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Tuesday
Apr162013

## Backwards and forwards reasoning

Most people, if you describe a train of events to them will tell you what the result will be. There will be few people however, that if you told them a result, would be able to evolve from their own consciousness what the steps were that led to that result. This is what I mean when I talk about reasoning backward.
— Sherlock Holmes, A Study in Scarlet, Sir Arthur Conan Doyle (1887)

Reasoning backwards is the process of solving an inverse problem — estimating a physical system from indirect data. Straight-up reasoning, which we call the forward problem, is a kind of data collection: empiricism. It obeys a natural causality by which we relate model parameters to the data that we observe.

### Modeling a measurement

Where are you headed? Every subsurface problem can be expressed as the arrow between two or more such panels.Inverse problems exists for two reasons. We are incapable of measuring what we are actually interested in, and it is impossible to measure a subject in enough detail, and in all aspects that matter. If, for instance, I ask you to determine my weight, you will be troubled if the only tool I allow is a ruler. Even if you are incredibly accurate with your tool, at best, you can construct only an estimation of the desired quantity. This estimation of reality is what we call a model. The process of estimation is called inversion.

### Measuring a model

Forward problems are ways in which we acquire information about natural phenomena. Given a model (me, say), it is easy to measure some property (my height, say) accurately and precisely. But given my height as the starting point, it is impossible to estimate the me from which it came. This is an example of an ill-posed problem. In this case, there is an infinite number of models that share my measurements, though each model is described by one exact solution.

Solving forward problems are nessecary to determine if a model fits a set of observations. So you'd expect it to be performed as a routine compliment to interpretation; a way to validate our assumptions, and train our intuition.

### The math of reasoning

Forward and inverse problems can be cast in this seemingly simple equation.

where d is a vector containing N observations (the data), m is a vector of M model parameters (the model), and G is a N × M matrix operator that connects the two. The structure of G changes depending on the problem, but it is where 'the experiment' goes. Given a set of model parameters m, the forward problem is to predict the data d produced by the experiment. This is as simple as plugging values into a system of equations. The inverse problem is much more difficult: given a set of observations d, estimate the model parameters m.

I think interpreters should describe their work within the Gm = d framework. Doing so would safeguard against mixing up observations, which should be objective, and interpretations, which contain assumptions. Know the difference between m and d. Express it with an arrow on a diagram if you like, to make it clear which direction you are heading in.

Illustrations for this post were created using data from the Marmousi synthetic seismic data set. The blue seismic trace and its corresponding velocity profile is at location no. 250.

Thursday
Apr112013

## How to get paid big bucks

Yesterday I asked 'What is inversion?' and started looking at problems in geoscience as either forward problems or inverse problems. So what are some examples of inverse problems in geoscience? Reversing our forward problem examples:

• Given a suite of sedimentological observations, provide the depositional environment. This is a hard problem, because different environments can produce similar-looking facies. It is ill-conditioned, because small changes in the input (e.g. the presence of glaucony, or Cylindrichnus) produces large changes in the interpretation.
• Given a seismic trace, produce an impedance log. Without a wavelet, we cannot uniquely deduce the impedance log — there are infinitely many combinations of log and wavelet that will give rise to the same seismic trace. This is the challenge of seismic inversion.

To solve these problems, we must use induction — a fancy name for informed guesswork. For example, we can use judgement about likely wavelets, or the expected geology, to constrain the geophysical problem and reduce the number of possibilities. This, as they say, is why we're paid the big bucks. Indeed, perhaps we can generalize: people who are paid big bucks are solving inverse problems...

• How do we balance the budget?
• What combination of chemicals might cure pancreatic cancer?
• What musical score would best complement this screenplay?
• How do I act to portray a grief-stricken war veteran who loves ballet?

What was the last inverse problem you solved?

Wednesday
Apr102013

## What is inversion?

Inverse problems are at the heart of geoscience. But I only hear hardcore geophysicists talk about them. Maybe this is because they're hard problems to solve, requiring mathematical rigour and computational clout. But the language is useful, and the realization that some problems are just damn hard — unsolvable, even — is actually kind of liberating.

### Forwards first

Before worrying about inverse problems, it helps to understand what a forward problem is. A forward problem starts with plenty of inputs, and asks for a straightforward, algorithmic, computable output. For example:

• What is 4 × 5?
• Given a depositional environment, what sedimentological features do we expect?
• Given an impedance log and a wavelet, compute a synthetic seismogram.

These problems are solved by deductive reasoning, and have outcomes that are no less certain than the inputs.

### Can you do it backwards?

You can guess what an inverse problem looks like. Computing 4 × 5 was pretty easy, even for a geophysicist, but it's not only difficult to do it backwards, it's impossible:

20 = what × what

You can solve it easily enough, but solutions are, to use the jargon, non-unique: 2 × 10, 7.2 × 1.666..., 6.3662 × π — you get the idea. One way to deal with such under-determined systems of equations is to know about, or guess, some constraints. For example, perhaps our system — our model — only includes integers. That narrows it down to three solutions. If we also know that the integers are less than 10, there can be only one solution.

Non-uniqueness is a characteristic of ill-posed problems. Ill-posedness is a dead giveaway of an inverse problem. Proposed by Jacques Hadamard, the concept is the opposite of well-posedness, which has three criteria:

• A solution exists.
• The solution is unique.
• The solution is well-conditioned, which means it doesn't change disproportionately when the input changes.

Notice the way the example problem was presented: one equation, two unknowns. There is already a priori knowledge about the system: there are two numbers, and the operator is multiplication. In geoscience, since the earth is not a computer, we depend on such knowledge about the nature of the system — what the variables are, how they interact, etc. We are always working with a model of nature.

Tomorrow, I'll look at some specific examples of inverse problems, and Evan will continue the conversation next week.

Friday
Jan112013

If you're anything like me before Agile, you don't read a lot of blogs. At least, not ones about geophysics. But they do exist! Get these in your browser favourites, or use a reader like Google Reader (anywhere) or Flipboard (on iPad).

### Seismos

Chris Liner, a geophysics professor at the University of Arkansas, recently moved from the University of Houston. He's been writing Seismos, a parallel universe to his occasional Leading Edge column, since 2008.

### MyCarta

Matteo Niccoli (@My_Carta on Twitter) is an exploration geoscientist in Stavanger, Norway, and he recently moved from Calgary, Canada. He's had MyCarta: Geophysics, visualization, image processing and planetary science, since 2011. This blog is a must-read for MATLAB hackers and image processing nuts. Matteo was one of our 52 Things authors.

### GeoMika

Mika McKinnon (@mikamckinnon), a geophysicist in British Columbia, Canada, has been writing GeoMika: Fluid dynamics, diasters, geophysics, and fieldwork since 2008. She's also into education outreach and the maker-hacker scene.

### The Way of the Geophysicist

Jesper Dramsch (@JesperDramsch), a geophysicist in Hamburg, Germany has written the wonderfully personal and philosophical The Way of The Geophysicist since 2011. His tales of internships at Fugro and Schlumberger provide great insights for students.

### VatulBlog

Maitri Erwin (@maitri), an exploration geoscientist in Texas, USA. She has been blogging since 2001 (surely some kind of record), and both she and her unique VatulBlog: From Kuwait to Katrina and beyond defy categorization. Maitri was also one of our 52 Things authors.

There are other blogs on topics around seismology and exploration geophysics — shout outs go to Hypocentre in the UK, the Laboratoire d'imagerie et acquisition des mesures géophysiques in Quebec, occasional seismicky posts from sedimentologists like @zzsylvester, and the panoply of bloggery at the AGU. Stick those in your reader!

Tuesday
Jan082013

If you had to choose your three favourite, most revisited, best remembered papers in all of exploration geophysics, what would you choose? Are they short? Long? Full of math? Well illustrated?

### Keep it honest

Barnes, A (2007). Redundant and useless seismic attributes. Geophysics 72 (3). DOI:10.1190/1.2716717
Rarely do we see engaging papers, but they do crop up occasionally. I love Art Barnes's Redundant and useless seismic attributes paper. In this business, I sometimes feel like our opinions — at least our public ones — have been worn down by secrecy and marketing. So Barnes's directness is doubly refreshing:

There are too many duplicate attributes, too many attributes with obscure meaning, and too many unstable and unreliable attributes. This surfeit breeds confusion and makes it hard to apply seismic attributes effectively. You do not need them all.

### And keep it honest

Blau, L (1936). Black magic in geophysical prospecting. Geophysics 1 (1). DOI:10.1190/1.1437076
I can't resist Ludwig Blau's wonderful Black magic geophysics, published 77 years ago this month in the very first issue of Geophysics. The language is a little dated, and the technology mostly sounds rather creaky, but the point, like Blau's wit, is as fresh as ever. You might not learn a lot of geophysics from this paper, but it's an enlightening history lesson, and a study in engaging writing the likes of which we rarely see in Geophysics today...

### And also keep it honest

Bond, C, A Gibbs, Z Shipton, and S Jones (2007), What do you think this is? "Conceptual uncertainty" in geoscience interpretation. GSA Today 17 (11), DOI: 10.1130/GSAT01711A.1
I like to remind myself that interpreters are subjective and biased. I think we have to recognize this to get better at it. There was a wonderful reaction on Twitter yesterday to a recent photo from Mars Curiosity (right) — a volcanologist thought it looked like a basalt, while a generalist thought it more like a sandstone. This terrific paper by Clare Bond and others will help you remember your biases!

My full list is right here. I hope you think there's something missing... please edit the wiki, or put your personal favourites in the comments.

The attribute figure is adapted from from Barnes (2007) is copyright of SEG. It may only be used in accordance with their Permissions guidelines. The Mars Curiosity figure is public domain.

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