Meteoritics & Planetary Science 48, Nr 6, 976–996 (2013) doi: 10.1111/maps.12118
A Bayesian statistical assessment of representative samples for asteroidal or meteoritical material Jonathan N. CARTER and Mark A. SEPHTON* Department of Earth Science and Engineering, Impacts and Astromaterials Research Centre, Imperial College London, London SW7 2AZ, UK * Corresponding author. E-mail:
[email protected] (Received 18 February 2012; revision accepted 27 March 2013)
Abstract–Primitive substances in asteroid and meteorite materials represent a record of early solar system evolution. To allow the study of these materials, they must be collected and transferred to the laboratory. Collection during sample return missions requires an assessment of the size of samples needed. Meteorite falls or finds must be subdivided into appropriate subsamples for analysis by successive generations of scientists. It is essential, therefore, to determine a representative mass or volume at which the collected or allocated sample is representative of the whole. For the first time, we have used a Bayesian statistical approach and a selected meteorite sample, Murchison, to identify a recommended smallest sample mass that can be used without interferences from sampling bias. Enhancing background knowledge to inform sample selection and analysis is an effective means of increasing the probability of obtaining a positive scientific outcome. The influence of the subdivision mechanism when preparing samples for distribution has also been examined. Assuming a similar size distribution of fragments to that of the Murchison meteorite, cubes can be similarly representative as fragments, but at orders of magnitude smaller sizes. We find that: (1) at all defined probabilities (90%, 95%, and 99%), nanometer-sized particles (where the axes of a three-dimensional sample are less that a nanometer in length) are never representative of the whole; (2) at the intermediate and highest defined probabilities (95% and 99%), micrometer-sized particles are never representative of the whole; and (3) for micrometer-sized samples, the only sample that is representative of the whole is a cube and then only at a 90% probability. The difference between cubes and fragments becomes less important as sample size increases and any >0.5 mm-sized sample will be representative of the whole with a probability of 99.9%. The results provide guidance for sample return mission planners and curators or advisory boards that must distribute valuable samples for analysis.
INTRODUCTION The present-day solar system contains materials that have remained relatively unchanged since the formation of the planets. These primitive materials include asteroids and comets. Accessing these objects allows the study of the earliest stages of solar system history. If these materials can be delivered to terrestrial laboratories, then the world’s most advanced instrumentation can be brought to bear on the most fundamental scientific problems. Transfer to Earth can occur naturally as with meteorite falls or can be
© The Meteoritical Society, 2013.
facilitated by human activities such as sample return space missions. Traditionally, primitive materials have been collected following fall to Earth. These highly visible occurrences provide documentable events and easily identifiable materials. Over a thousand meteorites have been collected in this way. It is generally accepted that most meteorites are fragments of asteroids propelled to Earth by collisions in the asteroid belt. In addition to falls, a large number of meteorites have been recovered as finds in areas of the Earth predisposed to extraterrestrial sample preservation,
976
Statistically representative samples
concentration, and identification. Antarctica is one such location (Cassidy et al. 1992) and over half the world’s collected meteorite population is derived from Antarctica, where conditions are conducive to sample survival and collection. In an attempt to access materials unavailable on Earth and/or derived from areas of known context and unadulterated by Earth processes, sample return missions transport distant primitive extraterrestrial materials to Earth. Successful examples include NASA’s Stardust mission that returned coma dust samples from Comet Wild 2 (Brownlee et al. 2006) and JAXA’s Hayabusa mission that returned samples from its asteroidal target 25143 Itokawa (Kawaguchi et al. 2008). Future sample return missions to asteroids include JAXA’s Hayabusa 2 (Hasegawa et al. 2008) and NASA’s OSIRIS-Rex (Lauretta and The OSIRIS-REx Team 2012). Some of these samples are, or are expected to be, extremely small. Once samples have been collected, there is an increasingly regimented process of curation, preservation, and distribution (Allton et al. 1998). For sample return, this involves a sample receiving facility. Proportions of samples must be allocated for posterity and remote storage in case of a catastrophe at one storage site. Subsequent distribution of samples for nondestructive and destructive analysis by scientists is the most fundamental use of curated primitive materials. Amounts of samples released to scientists depend on the size of sample required for the relevant analytical method and the amount of sample remaining. Decisions are made by advisory committees and collection curators. Based on over 40 yr of experience of extraterrestrial sample curation, it is a stated recommendation that effective curation should start with mission design (Allen et al. 2011). This places scientists and mission planners in the uncomfortable position of making decisions based on the nature of samples before they are actually observed or collected. Past experience is perhaps the best guide to future practice. An additional approach is to use representative samples to inform statistical assessments. Perhaps the most useful outcome of these evaluations is to determine the minimum aliquot size that is representative of the whole sample. When primitive materials are present on the Earth and selected for analysis, present-day methods can provide data at sensitivities that range from percentage levels down to parts-per-billion and beyond (e.g., dynamic secondary ion mass spectrometry, resonance ionization mass spectrometry). The resolution of measurements can achieve spot sizes down to micron or submicron levels (e.g., scanning electron microscopy, time of flight-
977
secondary ion mass spectrometry, nanoscale-secondary ion mass spectrometry) or even the level of atoms (scanning transmission electron microscopy, atom probe tomography). These great sensitivity and high-resolution capabilities allow meaningful information to be obtained from increasingly smaller samples (Davis 2011). It is at the most extreme levels of small sample analysis that questions about the representative nature of samples arise. In this article, we present an unprecedented assessment of the utility of a Bayesian statistical approach to identify the smallest representative sample of a carbonaceous chondrite type sample similar to Murchison. To accommodate the significant unknowns associated with making predictions before sample analysis has taken place, we use a Bayesian method to produce degrees of belief or “Bayesian probabilities.” The Bayesian approach is the most appropriate technique under these circumstances (e.g., MacKay 2003). The results can help decide the amounts of sample needed to be collected during sample return missions to carbonaceous asteroids. The results will also aid planning by museum curators and collection advisory committees on sample storage, distribution, and analysis for samples obtained by space mission sample returns, Antarctic expeditions, and conventional collection after meteorite falls. STATISTICAL ANALYSIS Representative Sample A representative sample of carbonaceous chondrite type material is provided by the Murchison meteorite. Murchison is a CM2 chondrite and this group is highly homogenous with regard to its mineralogy (Howard et al. 2009). We will assume that the target material being searched for is organic matter, although our statistical approach can be applied to any organic or inorganic material. Murchison is a particularly useful reference for organic analysis because, since its approximately 100 kg fall in 1969, it has been extensively characterized. The relative abundances and sizes of various constituents in Murchison are presented in Table 1. Statistical Analysis The objective is to take a piece of an extraterrestrial object and to determine if it is representative of the whole. We do not expect to know the degree of heterogeneity of a sample before its collection, hence the need for a statistical approach. In this section, we summarize the detailed statistical analysis and modeling that is presented in Appendices A–E.
978
J. N. Carter and M. A. Sephton
Table 1. Constituents in the Murchison (CM2) meteorite, their weight percent abundance, morphology, and size. Major- and subconstituent
%
Morphology/size
References
Matrix
86.3
King and King (1978); Buseck and Hua (1993)
Carbon Graphite Diamonds Silicon carbide Organic mater Sulfur Water (free and bound) H2O Metal Inclusions Refractory inclusions Chondrules
1.6–2.2 0.0002 0.04 0.0006 2 1.6–4.1 12.1
Finely comminuted opaque material. Comprises anhydrous silicates such as olivine and pyroxene, as well as lesser oxides, carbonates, and sulfates. Alteration products include serpentines and other layer silicates. Sizes range from 5 lm down to the limit of optical resolution – 0.8–28 lm 2 nm 0.1–1 lm Small compounds to macromolecules, 0.1 mm size fraction in the Murchison (CM2) meteorite (King and King 1978). Characteristic
Data
Percentile Grain size (phi) Grain size (mm)
5 1.52 0.112
16 2.10 0.125
25 2.31 0.133
Where Is the Target Material to Be Found? We now have descriptions of sample sizes and constituents for two different ways of creating the samples. The next step is to consider how the target material might be distributed. There are three possibilities to consider: 1. That the target material will be associated only with the clasts in the sample, P(M1|L,1); 2. That the target material is associated with both the clasts and the clay-rich matrix, P(M2|L,1); 3. That the target material is associated only with clayrich matrix present, P(M3|L,1). Bringing it All Together We have now introduced everything necessary to calculate the two original probabilities of interest, i.e., PðLjT;IÞ representing the probability that the target material does not exist even though the result was positive and PðLjT;IÞ representing the probability that the target material exists within the sample, for both possible methods of obtaining a sample, under the assumption that the two models for the distribution of the target material are sufficient. In doing so, we have introduced two parameters that we would wish to control, the method of sample preparation C and the sample volume V; five probabilities that need to be defined, P(L|I), PðTjS;IÞ, PðTjS;IÞ, P(M1|L,I), and P(M2|L,I); and three noncontrol parameters, v1, v2, and f1 that describe details of the extraterrestrial object and may be uncertain. The noncontrol parameters can be fixed if they are known in a particular case, or treated as unknown nuisance parameters. Nuisance parameters can be handled by integrating them out using the marginalization theorem. For example, if we were considering a class of extraterrestrial object where the unknown was only the fraction of clasts, then we can write Z PðLjT;IÞ ¼
Z PðL; f1 jT;IÞdf1 ¼
PðLjf1 ;T;IÞpðf1 jIÞdf1 (B11)
50 2.67 0.157
75 2.91 0.202
84 3.00 0.233
95 3.16 0.349
where p(f1|I) is a prior pdf for f1. In Appendix C (Calculating P(S|L,I)) we calculate P(S|L,I) using data for the Murchison meteorite.
APPENDIX C
ANALYSIS OF THE MURCHISON METEORITE When we come to estimate P(S|L,I) for the two methods of sample creation that we are considering, we will need to produce estimates for properties that characterize the target extraterrestrial object. For the purposes of this paper, we use data from the Murchison meteorite to estimate: 1. The typical volume of a clast 2. The fraction of the whole that is clast by volume 3. The typical volume of a clay-rich matrix fragment The distribution of clast and clay-rich matrix particle sizes is characterized by v1 and v2 (equation B7). We choose to use the mean volume of particles of each type to characterize the distributions; other choices are possible. In this appendix, we also assign values to the other probabilities that are required. The Mean Volume of a Clast (v1) Data on the grain sizes observed in the Murchison meteorite are available in the published literature (King and King 1978) and contain seven data points taken from the tail of a distribution (Table 3). We wish to estimate the distribution of clast volumes. For these data points, we know their ordinate and their relative cumulative probability within the tail of the measured distribution. We do not know the parameters of the distribution, or exactly where the tail starts (although it is reported to be 0.1 mm, but the error on this is unknown). After some experimentation, we have concluded that the distribution of particles described in the literature (King and King 1978) was best represented by an exponential distribution.
Statistically representative samples
989
106
Cumulative Number
105
Shot 051222 (Murchison)
104 Foil penetration data 103 102 weighed fragments
101
Fig. 6. A plot of the probability of the grain size (mm) being less than the given value for the Murchison meteorite.
pðdjkc ¼ 14:8454Þ ¼ kc ekc d
(C1)
with only particles with diameter greater than 0.111 mm being recorded in the data. This corresponds to 19.202% of all of the clasts that exist. A plot representing the probability of the grain size being less than a specified value from our model, along with the corresponding measured data, is given in Fig. 6. If we assume that the clasts are spheres, then we can calculate the volume, given the diameter.
100
10–10
f1 As ¼ NA
(C2)
with f1 being the fractional volume of clasts and N being the number of clasts. We deduce from the published data (King and King 1978) that the fraction of the sample area covered by large clasts is 10.5%. If Ac is the average area for these large clasts and fc is the fraction of large clasts, then Nfc Ac ¼ As 0:105
(C3)
Therefore, we have f1 ¼ It can be shown that
0:105 A fc Ac
(C4)
10–6
10–4
10–2
100
102
Fragment Mass (grams)
Fig. 7. Cumulative mass distribution for fragments of the Murchison meteorite, redrawn from Flynn et al. (2009).
R 1 pkc d2 k d e c dd R1 6 Ac ¼ 0:111 kc d dd 0:111 kc e p k2c ð0:111Þ2 þ kc ð0:111Þ þ 1 ¼ 2 2 3kc
! (C5)
And that A ¼
The Fraction of the Whole That Is Clast by Volume (f1) If As is a sample area and A is the average area of a clast within As, then provided the area is large enough so that boundary effects are unimportant, we have
10–8
Z1 0
pkc d2 kc d p e dd ¼ 2 6 3kc
(C6)
Therefore,
f1 ¼
0:105 1 ¼ 0:1363 2 kc ð0:111Þ2 0:19202 þ kc ð0:111Þ þ 1 2 (C7)
The Mean Volume of a Clay-Rich Matrix Fragment (v2) Published data (Flynn et al. 2009) give the distribution of mass for fragmentation from the Murchison meteorite; the key data are reproduced in Fig. 7. The curve was digitized and the data in Table 4 were obtained. These data contain information about the masses of all of the particles created, both clasts and clayrich matrix, that reached the detectors. We can assume that the very smallest particles failed to be collected and that there is an unknown offset, g, that needs to be accounted for in the analysis; the cumulative probability data are also to be found in Table 4.
990
J. N. Carter and M. A. Sephton
Table 4. Digitized data for the mass distribution curve given in Fig. 4. Mass (g) 1.43 1.51 6.64 2.51 1.05 3.41 1.05 2.78 6.99 2.64
9 9 9 9 9 9 9 9 9 9
Cumulative number
Cumulative probability
144351 139613 135031 130599 122167 106901 90473 62676 39282 19491
0.00000 0.03282 0.06456 0.09527 0.15368 0.25944 0.37324 0.56581 0.72787 0.86497
1010 109 109 108 107 107 106 106 106 105
Mass (g) 1.05 9 104 3.08 9 104 1.00 9 103 2.27 9 103 6.99 9 103 3.08 9 102 1.00 9 101 2.39 9 101 5.70 9 101 1.29 9 100
Cumulative number
Cumulative probability
8750 4199 1763 904 419 110 36 11 3 1
0.93938 0.97091 0.98779 0.99374 0.99710 0.99924 0.99975 0.99992 0.99998 0.99999
After some experimentation, we concluded that the clay-rich matrix fragments could be modeled with an exponential distribution. So the distribution of mass components is given by pðdjkc ¼ 14:8454; km ; g1 Þ ¼ g1 kc ekc d þ ð1 g1 Þkm ekm d (C8) The fractional mass of clasts is related to the fractional volume of clasts by g1 ¼
f1 q1 f1 q1 þ ð1 f1 Þq2
(C9)
where q1 = 3.32 9 103 g mm3 is the density of the clasts and q2 = 2.27 9 103 g mm3 is the density of the clay-rich matrix, so g1 = 0.1875. The data and the model were matched using a sum of weighted errors squared for the probability that the fragment mass is less than a specified value; the resulting match is shown in Fig 8. We estimate that km = 5.6520 and that the data represent 95.38% of the particles that were created in the experiment. Probability Assignments In this section, we assign several probabilities that are needed for the calculation of the indicative results. We do not have specific data to inform the choices that we make and so they are based on the background information that we have and which will be different from that of other people. P(L|I): The Probability that the Target Material Exists in the Original Extraterrestrial Object This probability appears in equations A12 and A13. One should not choose values of zero or one, and if we are in total ignorance we would set this to a value of 0.5.
Fig. 8. The match between the probabilities that the fragment mass is lesser than a given value as obtained from the data shown in Fig. 7 and the proposed model. The horizontal axis is the mass in grams, on a log scale, and the vertical axis is the probability, also on a log scale.
We will assume we are planning for a future meteorite collection expedition or sample return mission and are not sure of the composition of the extraterrestrial object. Taking meteorite fall data as an example, we know that the fraction of total falls that are chondrites is 0.86 and the fraction of these that are carbonaceous chondrites is 0.05 (Bischoff and Geiger 1995). Therefore, we can generate the value PðLjIÞ ¼ 0:86 0:05 ¼ 0:043
(C10)
P(TjS;I): The Probability of a False Positive PðTjS;IÞ appears in equations A12 and A13; it is the probability that the test procedure, given a sample that at the time the sample was created did not contain the target material, produces a positive result. This can
Statistically representative samples
either be through contamination or experimental error. A value of zero or one is not realistic, and its value may depend on the method of sample preparation. The probability of contamination is high; the likelihood of any organic contamination being mistaken for the indigenous material is less likely (Sephton et al. 2001). As an estimate, we have used a constant value, for both methods of sample preparation, of ¼ 0:10 PðTjS;IÞ
then any clast particle can have the target material associated with it. Using these considerations, we can assign the following probabilities. PðM1 jL;IÞ ¼ 0:00
(C13)
PðM2 jL;IÞ ¼ 0:20
(C14)
PðM3 jL;IÞ ¼ 0:80
(C15)
(C11)
PðTjS,I): The Probability of a False Negative PðTjS;IÞ appears in equations A12 and A13; it is the probability that the test procedure, given a sample that at the time the sample was created did contain the target material, produces a negative result. This can either be through contamination or experimental error. A value of zero or one is not realistic, and its value may depend on the method of sample preparation. The likelihood that the organic matter in the sample is below the sensitivity of the instrument is moderate. As an estimate, we have used a constant value, for both methods of sample preparation, of PðTjS;IÞ ¼ 0:10
991
(C12)
P(M|L): The Probability of the Test Target Material Being Associated with the Various Sample Constituents This probability appears in equation B1 as P(M|L,C,V,I), the dependence on C and V has been dropped as we do not expect the probability to depend on the method of sample preparation or sample volume. For carbonaceous chondrites, the target material is experimentally proven to be associated with the clay-rich matrix and, specifically, clay minerals (Pearson et al. 2002, 2007; Kebukawa et al. 2010). There is, however, a possibility of the target material being present in the clasts, specifically where they have been extensively altered during aqueous processing and the clay-rich matrix type material now penetrates the clasts (Pearson et al. 2007). The organic material can cover all of the clay-rich matrix surface. Terrestrial studies have suggested that monolayers of the organic target material actually cover all clay-rich surfaces (Mayer 1994) and studies of carbonaceous chondrites recognize the coating of small matrix grains with carbon (Amri et al. 2005), which is probably organic in nature (Alpern and Benkheiri 1973), so we can argue that any clay-rich matrix particle, no matter how small, contains the target material. By extension, if the substance has percolated into the clasts by means of incipient aqueous alteration,
Note that there is an additional probability, namely that the target material is associated with neither the clayrich matrix materials nor the clasts. In this instance, the target material would be present as discrete, unassociated entities. We do not consider this possibility in our calculations, but by expanding the options considered in Appendix B, this situation could be handled within the framework described in this paper. P(S|b, M,L,C,V,I) for Cutting and Fragmentation This probability appears in equation B1 and describes whether we expect the test material to be present in the sample, given assumptions about what material is present in the sample and if the material is associated with the material. We could have different assignments depending on whether we are considering sample preparation by cutting or by fragmentation. Cutting
PðSjb;M;L;C1 ;V;IÞ ¼ 2 b1 ; M1 ; P ¼ 1 b1 ; M2 ; P ¼ 1 6 4 b2 ; M1 ; P ¼ 1 b2 ; M2 ; P ¼ 1 b 3 ; M1 ; P ¼ 0
b3 ; M2 ; P ¼ 1
3 b1 ; M3 ; P ¼ 0 7 (C16) b2 ; M3 ; P ¼ 1 5 b3 ; M3 ; P ¼ 1
Fragmentation
PðSjb;M;L;C2 ;V;IÞ ¼ 2 b1 ; M1 ; P ¼ 1 b1 ; M2 ; P ¼ 1 6 4 b2 ; M1 ; P ¼ 1 b2 ; M2 ; P ¼ 1 b3 ; M1 ; P ¼ 0 b3 ; M2 ; P ¼ 1
b1 ; M3 ; P ¼ 0
3
7 (C17) b2 ; M3 ; P ¼ 1 5 b3 ; M3 ; P ¼ 1
Calculating P(S|L,I) The analysis presented in Appendix B makes assumption that the extraterrestrial object can represented using some simple models, although have recommended that it would be better to
the be we use
992
J. N. Carter and M. A. Sephton
stochastic models to estimate the required probabilities. These models contain constants, which, it is assumed, are known. The reality is that the model is only an approximation to the extraterrestrial object and the appropriate values of these constants are uncertain. Within the Bayesian statistical framework, this uncertainty can be formally handled and so allow us to make estimates of P(S|L,I). P(S|L,I) for Cubes To calculate P(S|L,I) the model developed in Appendix B, equations B3–B5, requires us to specify the size of the clasts d1. This is not something that we know and so needs to be treated as a nuisance parameter, so Z PðSjL; IÞ ¼ ¼
d1 ¼1
Zd1d¼0 1 ¼1 d1 ¼0
PðS; d1 jL; IÞdd1 PðSjd1 ; L; IÞPðd1 jL; IÞdd1
(C18)
X ðPðSjb;M;L;C1 ;V;IÞPðMjL;C1 ;V;IÞ M;b
Z
d1 ¼1
d1 ¼0
PðbjL;C1 ;V;IÞPðd1 jL;IÞdd1 Þ (C19)
The P(b|L,C1,V,I) are defined in Appendix B (Cubes) and in Appendix C (The Mean Volume of a Clast), we show that for the Murchison meteorite PðdjL; IÞ ¼ kc ekc d , with kc = 14.8454. The two pdf are related by the requirement that the clast should pffiffiffiffiffiffiffiffiffiffihave the same volume in both descriptions, so d1 ¼ 3 p=6d. P(S|L,I) for Fragments To calculate P(S|L,I), we need to specify P(S|b,M, L,C2,V,I) and to link the distributions of grain sizes that we have identified to the distributions of grain volumes required in the formulations given in Appendix C. 1=3 ! 2 6V 2=3 6V pðVjv1 ; F1 ; C2 ;IÞ ¼ P jkc p p p 2 6V 2=3 kc ð6Vp Þ1=3 ¼ kc e p p
(C21)
APPENDIX D
ADDITIONAL RESULTS AND SENSITIVITY ANALYSIS In this appendix, we include various additional results and sensitivity analyses that has been carried out on this example. Sensitivity of Prior Probabilities
It can be shown that PðSjL;IÞ ¼
! 1=3 2 6V 2=3 6V P jkm pðVjv2 ; F2 ; C2 ;IÞ ¼ p p p 6V 1=3 2 6V 2=3 ¼ km ekm ð p Þ p p
(C20)
In the section Dependency of PðLjT;IÞ and PðLjT;IÞ, on P(S|L,I), we considered how PðLjT;IÞ and PðLjT;IÞ depended on P(S|L,I). From equations A12 and A13, in Appendix A, we know that we need to also specify values for PðTjS;IÞ, PðTjS;IÞ; and P(L|I). These other three probabilities were assigned values in Appendix C (P(L|I): The Probability that the Target Material Exists in the Original Extraterrestrial Object, PðTjS;IÞ: The Probability of a False Positive, PðTjS;IÞ: The Probability of a False Negative as follows 1. PðTjS;IÞ ¼ 0:1 ¼ 0:1 2. PðTjS;IÞ 3. P(L|I) = 0.043 Figure 9 illustrates the sensitivity of the two probabilities to the other three prior probabilities, PðTjS;IÞ, PðTjS;IÞ, and P(L|I). In each case, we increase and decrease the prior probability by a factor of five. The largest effect on both probabilities is from increasing P(L|I); this cannot be reasonably done for a specific extraterrestrial object, but if we have yet to select the extraterrestrial object to be sampled, then we might achieve this by careful inspection. An example of such an inspection could be the remote observation or spectroscopic analysis of the object, the data from which enhance our background knowledge. Again, the importance of background knowledge on a positive scientific outcome is evident. The next biggest impact is obtained on PðLjT;IÞ by decreasing the probability of PðTjS;IÞ. This means improving our testing procedure so that the probability of obtaining a positive result when the sample does not contain the target material is reduced.
Statistically representative samples
993
to the three prior probabilities, Fig. 9. Plots illustrating the sensitivity of the two probabilities of interest PðLjT;IÞ and PðLjT;IÞ PðTjS;IÞ, PðTjS;IÞ, and P(LjI). In each case, we increase and decrease the prior probability by a factor of five. a) sensitivity of PðLjT;IÞ to variations in PðTjS;IÞ, b) sensitivity of PðLjT;IÞ to variations in PðTjS;IÞ, c) sensitivity of PðLjT;IÞ to variations in P to variations in PðTjS;IÞ, to variations in PðTjS;IÞ, (LjI), d) sensitivity of PðLjT;IÞ e) sensitivity of PðLjT;IÞ and f) sensitivity of to variations in P(LjI). PðLjT;IÞ
Sensitivities In this appendix, we consider the impact of where we think the target material is to be found and the fraction of a sample that is clast material. Sensitivity for Target Material Location We consider three different assignments for where we think the target material will be found 1. P(M1|L,I) = 1.0, P(M2|L,I) = 0.0, and P(M3|L, I) = 0.0, i.e., the target material is only associated with clasts. 2. P(M1|L,I) = 0.4, P(M2|L,I) = 0.2, and P(M3|L, I) = 0.4, i.e., the target material could be associated either clasts, clay-rich matrix, or both. 3. P(M1|L,I) = 1.0, P(M2|L,I) = 0.0, and P(M3|L, I) = 0.0, i.e., the target material is only associated with clay-rich matrix. Figure 10 shows how the three probabilities vary for the cutting method of sample preparation, as we change the probabilities for the location of the target material. The first line shows the situation when it is assumed that the target material is only associated with the clasts; in the second line, we have the case where the material could be associated with any of the meteorite components; and in the last line, we have the clay-rich matrix case. The general behavior observed is the same in each case as in the base case, only the magnitude of
the variations changes. It is notable that for the smallest sample sizes, when we assume that the target material is only associated with clasts, we see the probability of obtaining a representative sample decline to about 0.20. This is because for very small samples we may not have any clast material within the sample Figure 11 shows the same information for the case of sample preparation by fragmentation. When we assume that the target material is only associated with clasts, then for the largest samples, the probability that they contain the target material falls to zero. This is because the largest fragments are clay-rich matrix, which are assumed not to contain the target material. The probabilities for false-positive and false-negative results increase with sample volume. In the second line where we assume that the target material could be associated with either or both components we surmise that all three probabilities are constant. This behavior occurs because of the symmetry in our assignment of the probabilities. This shows that if we are ignorant as to where the target material is likely to be found, then sample size does not affect the probabilities of false positives and false negatives. Then, in the last line, where we assume that the target material only exists in the clay-rich matrix, we see the opposite behavior to that seen in the first line. From these sensitivity studies, we can see that the probabilities are sensitive to the method of sample
994
J. N. Carter and M. A. Sephton
IÞ; and PðLjT;IÞ Fig. 10. Plots illustrating the sensitivity of the three probabilities of interest P(SjL,I), PðLjT; with sample size in mm3, when cutting the extraterrestrial object into cubes, for variations in the probabilities P(MijL,I). First row: P(MijL,I) = (1.0, 0.0, 0.0), second row: P(MijL,I) = (0.4, 0.2, 0.4), and third row: P(MijL,I) = (0.0, 0.0, 1.0).
with sample size in Fig. 11. Plots illustrating the sensitivity of the three probabilities of interest P(SjL,I), PðLjT;IÞ, and PðLjT;IÞ mm3, when fracturing the extraterrestrial object, for variations in the probabilities P(MijL,I). First row: P(MijL,I) = (1.0, 0.0, 0.0), second row: P(MijL,I) = (0.4, 0.2, 0.4), and third row: P(MijL,I) = (0.0, 0.0, 1.0).
Statistically representative samples
995
with sample size in Fig. 12. Plots illustrating the sensitivity of the three probabilities of interest P(SjL,I), PðLjT;IÞ, and PðLjT;IÞ mm3, when cutting the extraterrestrial object into cubes, for variations in the assumed fraction of clasts in the extraterrestrial object. First row: f1 = 0.01, second row: f1 = 0.40, and third row f1 = 0.70.
preparation. If we wish to ensure that samples are representative, then we should be using cutting as the method for preparing samples. Sensitivity for Clast Volume Fraction The sensitivity studied here is the volume fraction of clast material within the extraterrestrial object. We consider three values: f1 = 0.01, f1 = 0.4, and f1 = 0.7. Figure 12 shows the variations in the three probabilities, assuming that the sample is prepared by cutting, as we change the fraction of the volume that is clasts from f1 = 0.01 in the top line, f1 = 0.4 in the second line, and f1 = 0.7 in the final line. As f1 increases, then sample size becomes more important. Figure 13 shows the variations in the three probabilities, assuming that the sample is prepared by fragmentation, as we change the fraction of the volume that is clasts from f1 = 0.01 in the top line, f1 = 0.4 in the second line, and f1 = 0.7 in the final line. As f1 increases, then sample size becomes more important. In general, for a given sample size, cutting is more likely to result in a representative sample.
APPENDIX E NOMENCLATURE Here, we list all of the parameters and variables used in this paper and Ac : areas used to calculate f1 for the 1. As, A, Murchison meteorite. 2. C,C1, and C2: the method by which a sample is prepared, C1 is cutting and C2 is fragmentation. 3. D: interclast spacing used in the cutting model for sample preparation. 4. d1: characteristic length of a clast of volume v1 used in the cutting model for sample preparation. 5. d: characteristic grain diameter for Murchison clasts. 6. f1: volume fraction of clasts in the extraterrestrial object. 7. fc: volume fraction of large clasts in the extraterrestrial object. 8. g1: mass fraction of clasts in the extraterrestrial object. 9. I: background knowledge.
996
J. N. Carter and M. A. Sephton
with sample size in Fig. 13. Plots illustrating the sensitivity of the three probabilities of interest P(SjL,I), PðLjT;IÞ, and PðLjT;IÞ mm3, when fracturing the extraterrestrial object, for variations in the assumed fraction of clasts in the extraterrestrial object. First row: f1 = 0.01, second row: f1 = 0.40, and third row f1 = 0.70.
the propositions that a particular material 10. L and L: does/does not exist in the extraterrestrial object. 11. M,M1, M2, and M3: location of the material whose existence is tested for, M1 clast only, M2 both, and M3 clay-rich matrix only. 12. P(|): probability of one proposition given another proposition. 13. p(|): probability distribution function of one variable given a proposition. the propositions that a particular 14. S and S: material does/does not exist in the sample. the propositions that a test for a particular 15. T and T: material does/does not result in a positive outcome.
16. V: volume of a sample. 17. v1 and v2: mean volumes of a clast and a clay-rich matrix fragment. 18. b, b1, b2, and b3: the material contained in a sample, b1 is clast only, b2 is a mixture, and b3 is clay-rich matrix only. 19. d: characteristic length for sample size in cutting model. 20. k, kc, and km: constants used in the description of particle size probability distribution functions. 21. q1 and q2: the densities of clast and clay-rich matrix.
