PoGO+ (Chauvin $et~{}al$., 2016; Friis $et~{}al$., 2018) is a balloon-borne
polarimetry mission, flown in 2016. It was optimized for pointing observation
of Crab and Cygnus X-1. PoGO+ utilizes Compton scattering within an array of
61 plastic scintillators (EJ-204), each 12 cm long and 3 cm cross in hexagonal
shape, read by individual PMTs (see Fig. 32).

Figure 32: Array of plastic scintillators for Compton polarimetry as used in
PoGO+ experiment. Picture courtesy $-$ Friis $et~{}al$. (2018) (reprinted
based on MDPI open access license policy).

The design is a matured version of the pathfinder balloon-born experiment,
PoGOLite, flown in 2013 (Chauvin $et~{}al$., 2016). The graded shield
collimators (graded shield of Copper, Tin and Lead layers) restrict the FOV to
$\sim$2∘ for pointed observations in 20 $-$ 180 keV. While the collimators
suppress most of the off-axis background events from front side, active BGO
shielding on the sides and back of the scintillators vetoes most of the
background events from other directions. In order to reduce the light leak
between the tightly packed adjacent scintillator rods, a special wrapping
material is used. PoGO+ type of instrument design is expected to provide a
sensitivity level of 11.1 % for Crab like sources for a 5-day balloon flight
mission.

####  5.3.4 Compton camera configuration

This configuration uses two separate layers of high Z detector planes with
Compton scattering in the first layer and subsequent absorption of the
scattered photon by the second layer. Therefore the instruments provide 3D
positional information of each event (2D position in x and y-direction from
individual layers and position in z from the known gap between the two layers)
which along with the deposited energy information can be used to reconstruct
the original photon direction. Illustration of the basic technique is shown in
Fig .29. This technique efficiently filters out most of the background events
which yields a high signal to noise ratio. From the location of 1st and 2nd
event from layer 1 and layer 2 respectively, obtained for each of the Compton
scattering event, can be used to extract the azimuthal angle of scattering.
Though Compton imager provides a better background rejection due to accurate
determination of the source direction (better positional resolution in
z-direction), because the geometry allows only forward scattering,
polarimetric efficiency and modulation factor are on the lower side (see Fig.
15). IBIS (Ubertini $et~{}al$., 2003) on board INTEGRAL, COSI (Yang
$et~{}al$., 2018), SGD (Tajima & et al., 2010) on board Hitomi use this type
of configuration to carry out spectro-polarimetry study in hard X-rays. IBIS
consists of a top layer of 128$\times$128 CdTe detector array
(4$\times$4$\times$2 mm3) with pixel pitch of 9.2 mm and a bottom layer of
64$\times$64 CsI detector bars (each 8.4$\times$8.4$\times$300 mm3). The two
layers are separated by 94 mm. Though primarily developed as a spectro-imager,
the same events which deposit energy in both the layers within a coincidence
window of 3.8 $\mu$s for reconstructing the source position, can be used in
extracting the azimuthal scattering angle distribution.

COSI, on the other hand, is a dedicated balloon borne Compton polarimeter and
spectro-imager sensitive in 200 keV to several MeV. It uses 12 cooled high-
purity Germanium double-sided strip detectors (each 8$\times$8$\times$1.5 cm3)
with a 2 mm strip pitch in 2$\times$2$\times$3 array to detect and reconstruct
the Compton events. With 37 orthogonal strips per side, the instrument
provides an excellent 2D and depth resolution (0.5 mm FWHM) which along with
excellent spectral resolution makes this instrument a sensitive spectro-
polarimeter. COSI-2 (Tomsick $et~{}al$., 2019), a similar design with an
additional layer of Germanium detector, has been selected for phase A study
for NASA SMEX mission. COSI-2 is expected to detect 20 GRBs a year with
polarization capability of 50 % MDP for 4$\times$106 erg/cm2 fluence, apart
from polarization measurements for bright galactic X-ray sources.

SGD Compton camera configuration consists of 32 layers of 0.6 mm thick Silicon
sensors and 8 layers of 0.75 mm thick CdTe detectors. These layers are
surrounded by 2 more layers of same CdTe detectors from all sides. Each
detector is pixelated with a pixel size of 3.2 mm. Total eight such units (210
cm2 geometric area) are collimated separately to a narrow FOV. Excellent
energy and positional resolution and optimized scattering geometry of SGD
allow sensitive polarimetry measurements in 50 $-$ 200 keV with MDP level of a
few percent for 100 mCrab sources in 100 ks exposure.

##  6 Summary and conclusion

X-ray polarization measurements provide two independent parameters $-$ degree
and angle of polarization characterizing the incoming radiation from any X-ray
source. Measurement of these parameters can provide a unique opportunity to
study the behavior of matter and radiation under extreme magnetic and
gravitational fields. Importance of X-ray polarimetry in astronomy has been
emphasized over the years, however its wealth has never really been utilized
until recently when a number of polarimetry and non-polarimetry instruments
provided some interesting polarization measurements in hard X-rays for Crab,
Cygnus X-1, V404-Cygni and a number of GRBs and solar flares. While some of
the instruments e.g. AstroSat, INTEGRAL will continue to provide new
measurements or improve upon the existing results for years to come, there is
a large number of confirmed and proposed dedicated hard X-ray polarimetry
experiments which ensures an extremely exciting time ahead in the field of
hard X-ray polarimetry. The primary reason behind these new measurements and
the urge of conducting new experiments at these energies is the higher level
of polarization that is expected in hard X-rays compared to that in soft
X-rays where the dominance of thermal emission dilutes the polarization level
to only a few percent. Consequently, a moderately sensitive polarimeter can
also provide interesting measurements for some of the bright sources. Hard
X-rays being generated close to the compact object and from fast cooling of
the highest energy particles, carry some unique insights of acceleration and
cooling mechanisms, signatures of high gravity and magnetic field close to the
black hole jets and low altitude magnetosphere of magnetars. We have already
seen a glimpse of such interesting science cases from AstroSat and Integral
measurements. With a significant advancement in the detection technology in
the last couple of decades in the field of scintillator readout methods, high-
speed readout Silicon sensors, 3D CZT and Germanium detectors and above of the
all, development of hard X-ray focusing optics, developing a sensitive compact
hard X-ray focal plane Compton polarimeter has come well within our reach. All
these new activities demand more focus in the field of hard X-ray polarimetry
to improve and develop sensitive polarimetry configurations using the new
detection and readout technologies, and develop robust methods to analyze the
existing and upcoming polarization data.

In this article, we reviewed the scientific potential of hard X-ray
polarimetry and some of the recent findings from various hard X-ray
polarimetry experiments. We discussed the existing polarimetry techniques and
compared them in hard X-rays. While Compton scattering is best suited in the
hard X-ray regime, photo-electron track imagers can also be explored in hard
X-rays in the future, in particular, the TPC detectors with higher gas depth
and density and the CdTe-Timepix detectors with smaller pixel sizes. Within
scattering based polarimetry technique, we found that a compact Compton
scattering polarimeter at the focal plane of a hard X-ray telescope can
provide an order of magnitude higher sensitivity in 25 $-$ 80 keV range. At
the lower energies, POLIX (Paul $et~{}al$., 2016) type Rayleigh polarimeters
are still the best option for polarimetry whereas at the most difficult photon
starved 100 $-$ 1000 keV region is probably best served by a polarimetry-
optimized large area wide FOV Compton camera.

With the growing realization of scientific potential of X-ray polarimetry, we
expect many more new approvals for dedicated X-ray polarimetry missions like
IXPE, the NASA mission and POLIX, the Indian next space mission, both set to
launch in 2021. In the hard X-rays, we have seen some proposals for dedicated
focal plane Compton polarimeters in 20 $-$ 80 keV range. The conventional
focal plane polarimeters employ plastic scintillator rod as scatterer and
sensitivity of the instruments is primarily limited by the lower energy
threshold of the plastic. In the future, we expect to see improvement in the
detection threshold with the advent of better readout technologies.
Improvisation in the scatterer configurations, e.g. replacing the plastic by a
hybrid passive (Be/Li) and active plastic scatterers in a single cylindrical
bundle might improve the overall energy threshold of the polarimeters by
utilizing the Rayleigh scattered X-rays at lower energies from the passive
scatterers. Thanks to the advancement in the detector technology like in
Double Sided Silicon Strip Detector (DSSD, ichiro $et~{}al$., 2008) or in the
Silicon based active pixel sensors, e.g. fast readout X-ray Hybrid CMOS
detectors (Bai $et~{}al$., 2008; Hull $et~{}al$., 2017; Chattopadhyay
$et~{}al$., 2018), in particular the SPEEDSTER-EXDs (Griffith $et~{}al$.,
2016), and Timepix detectors (Llopart $et~{}al$., 2007), it might also be
possible to replace the plastic scatterer by a position sensitive Silicon
sensor to enable simultaneous spectroscopic, timing, imaging and polarimetry
(see Vadawale $et~{}al$., 2012, for a conceptual design).

Unavailability of focusing optics in $>$80 keV warrants a large area detector
plane to be the obvious solution to build a sensitive polarimeter. Emergence
of Laue lens based on diffraction in crystals allowing concentration of X-rays
above 100 keV (Barrière $et~{}al$., 2011) or the newly proposed Stacked Prism
Lenses (SPL, Mi $et~{}al$., 2019) will provide an excellent solution in future
to develop sensitive detectors in the sub-MeV region (Roques $et~{}al$.,
2012). However, in absence of any real implementation of these techniques at
present, an effective approach might be to use a large area detector in wide
FOV configuration. The recently developed 3D CZT detectors can provide
excellent polarimetry and spectroscopic sensitivity in Compton camera mode in
this regime. Alternatively, a Compton camera made of a top layer of 2D CZT
detectors (like in CZTI) and a bottom layer of CeBr3 crystal with 2D SiPM
readout (recently shown by Goyal $et~{}al$., 2018) might be an effective way
to tackle this photon starved region.

With all these new developments in the hardware and upcoming dedicated
polarimetry missions, we expect many more new polarization measurements in
hard X-rays in the coming years.

## Acknowledgements

I would like to thank Prof. A. R. Rao, Prof. Santosh Vadawale and Prof. David
Burrows for the useful discussions while writing this article. Dr. Moszi Kiss
and Prof. Maxime Chauvin of PoGO+ collaboration provided some important inputs
about the PoGO+ payload and future hard X-ray polarimetry techniques. I am
thankful to Prof. Mark McConnell for sharing with me some useful contents
about GRAPE and LEAP polarimetry designs. My wife, Suravi, helped me with some
of the figures used in the article; a big thank to her. I would also like to
acknowledge Journal of Astrophysics and Astronomy and in particular, Prof.
Tushar P. Prabhu for inviting me to write this article. Some of the figures
and pictures used in this article are either adapted or taken from published
articles. I am grateful to the authors and the publishers of the articles for
providing me the necessary copyright permissions for the reuse of the
materials.

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# Student-t Networks for Melody Estimation

Udhav Gupta, Avi, Bhavesh Jain
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

(May 10, 2021)

##  1 Introduction

The task offered by problem of melody extraction is to obtain the frequency
values corresponding to the dominant audio signal in the input monophonic or
polyphonic audio data. This problem comprises of many sub-tasks, including,
selection of frequency values present as fundamental frequency, figuring out
the frequency corresponding to the most dominant voice.
The melody extraction problem from audio signals gets complicated when we
start dealing with polyphonic audio data. This is because in generalised audio
signals, the sounds are highly correlated over both frequency and time
domains. This complex overlap of many sounds, makes identification of
predominant frequency challenging.
In advancement to this, we can deal with filtering background noise in the
data, distinguish the components of audio data as voiced and non-voiced,
perform instrument recognition. Some mainstream applications are query-by-
humming, cover song identification, genre classification, automatic generation
of karaoke accompaniment and singer characterization.

##  2 Targeted Problem

Melody estimation or melody extraction refers to the extraction of the primary
or fundamental dominant frequency in a melody. This sequence of frequencies
obtained represents the pitch of the dominant melodic line from recorded music
audio signals. The music signal may be monophonic or polyphonic.
Monophonic Signal: Signal is generated from a single source.
Polyphonic Signal: Signal is a combination of signals generated by multiple
sources simultaneously.

####

Melody estimation in a monophonic audio is easier as compared to polyphonic
audio. In this paper we have targeted melody estimation for polyphonic audio.
We have compared 2 baseline models based on Signal Processing and patch-based
Convolutional Neural Networks in this paper. We will be implementing Student-
Teacher over the 2nd model for learning from less data using a semi-supervised
approach with reference to the following paper-

https://arxiv.org/pdf/2008.06358.pdf

##  3 Dataset Used

We have used the dataset MIR-1K for training the model. It contains 1000 audio
clips with pitch labeling for window size 40ms and hop size 20ms. Most audio
clips are extracted from Karaokes performed mostly by amateurs. Data comprises
of male and female audio sounds alike. Unvoiced sounds like friction sounds,
inhaling sound, sound of instruments, are also added to the dataset. The
datasets used for evaluating the models are MIREX-05 and adc2004.

##  4 Methods

###  4.1 Signal Processing

In this approach we have leveraged the basics of signal processing by
computing the auto-correlation of the STFT representation of the input signal.
Fundamental frequency is the number of samples in one time period. This time
period can be taken to be the distance between the first 2 peaks in the auto-
correlation.

###  4.2 Patch based CNN

We take a novel representation in time-frequency domain i.e CFP. This
spectrogram is split into patches of size $25\times 25$. These patches are
used as an input to CNN. The CNN comprises of two convolutional layers,
followed by three fully connected layers. The convolutional layers have
$8,5\times 5$ filters and $16,3\times 3$ filters. The number of units in fully
connected layers are 128, 64, 2 respectively. Now based on the highest
frequency present in a patch, it is classified as either vocal melody(labelled
1) or not (labelled 0). As only small portion of our input data contains vocal
melody, data imbalance occurs. Hence, $10\%$ of non-vocal peaks are randomly
selected into training data.
.The model is trained using the binary cross entropy loss between the CNN
output and the ground truth is minimized using batch gradient descent with the
Adam optimizer.
The CNN outputs the probability of the patch being a Vocal Melody. If the
output probability>0.5, then it is considered as vocal melody. Hence, we have
the required spectrogram, with vocal melodies set as 1 and others as 0.

Figure 1: CNN

###  4.3 Model Architecture

We propose convolutional recurrent neural network (CRNN) as the baseline
architecture. The CRNN architecture consists of 2 ResNet blocks and a bi-
directional long short-term memory layer. We first merge the audio waveforms
into a mono channel and downsample them to 8 kHz. We then calculate the
logarithmic-magnitude spectrogram using short-time Fourier transform with a
1024-point Hann window and an 80-point hop size.The CRNN architecture takes 31
consecutive frames of the spectrogram as input and predicts a pitch label
quantized with a resolution of 1/8 semitone. The size of the output layer is
442.

###  4.4 Proposed Teacher-Student Models

Initially we train out Teacher network in a supervised manner i.e with true
labels on a dataset $\mathcal{D}$ using the cross entropy loss function.
After this we train the Student network in a semi-supervised manner i.e. on a
different dataset $\mathcal{U}$, we generate the pseudo labels from our
Teacher network and use them as true labels for training the Student network
with the following cross-entropy function.

$\mathcal{L}_{b}=\frac{1}{M}\sum_{u=1}^{M}(H(\mathcal{y}_{u},p(y|x_{u};\Theta_{s}))+H(\mathcal{y}_{u},\mathcal{y}_{t}))$

Where $\mathcal{y}_{u}$ is the pseudo labels generated by Teacher Network on
$\mathcal{U}$ and $p(y|x_{u};\Theta_{s})$ are the predictions made by Student
network. And $\mathcal{y}_{t}$ are true labels.

Figure 2: Supervised Loss

Figure 3: Basic Teacher-Student Loss

##  5 Feature Extraction for patch based CNN

A patch captures events localised in time as well as frequency (or pitch). To
effectively localize a pitch event in frequency domain without interference
from the harmonics, we use the Combined Frequency and Periodicity
representation (CFP). The CNN model proposed in the baseline is then applied
on patches selected from this representation.

###  5.1 CFP representation:

The path-based CNN uses the Combined Frequency and Periodicity (CFP) approach
to represent data. This CFP representation is a combination of the Generalized
Cepsrtum (GC) and Generalized Cepstrun of Spectogram (GCoS). The GC is
obtained by removing the slow-varying portions in the time domain, resulting
in sub-harmonics in the lower frequency range. On the contrary, the GCoS
majorly represents the harmonics (higher frequency) as the slow varying-
portions in the frequency domain are removed. X:=STFT of input signal
amplitude
n, k:=Time and frequency indices respectively
q:=Quefrency
Wf, Wt:=High Pass Filters
F:=N-point Discrete Fourier Transform Matrix
$\sigma_{i}$:=Activation function

$\displaystyle
Spectogram:\textbf{Z}_{0}[k,n]=\sigma_{0}(\textbf{W}_{f}\textbf{X})$
$\displaystyle
GC:\textbf{Z}_{1}[q,n]=\sigma_{1}(\textbf{W}_{t}\textbf{F}^{-1}\textbf{Z}_{0})$
$\displaystyle
GCoS:\textbf{Z}_{2}[k,n]=\sigma_{2}(\textbf{W}_{f}\textbf{F}\textbf{Z}_{1})$
$\displaystyle
CFP:\textbf{Y}[p,n]=\tilde{\textbf{Z}}_{1}[p,n]\tilde{\textbf{Z}}_{2}[p,n]$

Before calculating the final CFP representation, we map GC and GCoS to log-
frequency domain (similar to pitch). The CFP representation mainly consists of
the fundamental frequency as the harmonics and sub-harmonics get suppressed on
combining GC and GCoS.

###  5.2 Patch Selection

We assume for every frame in Y, the peak is a vocal melody. We select a patch
around each such peak of size 25$\times$25\. These patches are the input for
the CNN model we have used above.

Figure 4: Data Representation

Figure 5: Patch Selection

##  6 Results and Conclusions

Figure 6: Accuracy for CRNN model

Figure 7: Accuracy for Teacher Student model

Figure 8: result1

Figure 9: result2

Figure 10: result3

Figure 11: result4

Method | ADC2004 | MIREX05
---|---|---
CRNN | 20.29/28.14 | 36.87/45.54
Teacher Student | 33.69/41.89 | 65.67/69.92
ResNet NS | 34.44/39.05 | 76.84/79.07
Patch CNN | -/- | 62.86/64.54
SP | 5.01/22.18 | 14.86/34.86

For 2 datasets ADC2004 and MIREX05, from different methods we have evaluated
RPA/RCA values. ResNet RS is the model mentioned in the paper, and hence is
used by us for comparison.

##  7 Acknowledgement

We thank Prof. Vipul Arora and all the Teaching Assistants of the course
EE698R, 2021 for their constant guidance.

## References

$[1]$ $http://mirlab.org/dataset/public/$
$[2]$ $https://ieeexplore.ieee.org/document/8462420$
$[3]$ $https://wp.nyu.edu/ismir2016/wp-
content/uploads/sites/2294/2016/07/072_{P}aper.pdf$ $[4]$
$https://ieeexplore.ieee.org/document/8462534$
$[5]$ $https://ieeexplore.ieee.org/document/6739213$
$[6]$ $https://arxiv.org/pdf/2008.06358.pdf$


# Influence of an environment changing in time on Crucial Events : the
earthquake prototype

Callum Muir  Center for Nonlinear Science, University of North Texas, P.O. Box
311427, Denton, Texas 76203-1427  Mauro Bologna  Departamento de Ingeniería
Eléctrica-Electrónica, Universidad de Tarapacá, Arica, Chile  Paolo Grigolini
Center for Nonlinear Science, University of North Texas, P.O. Box 311427,
Denton, Texas 76203-1427

###### Abstract

This paper is devoted to the study of the interaction between two distinct
forms of non-stationary processes, which we will refer to as non-stationarity
of first and second kind. The non-stationarity of first kind is caused by
criticality-generated events that we call _crucial events_. Crucial events
signal ergodicity breaking emerging from the interaction between the units of
the complex system under study, indicating that the non stationarity of first
kind has internal origin. The non-stationarity of second kind is due to the
influence on the system of interest of an environment changing in time,
thereby implying an external origin. In this paper we show that the non-
stationarity of first kind, measured by an inverse power law index $\mu$ is
characterized by singularities at $\mu=2$ and $\mu=3$. We realize the
interaction between the non-stationarity of first kind and the non-
stationarity of second kind with a model frequently adopted to study
earthquakes, namely, a system of main-shocks, assumed to be crucial events,
generating a cascade of after-shocks simulating the changing in time
environment. We prove that the after-shocks significantly affects the
detection of anomalous scaling, with this effect weakening as the value $\mu$
approaches $\mu=2.5$. We argue that this result is a consequence of the fact
that the states $\mu=2$ and $\mu=3$ are the borders between different
statistical regimes, where a sort of phase transition occurs, with $\mu=2.5$
being a state sufficiently far from both transition regimes. We conclude this
paper with the observation that the earthquakes should be interpreted as
resulting from the interaction between many geophysical units generating
criticality, with the non-stationary events of second kind affecting
conveniently short time regions between two consecutive crucial events.

##  I Introduction

The earthquake issue in the recent past has been the subject of an intense
debate concerning the power-law behavior in time distribution of earthquakes
doublesornette , helmstetter , bak , mega , turcotte , kagan . This problem
goes much beyond geophysics insofar as it is expected to be a property of
complex systems, including bacterial persistence bacterial , solar
radiophysics solar , stock price fluctuations stok , tree-limb branching tree
and the time duration of dictatorships romanempire . The main source of this
debate is the fact that the earthquakes are supposed to be generated by main
shocks activating a cascade of aftershocks. The authors of Ref. mega  made the
conjecture that the main shocks are crucial events. This assumption implies
that they are not stationary.

The concept of deviation from stationarity is an issue of increasing interest
in the field of complexity. We focus on experimental data expressed under the
form of a time series, $\\{\xi(t)\\}$. The autocorrelation function

$\Phi_{\xi}(t_{1},t_{2})=<\xi(t_{1})\xi(t_{2})>$ (1)

in general depends on both $t_{1}$ and $t_{2}$. The symbol $<...>$ represents
a Gibbs average of the quantity indicated by the three dots,
$\xi(t_{1})\xi(t_{2})$ in this case. The stationarity condition is the
idealized mathematical prescription:

$\Phi_{\xi}(t_{1},t_{2})=\Phi(|t_{1}-t_{2}|),$ (2)

with the symbol $\Phi(t)$ denoting a stationary correlation function. This
stationarity condition can be violated leading to an extended time regime
where the correlation function requires an explicit dependence on both $t_{1}$
and $t_{2}$. The deviation from the stationarity condition can be due to two
distinct reasons: (1) Non-stationarity of first kind. The complex system under
study is characterized by aging, namely, it has a dynamics strongly dependent
on the initial condition and on the time distance from the initial condition.
The dependence on the initial condition vanishes very slowly as time
increases; (2) Non-stationarity of second kind. The system might move towards
the stationary condition, but the external environment influencing the
dynamics of the system of interest changes in time.

It is interesting to notice that the joint action of these forms of non-
stationarity was recently discussed by Kataoka et al akimoto , although
reversing our definition, these authors denote the non-stationarity of first
kind as non-stationarity of second kind and the non-stationarity of second
kind as a non-stationarity of first kind.

We note that the non-stationarity of first kind is closely connected to the
engineering view of Cox cox , who defined the age dependent failure rate

$g(t)=\frac{\psi(t)}{\Psi(t)},$ (3)

where $\psi(t)$ is the waiting time distribution density between the birth of
a machine and the occurrence of its first failure, and $\Psi(t)$ is the
probability that no failure occurs up to time $t$.

Since

$\psi(t)=-\frac{d}{dt}\Psi(t),$ (4)

the use of Eq. (3) yields

$\Psi(t)=exp{\left\\{-\int_{0}^{t}dt^{\prime}g(t^{\prime})\right\\}}.$ (5)

It is straightforward to prove that the assumption

$g(t)=\frac{r_{0}}{1+r_{1}t}$ (6)

yields

$\Psi(t)=\left(\frac{T}{t+T}\right)^{\mu-1},$ (7)

where

$\mu=1+\frac{r_{0}}{r_{1}},$ (8)

with the corresponding survival probability

$\psi(t)=(\mu-1)\frac{T^{\mu-1}}{(t+T)^{\mu}}.$ (9)

It is important to notice that if we generate a large number of sequences with
the first event occurring at time $t=0$ the rate of events at a generic time
$t>0$, called $R(t)$ is feller

$R(t)=\frac{1}{t^{2-\mu}},$ (10)

if $1<\mu<2$ and bellazzini

$R(t)=\frac{1}{<\tau>}\left[1+\frac{1}{3-\mu}\left(\frac{T}{t}\right)^{\mu-2}\right]$
(11)

with

$<\tau>=\frac{T}{\mu-2},$ (12)

if $2<\mu<3$.

The condition $\mu=2$ corresponds to a transition from the region where the
rate of crucial events is never constant to the region where the rate of
crucial events becomes constant in the long time limit. This transition has
been widely discussed in literature. See, for example, barkai1 ; barkai2 ;
barkai3 ; barkai4 ; barkai5 . The transition from $\mu<3$ to $\mu>3$
corresponds to the transition from the regime characterized by Lévy statistics
to the regime characterized by Poisson statistics. It is an important
transition reminiscent of phase transition processes annunziato .

The main goal of this manuscript is to study the interaction between non-
stationarity of first kind and non-stationarity of second kind.

The outline of this paper is as follows. We devote Section II to discuss the
physical meaning of $\mu=2$ and $\mu=3$. In Section III we illustrate a model
of aftershock that we use in this paper to realize the non-stationarity of
second kind and in Section IV we make a statistical analysis of the
interaction between the two forms on non-stationarity. We devote Section V to
concluding remarks emphasizing that the results of this paper should be an
incentive to explore the still poorly understood terrestrial underground
dynamics.

##  II Review of the special properties of the condition $\mu=2$ and $\mu=3$

###  II.1 From $\mu<2$ to $\mu>2$

There exist two main proposals to describe the waiting time distribution
density with $\mu<2$. The first is the waiting time distribution density
$\psi(t)$ corresponding to the survival probability of Eq. (7). The second is
the Mittag-Leffler survival probability $\Psi_{ML}(t)$ defined by its Laplace
transform

$\hat{\Psi}_{ML}(s)=\frac{1}{s+\lambda^{\alpha}s^{1-\alpha}},$ (13)

with $0<\alpha<2$

The corresponding waiting time distribution density reads

$\hat{\psi}_{ML}(s)=\frac{1}{1+\left(\frac{s}{\lambda}\right)^{\alpha}}.$ (14)

Note that the Laplace transform of $\psi(t)$ of Eq. (9) for $s\rightarrow 0$
is:

$\hat{\psi}(s)=1-\Gamma(1-\alpha)(Ts)^{\alpha}+.....,$ (15)

which for $s\rightarrow 0$ is indistinguishable from Eq. (14) if

$T=\frac{1}{\lambda}\frac{1}{\Gamma(1-\alpha)^{1/2}}.$ (16)

Moving to $\alpha>1$ the ML becomes a correlation function with values non-
necessarily positive, while Manneville keeps the survival probability
condition.

If we limit our attention to $\Psi(t)$ of Eq. (7) and to corresponding waiting
time distribution density of Eq. (9) we are apparently led to believe that
this is a transition from perennial aging of Eq. (10) to the condition of Eq.
(11), where non-stationary of first kind is temporary. However, the main focus
of this paper is the region $2<\mu<3$, where the correlation function becomes
stationary in the long-time limit. However, despite the recovery of
stationarity in the long-time limit, this regime is characterized by
deviations from ordinary statistical physics, including multifractality
tenyearsearlier , ergodicty breaking and infinite density hanngy ; metzler .

In this paper we find an additional anomalous property of the region
$2<\mu<3$. In fact, the property $R(t)$ defined by Eq. (11) and Eq. (12) for
$\mu$ tending to the singularity of $\mu=3$ is characterized by an important
property. Eq.(11) is an approximated expression of the exact expression

$R(t)=\sum_{n=1}^{\infty}\psi_{n}(t),$ (17)

where $\psi_{n}(t)$ is the probability that the $n-th$ crucial event following
the first event occurring at time $t=0$, corresponding to
$\psi_{0}(t)=\delta(t)$, occurs at time $t>0$.

It is important to note that crucial events are renewal events, whose
occurrence at a given time $t$ does not have any memory of the times of
occurrence of the earlier crucial events, with the nature of those crucial
events being taken into account by:

$\psi_{n}(t)=\int_{0}^{t}dt^{\prime}\psi_{1}(t^{\prime})\psi_{n-1}(t-t^{\prime}).$
(18)

Using the property that the Laplace transform of a convolution between two
time function is the product of the Laplace transform of the two functions, we
get

$\hat{R}(s)=\sum_{n=1}^{\infty}\hat{\psi}_{n}(s)=\sum_{n=0}^{\infty}\hat{\psi}_{n}(s)-1=\frac{\hat{\psi}(s)}{1-\hat{\psi}(s)},$
(19)

leading to

$\hat{R}(s)=\frac{\hat{\psi}(s)}{1-\hat{\psi}(s)}.$ (20)

Note that $\psi_{0}(t)=\delta(t)$ and $\psi_{1}(t)=\psi(t)$.

Using the method illustrated in Appendix A to bypass the singularity of the
Gamma function at $\alpha=1$, we get:

$R(t)\approx\frac{1}{ln(\frac{t}{T})}.$ (21)

This implies $R(t)$ does not become constant but undergoes an extremely slow
decay.

###  II.2 From $\mu<3$ to $\mu>3$

There exists another important effect that has been pointed out in Ref.
annunziato . Let us assume that the time intervals generated by waiting time
distribution density of Eq. (9) are filled with either $W$ or $-W$ by tossing
a fair coin. We generate a time series $\xi(t)$ and we use the time series to
create the diffusion process $x(t)$, according to the prescription

$\dot{x}=\xi(t).$ (22)

The Fourier transform of the probability distribution density $p(x,t)$,
denoted by the symbol $\hat{p}(k,t)$ is described by the Lévy prescription

$\hat{p}(k,t)=e^{-b|k|^{\alpha}t},$ (23)

where

$\alpha=\mu-1$ (24)

and

$b=W\left[(\mu-2)TW)^{\mu-2}sin(\frac{\pi(\mu-2)}{2})\right]\Gamma(3-\mu).$
(25)

We see that if we approach $\mu=3$ from the side of $\mu<3$, there a
singularity generated by the Gamma function.

Let us now consider the condition $\mu>3$. In this case the mean waiting time
given by the waiting time distribution density of Eq. (9) if finite and it is
given by

$<t>=\frac{T}{(\mu-2)}.$ (26)

However, also for $\mu>3$ there are signs of a phase transition generating
divergences. This was discussed in Ref. annunziato  studying the effect of
friction through equation

$\dot{x}=-\lambda x+\xi(t),$ (27)

which for $\lambda=0$ recovers Eq. (22). In this case we get an equilibrium
distribution with variance

$\sigma^{2}_{\lambda}=\frac{(\mu-2)W^{2}T}{\lambda(\mu-3)}.$ (28)

Notice according to the Appendix,

$R(t)=1+\frac{T}{t}.$ (29)

This is a very slow transition to the condition of constant rate of events.
However, according to Eq. (11) the condition corresponding to $\mu=2.5$
generates a regression to constant rate even slower than $1/2$. Therefore, we
conclude that the singularity condition generating the result of this paper,
concerning the detection of crucial events, perturbed by the Omori cascade is
due to the variance singularity of Eq. (28), which makes it difficult to make
a distinction between crucial events and non-stationarity of second kind.

##  III A model for earthquake cascade generated by crucial events

For our model we combine the methods of generating aftershock sequences,
assigning an intensity (magnitude) to them, and then spacing both the
aftershocks and main shocks by their own unique time distances generated by
non-stationary waiting time distributions. For the aftershock generation
portion of the model we choose to use the equation from param

$\lambda(t,M,M_{m})=\frac{10^{a(M_{m})-b(M_{m})*M}}{(t+c)^{p(M_{m})}},$ (30)

which is based on the original equation of gutenomori .

This equation combines the aftershock rate given by the Omori Law omori , utsu
and the exponential distribution of aftershock intensities given by the
Gutenberg-Richter Law gutenberg . The parameters $a{(M_{m})}$, $b(M_{m})$, and
$p(M_{m})$ are equations that can be fitted from real data. The parameter t
represents the time after a main shock occurs, M is an earthquake magnitude
lower than that of the initial main shock, and $M_{m}$ is the intensity of the
main shock that caused the aftershock sequence. This portion gives us
information on how many aftershocks of intensity M should occur after a main
shock and how they should be distributed. For our analysis we limit the
magnitude of a main shock $M_{m}$ to be between the values of 6 and 9. We use
the fittings from the authors of param  in order to get values for the
equations in the model. These parameters are given by,

$\begin{matrix}a(M_{m})=(0.58+.12M_{o})*M_{m}-(2.34+.063*M_{o})\\\
b(M_{m})=0.12*M_{m}-0.063\\\ p(M_{m})=-0.06*M_{m}+1.44,\end{matrix}$ (31)

where these linear equations are fitted from data from their analysis of real
earthquake data with $M_{o}$ representing a lower threshold where a shock
intensity is no longer considered an aftershock. For our analysis we use the
value of 0.1 for $M_{o}$ and keep the value of c fixed at $10^{4}$. We
describe the time distances between aftershocks with a non-stationary Poisson
process, noting that our rate $\lambda$ changes in time due to the
environment. After we substitute the fittings into equation (30) we can create
multiple of these avalanches and space them by a time $\tau$, which is derived
from our waiting time distribution density of eq. 9. We get our times by
picking a random number y, which is randomly generated between 0 and 1, and
use the equation

$\tau=T*(\frac{1}{y^{\frac{1}{\mu-1}}}-1)$ (32)

to generate our waiting times. We can create as many of these waiting times as
needed in order to space the amount of main shocks we choose to generate.
Additionally we can adjust the value of T to space them so that the aftershock
sequences can overlap or not (as shown in FIG. 1), allowing us to create a
dense combination of non-stationary processes or allow the processes to remain
separated to an extent. For this study we create 100 main shocks with randomly
generated magnitudes within our set range to have a sufficient amount for our
analysis. Additionally, we vary the value of $\mu$ for both methods of
creating our sequences.

Figure 1: representation of series with overlapping and non-overlapping
aftershocks of the model used.

##  IV Interaction between non-stationarity of the first kind and non-
stationarity of the second kind

We analyze the series constructed by our model in both the case of overlapping
and non-overlapping Omori cascades. Both cases have the effect of hiding the
occurrence of main shocks, representing non-stationarity of first kind.
However, with non-overlapping cascades we find that the detection of events
can be recovered for regions away from the borders of statistical phase
changes. To discover the ”invisible” crucial events we adopt the technique of
Diffusion Entropy Analysis (DEA) with stripes dea . Since this method was
created to detect potentially invisible crucial events from real data sets
that may contain many other non-crucial events. This analysis takes advantage
of creating many diffusion trajectories through use of moving a mobile window
of size l and forming the probability density function

$p(x,l)=\frac{1}{l^{\delta}}F(\frac{x}{l^{\delta}})$ (33)

from data and evaluating the Shannon Entropy given by

$S(l)=\int_{-\infty}^{\infty}p(x,l)\ln{p(x,l)}dx.$ (34)

Integration of this equation yields

$S(l)=A+\delta\ln{(l)},$ (35)

where A is a constant and we get our anomalous scaling $\delta$ which is
fitted with a straight line in a log-log representation. We choose to use 11
stripes for our analysis with DEA on all of our series generated by the model.
Since we focus on the region $2<\mu<3$ for the analysis of our model we use
the scaling relation

$\delta=\frac{1}{\mu-1}$ (36)

to relate the scaling $\delta$ given by the DEA to the value of $\mu$ we
select for our model. In our analysis we select an initial $\mu$ so that we
know what scaling the DEA should recover when searching for crucial events.
This is the same value of $\mu$ that we use for spacing the main shocks in
equation (32) in our model. For the case of overlapping sequences of
aftershocks we find that mixing both kinds of non-stationarity in this way
causes the analysis to give an inaccurate scaling regardless of the value of
$\mu$. This dense mixing of non-stationarity creates a process that causes
errors in recovering the proper scaling. While the DEA should give the scaling
due to the presence of crucial events we find that a strong introduction of
other forms of non-stationarity can create a process that hides the true
scaling of the series. This discrepancy is shown in FIG. 2, which shows the
scaling found when analyzing overlapping Omori sequences have unexpected
values. These values diverge from those that should be found given our
relation in equation (36).

Figure 2: Analysis of various initial values of $\mu$ for overlapping
sequences of aftershocks. Titles include the initial value of $\mu$ and the
legend provides the resultant scaling in a log-log plot. All of these produce
a scaling inconsistent with the initial $\mu$ that should be reproduced
through the analysis.

However, in the case of Omori sequences that are not allowed to overlap we
find that we can recover the correct scaling for values of $\mu$ that are not
close to the borders of our region of study. While the non-stationary elements
are still both present, the intensity of the interaction is significantly
weaker in this condition. This is shown in our Fig 3 for various values of
$\mu$ where we see a perfect agreement at $\mu=2.5$ and the divergences at
values of $\mu$ close to the borders.

This effect can be explained by noticing that both $\mu=2$ and $\mu=3$ are
singularity conditions preventing the detection of crucial events in the
presence of the perturbing action of Omori cascades, not only in the
overlapping case but also in the non-overlapping one, if $\mu$ is not
sufficiently far away from $\mu=2$ and $\mu=3$.

In the case $\mu=2$ we must recall the direction of raffaelli . If we turn
crucial events into a walk generating scaling $\delta$, the value of the
scaling depends on the rule adopted to converting crucial events into a walk.
In the case when the walker makes a step ahead of constant intensity when a
crucial event occurs and it does not move if a crucial event does not occur,
then for $\mu<2$

$\delta=\frac{\mu-1}{2}.$ (37)

If $2<\mu<3$ we must use Eq. (36). We notice that the condition $\mu=2$ is a
kind of singularity that, in the presence of the perturbation exerted by the
occurrence of an Omori cascade, is very difficult to establish. In fact, in
the long time limit we should find a distance from two consecutive crucial
events leading to the correct value of $\mu$. The presence of events generated
by an Omori cascade makes it impossible to find the correct value.

In the appendix A we prove that the same additional term is present also for
$\mu>2$. This makes it difficult to establish the value at which the
transition from the Lévy to the Gauss regime occurs. Actually, as proved in
annunziato , a sort of phase transition occurs at $\mu=3$, forcing the width
of the distribution density to generate a singularity. See Fig. 3 of Ref.
annunziato . Under the presence of an even weaker perturbation this
singularity is replaced by a smooth transition. The Omori cascade exerts the
same role as a weak perturbation making it impossible to detect crucial
events, signalled by the slow tails of the Lévy distribution density.

This effect is in agreement with the results found in annunziato  at the
border $\mu=3$ and showing the discrepancy of theoretical prediction and
numerical results in the region. We can also refer back to our discussion of
section II and see that our results match the observance of divergences at the
borders of $\mu=2$ and $\mu=3$ as shown in equation (11). The discrepancy at
the borders can also be characterized by a phase change as well since at
$\mu=3$ there is a shift from Le _v_ y statistics to Gaussian as well as
moving to a region characterized by perennial aging beyond $\mu=2$. Ultimately
we find that the strength of non-stationary interaction causes a shift in the
detection of crucial events. This interaction strength along with the
amplified effects of divergences at the borders show the optimal values of
recovering correct information in areas defined by Lévy statistics.

Figure 3: Analysis of various initial values of $\mu$ for non-overlapping
sequences of aftershocks, making note of the perfect agreement for the value
of $\mu=2.5$. Titles include the initial value of $\mu$ and the legend
provides the resultant scaling in a log-log plot.

##  V concluding remarks

###  V.1 Theoretical problems with the Omori law

The surprising result of this paper is illustrated in Fig.( 3) which shows
that the adoption of DEA with stripes applied to the case of main shocks with
$\mu=2.5$ yields $\delta=0.66$ which is the prediction of Eq. (36). In this
case the detection of crucial events leads to an extremely accurate result.
One may make the conjecture that the distance of $\mu=2.5$ from the
singularity of the state $\mu=2$ is large enough as to overcome the
singularity effects of the phase transition at $\mu=2$. However, we find that
as we approach the value $\mu=3$ this accuracy is lost and the value
$\delta=0.7$ that we find at $\mu=3$ should correspond to a value of
$\mu=2.43$ according to Eq. (36). This surprising result is explained by the
arguments of Section (II.2). In fact, in this section we find the quite
surprising result of Eq. (29) which shows that at $\mu=3$ the time necessary
to reach the stationary condition is infinitely large, because $1/t$ is at the
border of integrability. These discrepancies can be caused by the assumption
that an Omori cascade may last for an extended time period. While not long
enough to reach a stationary condition, the long time correlation causes
effects that result in the inability to properly detect the correct scaling of
the sequence.

###  V.2 Shedding light into a long lasting controversy

To make easier for the readers to understand the importance of the conclusions
of this paper we invite them to read Ref. comment  and reply . In the model of
comment  the authors found the same scaling value of $\delta=0.9$ of mega
without making the assumption that the main shocks are crucial events. The
authors of reply  proved that this interesting result requires the action of
an Omori cascade extremely extended in time, thereby involving again the
action of a non-stationary process of second kind. We note that $\mu=2.1$ is a
property shared by the neurophysiology of healthy brains 2009 . On one side,
we find it difficult to support the universality of non-stationary processes
of the second kind to explain the observation of $\mu=2.1$ in the dynamics of
the underground, see mega  supplemented by the discussion of this paper on the
interaction between non-stationary processes of first kind and non-stationary
processes of the second kind. We also note that, in paper1 , if one does not
focus on a non-stationary process of the second kind and instead on an
interaction between the faults, one can achieve a scaling value closer to that
found in real earthquake data. On the other side, having in mind also Ref.
2009 , we are inclined to believe that $\mu=2.1$ is the manifestation of self-
organized transitions to criticality, ranging from neurophysiology to fault
interaction as originally proposed by Turcotte Turcotte . A rigorous proof of
this property that may be a remarkable contribution to current research work
on the origin of ”intelligence” would imply that the connection between main
shocks and aftershocks is not yet understood, in agreement with the view of
professional geophysicist poorlyunderstood .

##  Appendix A Rate event function

In this appendix we study the asymptotic behavior of the rate event function
$R(t)$ in the particular case of the critical values of the power law
$\mu=2,3$ corresponding to the appearance of the first and second finite
moment. The Laplace transform of $R(t)$ is

$\hat{R}(s)=\frac{\hat{\psi}(s)}{1-\hat{\psi}(s)}$ (38)

with $\hat{\psi}(s)$ Laplace transform of $\psi(t)$. We consider the
convenient general expression in terms of $s$ power given by mauro

$\hat{\psi}(s)=-\Gamma(2-\mu)(sT)^{\mu-1}\left(\exp[sT]-E_{\mu-1}^{sT}\right)$
(39)

where

$E_{\alpha}^{sT}=\sum_{n=0}^{\infty}\frac{(sT)^{n-\alpha}}{\Gamma(n+1-\alpha)}$
(40)

Let us consider $\mu=2$ Eq. (39) is indeterminate. Using Taylor expansion for
$\mu=2$ and taking into account that

$\displaystyle\lim_{\alpha\to
0}\frac{\Gamma^{\prime}(\alpha)}{\Gamma(\alpha)^{2}}=-1,\,\,\,\,\Gamma(2-\mu)\approx\frac{1}{2-\mu},$
(41)

we have

$\displaystyle
E_{\mu-1}^{sT}\\!\approx\\!E_{1}^{sT}\\!+\\!\frac{\partial}{\partial\mu}E_{\mu-1}^{sT}\mid_{\mu=2}\\!(\mu-2)\\!=\\!\exp[sT]\\!+\\!\frac{\partial}{\partial\mu}E_{\mu-1}^{sT}\mid_{\mu=2}\\!(\mu-2)$
$\displaystyle=\exp[sT]+\left(\sum_{n=0}^{\infty}\frac{(sT)^{n-1}}{\Gamma(n)}\left[\frac{\Gamma^{\prime}(n)}{\Gamma(n)}-\log(sT)\right]\right)(\mu-2)=$
$\displaystyle\exp[sT]-\frac{1}{s}+\left(\sum_{n=0}^{\infty}\frac{(sT)^{n}}{\Gamma(n+1)}\left[\frac{\Gamma^{\prime}(n+1)}{\Gamma(n+1)}-\log(sT)\right]\right)(\mu-2)$
(42)

Taking the limit for $\mu\to 2$ in Eq. (39) and keeping the first two terms of
the series we have

$\hat{\psi}(s)\approx
1+sT\left[\log(sT)+\gamma\right],\,\,\,\,\gamma\equiv-\Gamma^{\prime}(1)$ (43)

Consequently, in the limit $s\to 0$, we have

$\hat{R}(s)\approx\frac{\hat{\psi}(s)}{1-\hat{\psi}(s)}\approx-\frac{1}{sT(\log(sT)+\gamma)}\approx\frac{1}{sT\log\left(\frac{1}{sT}\right)}$
(44)

Applying the tauberian theorem weiss  in general we have

$\displaystyle\hat{f}(s)=\frac{L\left(\frac{1}{s}\right)}{s^{\alpha}}\Rightarrow
f(t)=\frac{d}{dt}\left[\frac{t^{\alpha}L\left(t\right)}{\Gamma(\alpha+1)}\right]$
(45)

with $L(t)$ a slow function. Using the result for $R(t)$ we have that

$R(t)\approx\frac{d}{dt}\left(\frac{t}{\log
t}\right)=\frac{1}{T\log\left(\frac{t}{T}\right)}-\frac{1}{T\log^{2}\left(\frac{t}{T}\right)}\approx\frac{1}{T\log\left(\frac{t}{T}\right)}$
(46)

Analogously, for $\mu=3$, we have (for $s\to 0$)

$\hat{\psi}(s)\approx 1-sT-(sT)^{2}(\log(sT)+\gamma)$ (47)

and for the rate event

$\hat{R}(s)\approx\frac{1}{sT}-\log(sT)=\frac{1}{sT}+\log\left(\frac{1}{sT}\right)$
(48)

Applying the tauberian theorem

$R(t)\approx\frac{1}{T}\left(1+\frac{T}{t}\right)$ (49)

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