38
Statistical Methods for Estimating Petroleum Resources
the unknown inclusion probabilities. An estimate of N is obtained
by an approximate Horvitz–Thompson-type estimator. This procedure requires solving a pair of symmetrical transcendental equations.
Barouch et al. (1985) proposed an alternate pair of asymmetrical transcendental equations to solve the problem.
The fourth method postulates that N also has a superpopulation
probability function, P(•|γ), indexed by a vector of parameters, γ, that is
independent of the variate, the pool sizes x. The posterior distribution
of γ is then used to make inferences about N. Here, the observations
consist of x n and N > n. The probability function, P(N |γ), may be interpreted as a model describing a random mechanism of how N is generated, or it might be considered as a prior distribution in an empirical
Bayesian context. Wang and Nair (1988) presented a lognormal case,
which was extended as a generalized procedure (Lee, 1997).
The BDSCV Model
Four statistical assumptions are inherent in the BDSCV model:
The probability of discovering a pool is proportional to its size
1.
with an exponential β (i.e., a large pool has a better chance of
being discovered).
Sampling occurs without replacement (i.e., a pool will not be
2.
discovered twice).
The pool-size distribution is approximated by a lognormal or
3.
nonparametric distribution.
The prior distribution of the number-of-pools distribution
4.
is approximated by a Poisson distribution or is assigned by
geologists.
The fi rst two assumptions are the same as NDSCV, and the fi rst three
assumptions are the same as LDSCV. The posterior number-of-pools
probability distribution can be any type of distribution. BDSCV provides a probability statement about the N value and also provides a
probability measure for each individual pool size (Lee and Wang,
1983b).
Now we use the lognormal hypothetical population and two discovery sequences to demonstrate the advantages of BDSCV. NDSCV was
used to make the point estimate about the N value, the nonparametric
pool-size distribution, and the exploration effi ciency, β. These estimates were entered into BDSCV for estimating the number-of-pools
Statistical Methods for Estimating Petroleum Resources
the unknown inclusion probabilities. An estimate of N is obtained
by an approximate Horvitz–Thompson-type estimator. This procedure requires solving a pair of symmetrical transcendental equations.
Barouch et al. (1985) proposed an alternate pair of asymmetrical transcendental equations to solve the problem.
The fourth method postulates that N also has a superpopulation
probability function, P(•|γ), indexed by a vector of parameters, γ, that is
independent of the variate, the pool sizes x. The posterior distribution
of γ is then used to make inferences about N. Here, the observations
consist of x n and N > n. The probability function, P(N |γ), may be interpreted as a model describing a random mechanism of how N is generated, or it might be considered as a prior distribution in an empirical
Bayesian context. Wang and Nair (1988) presented a lognormal case,
which was extended as a generalized procedure (Lee, 1997).
The BDSCV Model
Four statistical assumptions are inherent in the BDSCV model:
The probability of discovering a pool is proportional to its size
1.
with an exponential β (i.e., a large pool has a better chance of
being discovered).
Sampling occurs without replacement (i.e., a pool will not be
2.
discovered twice).
The pool-size distribution is approximated by a lognormal or
3.
nonparametric distribution.
The prior distribution of the number-of-pools distribution
4.
is approximated by a Poisson distribution or is assigned by
geologists.
The fi rst two assumptions are the same as NDSCV, and the fi rst three
assumptions are the same as LDSCV. The posterior number-of-pools
probability distribution can be any type of distribution. BDSCV provides a probability statement about the N value and also provides a
probability measure for each individual pool size (Lee and Wang,
1983b).
Now we use the lognormal hypothetical population and two discovery sequences to demonstrate the advantages of BDSCV. NDSCV was
used to make the point estimate about the N value, the nonparametric
pool-size distribution, and the exploration effi ciency, β. These estimates were entered into BDSCV for estimating the number-of-pools