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Re: [ai-geostats] Pareto vs Lognormal distribution

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  • Syed Shibli
    May I suggest that you look at other analogous datasets with n 25, e.g. the North Sea basin or Gulf of Mexico, before making some firm conclusions about
    Message 1 of 6 , Sep 1, 2005
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      May I suggest that you look at other analogous datasets with n > 25, e.g. the North Sea basin or Gulf of Mexico, before making some firm conclusions about whether Pareto or Lognormal works best. A lot of this information is in the public domain, one can browse the Websites of the UK DTI, Norwegian Petroleum Directorate, or the Danish Energy Agency for public domain field information. Certainly, at first blush, the log normal seems to make more sense than the other, you have your few giant fields (Brent, Statfjord, Ekofisk, etc), lots of middle sized fields, and many more small pimples in the North Sea that have yet to be developed.

      Cheers

      Syed

      On Wednesday, August 31, 2005, at 11:33PM, Beatrice Mare-Jones <B.Mare-Jones@...> wrote:

      >Hello list
      >
      >I am a PhD student looking at developing a statistical model to predict
      >the size-distribution of an area's oil and gas fields.
      >
      >It is clear that previous investigators prefer either a Pareto power law
      >or a lognormal distribution to approximate field-size distributions.
      >
      >The data I am using does not look like it comes from a Pareto distribution
      >- which I explain as being a result of undersampling - which previous
      >investigators have reported - that undersampling occurs because the small
      >fields are not sampled or recoded. However by using basin-modelling
      >software to simulate oil and gas fields (for the same basin that my
      >discovered empirical data comes from) I notice that this sample is also
      >undersampled - that is fields under a certain size are not being simulated
      >- which is probably due to the resolution of my input data but what is
      >interesting is that the undersampling actually occurs throughout all the
      >size ranges - including the medium to larger sizes - which I would not
      >have expected. Like the discovery dataset (n = 25) the simulated dataset
      >(n = 140) looks like it is more from a lognormal distribution than a
      >Pareto distribution.
      >
      >My conclusion is that without being able to say that a Pareto is better
      >than a lognormal and vise-versa it appears only logical to use both
      >distributions.
      >
      >Geologically there does not seems to be a reason why a modal size (greater
      >than what is detectable by exploration methods) of fields should exist -
      >which would be the case if the data was from a lognormal distribution -
      >except if the distribution is highly right skewed (at the small field
      >size) and the mode is actually just below the detection of size.
      >
      >Geologically there does seem reason for fields to become so small that
      >they become entities (that trap oil and gas) - and this relationship may
      >be better approximated by a Pareto.
      >
      >
      >The Pareto and lognormal form is similar but maybe one is better to
      >approximate field sizes than the other.
      >My question is do you think a Pareto distribution better approximates an
      >oil and gas size distribution than a lognormal (or vise-versa) and if so
      >why.
      >
      >
      >I am currently working on goodness of fit test to throw some more light on
      >this - but if anyone has any thing to say I'd appreciate some comments.
      >
      >Thank you,
      >
      >Kind regards
      >
      >Beatrice
      >
      >Geological and Nuclear Sciences
      >New Zealand
      >www.gns.cri.nz
      >
      >
      >* By using the ai-geostats mailing list you agree to follow its rules
      >( see http://www.ai-geostats.org/help_ai-geostats.htm )
      >
      >* To unsubscribe to ai-geostats, send the following in the subject or in the body (plain text format) of an email message to sympa@...
      >
      >Signoff ai-geostats
      >
    • Chris Hlavka
      Beatrice - This is a vexing problem that I ve tried to deal with in sizes of features in satellite imagery (Hlavka, C. A. and J. L. Dungan, 2002. Areal
      Message 2 of 6 , Sep 1, 2005
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        Re: [ai-geostats] Pareto vs Lognormal distribution
        Beatrice - This is a vexing problem that I've tried to deal with in sizes of features in satellite imagery (Hlavka, C. A. and J. L. Dungan, 2002.  Areal estimates of fragmented land cover - effects of pixel size and model-based corrections. International Journal of Remote Sensing23(4): 711-724.)  The affine (count versus continuous) nature of the digital imagery is at least part of the problem.  I've used probability plots to assess type of distribution.

        In gas field work, there is evidence that the apparent lognormality of field-sizes is due to lower rates of discovery of smaller fields than larger fields - especially for older surveys.   It has been noted that newer field data was closer to Pareto than older data and thus inferred that the actual distribution is Pareto.  -- Chris



        Hello list

        I am a PhD student looking at developing a statistical model to predict
        the size-distribution of an area's oil and gas fields.

        It is clear that previous investigators prefer either a Pareto power law
        or a lognormal distribution to approximate field-size distributions.

        The data I am using does not look like it comes from a Pareto distribution
        - which I explain as being a result of undersampling - which previous
        investigators have reported - that undersampling occurs because the small
        fields are not sampled or recoded.  However by using basin-modelling
        software to simulate oil and gas fields (for the same basin that my
        discovered empirical data comes from) I notice that this sample is also
        undersampled - that is fields under a certain size are not being simulated
        - which is probably due to the resolution of my input data but what is
        interesting is that the undersampling actually occurs throughout all the
        size ranges - including the medium to larger sizes - which I would not
        have expected.  Like the discovery dataset (n = 25)  the simulated dataset
        (n = 140) looks like it is more from a lognormal distribution than a
        Pareto distribution.

        My conclusion is that without being able to say that a Pareto is better
        than a lognormal and vise-versa it appears only logical to use both
        distributions.

        Geologically there does not seems to be a reason why a modal size (greater
        than what is detectable by exploration methods) of fields should exist  -
        which would be the case if the data was from a  lognormal distribution -
        except if the distribution is highly right skewed (at the small field
        size) and the mode is actually just below the detection of size.

        Geologically there does seem reason for fields to become so small that
        they become entities (that trap oil and gas)  - and this relationship may
        be better approximated by a Pareto.


        The Pareto and lognormal form is similar but maybe one is better to
        approximate field sizes than the other.
        My question is do you think a Pareto distribution better approximates an
        oil and gas size distribution than a  lognormal (or vise-versa) and if so
        why.


        I am currently working on goodness of fit test to throw some more light on
        this - but if anyone has any thing to say I'd appreciate some comments.

        Thank you,

        Kind regards

        Beatrice

        Geological and Nuclear Sciences
        New Zealand
        www.gns.cri.nz


        * By using the ai-geostats mailing list you agree to follow its rules
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        -- 
        
        ***************************************
        Chris Hlavka
        NASA/Ames Research Center 242-4
        Moffett Field, CA 94035-1000
        (650)604-3328  FAX 604-4680
        Christine.A.Hlavka@...
        ***************************************
      • Ted Harding
        I m intruding into foreign territory here, since I don t have experience in exploring for gas fields etc., (though I have had to deal with patches of
        Message 3 of 6 , Sep 2, 2005
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          I'm intruding into foreign territory here, since I don't have
          experience in exploring for gas fields etc., (though I have had
          to deal with patches of contamination, which is the same sort
          of thing on a small scale). So apologies if I blunder around
          and tread on toes!

          Be that as it may, a point which has occurred to me in reading
          this thread is that the distribution being observed is the
          distribution of size conditional on being discovered, and the
          probability of being discovered may be expected to increase
          with size.

          So the frequency f(x) of occurrence of a size x in nature is
          attenuated by a factor equal to the probability that an item
          of size x will be discovered -- g(x) say. The frequency of x
          in the observed data is f(x | D), say, and so

          f(x | D) = f(x)*g(x)

          from which f(x) = f(x | D)/g(x). From this point, provided
          there is a reasonaboly justifiable model for g(x) (to within
          a constant of proportionality, e.g. simply g(x) = x), you can
          "demodulate" the observed data to infer the "wild" data.

          There has for many decades been a similar problem in classical
          geometrical probability (the ancestor of spatial statistics
          and morphometry), namely to infer the distribution of (e.g.)
          areas of cells given the observed distribution of the sizes
          of transects by lines, or of counts of sampling points intersecting
          them, leading to an integral equation.

          Maybe all this is old hat in the areas you are investigating,
          but since it did not seem to be even implicit in the discussion
          so far I thought I would bring it to the surface.

          Best wishes to all,
          Ted.

          On 01-Sep-05 Chris Hlavka wrote:
          > Beatrice - This is a vexing problem that I've tried to deal with in
          > sizes of features in satellite imagery (Hlavka, C. A. and J. L.
          > Dungan, 2002. Areal estimates of fragmented land cover - effects of
          > pixel size and model-based corrections. International Journal of
          > Remote Sensing23(4): 711-724.) The affine (count versus continuous)
          > nature of the digital imagery is at least part of the problem. I've
          > used probability plots to assess type of distribution.
          >
          > In gas field work, there is evidence that the apparent lognormality
          > of field-sizes is due to lower rates of discovery of smaller fields
          > than larger fields - especially for older surveys. It has been
          > noted that newer field data was closer to Pareto than older data and
          > thus inferred that the actual distribution is Pareto. -- Chris
          >
          >
          >
          >>Hello list
          >>
          >>I am a PhD student looking at developing a statistical model to predict
          >>the size-distribution of an area's oil and gas fields.
          >>
          >>It is clear that previous investigators prefer either a Pareto power
          >>law
          >>or a lognormal distribution to approximate field-size distributions.
          >>
          >>The data I am using does not look like it comes from a Pareto
          >>distribution
          >>- which I explain as being a result of undersampling - which previous
          >>investigators have reported - that undersampling occurs because the
          >>small
          >>fields are not sampled or recoded. However by using basin-modelling
          >>software to simulate oil and gas fields (for the same basin that my
          >>discovered empirical data comes from) I notice that this sample is also
          >>undersampled - that is fields under a certain size are not being
          >>simulated
          >>- which is probably due to the resolution of my input data but what is
          >>interesting is that the undersampling actually occurs throughout all
          >>the
          >>size ranges - including the medium to larger sizes - which I would not
          >>have expected. Like the discovery dataset (n = 25) the simulated
          >>dataset
          >>(n = 140) looks like it is more from a lognormal distribution than a
          >>Pareto distribution.
          >>
          >>My conclusion is that without being able to say that a Pareto is better
          >>than a lognormal and vise-versa it appears only logical to use both
          >>distributions.
          >>
          >>Geologically there does not seems to be a reason why a modal size
          >>(greater
          >>than what is detectable by exploration methods) of fields should exist
          >>-
          >>which would be the case if the data was from a lognormal distribution
          >>-
          >>except if the distribution is highly right skewed (at the small field
          >>size) and the mode is actually just below the detection of size.
          >>
          >>Geologically there does seem reason for fields to become so small that
          >>they become entities (that trap oil and gas) - and this relationship
          >>may
          >>be better approximated by a Pareto.
          >>
          >>
          >>The Pareto and lognormal form is similar but maybe one is better to
          >>approximate field sizes than the other.
          >>My question is do you think a Pareto distribution better approximates
          >>an
          >>oil and gas size distribution than a lognormal (or vise-versa) and if
          >>so
          >>why.
          >>
          >>
          >>I am currently working on goodness of fit test to throw some more light
          >>on
          >>this - but if anyone has any thing to say I'd appreciate some comments.
          >>
          >>Thank you,
          >>
          >>Kind regards
          >>
          >>Beatrice
          >>
          >>Geological and Nuclear Sciences
          >>New Zealand
          >>www.gns.cri.nz
          >>
          >>
          >>* By using the ai-geostats mailing list you agree to follow its rules
          >>( see http://www.ai-geostats.org/help_ai-geostats.htm )
          >>
          >>* To unsubscribe to ai-geostats, send the following in the subject
          >>or in the body (plain text format) of an email message to
          >>sympa@...
          >>
          >>Signoff ai-geostats
          >
          >
          > --
          > ***************************************
          > Chris Hlavka
          > NASA/Ames Research Center 242-4
          > Moffett Field, CA 94035-1000
          > (650)604-3328 FAX 604-4680
          > Christine.A.Hlavka@...
          > ***************************************


          --------------------------------------------------------------------
          E-Mail: (Ted Harding) <Ted.Harding@...>
          Fax-to-email: +44 (0)870 094 0861
          Date: 02-Sep-05 Time: 09:16:02
          ------------------------------ XFMail ------------------------------
        • Beatrice Mare-Jones
          Hi Chris Thank you for your reply. And thank you for your paper reference - I ll take a look at your probability plots. Yes the apparent lognormality of oil
          Message 4 of 6 , Sep 2, 2005
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            Hi Chris

            Thank you for your reply. And thank you for your paper reference - I'll
            take a look at your probability plots.

            Yes the apparent lognormality of oil and gas fields which moves more to a
            Pareto form with progressive and mature exploration is explained by
            undersampling at the low end that eventually gets sampled as the economics
            of an area and technology make smaller fields viable.

            However I am surprised that my simulated oil and gas fields - based on
            basin modelling - and no economic and exploration-process involvement also
            produces lognormal populations of fields. And that teh undersampling is
            obvious throughout most of the size ranges - not just the small size end,

            I think Syed's suggesting to use a larger dataset from a mature area is a
            good way of seeing what the distribution is more like.


            Kind regards


            Beatrice

            Hydrocarbons Group
            Institute of Geological and Nuclear Sciences Limited
            69 Gracefield Road, Lower Hutt, WELLINGTON
            NEW ZEALAND
            64 4 570 4821
            b.mare-jones@...
            www.gns.cri.nz
          • Beatrice Mare-Jones
            HI Ted Thanks for your reply. Yes you are correct. The probability that a field will be discovered is the product of f(x) that it is from a natural abundance
            Message 5 of 6 , Sep 2, 2005
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              HI Ted

              Thanks for your reply.

              Yes you are correct. The probability that a field will be discovered is
              the product of f(x) that it is from a natural abundance (of its parent
              population) and as a function of its size g(x). And yes the larger the
              field is the greater the probability that it is found.

              I looked at described the probability that a field will be discovered
              conditional to its size for my empirical dataset as a discovery sequence -
              and although there is a first order trend of decreasing size with
              increasing discovery sequence there were lots of perturbations - for
              example the 3rd largest field was discovered 23rd in the discovery
              sequence. therefore describing the discovery sequence may not be a
              straightforward function. I may start off with a simplistic model to
              demodulate the observed data but may have more success with an integral as
              you have mentioned for the early geometrical work.

              The small dataset I have is probably not suitable to describe the
              discovery sequence theoretically - and I will use an analogue area to
              establish the discovery sequence function .

              Kind regards


              Beatrice


              Hydrocarbons Group
              Institute of Geological and Nuclear Sciences Limited
              69 Gracefield Road, Lower Hutt, WELLINGTON
              NEW ZEALAND
              64 4 570 4821
              b.mare-jones@...
              www.gns.cri.nz





              (Ted Harding) <Ted.Harding@...>
              02/09/2005 20:56
              Please respond to ted.harding


              To: Chris Hlavka <chlavka@...>
              cc: ai-geostats@..., Beatrice Mare-Jones <B.Mare-Jones@...>
              Subject: Re: [ai-geostats] Pareto vs Lognormal distribution


              I'm intruding into foreign territory here, since I don't have
              experience in exploring for gas fields etc., (though I have had
              to deal with patches of contamination, which is the same sort
              of thing on a small scale). So apologies if I blunder around
              and tread on toes!

              Be that as it may, a point which has occurred to me in reading
              this thread is that the distribution being observed is the
              distribution of size conditional on being discovered, and the
              probability of being discovered may be expected to increase
              with size.

              So the frequency f(x) of occurrence of a size x in nature is
              attenuated by a factor equal to the probability that an item
              of size x will be discovered -- g(x) say. The frequency of x
              in the observed data is f(x | D), say, and so

              f(x | D) = f(x)*g(x)

              from which f(x) = f(x | D)/g(x). From this point, provided
              there is a reasonaboly justifiable model for g(x) (to within
              a constant of proportionality, e.g. simply g(x) = x), you can
              "demodulate" the observed data to infer the "wild" data.

              There has for many decades been a similar problem in classical
              geometrical probability (the ancestor of spatial statistics
              and morphometry), namely to infer the distribution of (e.g.)
              areas of cells given the observed distribution of the sizes
              of transects by lines, or of counts of sampling points intersecting
              them, leading to an integral equation.

              Maybe all this is old hat in the areas you are investigating,
              but since it did not seem to be even implicit in the discussion
              so far I thought I would bring it to the surface.

              Best wishes to all,
              Ted.

              On 01-Sep-05 Chris Hlavka wrote:
              > Beatrice - This is a vexing problem that I've tried to deal with in
              > sizes of features in satellite imagery (Hlavka, C. A. and J. L.
              > Dungan, 2002. Areal estimates of fragmented land cover - effects of
              > pixel size and model-based corrections. International Journal of
              > Remote Sensing23(4): 711-724.) The affine (count versus continuous)
              > nature of the digital imagery is at least part of the problem. I've
              > used probability plots to assess type of distribution.
              >
              > In gas field work, there is evidence that the apparent lognormality
              > of field-sizes is due to lower rates of discovery of smaller fields
              > than larger fields - especially for older surveys. It has been
              > noted that newer field data was closer to Pareto than older data and
              > thus inferred that the actual distribution is Pareto. -- Chris
              >
              >
              >
              >>Hello list
              >>
              >>I am a PhD student looking at developing a statistical model to predict
              >>the size-distribution of an area's oil and gas fields.
              >>
              >>It is clear that previous investigators prefer either a Pareto power
              >>law
              >>or a lognormal distribution to approximate field-size distributions.
              >>
              >>The data I am using does not look like it comes from a Pareto
              >>distribution
              >>- which I explain as being a result of undersampling - which previous
              >>investigators have reported - that undersampling occurs because the
              >>small
              >>fields are not sampled or recoded. However by using basin-modelling
              >>software to simulate oil and gas fields (for the same basin that my
              >>discovered empirical data comes from) I notice that this sample is also
              >>undersampled - that is fields under a certain size are not being
              >>simulated
              >>- which is probably due to the resolution of my input data but what is
              >>interesting is that the undersampling actually occurs throughout all
              >>the
              >>size ranges - including the medium to larger sizes - which I would not
              >>have expected. Like the discovery dataset (n = 25) the simulated
              >>dataset
              >>(n = 140) looks like it is more from a lognormal distribution than a
              >>Pareto distribution.
              >>
              >>My conclusion is that without being able to say that a Pareto is better
              >>than a lognormal and vise-versa it appears only logical to use both
              >>distributions.
              >>
              >>Geologically there does not seems to be a reason why a modal size
              >>(greater
              >>than what is detectable by exploration methods) of fields should exist
              >>-
              >>which would be the case if the data was from a lognormal distribution
              >>-
              >>except if the distribution is highly right skewed (at the small field
              >>size) and the mode is actually just below the detection of size.
              >>
              >>Geologically there does seem reason for fields to become so small that
              >>they become entities (that trap oil and gas) - and this relationship
              >>may
              >>be better approximated by a Pareto.
              >>
              >>
              >>The Pareto and lognormal form is similar but maybe one is better to
              >>approximate field sizes than the other.
              >>My question is do you think a Pareto distribution better approximates
              >>an
              >>oil and gas size distribution than a lognormal (or vise-versa) and if
              >>so
              >>why.
              >>
              >>
              >>I am currently working on goodness of fit test to throw some more light
              >>on
              >>this - but if anyone has any thing to say I'd appreciate some comments.
              >>
              >>Thank you,
              >>
              >>Kind regards
              >>
              >>Beatrice
              >>
              >>Geological and Nuclear Sciences
              >>New Zealand
              >>www.gns.cri.nz
              >>
              >>
              >>* By using the ai-geostats mailing list you agree to follow its rules
              >>( see http://www.ai-geostats.org/help_ai-geostats.htm )
              >>
              >>* To unsubscribe to ai-geostats, send the following in the subject
              >>or in the body (plain text format) of an email message to
              >>sympa@...
              >>
              >>Signoff ai-geostats
              >
              >
              > --
              > ***************************************
              > Chris Hlavka
              > NASA/Ames Research Center 242-4
              > Moffett Field, CA 94035-1000
              > (650)604-3328 FAX 604-4680
              > Christine.A.Hlavka@...
              > ***************************************


              --------------------------------------------------------------------
              E-Mail: (Ted Harding) <Ted.Harding@...>
              Fax-to-email: +44 (0)870 094 0861
              Date: 02-Sep-05 Time: 09:16:02
              ------------------------------ XFMail ------------------------------

              * By using the ai-geostats mailing list you agree to follow its rules
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