Отправлено: 31.12.08 10:41. Заголовок: HUME , POPPER, BAYES AND GEOPHYSICAL INVERSE PROBLEMS

B020 HUME , POPPER, BAYES AND GEOPHYSICAL INVERSE PROBLEMS
G .G . DRIJKONINGEN and A .M . ZiOLI<OWSI<I
Technical . University Delft, Dept . of Minieg and Petroleum Engineering, Mijnbouwstraat
120, 2628 RX Delft, The Netherlands

We distinguish two approaches to geophysical inverse problems . One is the
opposite, or inverse, of the forward problem . Signature deconvolution and migration
are examples of this kind of inversion. The data are inverted to arrive at the Ba rth
model. The other approach is iterative forward modeping, in which we try to match the
measured data with synthetic data created with a model and the given theo ry. In recent
years Bayes's Tule has been a popular way to measure the quality of the result obtained
by iterative forward modelleng . We focus on the philosophical arguments for the two
approaches and on the difficulties in applying them to geophysical inverse problems .
The differences between these two approaches are related to the problem of
induction. In 1739, the philosopher David Hume (see, e .g., 1984) split the problem of
induction into two: a logical problem and a psychological problem. The logical problem
of induction is expressed by the question : Can we prove that a universal theo ry is tree?
Hume proved that the answer is no. The instances of our experience are always an
infinitessimally small fraction of the total number of possible instances . For example,
all our experiences of gravitational attraction at non-relativistic velocities corroborate
Newton's law of gravitation. We are confident that Newton's law applies throughout the
universe. But the law has not been proved . In fact, compared with the total amount of
evidence we should need to prove Newton's law throughout the whole universe, our
experience is infinitessimal . The probability (in the sense of the calculus of probability)
of Newton's law of gravitation being tree is then infinitesimally small .
Although Hume .solved the logical problem of induction, there stip remained the
so-called psychological problem of induction : Why do people have confidence in their
theories? Why do people believe that their past experience has relevance to the future?
For example, why do we believe the sun will rise tomorrow? In the first half of this
century, Popper (1934, 1959, 1969, 1986) solved this problem . Although Hume is
quite right that we can never prove a universal theory to be tree, we are able to prove
that a theory is fake . This can be established via some test against reality . If we test a
theory, exposing it to risks designed to question the axioms on which it rests, there is
the possibility that the theory will fail. If it does fail, we have to firid new axioms, or
laws. If it survives the tests, and continues to survive new tests, our con fidence in it is
increased . However, since the evidence from these tests is infinitessimal compared
with the amount of evidence we should need to prove the theo ry , the absolute
probability of the theory being true is stip always zero . Not all the theories we use can
be put at risk in the way suggested by Popper . Popper uses this distinction as the
demarcation between scientific and non-scientific theories . According to Popper,
theories are scientific if they are framed in such a way that they may be put at risk .
Theories are not scientific if they cannot be put at risk.
How do we decide whether a geven test has refuted a (scientific) theory? This is not
always clean-cut. If there is уnly onй wйll-established and accepted theory, it is very
difficult, psychologicбlly, to accept the evidence from a geven test that the theory has
been refuted. As Lakatos (1970) argued, in this situation ad hoc explanations are then
always proposed to account for the conflict of the evidence with the theo ry. He argued
that you really need to have compe ting theories. If a rival theory is able to explain the
118
new evidence as well as everything that was explained by the first theory, it is more
powerful and will displace the first theory .
There are geophysicists who take a probabilistic view of the world and express this
in mathematica) terms using Bayes's Tule (See; for example, Howson and Urbach,
1989). These geophysicists do not put the theory at risk, but they do question the model
which, together with the theory, yields synthetic data which are a match to the
measured data. They ask : What is the probability of the model being tree, given the
data? Bayes's Tule allows them to answer the question as follows :
P(model 1 data) = P
(data 1 model) P(model)
P(data)
in which P denotes the probability . The a priori information is given by P(model), and
we "learn" from the data via the likelihood P(data 1 model). The probability of the data
P(data) is usually considered to be a constant in order to make P(model 1 data) also a
probability function .
In our opinion the mais objection to the Bayesian approach as a philosophy, is the
use of probabilities, in the sense of the calculus of probability. The presence of
P(model), the a priori distribution, poses the mais difficulty . We do not know whether
the chosen parameterisation of the model is tree . There are, in principle, an infinite
number of ways in which we can parameterise the model . The probability of a given
parameterisation being tree is therefore zero . But we cas use Bayes's Tule only if the
probability of the parameterisation is finite . In fact, Bayes's Tule applied in the sense in
which it is used here only defines a given optimisation problem. Of course, the
optimisation problem is very interenting and geophysicists need methods to solve this
problem, but it is not the same as determining the probability that the solution to the
geophysical inverse problem han been found .
So, what cas we do then as geophysicists? We should certainly question our
parameterisations . How many significant parameters do we have? How many
significant data do we have? How do we define significant data? In order to solve
geophysical inverse problems witti scientific theories, we believe we should have at
least three requirements. First, we should put our theories at risk and test every step in a
given theory. Secondly, we should always have fewer parameters than significant data.
In this way we have anoverdetermined system and can quantify our errors . Finally, we
should always pose the question : What would we regard as a refutation of our
solution? We illustrate these points using examples taken from the field of geophysics .
References
Howson, C. and P. Urbach, 1989. Scientific Reasoning, The Bayesian Approach, Open
Court Publ . Co., La S alle, Illinois .
Hume, D., 1984. A Treatise of Human Nature, Penguin Books Ltd., London, U.K.
Lakatos, 1., 1970. Falsification and the methodology of scientific research programs. In:
Criticism and the Growth of Knowledge, I. Lakatos and A. Musgrave (ed),
Cambridge University Press, Cambridge, U.K.
Popper, K.R., 1934. Logik der Forschung, Julius Springer Verlag, Vienna.
Popper, K.R., 1959 . The Logic of Scientific Discovery, Hutchinson, London.
Popper, K.R., 1969 . Conjectures and Refutations, Roudedge and Kegan Paul, London.
Popper, K.R., 1986. Objective Knowledge, An Evolutionary Approach (revised
edition), Clarendon Press, Oxford, U.K.
119