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Friedrich-Alexander-Universität Erlangen-Nürnberg

Naturwissenschaftliche Fakultät

Bachelorarbeit

Vorgelegt von Niklas Hohmann

aus Erlangen

Betreuerin: Dr. Emilia Jarochowska

Bearbeitungszeit: 01.04.2018 - 07.06.2018

GeoZentrum

Nordbayern

Quantifying the Effects of Changing

Deposition Rates and Hiatii on the

Stratigraphic Distribution of Fossils

Contents

1 Introduction 2

2 The Basic Model 3

3 Expanding the Model 5

3.1 Incorporating Diastems, Hiatii and Erosion . . . . . . . . . . . . 5

3.2 First and Last Fossil occurrences in the Presence of Gaps . . . . 6

4 Evaluating the Eﬀects of Diﬀerent Deposition Models 7

5 Examples 9

5.1 The Eﬀect of Diﬀerent Deposition Models on the Recognition of

ExtinctionRates ........................... 9

5.1.1 The Relation of Extinction Rates and Rates of Last Ap-

pearanceDates........................ 9

5.1.2 Application: Analysis of the Mass Extinction at the K-Pg

Boundary on Seymour Island, Antarctica . . . . . . . . . 10

5.2 The Eﬀects of Binning on the Distinguishability of Extinction

Hypotheses .............................. 12

6 Discussion 14

6.1 Limits of the Model . . . . . . . . . . . . . . . . . . . . . . . . . 14

6.2 Obtaining Deposition Rates . . . . . . . . . . . . . . . . . . . . . 14

7 Summary 15

8 Acknowledgements 16

9 Appendix 16

10 Figures 17

11 Tables 35

1

1 Introduction

The stratigraphic distribution of fossils is an important source of information

on paleoecological processes such as origination or extinction rates [1], but it is

aﬀected by a number of biases that can complicate the reconstruction of the un-

derlying process. This is most clearly demonstrated by the Signor-Lipps eﬀect,

which shows that even a sudden extinction will lead to a ”backwards smearing”

of the last fossil occurrences of the taxa going extinct [2]. This makes the extinc-

tion appear more gradual than it originally was. Many other eﬀects that alter

the relation between the stratigraphic distribution of fossils and the underlying

processes are known. They include destructive eﬀects (diagenetic dissolution

due to shell chemnisty, destruction by bioturbation, any type of physical shell

destruction) and eﬀects of mixing and unmixing (any type of time-averaging,

hydrodynamic sorting) and changes in deposition rates as well as the presence

of hiatii and diastems [3] [4] [5] [6]. Understanding and quantifying these eﬀects

is crucial for any analysis that is based on the stratigraphic position of fossils,

since every fossil occurrence will to some extend be aﬀected by them.

In this paper, I will focus on studying the eﬀect of changing deposition rates,

hiatii, diastems, and erosion on the stratigraphic distribution of fossils. This

eﬀect is based on the observation that diﬀerent deposition rates place diﬀerent

amounts of sediment between fossils occurrences that were placed in the sedi-

ment after ﬁxed periods of time. This changes the distance between these fossils

occurrences and thereby alters the stratigraphic distribution of fossils [7] [8].

This eﬀect has been known for a long time and has been studied in the ﬁeld

as well as by means of simulations [7] [9] [10] [11]. Changing deposition rates

are also a big contribution to the eﬀects of sequence stratigraphy on the strati-

graphic distribution of fossils. Simulations show that these eﬀects can change

biostratigraphic accuracy up to 10000 % [9] and generate clusters of last fossil

occurrences [11]. These clusters can easily be mistaken for signs for elevated

extinction rates, especially since geochemical proxies of extinction causes can

also be aﬀected by these processes to match the pattern of increased extinction

rates [11] [12]. This emphasises the importance of studying the eﬀects of chang-

ing deposition rates and incorporating them into any statistical evaluation of

the stratigraphic distribution of fossils.

In this thesis, I present a model describing the eﬀects of changing deposition

rates and hiatii on the stratigraphic distribution of fossils. This model allows

to exactely determine the eﬀects of changing deposition rates on paleontological

2

hypotheses and can be naturally embedded into a propability theoretic context.

In this context, the eﬀects of diﬀerent deposition rates on paleontological hy-

pothesis can be approached with statistical methods. This allows, for example,

to correct data for eﬀects of changing deposition rates, to assess the eﬀect of dif-

ferent deposition models on the interpretation of extinction rates (section 5.1)

and to evaluate the distinguishability of hypothesis under diﬀerent sampling

procedures (section 5.2).

2 The Basic Model

For the beginning, assume a nonzero deposition rate and no erosion. These

model assumptions will be generalized in section 3. Under these assumptions,

every point in a section can be uniquely identiﬁed by the age it has been de-

posited. Conversely every point in time corresponds to one and only one point

in the section that has been deposited at this time. This type of correspondence

is called a 1-1 correspondence, which in this case connects points in the section

with points in time. For a given sedimentation rate r, this 1-1 correspondence

is determined by the integrated sedimentation rate:

I(t) = Zt

r(x) dx. (1)

This function assigns to each moment in time tthe amount of sediment that has

been accumulated up to this moment. This is equal to the height in the section

that has been deposited at this moment (ﬁg. 1). The inverse function I−1,

mapping each point in the section yto the time which it has been deposited is

given by the the integrated inverted deposition rate:

I−1(y) = Zy1

r(x)dx. (2)

This correspondence between the section and time can be used to transform

stratigraphic positions of fossils in a section back into time to reconstruct the

time of deposition of said fossils. It can also be used to transform this data fur-

ther into an imaginary section with constant deposition rate. In this imaginary

section, simple statistical methods based on the assumption of constant depo-

sition rates can be applied. This can for example be used to adapt simple ﬁrst

generation methods [13] to situations with complex sedimentological settings.

3

The results of these methods can then be transformed back into the original

section. This procedure allows to adapt already existing statistical methods to

complex sedimentological settings and therefore ensures backward compatibility

of the presented method.

The correspondence between time and section can be expanded further. For

this, denote the number of fossils that can be found in the section by Fsand

the number of individuals that die, are deposited in the sediment, survive the

taphonomic process and form fossils by Ft. For convenience, i will call these

individuals designated fossils. For a discussion of this concept see section 6.1.

Next, note that the number of fossils found in an interval in the section is de-

termined by the number of designated fossil that are buried during the time in

which the sediment forming this interval of the section has been deposited. This

can be formalized as

Fs([x1, x2]) = Ft(I−1([x1, x2])),(3)

with [x1, x2] being an interval in the section. In a measure theoretic context, the

transformation in equation 3 is known as the pushforward measure or the image

measure ( [14], p. 129; [15], p. 42), here with the number of designated fossils

Ftbeing pushed forward by the integrated sedimentation rate Ito determine

the number of fossil occurrences Fsin the section.

In a more general context, the number of designated fossils Ftcan be replaced

by any function of time f.1For clarity, i will call these functions temporal

rates. In the context discussed here, these functions can have multiple meanings.

The most important temporal rates are the number of individuals of a taxon

(assuming this is proportional to the number of designated fossils generated) as

well as rates of ﬁrst appearance dates (FAD) or rates of last appearance dates

(LAD) of taxa. These temporal rates can be pushed forward as a whole by the

analytical transformation ( [15], p. 42; [14], p. 132)

fr(y) = f(I−1(y))

r(I−1(y)) ,(4)

where yis a point in the section and fris the transformed temporal rate and

the result of the push forward. I will refer to these transformed temporal rates

or to any rate function deﬁned in the section as observable rates. With this

1This is justiﬁed by taking the function fas a density of some measure µwith respect to

the Lebesgue measure λ.

4

transformation, any of the temporal rates mentioned above can be transformed

from time into the section. Here they represent the corresponding observable

rate, given that the sedimentation rate ris the case:

•The number of individuals of a taxon throughout time corresponds to the

rate of fossil occurrences

•The rate of last appearance dates corresponds to the rate of last fossil

occurrences

•The rate of ﬁrst appearance dates corresponds to the rate of ﬁrst fossil

occurrences

The transformation from time to the section is displayed in the ﬁgures 2, 3, and

4). It is implemented for R and can be found in the appendix under the name

timetostratbasic.R

Since the connection between time and the section is a 1-1 correspondence,

the same principle can be used to transform observable rates from the section

into their corresponding temporal rates in time. In this case, the function used

for the push forward is I−1, the inverse of the integrated sedimentation rate I.

This transformation is also implemented for R and can be found in the appendix

under the name. strattotimebasic.R.

3 Expanding the Model

3.1 Incorporating Diastems, Hiatii and Erosion

The model is based in the assumption of no erosion and nonzero deposition.

This assumption is unrealistic, since it is known that the rock record is not

complete and consists of long periods of nondeposition [3] [4]. Therefore di-

astems, hiatii and erosion must be incorporated in the model.

To do this, the strictly positive sedimentation rate ris expanded to a depo-

sition rate (also denoted as r) that can be positive, zero or negative. Here a

deposition rate of zero corresponds to nondeposition, a negative deposition rate

corresponds to the removal of sediment and a positive deposition rate corre-

sponds to actual sedimentation.

The implementation of the transformation from time to the section can be

adapted to this more general framework. For this, the integrated deposition

5

rate is calculated as before. Then sediment from intervals with erosion is re-

moved, which is done by replacing intervals that are eroded with intervals of

nondeposition. After that, the temporal rate of interest is transformed for each

interval with strictly positive deposition rate separately by applying the analyt-

ical transformation described above. In a last step, all transformed parts of the

temporal rate are joined together to form the observable rate. This procedure

is implemented in R and can be found in the appendix under the name time-

tostrat.R. For more details see the comments in the code and the documentation

of the R code in the appendix.

The transformation from the section into time can be adapted to the more gen-

eral framework in a similar manner. Here the implementation is designed in a

way that hiatii and diastems of any duration can be inserted at points in the

section, which leads to gaps in the resulting temporal rate. This implementation

is named strattotime.R and can again be found in the appendix.

It must be noted that both procedures create discontinuities in the temporal or

observable rate at the points where the hiatus is introduced or the transformed

temporal rates are merged. This is unavoidable, since equation (4), which is

the basis for the transformation, does not hold for deposition rates of zero. The

more general formulation on the basis of the pushforward measure is still valid

in this case, but has not been implemented.

3.2 First and Last Fossil occurrences in the Presence of

Gaps

It is important to note that the transformations of rates of ﬁrst/last appearance

datums and fossil occurrences in the presence of diastems, hiatii and erosion

with the implementations described above will lead to erroneous results. This is

because rates of ﬁrst and last occurrences are deﬁned a posteriori in the section

based in the fossils that are present in the section and are therefore not aﬀected

by erosion or nondeposition. This means that the designated ﬁrst/last fossil

does not necessarily correspond to the ﬁrst/last fossil occurrence, since it might

be destroyed by erosion.

To compensate for the lack of an analytical solution, an implementation on

the basis of a Monte Carlo approach is used to generate rates of LFO and

FFO in the presence of hiatii. It is attached in the appendix under the name

timetostratlastﬁrstocc.R and is based on the following scheme:

1. Simulate one last fossil occurrence based on the assumption that they

6

follow an inhomogeneous Poisson point process with the LFO rate as the

rate function. This is done using code from [16]

2. Simulate all appearances of designated fossils before the LFO. This is done

by assuming that they follow a homogeneous Poisson point process with

a given rate. This rate determines the number of appearances per time

unit. This is equivalent to the assumption that the number of individuals

in one taxon is constant throughout time and that all taxa have the same

number of individuals.

3. Transform all appearances into the section, removing the ones that appear

in periods of nondeposition or are deposited in sediment that is eroded

4. Determine the last (highest) fossil occurrence that can be found in the

section

5. Repeat this multiple times to approximate the distribution of LFOs

The same procedure (mutatis mutandis) is also used for ﬁrst fossil occurrences.

4 Evaluating the Eﬀects of Diﬀerent Deposition

Models

The model created above can be used to evaluate the eﬀects of diﬀerent deposi-

tion models on the interpretation of fossil occurrence patterns. For this, the rate

functions introduced above are taken as representatives of probability distribu-

tions. This is done by interpreting them as density functions (with respect to

the Lebesgue measure) of an intensity measure EX that uniquely determines an

inhomogeneous Poisson point process (IPPP) ( [15], p. 530; [17], p. 77). This is

consistent with modeling fossil occurrences as rare and independent events that

occur with changing rates ( [18], p. 153-158).

Within this context, it is possible to deﬁne the relative entropy (RE) of two rate

functions, which is given by

E(f0;f1) = Zf1(x)−f0(x) + ln f0(x)

f1(x)f0(x) dx. (5)

[16]. It can be used as a measure of how good the two rate functions can be

distinguished by the samples they generate. This is directly supported by Stein’s

7

theorem ( [19], p. 425), which states that under increasing sample sizes, the rate

of convergence of the type two error is determined by the relative entropy. In

the context described here, the RE can be used in two ways.2

One is to look at two rate functions f0, f1and transform them with a ﬁxed

deposition rate r. Then the relative entropy E(f0

r;f1

r) of the transformed rate

functions is a measure for the distinguishability of the rate functions under the

deposition rate r: A high relative entropy allows to tell the two transformed

rate functions more easily apart. This can for example be used to assess the

distinguishability of two hypotheses on extinction rates under a given deposition

model and to adapt the sampling procedure accordingly. A slightly diﬀerent

application of this approach can be found in section 5.2.

The second possibility is to ﬁx a rate function fand transform it under diﬀerent

deposition rates ri, rj. For the resulting functions fri;frj, the relative entropy

E(fri;frj) can be determined. Here, a high relative entropy corresponds to a

higher chance of misinterpreting the underlying rate function f. This is since a

high relative entropy corresponds to friand frjbeing diﬀerent. But since they

are both created by the same underlying rate function, but look very diﬀerent

in the ﬁeld, the outcome of the transformation is sensitive to changes in the

deposition rates and can therefore be mistaken easily. This can for example be

used to asses the sensitivity of the interpretation of an extinction rate under

diﬀerent deposition models. This is demonstrated in section 5.1.

These two approaches allow the quantiﬁcation of the eﬀects of deposition rates

on the interpretation of fossil data. One example for this quantiﬁcation is the

application of ordination methods such as NMDS using the symmetrized relative

entropy (SRE)

˜

E(f0;f1) = E(f0;f1) + E(f1;f0) (6)

as a generalized notion of distance. The SRE can be used to identify rate

functions that generate very similar or very diﬀerent patterns in the section

under a given deposition model or to identify deposition models that can lead

to diﬀerent interpretations of the same geological event. This is demonstrated

in the examples below.

2note that the only diﬀerence between these approaches is that one ﬁxes a rate function,

while the other one ﬁxes a deposition rate

8

5 Examples

5.1 The Eﬀect of Diﬀerent Deposition Models on the Recog-

nition of Extinction Rates

As a ﬁrst example, the framework created above is used to study the eﬀects

of diﬀerent deposition models on the recognition of extinction rates.3. Since

until now, only rates of last appearance dates were discussed, it is necessary to

discuss the relation between extinction rates and rates of last appearance dates.

5.1.1 The Relation of Extinction Rates and Rates of Last Appear-

ance Dates

It should ﬁrst be noted that all information about extinction rates is indirect,

since it can only be derived from the last appearance dates. These are however

dependent on the rate of designated fossils, which determines the time between

the actual extinction of a taxon and its last appearance date. If the rate of

designated fossils is high, the time between the actual extinction and the last

appearence date of the taxon, being the last designate fossil before the extinc-

tion, will be shorter. Therefore rates of last appearance dates become more and

more similar to the extinction rate if the rate of designated fossils is higher (ﬁg.

6). So every rate of last appearance dates is a combination of an extinction rate

and a rate4of designated fossils, corresponding to an occurrence rate of some

taxa.

To be able to embed extinction rates into the framework built above, their cor-

responding rates of last appearance dates have to be determined. For this, the

following procedure is used:

1. Simulate an extinction based on the assumption that extinctions follow

an inhomogeneous Poisson point process with the extinction rate as the

rate function (ﬁg. 5). This is done using code from [16]

2. Determine a random number following an exponential distribution with

parameter λ. This number corresponds to the distance between the last

appearance date and the extinction

3All code used for this example can be found in the appendix under the name example1.R.

It uses the R packages MASS [20] and numDeriv [21]

4In a more general setting, every taxon will have its own rate of designated fossils. This

case will not be considered here

9

3. Subtract the random number from the time of extinction to determine the

last appearance date.

4. Repeat this multiple times to approxinmate the distribution of last ap-

pearance dates

Here, the parameter λdetermines the rate of designated fossils. This approach is

based on the model assumptions that the designated fossils appearances follow

a homogeneous Poisson point process with rate λ. This translates into the

assumption that all taxa have the same number of individuals and go extinct

independently of each other.

5.1.2 Application: Analysis of the Mass Extinction at the K-Pg

Boundary on Seymour Island, Antarctica

As a real world example for the methods developed, the K-Pg boundary on

Seymour Island in Antarctica is chosen. On this island, deposits from the Late

Cretaceous to the early Paleogene crops out. It is rich in macrofossils such as

ammonites, bivalves, and gastropods and well sampled [22] [23] [24] [25]. There-

fore data from this locality has been frequently used to derive information about

the nature of the mass extinction at the K-Pg boundary [26] [27] [28] [29] [30].

Hypotheses that were discussed using data from this section include a gradual

extinction [25] as well as one or multiple extinction pulses [31] [26] [32]. Since this

locality is well sampled, data from it is also used as a benchmark for new statis-

tical methods that derive information about extinction events [26] [27] [33] [29].

This is based on the assumption that the fossils in the section on Seymour island

have uniform recovery potential [34] and has been deposited constantly [25].

As the extinction rate whose recognition under diﬀerent deposition models will

be discussed, a single extinction pulse at the K-Pg boundary as was proposed

by [31] is chosen. This extinction rate is displayed in ﬁg. 8, the rate of last

appearance datums that was derived with the procedure described in section

5.1.1 (with λ= 20 designated fossils per ma) is displayed in ﬁg. 9.

The assumption of a constant deposition rate that is made by many authors is

contrasted by a sequence stratigraphic interpretation by Macellari (1988) [35]

with a maximum ﬂooding surface and a stillstand at the K-Pg boundary(ﬁg. 7).

This is is supported by the presence of hardgrounds [36], condensation horizons

( [24] section A, [35]) as well as lag deposits [37] and dinocyst data that suggests

a ﬂooding [38]. The lateral homogeneity is also questionable ( [24] section A vs.

10

B, [39]), which makes correlation between sections problematic and correlated

datasets less reliable. Additionally there are cold-seep carbonates present in the

lower parts of the section and cold-seep faunas in the higher parts [40]. This

can alter the speed of cementation, which increases preservation [6].

These are all indicators that contrast the assumption of a uniform recovery po-

tential. The aim of this example is to asses the eﬀects of deposition models that

do not assume a constant deposition rate on the recognition of a sudden spike

in the extinction rate at the K-Pg boundary. This is done by transforming the

rate of last appearance dates which was created above (ﬁg. 9) by ten diﬀerent

deposition models: one with a constant deposition rate and nine variations of

Macellaris sequence stratigraphic interpretation (ﬁg. 10, subﬁgs. 1 to 10).

These nine deposition models were created by using the temporal framework

provided by Witts et al. (2016) [24] to generate boundary conditions with abso-

lute age. Within this framework, the maximum ﬂooding surface was ﬁxed at the

stratigraphic height where the data from section A from Witts et al. displays a

spike in overall fossil occurrences as well as last and ﬁrst fossil occurrences. This

height roughly corresponds to the stratigraphic height of the maximum ﬂooding

surface proposed by Macellari in section F from Macellari. This section is closest

to section A from Witts et al. among all sections from Macellari. Then three

deposition rates for the proposed stillstand at the K-Pg boundary (200, 50, and

5 m per ma deposition) and three diﬀerent durations of the maximum ﬂooding

surface (maximum ﬂooding surface ending 0.199, 0.149, and 0.099 ma before

the K-Pg boundary) were chosen. Creating all possible combinations of these

parameters generates the nine diﬀerent depostion models that are displayed in

ﬁg. 10.

Three of the last fossil occurrence rates that were generated by pushing the rate

of last appearance dates forward are displayed in ﬁg. 11. For each pair out

of the ten of the transformed rates, the symmetrized relative entropy (SRE) is

calculated and used as a measure of dissimilarity for a NMDS [41] [42]. The raw

values of the SRE can be found in table 1.

The result of the NMDS, which was performed using the R package MASS [20],

is shown in ﬁg. 12. In this plot, the distance between two numbers corresponds

to how diﬀerent the rates of last fossil occurrences appear in the ﬁeld. Since

every rate of LFOs is generated by the same underlying extinction rate, but

transformed by a diﬀerent deposition model, the distance between two numbers

also corresponds to how diﬀerent the samples generated by the extinction rate

look like under diﬀerent deposition models. The deposition models that corre-

11

spond to the numbers can be found in ﬁg. 10.

Most relevant is the position of the constant deposition rate (denoted by the

number 1), since it is frequently used by diﬀerent authors [33] [29] [34] [25] [43].

It is located close to the numbers 2 and 3, but has a marginal position in the

plot. This shows that deposition rates 2 and 3 yield similar results compared

to the constant deposition rate, but results derived under the assumption of a

constant deposition rate will in general diﬀer from the results derived under the

other deposition models. The numbers 8,9 and 10 form a cluster at the right

side of the plot. This shows that results derived from these deposition models

will be very similar. So for a long duration of the maximum ﬂooding surface,

the results are robust in terms of changes of deposition rates at the K-Pg bound-

ary. This does not hold for medium and short durations of maximum ﬂooding

surfaces, since the numbers 2,3, and 4 as well as 5,6, and 7 do not form clusters.

5.2 The Eﬀects of Binning on the Distinguishability of

Extinction Hypotheses

As a second example, the eﬀects of binning on the distinguishability of hypothe-

ses on extinction rates are discussed.5For this, the dataset from Macellari

(1986) [22], consisting of ammonite occurrences collected on Seymour Island, is

used as an example. This dataset is frequently used to derive information about

the mass extinction at the K-Pg boundary (see ﬁrst example) and to demon-

strate the performance of new statistical methods that derive information about

extinction rates [26] [27] [28] [29] [30] [44] [33] [45] [43] [46]. This data is usually

obtained by digitizing ﬁgure 5 from Macellari (1986) [26] [28], whose way of

representation suggests that the data was sampled continuously, as is seen in

ﬁgure 13. But this is not the case, since the data from Macellari (1986) is based

on Macellari (1984) [47], where the raw data from diﬀerent sections is presented

in sampling bins with up to 39 meters length, and ﬁndings of ammonites are

recorded as present, abundant or very abundant within these bins. The data

presented by Macellari (1986) is also correlated across Seymour Island, which

might be problematic because of the lack lateral homogeneity that was noted in

section 5.1.

The goal of this example is to asses how the distinguishability of diﬀerent pairs

of extinction rates changes due to binning. For this, nine extinction rates are

5All code used for this example can be found in the appendix under the name example2.R.

It uses the R packages MASS [20] and numDeriv [21]

12

deﬁned. They consist of single extinction pulses of diﬀerent duration (extinc-

tion rates 1-3), gradually increasing extinction rates (extinction rate 4) or two

extinction pulses that are diﬀerently weighted(extinction rates 5-9) (ﬁg. 14).

For comparability, the extinction rates are chosen in a way that in average they

all generate the same number of extinctions. The ﬁrst extinction interval in the

extinction rates 5 to 9 is created by taking the extinction interval proposed by

in ﬁg. 5 from [26] and transforming it back into time under the assumption of a

constant deposition rate. The rates of last appearance dates derived from these

extinction rates are displayed in ﬁg. 15.

For simpliﬁcation, the deposition rate is assumed to be constant. Therefore the

pushforward by this deposition rate reduces to a rescaling of the rate functions

as they are displayed in ﬁg. 14. Then for each pair of these rates of last fossil

occurrences, the symmetricized relative entropy (SRE) is calculated. The re-

sults are displayed in table 2 and taken as input for a NMDS6. The result of

this NMDS is displayed in ﬁgure 16. In this ﬁgure, the distance between two

numbers is proportional to the distinguishability of the corresponding extinc-

tion rates shown in ﬁgure 14. This shows that the single pulses with diﬀerent

durations (numbers 1,2, and 3) are easy to distinguish, whereas the gradual ex-

tinction (4) and the two longer pulses are harder to distinguish. The extinctions

with two extinction pulses (5-9) are approximately equally hard to distinguish.

In a second step, each of the rate of last fossil occurrences is binned with the

bins taken from Macellari (1984), section A. One example for the continuous

as well as the binned version of a rate function can be found in ﬁg. 17. Then

the SRE is calculated for each pair of the binned rate functions. The results

are shown in table 3, which are used for another NMDS. The results of this

NMDS are displayed in ﬁgure 18. It shows, up to a rotation, the same pattern

as the NMDS of the continuous functions. This demonstrates that the relation

of distingushability between diﬀerent extinction rates does not change much due

to binning.

Last, for each pair of last fossil occurrence rate, the ratio of the SRE of the

continuous functions and the SRE of the binned functions is determined. These

ratios are presented in table 11. This ratio serves as a measure of how much

harder the diﬀerent extinction hypotheses are to distinguish after binning. The

results show a range of approximately 1.1 to approximately 2.4. So in the best

case, extinctions just get roughly 10 % harder to distinguish after binning. This

6using the R package MASS [20]

13

is for example the case for any pair of extinction rates with two pulses (5-9). In

the worst case, distinguishability increases by 140 % because of binning. This

is the case for the extinction rate with one pulse of intermediate duration (2)

and the gradually increasing extinction rate (4).

6 Discussion

6.1 Limits of the Model

There are certain limits to the model presented here.

One implicit model assumption that was not discussed or mentioned above is

that shells have no volume, so they do not contribute to the deposition rate.

This is neglectable in periods with low input of shell volume and suﬃciently

high sediment input. However in periods with high volume of shell input, com-

bined with intervals of nondeposition, this model assumption will lead to errors.

Another potential problem is the concept of the designated fossil. It was in-

troduced to ensure that the fossil occurrences in the section correspond to an

individual of some taxon in time. To make this correspondence meaningful, it

is necessary that the rate of designated fossils is proportional to the rate of the

actual individuals of the taxon. This basically translates to the model assump-

tion that all taphonomic processes are invariant. This model assumption can

be problematic, since there is a variety of processes like time-averaging, com-

paction, diagenetic dissolution and the eﬀects of bioturbation that can disturb

this invariance. It is possible to incorporate these processes into the framework

provided in this thesis, but this requires a more detailed discussion than is pos-

sible here. Everything said above applies mutatis mutandis to designated ﬁrst

fossils and designated last fossils.

6.2 Obtaining Deposition Rates

Deposition rates play a crucial role in the model presented here. It is however

hard to obtain reasonable estimates for these rates. This can be made easier by

the following considerations:

•Every age model is always also a deposition model. This is since both

connect point in time with point in the section and vice versa. So points in

the section with a ﬁxed absolute age can be used as a temporal framework

14

and as boundary conditions for the deposition rates. This is for example

done in section 5.1.

•Deposition rates do not change arbitrarily, but within certain restrictions

that are set by the sequence stratigraphic framework. This can be used

to derive points with minimal or maximal deposition rates or to identify

intervals having lower or higher intervals than other intervals. This is for

example done in section 5.1.

These approaches can be used as a rough framework for deposition rates that

can then be ﬁlled with more detailed observations. These observations can for

example be obtained by using a sampling procedure that does not only aim

at the stratigraphic position of fossils, but also at the reconstruction of the

depositional environment.

And even in cases where no unique or precise deposition rate can be derived,

the model can still be used to assess the robustness of the results under the

available competing deposition rates.

7 Summary

The presented model embeds the eﬀects of changing deposition rates and hiatii

on the stratigraphic distribution of fossils in a probability theoretical framework.

This allows, among others, to

•Correct data for eﬀects of changing deposition rates (section 2)

•Analytically study the eﬀects of changing deposition rates and hiatii on

paleontological patterns such as extinction rates and origination rates (sec-

tion 3)

•Analyse the sensitivity of results with regard to diﬀerent deposition models

(sections 4 and 5.1)

•Assess the distinguishability of hypotheses that are based on the strati-

graphic distribution of fossils and adapt sampling procedures accordingly

(sections 4 and 5.2)

15

8 Acknowledgements

I would like to thank Emilia Jarochowska for being a very patient and encour-

aging supervisor, and Rafal Nawrot for taking the time to give feedback on the

project.

9 Appendix

In the digital version of this thesis, the following ﬁles are provided:

•A R ﬁle named example1.R, containing all code that was used for the

example in section 5.1.

•A R ﬁle named example2.R, containing all code that was used for the

example in section 5.2.

•The R ﬁles timetostrat,timetostratbasic,strattotime,strattotimebasic, and

timetostratlastﬁrstocc.R. They perform the transformations described in

the sections 2 and 3

•The ﬁle docrcode.pdf, containing descriptions of the transformations that

complement the comments in the code

•The R ﬁle functionexamples.R, containing simple examples of the trans-

formations described in the sections 2 and 3

16

10 Figures

0 2 4 6 8 10 12

0 5 10 15 20 25 30 35

Time (Arbitrary Unit)

Stratigraphic Height (in m)

Figure 1: An integrated sedimentation rate I. The thin dashed black lines

indicate how points in time are connected to heights in the section and vice versa

by I. The sedimentation rate rthat generated this integrated sedimentation rate

is qualitatively indicated by the red dashed line. Time intervals with a high

sedimentation rate lead to a steep integrated sedimentation rate, whereas time

intervals with a low sedimentation rate lead to a ﬂat integrated sedimentation

rate.

17

0 2 4 6 8 10 12

0 5 10 15 20 25 30 35

Time (Arbitrary Unit)

Stratigraphic Height (in m)

Figure 3: The temporal rate ffrom ﬁg. 2(blue dashed line) being pushed

forward by the integrated deposition rate I(red thick line) from ﬁg. 1) to

form the corresponding observable rate fr(thick blue line). The thin black

dashed lines indicate the 1-1 correspondence between time and stratigraphic

height. Note that the maximum of the temporal rate has no corresponding

maximum in the observable rate, since it is accompanied by a high deposition

rate (compare with ﬁg. 2), which dilutes the rate function. This is contrasted

by the maximum in the observable rate, which is created by low deposition rates

(compare ﬁg. 2), although the temporal rate in the corresponding time interval

decreases.

19

0 5 10 15 20 25 30 35

0.0 0.5 1.0 1.5 2.0 2.5

Stratigraphic Height (in m)

Events per m Stratigraphic Height

Figure 4: The observable rate that was generated by the transformation dis-

played in ﬁg. 3. The sediment between two thin dashed black lines has been

deposited in two time units. This emphasizes that the sediment containing the

peak in the observable rate has been deposited over a longer period of time than

all of the sediment forming the lowest 15 meters of the section.

20

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0246810

Time (Arbitrary Unit)

Extinctions per Time Unit

Figure 5: An extinction rate (thick red line). On the six thin dotted lines,

diﬀerent simulated positions of extinctions (black crosses) are displayed. Note

that in general, not only the position of the extinctions is random, but also the

number of extinctions generated by an extinction rate. In the case displayed

here, the number of observed extinctions ranges from ﬁve (second dotted line

from below) to thirteen (third dotted line from below). The maximum in the

extinction rate does also not necessarily generate a cluster of extinctions (fourth

dotted line from below), since the position of extinctions is nondeterministic.

The rates of last appearance dates derived from this extinction rate for diﬀerent

parameters λare desplayed in ﬁg. 6.

21

0.0 1.0 2.0 3.0

048

Lambda = 2

Time (Arbitrary Units)

Rate

0.0 1.0 2.0 3.0

048

Lambda = 4

Rate

Time (Arbitrary Units)

0.0 1.0 2.0 3.0

048

Lambda = 8

Rate

Time (Arbitrary Units)

0.0 1.0 2.0 3.0

048

Lambda = 16

Rate

Time (Arbitrary Units)

Figure 6: The rate of last appearence dates (black, in last appearance dates per

time unit), derived from the extinction rate in ﬁg. 5 and assuming diﬀerent rates

of designated fossils λ(see section 5.1). As λincreases, the shape of the rate of

last appearance dates is approximating the extinction rate (red, in extinctions

per time unit). This is since for increasing rates, the random numbers following

an exponential distribution are getting smaller, which corresponds to more des-

ignated fossils per time unit. This reduces the distance between an extinction

and the corresponding last appearance date. Transforming this into the section,

this can be seen as a depiction of the Signor-Lipps eﬀect: as fossil occurrences

become rarer (corresponding to a lower rate λ), the last fossil occurrences show

more backwards smearing compared to the actual extinction rate. Note that

since the overall number of extinctions is constant, the overall number of last

fossil occurrences is also constant. Therefore the backwards smearing not only

spreads the the last fossil occurrences over a longer period of time, but also

makes the extinction appear less dramatic.

22

66.4 66.3 66.2 66.1 66.0

0 50 100 150 200

Age (ma)

Extinctions per ma

Figure 8: The extinction rate used for the ﬁrst example from Seymour island

(section 5.1). It consists of one spike in the extinction rate right at the K-Pg

boundary and a low background extinction rate. The background extinction

rate generates in average one extinction per ma. The duration of the extinction

pulse is chosen to generate in average as many extinctions as the background

extinction would in one million years.

24

66.4 66.1

940 1000

Age

Stratigraphic Height

Time

accumulated sediment

Time

accumulated sediment

Time

accumulated sediment

Time

accumulated sediment

Time

accumulated sediment

Time

accumulated sediment

Time

accumulated sediment

accumulated sediment

accumulated sediment

940 1000

940 1000

940 1000

940 1000

940 1000

940 1000

940 1000

940 1000

940 1000

66.4 66.1 66.4 66.1 66.4 66.1

66.4 66.1 66.4 66.1 66.4 66.1 66.4 66.1

66.4 66.1 66.4 66.1

10

9

8765

4321

Figure 10: An overview of the deposition models from the example in section

5.1, displayed as accumulated sediment/integrated deposition rate. The x-axis

is age, the y-axis is stratigraphic height, both are taken from [24]. The depo-

sition rates for the proposed stillstand at the K-Pg boundary are 200 (high),

50 (medium), and 5 (low) m deposition per ma, the three diﬀerent durations of

the maximum ﬂooding surface end 0.199 (short duration mfs), 0.149 (medium

duration mfs), and 0.099 ma (long duration mfs) before the K-Pg boundary.

The numbers display the following deposition models: 1: constant deposition

rate, 2: short maximum ﬂooding surface (mfs) duration and high deposition rate

(dr) at the K-Pg boundary, 3: short mfs duration and medium dr at the K-Pg

boundary, 4: short mfs duration and low dr at the K-Pg boundary, 5: medium

mfs duration and high dr at the K-Pg boundary, 6: medium mfs duration and

medium dr at the K-Pg boundary, 7: medium mfs duration and low dr at the

K-Pg boundary, 8: long mfs duration and high dr at the K-Pg boundary, 9:

long mfs duration and medium dr at the K-Pg boundary, 10: long mfs duration

and low dr at the K-Pg boundary.

26

940 960 980 1000

0.00 0.05 0.10 0.15 0.20

Stratigraphic Height (in m)

Last Fossil Occurrences per Meter

Figure 11: Three transformations of the rate of last appearance dates as dis-

played in ﬁg. 9. The black line corresponds to the rate of last fossil occurrences

under a constant deposition rate (deposition model 1 in ﬁg. 10), the red line to

the rate of last fossil occurrences under a short mfs and high deposition rates at

the K-Pg boundary (deposition model 2 in ﬁg 10) and the red line to the rate

of last fossil occurrences under a long mfs and low deposition rates at the K-Pg

boundary (deposition model 10 in ﬁg 10). The spike in the green rate of last

fossil occurrences reaches values of over two. Note that the overall number of

last fossil occurrences is constant, so more condensation at the mfs with more

last fossil occurrences automatically leads to less last fossil occurrences at the

K-Pg boundary.

27

1

2

3

4

5

6

7

8

9

10

−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0

−1.0 −0.5 0.0 0.5

NMDS Axis 1

NMDS Axis 2

Figure 12: The results of the NMDS from the example in section 5.1. The

numbers correspond to the deposition models as described in ﬁg. 10. The

distance between two numbers corresponds to how diﬀerent the signal of the

extinction described in the ﬁgures 8 and 9 looks if the diﬀerent deposition models

from ﬁgure 10 are used for the transformation from time into the section, see

section 4, approach 2. Therefore the distance between two numbers is a measure

of how strongly the interpretation of the extinction by an sudden extinction

pulse at the K-Pg boundary is altered by the choice of the deposition model.

28

0 50 150

Time

Ext. per ma

5 10 20

Time

Ext. per ma

2 4 6 8

Time

Ext. per ma

5 10 20

Time

Ext. per ma

0 50 150

Time

Ext. per ma

0 50 100

Time

Ext. per ma

0 40 80 140

Ext. per ma

0 40 80 120

Ext. per ma

0 40 80

Ext. per ma

123

456

9

87

66.4 66.25 66.1 66.4 66.25 66.1 66.4 66.25 66.1

66.4 66.25 66.1 66.4 66.25 66.1 66.4 66.25 66.1

66.4 66.25 66.1 66.4 66.25 66.1 66.4 66.25 66.1

Figure 14: The nine diﬀerent extinction rates used for the example in section

5.2. The x-axis is age, the y-axis is extinctions per ma. Note the diﬀerent scales

on the y-axis. All extinction rates are adjusted to generate in average the same

number of extinctions.

30

66.4 66.25 66.1

0 5 10 20

time

Rate of LAD

0 5 10 20

time

Rate of LAD

0 5 10 20

time

Rate of LAD

0 5 10 20

time

Rate of LAD

0 5 10 20

time

Rate of LAD

0 5 10 20

time

Rate of LAD

0 5 10 20

Rate of LAD

0 5 10 20

Rate of LAD

0 5 10 20

Rate of LAD

66.4 66.25 66.1 66.4 66.25 66.1

66.4 66.25 66.1 66.4 66.25 66.1 66.4 66.25 66.1

66.4 66.25 66.1 66.4 66.25 66.1 66.4 66.25 66.1

123

456

9

87

Figure 15: The nine continuous rates of last appearance dates derived from the

extinction rates shown in ﬁgure 14, using the procedure described in section

5.1.1 and a rate of λ= 20 designated fossils per ma. The x-axis is age, the

y-axis is last appearance dates per ma.

31

1

2

3

4

5

6

7

8

9

−0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6

−0.6 −0.4 −0.2 0.0 0.2 0.4

NMDS Axis 1

NMDS Axis 2

Figure 16: The result of the NMDS of the continuous rates of last fossil occur-

rences from section 5.2. The distance between the numbers corresponds to how

well the corresponding extinction rates in ﬁg. 10 can be distinguished via their

derived continuous rates of last fossil occurences.

32

66.40 66.30 66.20 66.10

0 5 10 15

Age (ma)

Last Appearance Dates per ma

Figure 17: The continuous rate of last appearance dates (red) and its binned

counterpart (blue), derived from the extinction rate number six in ﬁg. 14. The

bins from Macellari (1984) [47], section A were transformed into time assuming

a constant deposition rate. The thin dashed lines indicate the borders of these

bins. For each bin, the area of the bin and the area under the continuous rate

of last appearance dates are equal. Note that the two extinction pulses are well

visible in the continuous rate of last appearance dates, whereas they are not

visible at all in the binned rate of last appearance dates.

33

1

2

3

4

5

6

7

8

9

−0.5 0.0 0.5

−0.6 −0.2 0.2 0.4 0.6

NMDS Axis 1

NMDS Axis 2

Figure 18: The result of the NMDS of the binned rates of last fossil occurrences

from section 5.2. The distance between the numbers corresponds to how well the

corresponding extinction rates in ﬁg. 10 can be distinguished via their derived

binned rates of last fossil occurrences.

34

11 Tables

model 1 2 3 4 5 6 7 8 9 10

1 0 0.2460 0.8402 1.4690 2.5539 2.8857 3.1526 3.4248 3.5420 3.6339

2 0.2460 0 0.5469 1.2009 1.4084 1.6809 1.9647 1.7088 1.7914 1.8796

3 0.8402 0.5469 0 0.2680 1.8965 1.4445 1.5143 2.1448 1.8278 1.8100

4 1.4690 1.2009 0.2680 0 2.4968 1.7201 1.4810 2.6480 2.1168 1.9599

5 2.5539 1.4084 1.8965 2.4968 0 0.2460 0.5014 0.7538 0.8297 0.9089

6 2.8857 1.6809 1.4445 1.7201 0.2460 0 0.0816 0.9845 0.7866 0.7838

7 3.1526 1.9647 1.5143 1.4810 0.5014 0.0816 0 1.2004 0.8840 0.8096

8 3.4248 1.7088 2.1448 2.6480 0.7538 0.9845 1.2004 0 0.0755 0.1419

9 3.5420 1.7914 1.8278 2.1168 0.8297 0.7866 0.8840 0.0755 0 0.0151

10 3.6339 1.8796 1.8100 1.9599 0.9089 0.7838 0.8096 0.1419 0.0151 0

Table 1: The raw values of the symmetrized relative entropy (SRE) from the

example in section 5.1. The SRE of the transformation of the rate of last

appearance dates (ﬁg. 9) by the i-th and by the j-th deposition model (ﬁg. 10)

can be found in the i-th column and the j-th row. Note that the table, taken as

a matrix, is symmetrical, since the SRE is symmetrical.

model 1 2 3 4 5 6 7 8 9

1 0 0.4253 0.8857 0.5004 0.0460 0.1582 0.3273 0.5513 0.8254

2 0.4253 0 0.2432 0.0363 0.4116 0.4749 0.5938 0.7657 0.9787

3 0.8857 0.2432 0 0.0997 0.7923 0.7816 0.8235 0.9217 1.0541

4 0.5004 0.0363 0.0997 0 0.4643 0.5065 0.6036 0.7542 0.9442

5 0.0460 0.4116 0.7923 0.4643 0 0.0342 0.1279 0.2775 0.4795

6 0.1582 0.4749 0.7816 0.5065 0.0342 0 0.0305 0.1174 0.2582

7 0.3273 0.5938 0.8235 0.6036 0.1279 0.0305 0 0.0292 0.1127

8 0.5513 0.7657 0.9217 0.7542 0.2775 0.1174 0.0292 0 0.0282

9 0.8254 0.9787 1.0541 0.9442 0.4795 0.2582 0.1127 0.0282 0

Table 2: The values of the SRE of the continuous rates of last fossil occurences

from the example in section 5.2. The SRE of the i-th and the j-th continuous

rate of last fossil occurrences can be found in the i-th column and the j-th

row. Note that the table, taken as a matrix, is symmetrical, since the SRE is

symmetrical.

35

model 1 2 3 4 5 6 7 8 9

1 0 0.2020 0.6527 0.3242 0.0410 0.1431 0.2970 0.5023 0.7524

2 0.2020 0 0.1325 0.0150 0.1993 0.2619 0.3744 0.5398 0.7418

3 0.6527 0.1325 0 0.0582 0.5764 0.5733 0.6204 0.7222 0.8549

4 0.3242 0.0150 0.0582 0 0.2987 0.3408 0.4328 0.5784 0.7585

5 0.0410 0.1993 0.5764 0.2987 0 0.0307 0.1163 0.2539 0.4384

6 0.1431 0.2619 0.5733 0.3408 0.0307 0 0.0273 0.1074 0.2361

7 0.2970 0.3744 0.6204 0.4328 0.1163 0.0273 0 0.0264 0.1028

8 0.5023 0.5398 0.7222 0.5784 0.2539 0.1074 0.0264 0 0.0252

9 0.7524 0.7418 0.8549 0.7585 0.4384 0.2361 0.1028 0.0252 0

Table 3: The values of the SRE of the binned rates of last fossil occurences from

the example in section 5.2. The SRE of the i-th and the j-th binned rate of last

fossil occurrences can be found in the i-th column and the j-th row. Note that

the table, taken as a matrix, is symmetrical, since the SRE is symmetrical.

model 1 2 3 4 5 6 7 8 9

1 - 2.1058 1.3570 1.5434 1.1198 1.1054 1.1018 1.0976 1.0970

2 2.1058 - 1.8354 2.4148 2.0647 1.8132 1.5861 1.4185 1.3192

3 1.3570 1.8354 - 1.7141 1.3745 1.3633 1.3274 1.2762 1.2330

4 1.5434 2.4148 1.7141 - 1.5546 1.4862 1.3945 1.3040 1.2447

5 1.1198 2.0647 1.3745 1.5546 - 1.1111 1.0996 1.0932 1.0938

6 1.1054 1.8132 1.3633 1.4862 1.1111 - 1.1152 1.0931 1.0939

7 1.1018 1.5861 1.3274 1.3945 1.0996 1.1152 - 1.1078 1.0957

8 1.0976 1.4185 1.2762 1.3040 1.0932 1.0931 1.1078 - 1.1198

9 1.0970 1.3192 1.2330 1.2447 1.0938 1.0939 1.0957 1.1198 -

Table 4: The values of the ratio of the SREs of the continuous rate of last fossil

occurrences and the binned rate of last fossil occurrences from the example

in section 5.2. The ratio of the i-th and the j-th continuous and binned rate

function can be found in the i-th column and the j-th row. Note that the table,

taken as a matrix, is symmetrical, since the SRE is symmetrical.

36

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41

Friedrich-Alexander-Universität Erlangen-Nürnberg

GeoZentrum Nordbayern

Name: ………………………………………………….. Vorname: ……………………………………………….

Matrikelnummer:………………………………….

Selbständigkeitserklärung zur Bachelorarbeit

Ich erkläre ausdrücklich, dass es sich bei der von mir eingereichten schrilichen Arbeit

mit dem Titel

............................................................................................................................

............................................................................................................................

um eine von mir selbstständig und ohne fremde Hilfe verfasste Arbeit handelt.

Ich erkläre, dass ich sämtliche in der oben genannten Arbeit verwendeten fremden Quellen,

auch aus dem Internet (einschließlich Tabellen, Graﬁken u. Ä.) als solche kenntlich gemacht

habe. Insbesondere bestäge ich, dass ich ausnahmslos sowohl bei wörtlich übernommenen

Aussagen bzw. unverändert übernommenen Tabellen, Graﬁken u. Ä. (Zitaten) als auch bei in

eigenen Worten wiedergegebenen Aussagen bzw. von mir abgewandelten Tabellen,

Graﬁken u. Ä. anderer Autorinnen und Autoren (Paraphrasen) die Quelle angegeben habe.

Ich erkläre hiermit weiterhin, dass die vorgelegte Arbeit zuvor weder von mir noch – soweit

mir bekannt ist – von einer anderen Person an dieser oder einer anderen Hochschule

eingereicht wurde.

Mir ist bewusst, dass Verstöße gegen die Grundsätze der Selbstständigkeit als Täuschung

betrachtet und dass die Unrichgkeit dieser Erklärung eine Benotung der Arbeit mit der Note

nicht ausreichend zur Folge hat und dass Verletzungen des Urheberrechts strafrechtlich

verfolgt werden können.

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(Datum) (Unterschri)