Seismic Data Integration
Geostatistical Reservoir Characterization
Well Data vs. Seismic Data
l Well Data
- Accurate
- Good vertical resolution
- Sparse and biased sampling
l Seismic Data (Attributes)
- Dense areal coverage and equally sampled
- Poor vert

INDICATOR METHODS
The choice of multigaussian models imposes almost no
correlations for extreme values, both high and low. This is
geologically unrealistic, since for modeling fluid flow, we
are mostly interested in reproducing the high permeability
chann

CROSS VALIDATION
Purpose:
Cross validation is a technique to test the validity of an
approach or tool to be used for analysis of data. Cross
validation is aimed towards determining what could go
wrong; however, it does not ensure the success of the
techni

CONDITIONAL SIMULATION
Some important points about kriging :
Estimator is exact (reproduces zi at known points).
Kriging surfaces are generally too smooth and thus tend to
underestimate heterogeneity.
Kriging surface is deterministic (not stochastic).
Con

KRIGING
GEOSTATISTICAL ANALYSIS
Define a space A, deemed homogeneous enough to warrant
statistical averaging within it (assumption of stationarity).
Scan all available data within A to calculate the experimental
(raw) variogram, i.e. assumption of ergod

BIVARIATE, MARGINAL AND CONDITIONAL
DISTRIBUTIONS
Very often in earth sciences we are interested in the
simultaneous behavior of two random variables. More
importantly, we want to know the pattern of dependence
relating one variable to another. Knowledge

COKRIGING
Cokriging refers to kriging of more than one variable at a
time, e.g. permeability and porosity. The variable to be
estimated is called primary and the other secondary.
The estimator is given as
N
M
z* = z + y
i i
j j
i =1
j =1
z = primary vari

BLOCK KRIGING
Objective:
Estimate average of an attribute z over a block v centered at u.
Assume:
Linear averaging
zv (u) =
1
1 N
z(u) du i=1 z(ui )
v v(u)
N
Approach:
z v (u ) =
n (u )
=1
v ( u ) z ( u )
zv (u ) must be unbiased and minimum variance
cfw

MULTIDISCIPLINARY DATA
INTEGRATION
Geostatistical Reservoir Characterization
What is our Goal?
Honor well data and reproduce small scale
variability observed in well logs/cores.
Reproduce the large scale structure and
continuity observed in seismic data.

Modeling Zonal Anisotropy (After Andre Journel)
Consider the following two directional variograms, in the x
(East) and y (North) directions; they are fitted with nested
sums of spherical structures, and show zonal anisotropy.
Note also the anisotropic nug

VARIOGRAM
Variogram measures the degree of similarity between two
samples taken some distance apart. It is defined as follows:
r
r 2
2 (h ) = E z ( x) z ( x + h )
and can be computed as
(
)
r
r 2
1 N
(h ) =
z( x ) z( x + h )
i
2 N i = 1 i
where N is t

Nonparametric Regression:
Background and Applications
1
Outline
Parametric vs. Non-parametric Regression
Scatterplot Smoothers
Selection of Smoothing Parameters
ACE and GAM
Applications
2
Parametric Regression
E (Y | X ) = + X
E (Y | X ) = + X1 + X 2 2
X

MOMENTS AND EXPECTATIONS
NON-CENTERED MOMENTS
For a random variable Z and its associated p.d.f f ( z ) , the
r th non-centered moment will be:
r =
z r f ( z ) dz
For a discrete RV
1
r =
N
N
zir
i =1
There is no p.d.f for the discrete RV. The p.d.f is re

BACKGROUND IN
UNIVARIATE ANALYSIS
RANDOM VARIABLE (RV)
Definition
A random variable denotes a numerical quantity that is defined in
terms of the outcome of an experiment, e.g. the grade of an ore, the
outcome of a coin toss, etc.
The random variable, z,

COVARIANCE AND CORRELATIONS
Consider the variance of the sum of (or difference of ) two
random variables that are not necessarily independent.
cfw_
var( x y ) = E ( x y )
2
cfw_ E ( x y )
2
= var( x) + var( y ) 2cov( x, y )
where
cov( x, y )
cfw_
Some c

CONDITIONAL EXPECTATION: BASICS
Consider the data set
Here a linear regression may not be a good assumption
Instead, we can calculate the mean value of Y for
different ranges of X. This is a type of conditional
expectation curve.
The aim of regression