intuition than modeling, and modelr is a great place to start modeling.

Here's a challenge: take 3 minutes and see if you can generate...

 A wedge model & tuning curve An AVA gather for a Class 4 sand A stochastic AVA crossplot

### 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.

# The most important thing nobody does

A couple of weeks ago, we told you we were up to something. Today, we're excited to announce modelr.io — a new seismic forward modeling tool for interpreters and the seismically inclined.

Modelr is a web app, so it runs in the browser, on any device. You don't need permission to try it, and there's never anything to install. No licenses, no dongles, no not being able to run it at home, or on the train.

Later this week, we'll look at some of the things Modelr can do. In the meantime, please have a play with it.
Just go to modelr.io and hit Demo, or click on the screenshot below. If you like what you see, then think about signing up — the more support we get, the faster we can make it into the awesome tool we believe it can be. And tell your friends!

This will add you to the email list for the modeling tool. We never share user details with anyone. You can unsubscribe any time.

### 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.

# Transforming geology into seismic

Hart (2013). ©SEG/AAPGForward modeling of seismic data is the most important workflow that nobody does.

### Why is it important?

• Communicate with your team. You know your seismic has a peak frequency of 22 Hz and your target is 15–50 m thick. Modeling can help illustrate the likely resolution limits of your data, and how much better it would be with twice the bandwidth, or half the noise.
• Calibrate your attributes. Sure, the wells are wet, but what if they had gas in that thick sand? You can predict the effects of changing the lithology, or thickness, or porosity, or anything else, on your seismic data.
• Calibrate your intuition. Only by predicting the seismic reponse of the geology you think you're dealing with, and comparing this with the response you actually get, can you start to get a feel for what you're really interpreting. Viz Bruce Hart's great review paper we mentioned last year (right).

### Why does nobody do it?

Well, not 'nobody'. Most interpreters make 1D forward models — synthetic seismograms — as part of the well tie workflow. Model gathers are common in AVO analysis. But it's very unusual to see other 2D models, and I'm not sure I've ever seen a 3D model outside of an academic environment. Why is this, when there's so much to be gained? I don't know, but I think it has something to do with software.

• Subsurface software is niche. So vendors are looking at a small group of users for almost any workflow, let alone one that nobody does. So the market isn't very competitive.
• Modeling workflows aren't rocket surgery, but they are a bit tricky. There's geology, there's signal processing, there's big equations, there's rock physics. Not to mention data wrangling. Who's up for that?
• Big companies tend to buy one or two licenses of niche software, because it tends to be expensive and there are software committees and gatekeepers to negotiate with. So no-one who needs it has access to it. So you give up and go back to drawing wedges and wavelets in PowerPoint.

### Okay, I get it, how is this helping?

We've been busy lately building something we hope will help. We're really, really excited about it. It's on the web, so it runs on any device. It doesn't cost thousands of dollars. And it makes forward models...

That's all I'm saying for now. To be the first to hear when it's out, sign up for news here:

This will add you to the email list for the modeling tool. We never share user details with anyone. You can unsubscribe any time.

Seismic models: Hart, BS (2013). Whither seismic stratigraphy? Interpretation, volume 1 (1). The image is copyright of SEG 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.

# EAGE 2013 in a nutshell

I left London last night for Cambridge. On the way, I had a chance to reflect on the conference. The positive, friendly vibe, and the extremely well-run venue. Wi-Fi everywhere, espresso machines and baristas keeping me happy and caffeinated.

### Knowledge for sale

I saw no explicit mention of knowledge sharing per se, but many companies are talking about commoditizing or productizing knowledge in some way. Perhaps the most noteworthy was an update from Martyn Millwood Hargrave at Ikon's booth. In addition to the usual multi-client reports, PowerPoint files, or poorly architected database, I think service companies are still struggling to find a model where expertise and insight can be included as a service, or at least a value-add. It's definitely on the radar, but I don't think anyone has it figured out just yet.

### Better than swag

Yesterday I pondered the unremarkability of carrot-and-ginger juice and Formula One pit crews. Paradigm at least brought technology to the party. Forget Google Glass, here's some augmented geoscience reality:

This notion of 3D seismic vizualization and interpretation is finally coming to gathers. The message: if you are not going pre-stack, you are missing out. Pre-stack panels are being boasted in software demos by the likes of DUG, Headwave, Transform, and more. Seems like this trend has been moving in slow motion for about a decade.

Another bandwagon is modeling while you interpret. I see this as an unfeasible and potentially dangerous claim, but some technologies companies are creating tools and workflows to fast-track the seismic interpretation to geologic model building workflows. Such efficiencies may have a market, but may push hasty solutions down the value chain.

What do you think? What trends do you detect in the subsurface technology space?

# Fitting a model to data

In studying the earth, we can't afford to take enough observations, and they will never be free of noise. So if you say you do geoscience, I hereby challenge you to formulate your work as a mathematical inverse problem. Inversion is a question: given the data, the physical equations, and details of the experiment, what is the distribution of physical properties? To answer this question we must address three more fundamental ones (Scales, Smith, and Treitel, 2001):

• How accurate is the data? Or what does fit mean?
• How accurately can we model the response of the system? Have we included all the physics that can contribute signifcantly to the data?
• What is known about the system independent of the data? There must be a systematic procedure for rejecting unreasonable models that fit the data as well.

Setting up an inverse problem means coming up with the equations that contain the physics and geometry of the system under study. The method for solving it depends on the nature of the system of equations. The simplest is the minimum norm solution, and you've heard of it before, but perhaps under a different name.

### To fit is to optimize a system of equations

For problems where the number of observations is greater than the number of unknowns, we want to find which unknowns fit the best. One case you're already familiar with is the method of least squares — you've used it fitting a line of through a set of points. A line is unambiguously described by only two parameters: slope a and y-axis intercept b. These are the unknowns in the problem, they are the model m that we wish to solve for. The problem of line-fitting through a set of points can be written out like this,

$\large \mathbf{d}=\begin{pmatrix}y_{1} \\ y_{2} \\ ... \\ y_{n} \end{pmatrix}, \mathbf{G}=\begin{pmatrix} x_{1} & 1\\ x_{2} & 1\\ ... & ...\\ x_{n} & 1 \end{pmatrix}, \mathbf{m}=\begin{pmatrix} a\\ b \end{pmatrix}$

As I described in a previous post, the system of the problem takes the form d = Gm, where each row links a data point to an equation of a line. The model vector m (M × 1), is smaller than the data d (N × 1) which makes it an over-determined problem, and G is a N × M matrix holding the equations of the system.

Why cast a system of equations in this matrix form? Well, it turns out that the the best-fit line is precisely,

$\large \mathbf{m}=(G^{T}G)^{-1}G^{T}\mathbf{d}$

which are trivial matrix operations, once you've written out G.  T means to take the transpose, and –1 means the inverse, the rest is matrix multiplication. Another name for this is the minimum norm solution, because it defines the model parameters (slope and intercept) for which the lengths (vector norm) between the data and the model are a minimum.

One benefit that comes from estimating a best-fit model is that you get the goodness-of-fit for free. Which is convenient because making sense of the earth doesn't just mean coming up with models, but also expressing their uncertainty, in terms of the errors with which they are linked.

I submit to you that every problem in geology can be formulated as a mathematical inverse problem. The benefit of doing so is not just to do math for math's sake, but it is only through quantitatively portraying ambiguous inferences and parameterizing non-uniqueness that we can do better than interpreting or guessing.