1996MNRAS.282.1407M
Mon. Not. R. Astron. Soc. 282,1407-1412 (1996)
A statistical test for correlation between crater formation rate and mass
extinctions
Maki Matsumoto * t and Hirota Kubotani* t
Department of Physics, Ochanomizu University, Bunkyo, Tokyo 112, Japan
Accepted 1996 June 18. Received 1996 April 23; in original fonn 1995 December 11
ABSTRACT
Previous works have analysed the stationary periodicities of terrestrial crater
formation rate and mass extinctions, and have compared the periods to discuss their
causality. However, the coincidence of the periods does not necessarily mean that
the two are correlated. Thus, in order to estimate the correlation directly, we test the
null hypothesis that extinctions occurred independently of peaks in the crater
formation rate. This hypothesis is rejected with a low significance level (3.0-7.1 per
cent), even if the uncertainty in dating the craters is taken into account. Therefore,
the rhythm of the mass extinctions with the time-scale of 30 Myr is highly correlated
with that of the giant impacts.
Key words: methods: statistical - comets: general - Earth - meteors, meteoroids.
1 INTRODUCTION
Fossil records show that most biological species have not
always evolved monotonously; they have also experienced
extinctions. Furthermore, mass extinctions, i.e. events in
which numerous species in different regions on the Earth
disappeared simultaneously at certain horizons in the geological records, are observed. For example, we can find
bones of dinosaurs and fossils of ammonites up to just below
the Cretaceous-Tertiary (KIT) boundary, which was
formed 65 million years ago (Ma), but not above it. It is also
well known that a lot of marine animal families and genera
died out at the end of the Permian (250 Ma), which corresponds to the Permian-Triassic (PfTr) boundary. The cause
of the KIT boundary extinction has been researched most
widely and verified most frequently. Alvarez et al. (1980)
claimed that the mass extinction at the KIT boundary was
brought about catastrophically by the giant impact of an
asteroid. They based their argument on world-wide iridiumrich layers found in the K!f boundary clay, which might be
the vapourized remnant of the crash. It is called an iridium
anomaly, since iridium had been depleted in the crust and
mantle and concentrated in the core of the Earth. For the K/
*Present address: Asahi-Kasei Microsystems, Atsugi 243, Japan
(MM); Department of Mathematics, Nara University of Education, Takabatake, Nara 630, Japan (HK). '
tE-mail: [email protected] (MM),
[email protected] (HK)
T boundary, the discovery of an iridium anomaly and other
recent analyses consistently supports the extraterrestrial
scenario of the extinction (van den Bergh 1994; Rampino &
Haggerty 1995).
In 1984, Raup & Sepkoski (1984) counted extinctions of
marine animals over the past 250 million years (Myr) and
identified 12 significant mass extinction events. They found
out that the mass extinctions occurred with a 26-Myr period.
Their conclusion evoked controversy, since it had been
implicitly believed that mass extinctions were random
events driven by miscellaneous factors, but the observed
periodicity indicated the existence of a dominant unique
reason. After their work, some astronomers elaborately
analysed the ages of terrestrial impact craters in order to
seek evidence of bombardments with a period of the order
of 30 Myr, which might be followed by mass extinctions just
as in the KIT boundary. They also proposed causes for the
periodicity. According to various scenarios, the Oort cloud
is perturbed by a large interstellar cloud (Rampino &
Stothers 1984) and by gravitational tidal force (Bailery,
Clube & Napier 1990; Matese et al. 1995) which the Solar
system encounters periodically through the perpendicular
motion in our Galactic disc or through the influence of
Nemesis (Whitmire & Jackson 1984; Davis, Hut & Muller
1984), which is an unseen companion star of the Sun. As a
result, a significant number of comets were driven into the
orbit arriving at the planetary region. The bombardment in
a discrete shower called a 'comet shower', continued for 1-3
Myr (Hut et al. 1987). During each comet shower, several
comets were expected to collide with the Earth and change
© 1996 RAS
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1996MNRAS.282.1407M
1408 M. Matsumoto and H. Kubotani
the terrestrial ecosystem abruptly and drastically, through,
for example, the nuclear winter effect, the greenhouse
effect and/or acid rain. Alternatively, the impact might
induce global volcanism (Rampino 1987) or produce a geomagnetic reversal (Muller & Morris 1986). The extraterrestrial scenario that sudden changes in the environment
resulting from giant impacts from outer space periodically
extinguished a great number of species through some process has been suggested as a plausible one.
To verify the proposed causes of the periodicity, previous
works in this field have tried to estimate the correlation
between mass extinctions and (variously) peaks of the differential crater formation rate (Alvarez & Muller 1984),
magnetic reversal events (Raup 1985) or the Sun's oscillation about the Galactic plane (Rampino & Stothers 1984;
Schwartz & James 1984). These papers analysed cyclicity in
the event proposed as a cause of the extinction, assuming
that it had stationary periodicity, and compared the results
with those of Raup & Sepkoski. Then, on the grounds that
the periods of the two time series were nearly the same and
the last events happened almost at the same time, it was
concluded that the mass extinctions were certainly correlated with the suggested causes. However, this method of
estimating correlations is unsatisfactory in principle. The
cyclicity found in the adopted procedure simply means that
the successive events tend to be centred around an ideal
sequence predicted from a period and a phase position.
Consider the case in which each of two time series has a
large drift from the ideal events used as referential markers.
If they may drift contrary to each other, we will scarcely
recognize a correlation between them. Therefore, the correspondence in the period and the phase do not necessarily
signify a strong correlation between them. Practically, it is
not clear what we mean by the difference of the periods
derived from the stationary periodicity analysis, which picks
up only the time-scale of the rhythm of a series of events.
For mass extinctions, Raup & Sepkoski (1986) suggested a
26-Myr periodicity. On the other hand, Alvarez & Muller
found a dominant cyclicity of 28 Myr in the observed age
distribution of impact craters. The difference of the periods,
2 Myr, appears to show a marked discrepancy between the
time series, since 200 Myr later the total value of the delay
will accumulate to one entire cycle interval, even if the
initial events coincide with each other. Is this argument
justified?
In this paper, we re-examine whether mass extinctions
and crater formation rate have any correlation by using a
proper statistical method. We concentrate on this particular
correlation since it is the most fundamental one in the extraterrestrial scenarios. Considering that we have not yet been
able to specify the cause event for the KIT boundary, which
has been researched most widely, it is reasonable and still
significant to estimate the statistical correlation. We use the
method developed by Ertel (1994) to test the correlation
between the point time series, since traditional methods of
checking correlations between two time series do not apply
to them (Stothers 1994).
This paper is organized as follows. In Section 2, we
explain the statistical method used to analyse the correlation between the two time series. Next, the data sets of the
ages of the observed craters and the mass extinctions, which
are used for our analysis, are introduced. Section 4 is
devoted to reporting our results. Finally, we discuss the
implications.
2 METHOD
In this section, we review the method used to analyse the
correlation between two point series, {XI' X 2 , X 3 , ... ,xm } and
{YI' Y2, Y3,···,Yn}, following Ertel (1994). In using this
method, we must note that it does not matter which time
series fits the cause or result; what is important is the
number of points in each of them. The time series which
contains the smaller number of points fits {xJ. Therefore,
when the method is applied to our work, the time series {xJ
and {Yj} will correspond to the mass extinctions and the
peaks of the crater formation rate, respectively. We first
compute D; (i = 1, 2, 3, ... , m), the distance of each X; from
the nearest Yr Then we calculate Q; (i = 1, 2, 3, ... ,m):
Q=1-2 -
D,
(2.1)
Dmax'
l
where Dmax is the maximum value of D;, that is, the half of
the interval between Yj and Yj + I (yj _ I) which contains x;, The
values of Q; range from -1 (maximum possible distance
between X; and y) to + 1 (minimum possible distance, i.e.
D; = zero). The mean Q is then obtained using
i=1
Q- =
-.
m
(2.2)
Recall here that m is the number of points of the series {x;}.
A large value for Q means that the two series are correlated.
~or example, when the series {xJ and {~) coincide,
Q = + 1. On the other hand, if the value of Q is - 1, the
series {x;} avoids the series {Yj}' Q = 0 simply means that
the two series have no correlation.
To check the significance of the value of Q, we must
estimate the probability that the observed Q can be
obtained by chance. We make randomized sequences {y)
by using the following replacement technique. The real time
series is randomized by scrambling the intervals between Yj
and Yj + I ; the first and last are used as fixed endpoints and all
intervals in between are rearranged using standard Monte
Carlo techniques. Most properties of the randomized series
are the same as those of the original one, but each {y) is
located at an unexpected point because of the differences
between the intervals. We repeat randomizing the series of
{Yj} many times, and calculate Q every time. Those Q fo!
the randomized series that equal or surpass the value of Q
for the original series are indications that the original series
are not correlated, in the sense that the null hypothesis that
the series {xJ happens with no correlation to the series {y)
is not rejected. The number of these Qs, expressed in percentage, represents the 'chance' probability P. It points out
how often a random series outperforms the original series in
matching the observed {x;}.
3
DATA
In this section, we present the data regarding mass extinctions and the formation ages of terrestrial craters. For the
© 1996 RAS, MNRAS 282,1407-1412
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1996MNRAS.282.1407M
Crater formation rate and mass extinctions
ages of craters, we mainly use the data set of Grieve &
Shoemaker (1995), which consists of 139 craters. For some
craters, only an upper or lower limit is given for their ages.
We exclude these ambiguous segments from the present
analysis. If one divides the craters according to their ages, 68
have ages ::;; 300 Myr. This means that younger craters are
better preserved than the older ones. Therefore, we restrict
our attention to the events during the last 300 Myr. However, we exclude the most recent 5 Myr, since the interval
has a much stronger bias towards retaining craters than the
previous part. Furthermore, we also disregard those craters
that have an age determination error ;?: 20 Myr in order to
concentrate on the time-scale discussed by the previous
works in relation to the periodicity (Section 1). In consequence, 35 craters are available for our analysis.
Nowadays, several craters are recognized every year as
remnants of giant impacts, and the ages of some of the
craters have been re-analysed. We replace the old data with
the new. The age of the Manson Crater, which was thought
to be a candidate for the cause of the K!f boundary (65
Ma), has been re-estimated recently by Izett et al. (1993)
using the radio isotope 4°ArrAr method. We adopt the
larger age, 73.8 ± 0.15 Ma (Alvarez, Claeys & Kieffer 1995).
Further, for Aragunainha Crater, we adopt the new data
which were determined by Hammerschmidt & Von Engelhardt (1995) using the same method (245.5 ± 1.75 Ma).
Raup & Sepkoski (1986) made a data set of extinctions of
marine animal families during the past 250 Myr, and classified them by the stratigraphical stages. They judged eight of
the extinction peaks in the time sequence to be significant
mass extinction events and four peaks to be doubtful ones.
We adopt the mass extinctions that were identified by them
as being significant (Table 1).
4 RESULTS
In this section, we estimate the correlation between terrestrial crater formation rate and mass extinction events, folTable 1. The data set of the mass extinctions
adopted from Raup & Sepkoski (1984,1986). This
data set shows the significant peaks of the percentage of the extinction of the marine animal
families. The geological time-scale of Harland et
al. (1990) is used.
End of interval
System and stage
( 106 years ago
= Ma)
Tertiary, middle Miocene
10.4
Thrtiary, late Eocene
35.4
Cretaceous, Maestrichtian
65.0
Cretaceous, Cenomanian
90.4
Jurasric, Tithonian
145.6
Jurasric, Pliensbachian
187.0
'Triasric, Norian
209.0
Permian, Dzulfian
245.0
1409
lowing the statistical method introduced in Section 2. At
first, we ignore the uncertainty of the age determination of
craters. Fig. 1 shows a probability distribution of the ages of
the observed impact structures during the last 300 Myr. In
order for the variance of crater formation rate of the order
of 10 Myr to stand out, the probability distribution has been
smoothed out by a Gaussian window function whose dispersion (1' is 3.0, 4.0 or 5.0 Myr, which is ofthe same order as the
duration of a comet shower.! We identify the local maxima
of Fig. 1 as peaks ofthe crater formation rate; we find 15, 11
and 10 peaks in Figs l(a), (b) and (c), respectively. Following the procedure given in Section 2, the values of Q, 0.577,
0.692 and 0.606, are obtained for (1'=3.0, 4.0 and 5.0 cases,
respectively. The calculation of Q for 10000 randomized
series is performed. The distribution of Qfor (1' = 4.0, which
approximates to the Gaussian distribution whose standard
deviation is 0.240, is given in Fig. 2. These Qs are greater
than or equal to the original Q for 24, 5 and 45 cases out of
10 000 randomizations in the case of (1' = 3.0, 4.0 and 5.0
Myr, respectively. The results indicate the low probability of
a chance association between the peaks of the crater formation rate and the mass extinctions; P is equal to 0.24, 0.05 or
0.45 per cent for each case. In other words, their correlation
is significant. This result does not depend upon the width of
the window function.
We have calculated Q for the randomized series 10 000
times. The Monte Carlo simulation was performed until the
standard deviation of the distribution of these values converged to a single value. For (1'=4.0 Myr, these are 0.243,
0.240 and 0.240 for 1000, 5000 and 10000 times, respectively. Therefore, a 10 OOO-times simulation is enough for
our analysis.
Next we incorporate the effect of errors. The mean value
of the uncertainty of the age determination in our data set is
± 3.6 Myr. It is assumed that the uncertainty has a Gaussian
distribution, the deviation of which has the same value as
the error. For each crater that has an age determination
error, an error that is randomized (for each crater) is added
to the centre age. Especially for two pairs of craters (i.e.
Kara and Ust-Kara), which are placed very close together
and are thus believed to have been created simultaneously,
the same scale of errors is added. This operation is reiterated 10 000 times. The distribution of the obtained values of
Q and the 'chance' probability P are shown in Figs 3 and 4,
respectively. These results mean that we can still recognize
the correlation between the two series with a low significance level, although the consideration. of the determination error of crater ages raises the value of the significance
level; the expectation value of P, (P), is 2.97, 3.86 or 7.06
per cent for (1'=3, 4 or 5 Myr, respectively. Here we note
that we have quantified the effect of determination errors of
the ages; this effect cannot be conjectured from the error of
1If comets in a shower crash on the Earth, their ages are correlated
with each other. To estimate the correlation with extinctions
properly, the correlation between the comet-origin craters in a
shower must be excluded. The smoothing of the cratering rate
absorbs the correlation. We note that the procedure never does
erase the random component in the cratering rate (e.g. asteroids),
although the information regarding age is lost a little. To estimate
the validity of the smoothing, we must check its dependence on the
width of the window function.
© 1996 RAS, MNRAS 282, 1407-1412
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1996MNRAS.282.1407M
1410 M Matsumoto and H. Kubotani
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100
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200
250
300
Figure 1. Smoothed-out probability distribution of the ages of the
observed terrestrial impact structures during the last 300 Myr,
derived from the ages presented in Grieve & Shoemaker (1995).
The probability distributions (a), (b) and (c) have been smoothed
out by Gaussian window functions with 3-, 4- and 5-Myr dispersions, respectively. The triangles and the arrows mark the peaks,
which we identify as the peak of the crater formation rate, and the
mass extinctions (Table 1), respectively.
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the periodicity analysis. This is the result of the adoption of
our method.
Yabushita (1991) showed that the periodicity of crater
ages depends on their diameters. He found that the small
craters (D < 10 km), in contrast to the larger ones, satisfied
a period ranging from 30 to 50 Myr. Prior to this Shoe-
20
is
o
~--~~~~~--~~~~~~~--~~~~
0.001
0.1
0.Q1
p
Figure 4. Distribution of P which corresponds to Q shown in Fig.
3.
© 1996 RAS, MNRAS 282, 1407-1412
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1996MNRAS.282.1407M
Crater formation rate and mass extinctions
maker, Wolfe & Shoemaker (1990) suggested that large
craters tend to originate from the crash of comets rather
than asteroids. According to the comet shower scenario,
craters are necessarily composed of recursive comet
showers, and asteroids may contribute to random components of the crater formation. Therefore, Yabushita's claim
is worth consideration. From the point of view of our analysis, it is also significant to discuss which types of impacting
bodies are more closely correlated with mass extinction
events. To make this problem tractable, we investigate the
dependence of the correlation on the diameter of craters,
since the identification of the type of impacting bodies is still
ambiguous for a considerable number of craters [see table 4
in Grieve & Shoemaker (1995)]. First, we use the data from
26 large craters, with diameters greater than or equal to 10
km. We would not perform the statistical test for the six
small craters, since they are hard to preserve and their ages
are confined to the last 120 Myr in our data, and we can use
only three extinction events in Table 1 to evaluate Q. The
values of Q and the 'chance' probability P for the large
craters are 0.582 and 0.20 per cent, respectively. When the
uncertainty of the age of the crater is considered, the
'chance' probability P is expected to be 6.98 per cent. Here
we use 0"=4.0 Myr. At least the large craters (D~10 km)
contribute to the strong correlation with the mass extinction. Next, we divide the crater data by the threshold of 23
km in diameter. The values of Q, P and <P) are 0.356, 9.9
and 19.0 per cent, respectively, for 17 large craters. For 18
small ones, these are 0.211, 11.5 and 18.6 per cent respectively. Here we again use 0" =4.0 Myr. No difference is found
between the large and the small craters, although the degree
-4
-2
o
2
4
time delay a (Myr)
Figure 5. The values of Q for the time series {Yj} and {Xi + a},
where a is the delay interval. Here, {Yj} and {x,} are the peaks of
the crater formation rate and the mass extinction events, respectively. They are calculated to check the correlation where the incubation period is taken into account. The origin of the horizontal
axis is the time when the cause of the extinction can be expected to
occur. Q measures the correlation between these events and the
peaks of the crater formaton rate.
1411
of correlation is worse than in the total data because of the
smaller data set available for each of these cases.
We now examine whether the mass extinctions coincide
with the peaks of crater formation rate, or if they occurred
some million years later. To find this out, we determined
that the mass extinctions lag relative to the peak of the
crater formation rate in increments of 0.1 Myr, for a total of
5 Myr. If a mass extinction tends to occur a Myr after a peak,
the value of Q will be maximized when the lag is a Myr. Fig.
5 seems to indicate that the mass extinction event is delayed
about 1.2 Myr from the peak of the crater formation. However, if we pay attention to the age-determination errors,
the peak of the line fluctuates around the mean lag, 0.72
Myr, with a dispersion of 1.83 Myr. It shows that the effect
of the impact of large bodies on mass extinctions is not
delayed. If the width of this peak were narrower, then a
mass extinction would have a correlation with only the peak
of the crater formaton rate. This means that no body that
crashes before or after the peak of the cratering rate will
have an effect on the mass extinctions. This would not allow
the 'comet shower' picture, in which the bombardment continues for several Myr.
5 DISCUSSION
We have verified that giant impacts are indeed correlated
with mass extinctions by using the statistical method of Ertel
(1994). It was found that these two series were correlated.
Our results are not changed by the uncertainty of the crater
ages. The correlation has no dependence on the diameters
of the craters. In addition, we have checked the possibility of
a time lag between the two series, and the result indicates
that the extinctions occurred with no delay after the peaks
of crater formation rate.
We have used Q as a measure of the correlation between
two discrete series, i.e. the peaks of the cratering rate and
the mass extinction events, where one series is assumed to
be the cause of the other. Whether the former series
happens before or after the latter one does not reflect on
the measure. Indeed, when we seek to discover which
impact caused a mass extinction, the order is important.
However, our statistical approach is focused on whether the
eight mass extinctions were always accompanied by impacts.
Therefore, we do not require a measure that is sensitive to
whether a mass extinction event happened before or after
the peak point of the cratering rate.
There is no unique way to make a randomized time series
for the purpose of comparison with the original one put to
the statistical test. Stothers (1994) recommended a Poisson
process for the time series which corresponds to peaks of
the crater formation rate. In our case, however, previous
works reported that the series {xJ and {y) have almost the
same periodicity. Hence, if the series {y) is assumed to be
a Poisson process, the Monte Carlo simulation will reduce
only to the analysis of the periodicity of this series. This is
why we have used the method of randomization adopted by
Ertel (1994).
Errors in the determination of the ages of craters are
critical when analysing the uniform periodicity of giant
impacts. Grieve et al. (1985) showed that the possible
period ranged from 10 to 35 Myr, when dating errors were
included in their analysis. The results of the analysis are very
© 1996 RAS, MNRAS 282,1407-1412
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1996MNRAS.282.1407M
1412 M Matsumoto and H Kubotani
sensitive to errors. Further, it has also been emphasized that
a large uncertainty makes it impossible to distinguish
genuine periodic events from apparent ones, and that in
order to fix a period, the time series analysis needs an error
of less than 10 per cent of the observed period (Grieve et al.
1987; Heisler & Tremaine 1989). However, we are only
concerned with the correlation of craters and another time
series. The uncertainty affects the detection of correlation
less severely than it affects the detection of uniform periodicity, since as far as the correlation is concerned, the degree
of relative neighbourhood (D;) rather than the age itself is
of vital importance. Further, recently, the 4()ArrAr isotope
method has been developed (York, Hall & Hanes 1981),
and as a result we can routinely determine the age of a very
small amount of sample (;51 mg) with low systematic error.
The use of the more recent data has also reduced the ambiguity of the results of our analysis. Therefore, we could
apply the correlation analysis to the impact crater data in
question. Finally, we would like to remark that we can use
only 35 crater ages due to the uncertainty of the crater age
determination, although more than 150 craters had been
discovered by 1994 (Hodge 1994). Therefore, the re-analysis of the ages of the craters, evaluated by the isotope agedetermination method, will be important.
From a statistical point of view, the peaks of the cratering
rate are well correlated with the mass extinctions. However,
there remain some problems about details of the extinction
process. For instance, the Chicxulub Crater, which was
found a few years ago, is regarded as one of the reasons for
the KIT boundary (van den Bergh 1994). As for the P{T
boundary, terrestrial explanations are generally accepted
(Erwin 1993), but Aragunainha Crater was made almost at
the same time (Hammerschmidt & van Engelhardt 1995).
Intensive further research is required on the direct relationship between impacts and geological boundaries.
ACKNOWLEDGMENTS
We would like to thank Professor M. Morikawa for continuous encouragement. We also would like to thank Dr M.
Abmady for his careful reading of this manuscript. HK is
indebted to the Japan Society for Promotion for financial
aid. This work was partly supported by Grants-in-Aid for
the Encouragement of Young Scientists.
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