[PPT] - Spatial Resolution for How This Idea Is . . . Processing Seismic PowerPoint Presentation

SLIDE 1

Need for Seismic Data . . . How Seismic Inverse . . . Uncertainty vs. . . . How Traditional . . . How This Idea Is . . . Limitations of Ray . . . Angular Diversity: A . . . Limitations of the . . . Gauging Uncertainty: . . . Gauging Spatial . . . Title Page ◭◭ ◮◮ ◭ ◮ Page 1 of 18 Go Back Full Screen Close Quit

Spatial Resolution for Processing Seismic Data: Type-2 Methods for Finding the Relevant Granular Structure

Vladik Kreinovich, Jaime Nava, Rodrigo Romero, Julio Olaya, Aaron Velasco

Cyber-ShARE Center, University of Texas at El Paso 500 W. University, El Paso, Texas 79968, USA contact vladik@utep.edu

Kate C. Miller

Department of Geology and Geophysics Texas A&M University, College Station, Texas 77843, USA

SLIDE 2

Need for Seismic Data . . . How Seismic Inverse . . . Uncertainty vs. . . . How Traditional . . . How This Idea Is . . . Limitations of Ray . . . Angular Diversity: A . . . Limitations of the . . . Gauging Uncertainty: . . . Gauging Spatial . . . Title Page ◭◭ ◮◮ ◭ ◮ Page 2 of 18 Go Back Full Screen Close Quit

1. Need for Seismic Data Processing

It is very important to determine Earth structure:

– to find fossil fuels (oil, coal, natural gas), minerals, water; – to assess earthquake risk.

Data that we can use to determine the Earth structure:

– data from drilling boreholes, – gravity and magnetic measurements, – analyzing the travel-times and paths of seismic ways as they propagate through the earth.

Active and passive (= from earthquakes) seismic mea-

surements are usually the most informative: – they come from areas of different depth; – each sound wave travels along a narrow path, so it provides a detailed structure of the Earth.

SLIDE 3

Need for Seismic Data . . . How Seismic Inverse . . . Uncertainty vs. . . . How Traditional . . . How This Idea Is . . . Limitations of Ray . . . Angular Diversity: A . . . Limitations of the . . . Gauging Uncertainty: . . . Gauging Spatial . . . Title Page ◭◭ ◮◮ ◭ ◮ Page 3 of 18 Go Back Full Screen Close Quit

2. How Seismic Inverse Problem Is Solved

First, we discretize: divide region into cells, with con-

stant velocity vj in each cell j.

Once we know vj, we can determine the (fastest) paths
f the seismic waves – which leads to Snell’s law:

sin(α1) v1 = sin(α2) v2 .

❇ ❇ ❇ ❇ ❇ ❇ ❆ ❆ ❆ ❆ ❆✁ ✁ ✁ ✁ ✁ ✂ ✂ ✂ ✂ ✂ ✂

❅
❅

✻ ❄ ✻ ❄

d2 d1 α1 α2 v2 v1

SLIDE 4

Need for Seismic Data . . . How Seismic Inverse . . . Uncertainty vs. . . . How Traditional . . . How This Idea Is . . . Limitations of Ray . . . Angular Diversity: A . . . Limitations of the . . . Gauging Uncertainty: . . . Gauging Spatial . . . Title Page ◭◭ ◮◮ ◭ ◮ Page 4 of 18 Go Back Full Screen Close Quit

3. How Seismic Inverse Problem Is Solved (cont-d)

Main idea: the measured travel-time ti along the i-th

path is ti =

j

ℓij · sj, where sj

def

= 1 vj .

Algorithm: repeat the following two steps until the pro-

cess converges: – based on the current values of sj, find the shortest paths and thus, the values ℓij; – based on the current values ℓij, we solve the above system of linear equations, and get the updated sj.

Need to find spatial resolution (granularity): it is im-

possible to distinguish between sj at two nearby points.

Fact: some reconstructed value sj relate not to an in-

dividual cell but to a cluster of cells (spatial granule).

SLIDE 5

Need for Seismic Data . . . How Seismic Inverse . . . Uncertainty vs. . . . How Traditional . . . How This Idea Is . . . Limitations of Ray . . . Angular Diversity: A . . . Limitations of the . . . Gauging Uncertainty: . . . Gauging Spatial . . . Title Page ◭◭ ◮◮ ◭ ◮ Page 5 of 18 Go Back Full Screen Close Quit

4. Uncertainty vs. Spatial Resolution (Granularity)

There are two types of uncertainty:

– the “traditional” uncertainty – due to measurement inaccuracy and incomplete coverage. – the spatial resolution – each measured value repre- sents the “average” value over the region (granule).

Fact: methods of determining traditional uncertainty

have been traditionally more developed.

Corollary: the main ideas for determining spatial res-
lution comes from these more traditional methods.
In view of this,

– before we describe the existing methods for deter- mining spatial resolution, – let us describe the corresponding methods for de- termining more traditional uncertainty.

SLIDE 6

Need for Seismic Data . . . How Seismic Inverse . . . Uncertainty vs. . . . How Traditional . . . How This Idea Is . . . Limitations of Ray . . . Angular Diversity: A . . . Limitations of the . . . Gauging Uncertainty: . . . Gauging Spatial . . . Title Page ◭◭ ◮◮ ◭ ◮ Page 6 of 18 Go Back Full Screen Close Quit

5. How Traditional Uncertainty Is Determined: Main Idea

Main idea:

– First, we add simulated noise to the measured val- ues of traveltimes. – Then, we reconstruct the new values of the veloci- ties based on these modified traveltimes. – Finally, we compare the resulting velocities with the originally reconstructed ones.

Main assumption: we know the accuracy of different

measurements.

Fact: geophysical analysis usually also involves expert

knowledge.

Conclusion: it is also necessary to take into account

the uncertainty of the expert statements.

SLIDE 7

Need for Seismic Data . . . How Seismic Inverse . . . Uncertainty vs. . . . How Traditional . . . How This Idea Is . . . Limitations of Ray . . . Angular Diversity: A . . . Limitations of the . . . Gauging Uncertainty: . . . Gauging Spatial . . . Title Page ◭◭ ◮◮ ◭ ◮ Page 7 of 18 Go Back Full Screen Close Quit

6. How This Idea Is Applied to Determine Spatial Resolution

First, we add a perturbation of spatial size δ0 (e.g.,

sinusoidal) to the reconstructed field v(x).

Then, we simulate the new traveltimes based on the

perturbed values of the velocities.

Finally, we apply the same seismic data processing al-

gorithm to the simulated traveltimes and get vnew(x).

If the perturbations are visible in

vnew(x) − v(x), then details of spatial size δ0 can be reconstructed.

If the perturbations are not visible in

vnew(x) − v(x), then details of spatial size δ0 cannot be reconstructed.

In the geosciences, this method is known as a checker-

board method.

SLIDE 8

Need for Seismic Data . . . How Seismic Inverse . . . Uncertainty vs. . . . How Traditional . . . How This Idea Is . . . Limitations of Ray . . . Angular Diversity: A . . . Limitations of the . . . Gauging Uncertainty: . . . Gauging Spatial . . . Title Page ◭◭ ◮◮ ◭ ◮ Page 8 of 18 Go Back Full Screen Close Quit

7. Checkerboard Method: Main Limitation

Main limitation: Its running time is several times higher

than the original seismic data processing. Indeed, – in addition to applying the seismic data processing algorithm to the original data, – we also need to apply the same algorithm to the simulated data – and apply it several times.

The computation time drastically increases – and the

whole process slows down.

It is therefore desirable to develop faster techniques for

estimating spatial resolution, – techniques that will not require new processing of simulated seismic data – and will only use the results of the processing the

riginal seismic data.

SLIDE 9

Need for Seismic Data . . . How Seismic Inverse . . . Uncertainty vs. . . . How Traditional . . . How This Idea Is . . . Limitations of Ray . . . Angular Diversity: A . . . Limitations of the . . . Gauging Uncertainty: . . . Gauging Spatial . . . Title Page ◭◭ ◮◮ ◭ ◮ Page 9 of 18 Go Back Full Screen Close Quit

8. A Similar Problem Arises For Estimating Tradi- tional Uncertainty

As mentioned, the existing methods for determining

traditional uncertainty are also based on – simulating errors and – applying the (time-consuming) seismic data pro- cessing algorithms to the simulated traveltimes.

As a result, the existing methods for determining the

traditional uncertainty are also too time-consuming,

There is a similar need to developing faster uncertainty

estimation techniques.

SLIDE 10

Need for Seismic Data . . . How Seismic Inverse . . . Uncertainty vs. . . . How Traditional . . . How This Idea Is . . . Limitations of Ray . . . Angular Diversity: A . . . Limitations of the . . . Gauging Uncertainty: . . . Gauging Spatial . . . Title Page ◭◭ ◮◮ ◭ ◮ Page 10 of 18 Go Back Full Screen Close Quit

9. First Heuristic Idea For Estimating Uncertainty: Ray Coverage

The more measurements we perform, the more accu-

rately we can determine the desired quantity.

In particular, for each cell j, the value vj affects those

traveltime measurements ti for which for which ℓij > 0.

Thus, the more rays pass through the cell, the more

accurate the corresponding measurement.

The number of such rays – called a ray coverage – is

indeed reasonably well correlated with uncertainty.

Thus it can serve as an estimate for this uncertainty:

– the smaller the ray coverage, – the larger the uncertainty.

SLIDE 11

Need for Seismic Data . . . How Seismic Inverse . . . Uncertainty vs. . . . How Traditional . . . How This Idea Is . . . Limitations of Ray . . . Angular Diversity: A . . . Limitations of the . . . Gauging Uncertainty: . . . Gauging Spatial . . . Title Page ◭◭ ◮◮ ◭ ◮ Page 11 of 18 Go Back Full Screen Close Quit

10. Limitations of Ray Coverage and the DWS Idea

Counting the rays doesn’t take into account that some

rays barely go though the cell (i.e., ℓij is small).

A more adequate idea is thus to compute the sum of

the lengths D(j) =

i

ℓij.

This sum make sense if we assume that the slowness is

constant inside each cell: s(x) ≈ sj.

Often, we use a linear interpolation instead:

s(x) ≈ ωj(x) · sj.

In this case, instead of the sum of the lengths, it is

reasonable to take the sum of the integrals D(j) =

i
ωj(x) dγi.
This is actually the original form of the DWS.

SLIDE 12

Need for Seismic Data . . . How Seismic Inverse . . . Uncertainty vs. . . . How Traditional . . . How This Idea Is . . . Limitations of Ray . . . Angular Diversity: A . . . Limitations of the . . . Gauging Uncertainty: . . . Gauging Spatial . . . Title Page ◭◭ ◮◮ ◭ ◮ Page 12 of 18 Go Back Full Screen Close Quit

11. Angular Diversity: A Similar Approach To Spatial Resolution

Notation: let eij = (eij,1, eij,2, eij,3) be a unit vector in

the direction in which the i-th ray crosses the j-th cell.

Main idea: for each cell j, compute a 3 × 3 matrix

Rab(j) =

i

ℓij · eij,a · eij,b.

Plot, for each unit vector e = (e1, e2, e3), the value
a,b

Rab(j) · ea · eb in the corresponding direction.

Conclusions based on the resulting ellipsoid E:

– If E is close to a sphere, we have equally good spa- tial resolution in different directions. – If E is strongly tilted in one direction, spatial res-

lution in this direction is not good.

SLIDE 13

Need for Seismic Data . . . How Seismic Inverse . . . Uncertainty vs. . . . How Traditional . . . How This Idea Is . . . Limitations of Ray . . . Angular Diversity: A . . . Limitations of the . . . Gauging Uncertainty: . . . Gauging Spatial . . . Title Page ◭◭ ◮◮ ◭ ◮ Page 13 of 18 Go Back Full Screen Close Quit

12. Limitations of the Known Approaches

From the application viewpoint, the main limitation is

that these methods are, in effect, qualitative.

They do not give a geophysicist any specific guidance
n how to use these techniques:

– what exactly is the accuracy? – what exactly is the spatial resolution in different directions?

Additional limitation: the above methods are heuristic

techniques.

They are not justified – statistically or otherwise.
It is therefore desirable to provide justified quantitative

estimates: – for uncertainty – and for spatial resolution.

SLIDE 14

Need for Seismic Data . . . How Seismic Inverse . . . Uncertainty vs. . . . How Traditional . . . How This Idea Is . . . Limitations of Ray . . . Angular Diversity: A . . . Limitations of the . . . Gauging Uncertainty: . . . Gauging Spatial . . . Title Page ◭◭ ◮◮ ◭ ◮ Page 14 of 18 Go Back Full Screen Close Quit

13. Gauging Uncertainty: Gaussian Approach

For each cell j, each ray i that passes though it leads

to an equation ℓij · sj + . . . = ti.

Assumption: measurement errors are independent and

normally distributed, with standard deviation σ.

The probability of a given value sj is proportional to
i

exp

−(ℓij · sj + . . . − ti)2

2σ2

∼ exp
−(sj − . . .)2

2σ(j)2

,

where (σ(j))2 def = σ2 D2(j), and D2(j)

def

=

i

ℓ2

ij.

Conclusion: The resulting estimate for sj is normally

distributed, with standard deviation σ(j) = σ

D2(j)

.

This formula is similar for the formula for the DWS,

the only difference is that we add up ℓ2

ij instead of ℓij.

SLIDE 15

Need for Seismic Data . . . How Seismic Inverse . . . Uncertainty vs. . . . How Traditional . . . How This Idea Is . . . Limitations of Ray . . . Angular Diversity: A . . . Limitations of the . . . Gauging Uncertainty: . . . Gauging Spatial . . . Title Page ◭◭ ◮◮ ◭ ◮ Page 15 of 18 Go Back Full Screen Close Quit

14. Gauging Uncertainty: Robust Statistical Approach

For normal distr., we get

i

e2

i

σ2

i

→ min, where ei is the difference between the model and measured values.

Often, measurement and estimation errors are not nor-

mally distributed.

Moreover, we often do not know the shape of the cor-

responding distribution.

In this case, instead it makes sense to consider so-called

lp-methods

i

ep

i

σp

i

→ min.

For seismic data processing, the empirical value p is

close to 1.

Thus, we get σ(j) =

σ D(j), where D(j) =

i

ℓij is exactly the DWS expression.

SLIDE 16

Need for Seismic Data . . . How Seismic Inverse . . . Uncertainty vs. . . . How Traditional . . . How This Idea Is . . . Limitations of Ray . . . Angular Diversity: A . . . Limitations of the . . . Gauging Uncertainty: . . . Gauging Spatial . . . Title Page ◭◭ ◮◮ ◭ ◮ Page 16 of 18 Go Back Full Screen Close Quit

15. Gauging Spatial Resolution

Question: What is the accuracy with which we can

determine, e.g., the partial derivative ∂s ∂x1 ?

If the ray i is parallel to x1, then we can only determine

the average value (s + s′)/2.

The difference between the corresponding interpolation

coefficients at s and s′ is proportional to ℓij · sin(αij,1).

✻ ✁ ✁ ✁ ✁ ✁ ✁

ℓij αij,1 ∂s ∂x1

✲ ✛

SLIDE 17

Need for Seismic Data . . . How Seismic Inverse . . . Uncertainty vs. . . . How Traditional . . . How This Idea Is . . . Limitations of Ray . . . Angular Diversity: A . . . Limitations of the . . . Gauging Uncertainty: . . . Gauging Spatial . . . Title Page ◭◭ ◮◮ ◭ ◮ Page 17 of 18 Go Back Full Screen Close Quit

16. Gauging Spatial Resolution (cont-d)

So, in addition to a term ∼ (s+s′)/2, we also get a term

proportional to s′ − s, with a coefficient ℓij · sin(αij,1).

Similarly to uncertainty, the accuracy in ∂s

∂x1 is ∼ σ2

1 =

σ2 D11(j), where D11 =

i

ℓ2

ij · sin2(αij,1).

In general, the accuracy in the direction e = (e1, e2, e3)

is ∼ σ

De(j)

, where De(j) = Dab(j) · ea · eb, and Dab(j) = D2(j) · δab −

i

ℓ2

ij · eij,a · eij,b.

This formula is similar to the ray density tensor for-

mula, with ℓ2

ij instead of ℓij.

In the robust case, we get exactly the ray density ten-

sor.

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Need for Seismic Data . . . How Seismic Inverse . . . Uncertainty vs. . . . How Traditional . . . How This Idea Is . . . Limitations of Ray . . . Angular Diversity: A . . . Limitations of the . . . Gauging Uncertainty: . . . Gauging Spatial . . . Title Page ◭◭ ◮◮ ◭ ◮ Page 18 of 18 Go Back Full Screen Close Quit