The mathematical, statistical, and computational approaches used in the investigation of the interrelationship of lunar fragmental material, regolith, lunar rocks, and lunar craters are described. The first two phases of the work explored the sensitivity of the production model of fragmental material to mathematical assumptions, and then completed earlier studies on the survival of lunar surface rocks with respect to competing processes. The third phase combined earlier work into a detailed statistical analysis and probabilistic model of regolith formation by lithologically distinct layers, interpreted as modified crater ejecta blankets. The fourth phase of the work dealt with problems encountered in combining the results of the entire project into a comprehensive, multipurpose computer simulation model for the craters and regolith. Highlights of each phase of research are given

Marcus, A. H.

English

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THE JOHNS HOPKINS UNEUERSITY / 0//73
FINAL REPORT
N.A.S.A. GRANT NGR-21-OO1-080
"STATISTICAL MODELS OF LUNAR ROCKS AND REGOLITH" Allan H. Marcus, Principal
Investigator
INTRODUCTION.
The original research proposal for this grant suggested a variety
of mathematical, statistical, and computational approaches to an
investigation of the interrelationship of lunar fragmental material and
regolith, lunar rocks, and lunar craters. The project developed in
four phases. The first two phases were essentially mathematical,
exploring first the sensitivity of the production model of fragmental
Sa material to assumptions, and then completing earlier studies on the
o : w survival of lunar surface rocks with respect to competing processes.
The third phase (not yet completed) combines earlier work into a
detailed statistical analysis and probabilistic model of regolith
0 a formation by lithologically distinct layers interpreted as modified
" : crater ejecta blankets. The fourth phase has dealt with problems
I W W encountered in combining these results into a comprehensive multipurpose
!Z ! LO  computer simulation model for the craters and regolith.
PHASE 1. Sept. 1971 - April 1972.
Before any computer simulations could be attempted, it was necessary
H~ to explore the sensitivity to parameters and moa4 functions of the basic
MH
o n=t crater model. An extensive analysis of the gross rate of production of
lunar fragmental material was developed. The model dealt. exclusively
0 with the mean volume of material per unit area (translated into regolith
n t- thickness) as a function of time, without considering the variable
distribution of thickness spatially. Even so, difficult numerical
OE integrations were required. Fortunately, one of the research assistants,
o CD Mr. William Dupont, had very extensive experience in computation of
Oa t numerical integrals and was able to successfully complete this project.
- i The results were presented at the American Geophysical Union meeting
n F in April, 1972 and subsequently published in Icarus (1).
o The model proved to be distressingly sensitive to small differences
in parameters which were within the limits of uncertainly of the
Sexperiments from which the parameters were estimated. (It was almost
essential to overinterpret the laboratory data). Small changes in
scaling law exponents and fresh crater diameter distribution exponents
o induced order-of-magnitude variations in mean regolith thickness.
Variation of ad hoc diameter limits (the minimum and maximum crater
S _J diameters affecting the region) induced similarly large variations.
1n C The model produced very plausible values for mean regolith thickness,
W but so did most other models. The model made some clear but counter-
N intuitive predictions e.g. the possibility of a statistical reverse
Sbedding with older surfaces having a smaller surface density, but higher
proportion of very large blocks and boulders. This then became the
next major research effort.
https://ntrs.nasa.gov/search.jsp?R=19740022168 2020-03-23T04:21:52+00:00Z
PHASE 2. April, 1972 - December, 1972
In order to determine the lifetimes for destruction of lunar surface
rocks by erosion (microcratering) and shattering (macrocratering), it was
necessary to obtain recent empirical information on the returned lunar
samples. I accordingly visited the NASA Manned Spacecraft Center in
Houston on April 4-5, 1972 for informative discussions with Drs. Fred
Horz, David McKay, and Jack Hartung. They described their own work and
recent (as yet unpublished) correlated studies by Don Gault at NASA-Ames.
I subsequently developed an extensive analytical model by which the
survival lifetime distributions of lunar rocks could be compared for a
variety of possible fates - single-event burial, multiple-event burial,
shattering, and erosion. The shattering and erosion models were based
-on recent studies by Gault (2) greatly improving on both my own earlier
study (3) and that of Shoemaker (4). The burial model was extremely
sensitive to the effective crater volume distribution exponent a = (S-2)/h,
where S is the exponent of the fresh crater diameter distribution and h
the exponent of the crater rim height-diameter distribution. The value
a = 1 is a critical dividing line, and the experimental value of a was
believed to be close to 1.0. However, a "small-crater" model with a
much larger than 1.0 was also developed and seemed to give quite useful
results.
The results were reported at the American Geophysical Union meeting
in San Francisco in December, 1972. The paper (5) was subsequently
submitted for publication, but critically reviewed because of a failure
to provide comparisons of predicted and observed exposure lifetimes.
Such comparisons require access to information not readily available to
me: There are presently no plans to extend this research.
A side-effect of my discussions with Hartung was a careful reconsideration
of the concept of crater saturation, possibly applicable to densities of
microcraters on lunar rocks (6).
PHASE 3. January 1973 - April 1973.
After studying surface activities, I once again considered the
regolith dynamics. My earlier regolith model (7) was extended to mope
realistic crater blanket geometries (8). The existence of lunar core
tube samples opened up the possibility of studying a layered lunar
regolith in terms of its ejecta blanket layers (the basic element of the
crater model). A series of detailed statistical analyses were performed
on core tube layer data from Apollos 12, 14, 15, and 16. These were
reported at the American Geophysical Union meeting of April 19, 1973.
A number of critical comments were received: (1) Layers had little real
meaning or physical significance, unlike terrestrial geologic strata.
(2) Spatial cross-correlations of layers with similar properties were
needed. I subsequently reevaluated the data with respect to these two
points. The layers had a variety of consistent, operationally-well-
defined differences; (9) but it was not possible to decide if they
constituted lithologically distinct beds from the same ejecta blanket,
or were aggregates of different but lithologically similar blankets
which could not be individually distinguished. A number of techniques
were used to seek spatial correlation or coherency of layers from four
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Apollo 16 core sites, but no conclusive evidence was found. Some layers
resemble others in the same and other tubes, but the order is completely
scrambled due to subsequent cratering and mixing. The statistical
identification of coherent regional blankets would confirm both the
reality of layers as ejecta blankets and the validity of the crater model,
and the identity (10) of the layers.
The statistical methods used so far have been relatively model-free.
More powerful methods require a probabilistic model. Even the simplest
realistic models show unexpected complexity, due to: the fixed length
of the sample; the truncation of the bottom-most layer; regional lithologic
cross-contamination (11); the highly skewed nature of the layer thickness
distributions. The usual small-sample optimal statistical methods do not
fit this problem, and additional methods based on short samples from a
Markov renewal process are being developed. Fragments of a manuscript
are ready, but, because of the frustratingly inconclusive nature of these
results, it is not possible to issue a technical report of acceptable
scientific quality at this time. Additional studies are progressing and
may be reported at the April 1974 A. G. U. meeting.
PHASE 4. April 1973 - August 1973
As a result of the comments received at the AGU meeting, I returned
to the possibilities of computer simulation. Most of the computation
(both self-instruction and testing out of component submodels) was done
on the remote terminal installed in Maryland Hall across from my office.
I also attended the Fourth Pittsburgh Symposium on Modeling and Simulation
on April 24, 1973. The result of the extensive computations was that,
while I learned much about large-scale computer models, I learned little
about the Moon. During the course of this grant, I became familiar with
a number of unpublished studies (12) (13) and published studies (14) (15)
(16) which also investigated lunar fragmental thickness distributions by
deterministic and stochastic simulations. Of these studies, those by
Lindsay (13) and Oberbeck et. al. (15) (16) proved to be the most sophisticated
and together effected nearly all of the innovations in simulation described
in the proposal for this grant. The computational logic is detailed by
Lindsay (13). Oberbeck et. al. included the redistribution of materials by
subsequent primary and secondary craters, including the shielding effects
of the regolith (15) (16).
In general these simulations have similar unsatisfactory properties,
mainly: (a) Inability to deal with edge effects accurately, or to minimize
edge effects by modeling a sufficiently large region. (b) Strict additivity
or superposition of elevation changes without allowance for mass wasting
slump and down-slope movements of loose material. The real moon is smoother
than the model moons! (c) Devastating sensitivity to uncertain or ad hoc
parameters and unknown model functions. These include lower and (especially)
upper diameter limits for craters, shape of the ejecta blankets, and fresh
crater diameter exponent S. (d) Incorrectly chosen inputs which should be
revised on the basis of recent results on impact mechanics (17) and ejecta
blanket shape (18). (e) Insufficient attention to the fact that ejecta
blankets are clumpy and irregular, with oscillating radial and circumferential
patterns as well as a smooth radial variation. (f) Only Lindsay's simulation
deals with the probability of survival of blanket integrity. To trace
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turned-over layers in a computer simulation, one would have to digitize
layer thickness on a scale of 1 to 5 centimeters and mark each layer with
an identifying symbol, a task that greatly exceeds the memory of available
computers.
I do not wish to add to existing polemics on the failures of large-scale
models e.g. (19); merely to point out problems that emerged from the original
proposal.
CONCLUSIONS AND SUMMARY
1. Useful analytical methodology has been developed for determining the
gross rate of production of lunar fragmental material and for estimating
the lifetime of lunar surface rocks with respect to competing processes
of destruction or burial. The numerical results require extensive modification
in light of recent data and experiments on impact mechanics.
2. Serious analytical problems requiring further study remain for the
statistical analysis and theoretical interpretation of regolith layers
in lunar core tube samples.
3. Large-scale multipurpose computer simulations are unlikely to be
helpful in understanding regolith dynamics. Special-purpose small-scale
simulations and theoretical analyses (for all their shortcomings) are likely
to provide a more useful methodology, being less sensitive to incorrectly
or incompletely specified inputs and to accumulation of errors. This is
particularly true for intrinsically highly variable quantities such as
lifetimes with respect to competing processes.
4. The most helpful experiment in establishing the validity of an
impact-generated regolith model would be a network or grid of core-tube
samplers, to determine the spatial coherence and integrity of regolith
layers, their variability, and their potential utility as indicators of
lunar microstratigraphy.
REFERENCES
1. Marcus, A. H., Production of lunar fragmental material by meteoroid
impact. Icarus, Vol. 18 (1973), 621-633.
2. Gault, D. E., Horz, Friedrich, and Hartung, Jack B., Effects of
microcratering on the lunar surface. (1972) to appear. Proceedings
of the Third Lunar Science Conference.
3. Marcus, A. H., Stochastic models of lunar rocks and regolith. Math.
Geol., Vol. 2 (1970) 153-174.
4. Shoemaker, E. M., Origin of fragmental debris on the lunar surface
and the bombardment history of the Moon. Publ. Inst. Geol. Investig
(Barcelona) Vol. 24 (1970).
5. Marcus, A. H., The burial of lunar rocks by crater ejecta blankets.
Johns Hopkins Univ., Dept. Math Sciences Tech. Rept. 170, 1972.
6. Marcus, A. H., On the definition of crater saturation diameter.
Johns Hopkins Univ., Dept. Math Sciences Tech. Rept. 168, 1972.
7. Marcus, A. H., The distribution and covariance function of elevations
on a cratered planetary surface. The Moon, Vol. 1, (1970) 297-337.
8. Marcus, A. H., Statistical analysis of a layered lunar regolith.
Part I. Johns Hopkins Univ., Dept. Math Sciences Tech. Rept. 169,
1972.
9. Preliminary Science Reports, Apollo 12, Apollo 14, Apollo 15,
Apollo 16. Also: Stratigraphy of the Apollo 15 Drill Core, NASA
TM-X-58101, 1972.
10. Apollo 16 Special Samples, NASA-MSC-1973. Also: G. Arrhenius et. al.,
The exposure history of the Apollo 12 regolith. Proceedings of the
Second Lunar Science Conference, Vol. 3, 2583-2598. M.I.T. Press,
Cambridge, 1971.
11. Proceedings of the Third Lunar Science Conference, 1972.
a. D. Macdougall, B. Martinek, G. Arrhenius, Regolith dynamics.
b. J. B. Adams, T. B. McCord, Optical evidence for regional cross-
contamination of highland and mare soils.
12. Mendell, Wendell. Personal communication, March 28, 1972.
13. Lindsay, John F. A depositional model for post-mare debris on the
lunar surface. 1971.
14. Short, N. M. and Forman, M. L. Thickness of impact crater ejecta
on the lunar surface. Mod. Geol., Vol. 3 (1972) 69-91.
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15. Oberbeck, V. R., Quaide, W. L., Mahan, M. and Paulson, J. Monte
Carlo calculations of lunar regolith thickness distributions.
Icarus, Vol. 19 (1973) 87-107.
16. Oberbeck, V. R., Horz, F., Morrison, R. H. and Quaide, W. L.
Emplacement of the Cayley Formation. NASA-TM-X-62302, 1973.
17. Gault, D. E. Displaced mass, depth, diameter, and effects of
oblique trajectories for impact craters formed in dense crystalline
rocks. The Moon, Vol. 6 (1973) 32-44.
18. McGetchin, T. R., Settle, M. and Head, J. W. Radial thickness
variation in impact crater ejecta: Implications for lunar basin
deposits. Earth and Planet. Sci. Letters (1973) in press.
19. Lee, D. B. Jr. Requiem for large-scale models. J. Amer. Inst.
of Planners, May 1973, PP. 163-178.
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Statistical models of lunar rocks and regolith

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