$\mathbb{E}\,\textrm{qs}_{\alpha_{0}}(X,Y)\geq\mathbb{E}\,\textrm{qs}_{\alpha_{0}}(F^{-1}(\alpha_{0}),Y)$
since
$\mathbb{E}\,\textrm{qs}_{\alpha_{n}}(X,Y)\geq\mathbb{E}\,\textrm{qs}_{\alpha_{n}}(F^{-1}(\alpha_{n}),Y)$
for all $n\in\mathbb{N}$. This shows that
$q_{\alpha}\big{(}Y\mid\mathscr{L}(F^{-1}(\alpha))\big{)}$ is an
$\alpha$-quantile of $F$ for $\alpha\in(0,1)$. By construction in Section 6 of
Arnold and Ziegel (2023),
$q_{\alpha}\big{(}Y\mid\mathscr{L}(F^{-1}(\alpha))\big{)}$ is the smallest
possible minimizer of $\mathbb{E}\,\mathrm{qs}_{\alpha}(X,Y)$, so it coincides
with $F^{-1}(\alpha)$ for all $\alpha\in(0,1)$ and, hence, $N=\emptyset$.
Clearly, ${\rm DSC}_{\rm QS}=0$ if
$q_{\alpha}\big{(}Y\mid\mathscr{L}(F^{-1}(\alpha))\big{)}=q_{\alpha}(Y)$ for
$\alpha\in(0,1)$. Conversely, if ${\rm DSC}_{\rm QS}=0$ then
$q_{\alpha}\big{(}Y\mid\mathscr{L}(F^{-1}(\alpha))\big{)}=q_{\alpha}(Y)$ for
$\alpha\in(0,1)$. ∎

###### Proof of Corollary 4.6.

For any $z\in\mathbb{R}$, $P_{Y\mid F}(\cdot,(z,\infty))$ minimizes
$\mathbb{E}(p-\mathbbm{1}\\{Y>z\\})^{2}$ over all $\sigma(F)$-measurable
random variables $p$, and hence, also over all $\mathscr{L}(F)$-measurable
random variables since any $\mathscr{L}(F)$-measurable random variable is also
$\sigma(F)$-measurable, see Arnold and Ziegel (2023, Lemma 3.1). Thus, we
apply Fubini to derive

$\displaystyle\mathbb{E}\,{\rm crps}(P_{Y\mid F},Y)$
$\displaystyle=\int\mathbb{E}\,(P_{Y\mid
F}(\cdot,(z,\infty))-\mathbbm{1}\\{Y>z\\})^{2}\,\mathrm{d}z$
$\displaystyle\leq\int\mathbb{E}\,(P_{Y\mid\mathscr{L}(F)}(\cdot,(z,\infty))-\mathbbm{1}\\{Y>z\\})^{2}\,\mathrm{d}z=\mathbb{E}\,{\rm
crps}(P_{Y\mid\mathscr{L}(F)},Y),$

which implies ${\rm MCB}_{\rm CT}\geq{\rm MCB}_{\rm ISO}$. Moreover, for any
$z\in\mathbb{R}$ we know that
$\mathscr{L}(F(z))\subseteq\overline{\mathscr{L}(F)}$, where for any
$\sigma$-lattice $\mathcal{A}\subseteq\mathcal{F}$, $\bar{\mathcal{A}}$
denotes the $\sigma$-lattice which consists of all complements of elements in
$\mathcal{A}$. Hence, we may argue similarly that

$\displaystyle\mathbb{E}\,{\rm crps}(P_{Y\mid\mathscr{L}(F)},Y)$
$\displaystyle=\int\mathbb{E}(1-P_{Y\mid\mathscr{L}(F)}(\cdot,(z,\infty))-\mathbbm{1}\\{Y\leq
z\\})^{2}\,\mathrm{d}z$ $\displaystyle\leq\int\mathbb{E}(\mathbb{P}(Y\leq
z\mid\mathscr{L}(F(z)))-\mathbbm{1}\\{Y\leq z\\})^{2}\,\mathrm{d}z,$

which implies ${\rm MCB}_{\rm ISO}\geq{\rm MCB}_{\rm BS}$. Finally for any
$\alpha\in(0,1)$, we have that $P_{Y\mid\mathscr{L}(F)}^{-1}(\alpha)$
minimizes $\mathbb{E}\,\mathrm{qs}_{\alpha}(X,Y)$ over all
$\mathscr{L}(F)$-measurable random variables $X$. We use that
$\mathscr{L}(F^{-1}(\alpha))\subseteq\mathscr{L}(F)$, to derive that

$\displaystyle\mathbb{E}\,{\rm
crps}(P_{Y\mid\mathscr{L}(F)},Y)=\int_{0}^{1}\mathbb{E}\,\mathrm{qs}_{\alpha}(P_{Y\mid\mathscr{L}(F)}^{-1}(\alpha),Y)\,\mathrm{d}\alpha\leq\int_{0}^{1}\mathbb{E}\,\mathrm{qs}_{\alpha}(q_{\alpha}(Y\mid\mathscr{L}(F^{-1}(\alpha)),Y)\,\mathrm{d}\alpha$

and hence ${\rm MCB}_{\rm ISO}\geq{\rm MCB}_{\rm QS}$. ∎

###### Proof of Proposition 4.7.

The claim in part (a) follows from the definition of MS at (41). For part (b),
suppose that $F$ is auto-calibrated. Then $Y\in{\rm supp}(F)$ almost surely
and hence ${\rm MS}=0$ by part (a). The tower property implies for any
$A\in\mathcal{B}(0,1)$ that

$\displaystyle\tau(A)$
$\displaystyle=\mathbb{E}\left(\mathbb{E}\left(\int_{A}\mathbbm{1}\\{F(Y)\leq
p\\}\,\mathrm{d}\nu_{F}(p)\hskip 2.84526pt\Big{|}\hskip
2.84526ptF\right)\right)$
$\displaystyle=\mathbb{E}\left(\int_{A}\mathbb{E}\left(\mathbbm{1}\\{F(Y)\leq
p\\}\mid F\right)\,\mathrm{d}\nu_{F}(p)\right)$
$\displaystyle=\mathbb{E}\left(\int_{A}F(F^{-1}(p))\,\mathrm{d}\nu_{F}(p)\right),$

where the last equality follows since if $Y\in{\rm supp}(F)$, then $F(Y)\leq
p$ if and only if $Y\leq F^{-1}(p)$ and $\mathbb{P}(Y\leq F^{-1}(p)\mid
F)=F(F^{-1}(p))$ by auto-calibration. By the properties of generalized
inverses (Embrechts and Hofert, 2013), we have $F(F^{-1}(p))\geq p$ for all
$p\in(0,1)$. However, if $F(F^{-1}(p))>p$ for all $p\in B$ in some
$B\in\mathcal{B}(0,1)$, then $F^{-1}(B)=\\{x\in\mathbb{R}\mid F(x)\in
B\\}=\emptyset$ and hence $\nu_{F}(B)=0$ almost surely. That is,
$\nu_{F}(\\{p\in(0,1):F(F^{-1}(p))>p\\}=0$ almost surely and thus

$\displaystyle\tau(A)=\mathbb{E}\left(\int_{A}F(F^{-1}(p))\,\mathrm{d}\nu_{F}(p)\right)=\mathbb{E}\left(\int_{A}p\,\mathrm{d}\nu_{F}(p)\right)=\int_{A}p\,\mathrm{d}\mu(p).$

We conclude that $f(p)=p$ $\mu$-almost surely and hence ${\rm MCB}_{\rm
HB}=0$.

The condition in part (c) is equivalent to assuming that
$\frac{\,\mathrm{d}}{\,\mathrm{d}p}F^{-1}$ is almost surely constant for all
$p\in(0,1)$. Since $F$ is probabilistically calibrated, we have for any
$p\in(0,1)$,

$\displaystyle f(p)=\frac{1}{\gamma(p)}\mathbb{E}\left(\mathbbm{1}\\{F(Y)\leq
p\\}\frac{\,\mathrm{d}}{\,\mathrm{d}p}F^{-1}(p)\right)=\frac{\gamma(p)}{\gamma(p)}\mathbb{E}\left(\mathbbm{1}\\{F(Y)\leq
p\\}\right)=\mathbb{P}(F(Y)\leq p)=p$

and hence ${\rm MCB}_{\rm HB}=0$. ∎

##  Appendix E Analytic examples at the population level

In this section we compare the population level decompositions from Section 4
in a number of examples in the prediction space setting. Table 2 collects and
summarizes the analytic forms of the decomposition components in these
examples. Assumption 4.1 is satisfied throughout.

Table 2: Analytic form of the various different types of decomposition in population level examples E.1, …, E.5. For details and supporting calculations see the text.  Example | E.1 | E.2 | E.3 | E.4 | E.5
---|---|---|---|---|---
$\mathbb{E}\,{\rm crps}(F,Y)$ | $\sum_{i=1}^{n}w_{i}\frac{\sigma_{i}}{\sqrt{\pi}}$ | $\frac{1}{6}$ | 1 | $\frac{39}{80}$ | $\frac{5}{24}t$
${\rm UNC}_{0}$ | $\frac{1}{2}\sum_{i,j=1}^{n}w_{i}w_{j}\,A(\mu_{i}-\mu_{j},\sigma_{i}^{2}+\sigma_{j}^{2})$ | $\frac{2}{5}$ | $\frac{3}{4}$ | $\frac{3}{2}$ | $\frac{2}{9}t$
${\rm MCB}_{\rm CT}$ | 0 | $\frac{1}{30}$ | 1 | $\frac{7}{400}$ | $\frac{3}{200}t$
${\rm MCB}_{\rm ISO}$ | 0 | $\frac{1}{30}$ | 1 | $\frac{9}{2800}$ | $\frac{3}{200}t_{2}$
${\rm MCB}_{\rm QS}$ | 0 | $\frac{1}{30}$ | $\frac{13}{16}$ | $\frac{9}{2800}$ | 0
${\rm MCB}_{\rm BS}$ | 0 | $\frac{1}{30}$ | $\frac{1}{2}$ | $\frac{9}{2800}$ | 0
${\rm MCB}_{\rm HB}$ | 0 | 0 | $\frac{1}{8}$ | $\frac{1}{1600}$ | 0

###  E.1 Auto-calibrated Gaussian

In this example, the predictive distribution $F$ is Gaussian with mean
$\mu_{i}$ and standard deviation $\sigma_{i}>0$ with probability $w_{i}$ for
$i=1,\dots,n$, where $w_{i}+\dots+w_{n}=1$. Conditionally on $F$, the outcome
$Y$ has distribution $F$, so $F$ is auto-calibrated. We conclude that

$\displaystyle{\rm MCB}_{\rm CT}={\rm MCB}_{\rm ISO}={\rm MCB}_{\rm BS}={\rm
MCB}_{\rm QS}=0.$

Since auto-calibration implies probabilistic calibration, Proposition 4.7
yields ${\rm MCB}_{\rm HB}={\rm MS}_{\rm HB}=0$. Finally, we apply formulas in
Grimit et al. (2006) to obtain

$\displaystyle\mathbb{E}\,{\rm
crps}(F,Y)=\sum_{i=1}^{n}w_{i}\frac{\sigma_{i}}{\sqrt{\pi}}\quad\textrm{and}\quad{\rm
UNC}_{0}=\frac{1}{2}\sum_{i,j=1}^{n}w_{i}w_{j}A(\mu_{i}-\mu_{j},\sigma_{i}^{2}+\sigma_{j}^{2}),$

where
$A(\mu,\sigma^{2})=2\sigma\varphi(\frac{\mu}{\sigma})+\mu(2\Phi(\frac{\mu}{\sigma})-1)$,
with $\varphi$ and $\Phi$ denoting the density and the cdf of the standard
normal distribution, respectively.

###  E.2 Example in Candille and Talagrand (2005)

In this example of Candille and Talagrand (2005, p. 2145), the forecast $F$ is
$F_{1}$, which is uniform on $(-1,0)$, or $F_{2}$, which is uniform on
$(0,1)$, with equal probability. Given $F=F_{1}$, the conditional cdf of $Y$
is $Q_{1}(z)=1-z^{2}$ for $z\in(-1,0)$, and given $F=F_{2}$, the conditional
cdf of $Y$ is $Q_{2}(z)=z^{2}$ for $z\in(0,1)$.

For $i=1,2$, we denote by $G_{i}$ the isotonic conditional law of $Y$ given
$F=F_{i}$. Since $F_{1}\leq_{\textrm{st}}F_{2}$ and
$Q_{1}\leq_{\textrm{st}}Q_{2}$ it follows that $Q_{i}=G_{i}$ for $i=1,2$ and
the isotonicity-based decomposition coincides with the Candille–Talagrand
decomposition. For any $z\in(-1,1)$, $F_{1}(z)$ and $F_{2}(z)$ strictly order
and hence the random variable $F(z)$ already reveals the value of $F$. That
is, $\sigma(F(z))=\sigma(F)$ and hence $\mathbb{P}(Y\leq z\mid
F(z))=\mathbb{P}(Y\leq z\mid F)=P_{Y\mid F}(z)$. Since this conditional
probability is already an increasing function of $F(z)$, we may conclude by
Proposition 3.2. in Arnold and Ziegel (2023) that $\mathbb{P}(Y\leq
z\mid\mathscr{L}(F(z)))=P_{Y\mid F}(z)$ for all $z\in\mathbb{R}$ and hence the
Brier score based decomposition correspond with the Candille–Talagrand
decomposition. Analogously the claim can be shown for the quantile score based
decomposition. Thus the isotonicity-based, Brier score based, and quantile
score based decompositions coincide with the Candille–Talagrand decomposition,
where $\mathbb{E}\,{\rm crps}(F,Y)=1/6$, ${\rm MCB}_{\rm CT}=1/30$, and ${\rm
UNC}_{0}=2/5$.

The forecasts satisfy the conditions in part (c) of Proposition 4.7, therefore
${\rm MCB}_{\rm HB}=0$. Since $Y\in{\rm supp}(F)$ almost surely, we have ${\rm
MS}=0$.

###  E.3 Example with two atoms

This simple example illustrates that the Brier score and quantile score based
decompositions do not coincide in general, that the corresponding calibration
methods do not necessarily produce valid cdfs or quantile functions,
respectively, and that ${\rm DSC}_{\rm HB}$ can be negative.

Consider the distributions $F_{1}=(\delta_{1}+\delta_{2})/2$ and
$F_{2}=(\delta_{0}+\delta_{3})/2$, where $\delta_{z}$ denotes the Dirac
measure at $z\in\mathbb{R}$. Assume that $F$ is $F_{1}$ and $F_{2}$ with equal
probability and that $Y=y_{1}$ if $F=F_{1}$ and $Y=y_{2}$ if $F=F_{2}$. Let
$y_{1}=3$ and $y_{2}=0$, so the marginal law $F_{\rm mg}$ of $Y$ is $F_{2}$.
We readily compute $\mathbb{E}\,{\rm crps}(F,Y)=1$ and $\mathbb{E}\,{\rm
crps}(F_{\rm mg},Y)={\rm UNC}_{0}=3/4$.

An application of the PAV algorithm for the mean functional on
$(\mathbbm{1}\\{y_{1}\leq z\\},\mathbbm{1}\\{y_{2}\leq z\\})$ with respect to
the order induced by $(F_{1}(z),F_{2}(z))$ at threshold $z\in\mathbb{R}$
results in

$\displaystyle\textstyle{\acute{F}}_{1}(z)=\frac{1}{2}\mathbbm{1}_{[1,3)}(z)+\mathbbm{1}_{[3,\infty)}(z)\quad\textrm{and
}\quad{\acute{F}}_{2}(z)=\mathbbm{1}_{[0,1)}(z)+\frac{1}{2}\mathbbm{1}_{[1,3)}(z)+\mathbbm{1}_{[3,\infty)}(z),$

and we see that ${\acute{F}}_{2}$ fails to be increasing. Similarly, an
application of the PAV algorithm for the $\alpha$-quantile on $(y_{1},y_{2})$
with respect to the order induced by $(F_{1}^{-1}(\alpha),F_{2}^{-1}(\alpha))$
at level $\alpha\in(0,1)$ results in

$\displaystyle\textstyle{\grave{F}}^{-1}_{1}(\alpha)=3\quad\textrm{and
}\quad{\grave{F}}^{-1}_{2}(\alpha)=3\mathbbm{1}_{(\frac{1}{2},1]}(\alpha),$

so ${\grave{F}}^{-1}_{2}$ fails to be increasing. Furthermore, it follows
easily that ${\rm MCB}_{\rm BS}=1/2\neq 13/16={\rm MCB}_{\rm QS}$. As the
conditional law of $Y$ given $F$ is a Dirac measure, $\mathbb{E}\,{\rm
crps}(P_{Y\mid F},Y)=0$ and ${\rm MCB}_{\rm CT}=1$. Similarly, ${\rm MCB}_{\rm
ISO}=1$ since $F_{1}$ and $F_{2}$ do not order.

According to the formulas in Section 2.5, $\bar{g}_{1}=2$ and
$\bar{f}_{1}=(\mathbbm{1}\\{F_{1}(y_{1})\leq\frac{1}{2}\\}+3\mathbbm{1}\\{F_{2}(y_{2})\leq
1/2\\})/(2\bar{g}_{1})=3/4$ and thus ${\rm MCB}_{{\rm
HB}}=(p_{1}-\bar{f}_{1})^{2}\,\bar{g}_{1}=1/8$, whence we conclude that ${\rm
DSC}_{{\rm HB}}={\rm MCB}_{{\rm HB}}+{\rm UNC}_{0}-\mathbb{E}\,{\rm
crps}(F,Y)=-1/8$.

###  E.4 Example 2.4 a) in Gneiting and Resin (2023)

Let $F$ be a mixture of uniform distributions on $[0,1]$, $[1,2]$, and $[2,3]$
with weights $p_{1},p_{2}$, and $p_{3}$, respectively, and let $Y$ be drawn
from a mixture of these distributions with weights $q_{1},q_{2}$, and $q_{3}$,
respectively, where the tuple $(p_{1},p_{2},p_{3};q_{1},q_{2},q_{3})$ attains
each of the values

$\displaystyle\textstyle\left(\frac{1}{2},\frac{1}{4},\frac{1}{4};\frac{5}{10},\frac{1}{10},\frac{4}{10}\right),\quad\left(\frac{1}{4},\frac{1}{2},\frac{1}{4};\frac{1}{10},\frac{8}{10},\frac{1}{10}\right),\quad\left(\frac{1}{4},\frac{1}{4},\frac{1}{2};\frac{4}{10},\frac{1}{10},\frac{5}{10}\right)$

with equal probability. We note that $F$ is probabilistically calibrated, and
still we find that ${\rm MCB}_{\rm HB}\not=0$.

Let $F_{1},F_{2}$, and $F_{3}$ denote the distributions that $F$ attains. For
$i=1,2,3$, let $Q_{i}$ be the conditional law of $Y$ given $F=F_{i}$, and let
$G_{i}$ be the isotonic conditional law of $Y$ given $F=F_{i}$. The marginal
law $F_{\rm mg}$ of $Y$ is uniform on $[0,3]$ and, hence,

$\displaystyle\textstyle{\rm UNC}_{0}=\mathbb{E}\,{\rm crps}(F_{\rm mg},Y)$
$\displaystyle=\int\int(F_{\rm mg}(x)-\mathbbm{1}\\{y\leq
x\\})^{2}\,\mathrm{d}x\,\mathrm{d}F_{\rm mg}(y)$
$\displaystyle=\frac{1}{3}\int_{0}^{3}\int_{0}^{3}\left(\frac{x}{3}-\mathbbm{1}\\{y\leq
x\\}\right)^{2}\,\mathrm{d}x\,\mathrm{d}y=\frac{3}{2}.$

It holds that $F_{1}\leq_{\textrm{st}}F_{2}\leq_{\textrm{st}}F_{3}$ but only
$Q_{1}\leq_{\textrm{st}}Q_{3}$, hence $P_{Y\mid
F}\not=P_{Y\mid\mathscr{L}(F)}$. Let $r=10/7$, $s=11/7$. On $(-\infty,r]$, we
have the pointwise inequalities $Q_{2}\leq Q_{3}\leq Q_{1}$; on $[r,s]$, we
have $Q_{3}\leq Q_{2}\leq Q_{1}$; and on $[s,\infty)$, we have $Q_{3}\leq
Q_{1}\leq Q_{2}$. Consider the pooled cdfs $Q_{12}=(Q_{1}+Q_{2})/2$ and
$Q_{23}=(Q_{2}+Q_{3})/2$. The $G_{i}$’s may be derived by pooling the
$Q_{i}$’s according to the given order constraint
$G_{1}\leq_{\textrm{st}}G_{2}\leq_{\textrm{st}}G_{3}$, namely,

$\displaystyle
G_{1}(z)=Q_{1}(z)\mathbbm{1}_{(-\infty,s]}(z)+Q_{12}(z)\mathbbm{1}_{[s,\infty)}(z),$
$\displaystyle
G_{2}(z)=Q_{23}(z)\mathbbm{1}_{(-\infty,r]}(z)+Q_{2}(z)\mathbbm{1}_{[r,s]}(z)+Q_{12}(z)\mathbbm{1}_{[s,\infty)}(z),$
$\displaystyle
G_{3}(x)=Q_{23}(z)\mathbbm{1}_{(-\infty,r]}(x)+Q_{3}(z)\mathbbm{1}_{[r,\infty)}(z).$

By the law of total expectation and Fubini’s theorem,

$\displaystyle\textstyle\mathbb{E}\,{\rm crps}(F,Y)$
$\displaystyle=\frac{1}{3}\sum_{i=1}^{3}\mathbb{E}\,\big{(}{\rm crps}(F,Y)\mid
F=F_{i}\big{)}$
$\displaystyle=\frac{1}{3}\sum_{i=1}^{3}\int\int\big{(}F_{i}(x)-\mathbbm{1}\\{y\leq
x\\}\big{)}^{2}\,\mathrm{d}x\,\mathrm{d}Q_{i}(y)$
$\displaystyle=\frac{1}{3}\sum_{i=1}^{3}\int\int\big{(}F_{i}(x)-\mathbbm{1}\\{y\leq
x\\}\big{)}^{2}\,\mathrm{d}Q_{i}(y)\,\mathrm{d}x$
$\displaystyle=\frac{1}{3}\sum_{i=1}^{3}\int\left(F_{i}^{2}(x)-2F_{i}(x)Q_{i}(x)+Q_{i}(x)\right)\,\mathrm{d}x.$

Similarly, we find that $\mathbb{E}\,{\rm
crps}(G,Y)=(1/3)\sum_{i=1}^{3}\int(G_{i}^{2}(x)-2G_{i}(x)Q_{i}(x)+Q_{i}(x))\,\mathrm{d}x$
and $\mathbb{E}\,{\rm
crps}(Q,Y)=(1/3)\sum_{i=1}^{3}\int(Q_{i}(x)-Q_{i}^{2}(x))\,\mathrm{d}x$; hence
$\mathbb{E}\,{\rm crps}(F,Y)=39/80$, $\mathbb{E}\,{\rm crps}(G,Y)$ $=339/700$,
and $\mathbb{E}\,{\rm crps}(Q,Y)=47/100$. We conclude that

$\displaystyle\textstyle{\rm MCB}_{\rm
CT}=\frac{39}{80}-\frac{47}{100}=\frac{7}{400}\quad\textrm{and }\quad{\rm
MCB}_{\rm ISO}=\frac{39}{80}-\frac{339}{700}=\frac{9}{2800}.$

Since the predictive distributions are ordered with respect to
$\leq_{\textrm{st}}$, it follows that for every threshold $z$, the ordering of
$F_{i}(z)$ is the same. For $z\in(-\infty,1]$, $F_{2}(z)$ and $F_{3}(z)$
coincide but this also holds for $G_{2}(z)$ and $G_{3}(z)$. Similarly, for
$z\in[2,\infty)$, $F_{1}(z)$ and $F_{2}(z)$ coincide but this also holds for
$G_{1}(z)$ and $G_{2}(z)$. This implies that the Brier score based and the
isotonocity-based decompositions coincide. Since the stochastic order is
equivalently characterized by pointwise orderings of lower quantile functions,
the quantile score based and the isotonicity-based decompositions also
coincide.

As all $F_{i}^{-1}$’s are absolutely continuous, we may apply Corollary 4.3 to
compute ${\rm MCB}_{\rm HB}$. For $p\in(0,1)\setminus\\{1/4,1/2,3/4\\}$ we
find that

$\displaystyle\frac{\,\mathrm{d}}{\,\mathrm{d}p}F_{1}^{-1}(p)=2\mathbbm{1}_{(0,\frac{1}{2})}(p)+4\mathbbm{1}_{(\frac{1}{2},1)}(p),\quad\frac{\,\mathrm{d}}{\,\mathrm{d}p}F_{3}^{-1}(p)=4\mathbbm{1}_{(0,\frac{1}{2})}(p)+2\mathbbm{1}_{(\frac{1}{2},1)}(p),$
$\displaystyle\frac{\,\mathrm{d}}{\,\mathrm{d}p}F_{2}^{-1}(p)=4\mathbbm{1}_{(0,\frac{1}{4})}(p)+2\mathbbm{1}_{(\frac{1}{4},\frac{3}{4})}(p)+4\mathbbm{1}_{(\frac{3}{4},1)}(p),$

hence

$\displaystyle\textstyle\gamma(p)=\frac{1}{3}\sum_{i=1}^{3}\mathbb{E}\left(\frac{\,\mathrm{d}}{\,\mathrm{d}p}F^{-1}(p)\Big{|}F=F_{i}\right)=\frac{10}{3}\mathbbm{1}_{(0,\frac{1}{4})}(p)+\frac{8}{3}\mathbbm{1}_{(\frac{1}{4},\frac{3}{4})}(p)+\frac{10}{3}\mathbbm{1}_{(\frac{3}{4},1)}(p).$

The law of total expectation implies

$\displaystyle\textstyle\mathbb{E}\left(\mathbbm{1}\\{F(Y)\leq
p\\}\frac{\,\mathrm{d}}{\,\mathrm{d}p}F^{-1}(p)\right)$
$\displaystyle=\frac{10}{3}p\mathbbm{1}_{(0,\frac{1}{4})}(p)+\left(\frac{3}{15}+\frac{34}{15}p\right)\mathbbm{1}_{(\frac{1}{4},\frac{3}{4})}(p)+\frac{10}{3}p\mathbbm{1}_{(\frac{3}{4},1)}(p),$

and hence,

$\displaystyle\textstyle
f(p)=p\mathbbm{1}_{(0,\frac{1}{4})}(p)+\left(\frac{3}{40}+\frac{17}{20}p\right)\mathbbm{1}_{(\frac{1}{4},\frac{3}{4})}(p)+p\mathbbm{1}_{(\frac{3}{4},1)}(p).$

Finally, we obtain

$\displaystyle\textstyle{\rm MCB}_{\rm
HB}=\int\left(p-f(p)\right)^{2}\gamma(p)\,\mathrm{d}p=\int_{\frac{1}{4}}^{\frac{3}{4}}\left(\frac{3}{20}p-\frac{3}{40}\right)^{2}\frac{8}{3}\,\mathrm{d}p=\frac{1}{1600}.$

###  E.5 Example 2.14 b) in Gneiting and Resin (2023)

For $y_{1}<y_{2}<y_{3}$, let $F$ be a mixture of the Dirac measures on
$y_{1},y_{2}$, and $y_{3}$ with weights $p_{1},p_{2}$, and $p_{3}$, and let
$Y$ be drawn from a mixture of the same Dirac measures with weights
$q_{1},q_{2}$, and $q_{3}$, respectively. Suppose that the tuple
$(p_{1},p_{2},p_{3};q_{1},q_{2},q_{3})$ attains each of the values

$\displaystyle\textstyle\left(\frac{1}{2},\frac{1}{4},\frac{1}{4};\frac{5}{10},\frac{4}{10},\frac{1}{10}\right),\quad\left(\frac{1}{4},\frac{1}{2},\frac{1}{4};\frac{1}{10},\frac{5}{10},\frac{4}{10}\right),\quad\left(\frac{1}{4},\frac{1}{4},\frac{1}{2};\frac{4}{10},\frac{1}{10},\frac{5}{10}\right)$

with equal probability. Let $t_{1}=y_{2}-y_{1}>0$, $t_{2}=y_{3}-y_{2}>0$, and
$t=t_{1}+t_{2}$. It is immediate that $\mathbb{E}\,{\rm crps}(F,Y)=5t/24$ and
${\rm UNC}_{0}=\mathbb{E}\,{\rm crps}(F_{\rm mg},Y)=2t/9$. As Gneiting and
Resin (2023) show, $F$ is threshold and quantile calibrated, hence ${\rm
MCB}_{\rm BS}={\rm MCB}_{\rm QS}=0$.

Let $F_{1},F_{2}$, and $F_{3}$ denote the three discrete distributions that
$F$ may attain. For $i=1,2,3$, denote by $Q_{i}$ the conditional law of $Y$
given $F=F_{i}$ and by $G_{i}$ the isotonic conditional law of $Y$ given
$F=F_{i}$, namely,

$\displaystyle\textstyle
G_{1}=\frac{1}{2}\delta_{y_{1}}+\frac{4}{10}\delta_{y_{2}}+\frac{1}{10}\delta_{y_{3}},\quad
G_{2}=\frac{1}{4}\delta_{y_{1}}+\frac{7}{20}\delta_{y_{2}}+\frac{4}{10}\delta_{y_{3}},\quad
G_{3}=\frac{1}{4}\delta_{y_{1}}+\frac{1}{4}\delta_{y_{2}}+\frac{1}{2}\delta_{y_{3}}.$

Since the image of the random vector $(F,Y)$ is finite and ICL is the
population version of IDR (Arnold and Ziegel, 2023, Proposition 4.1), one
obtains the $G_{i}$’s alternatively by applying IDR on the finite sample of
size $n=30$ with five occurrences of $(F_{1},y_{1})$, four of $(F_{1},y_{2})$,
one each of $(F_{1},y_{3}$ and $(F_{2},y_{1})$, five of $(F_{2},y_{2})$, four
each of $(F_{2},y_{3})$ and $(F_{3},y_{1})$, one of $(F_{3},y_{2})$, and five
of $(F_{3},y_{3})$. The ${\rm MCB}_{\rm CT}$ and ${\rm MCB}_{\rm ISO}$
components may be calculated in analogy to previous examples. We obtain ${\rm
MCB}_{\rm CT}=3t/200$ and ${\rm MCB}_{\rm ISO}=3t_{2}/200$.

To compute the Hersbach decomposition, let $\nu_{i}$ be the image of the
Lebesgue measure on $(0,1)$ under $F_{i}$ where $i=1,2,3$. We have
$\nu_{1}=t_{1}\delta_{1/2}+t_{2}\delta_{3/4}$,
$\nu_{2}=t_{1}\delta_{1/4}+t_{2}\delta_{3/4}$, and
$\nu_{3}=t_{1}\delta_{1/4}+t_{2}\delta_{1/2}$, and hence,
$\mu=(1/3)(2t_{1}\,\delta_{1/4}+t\,\delta_{1/2}+2t_{2}\,\delta_{3/4})$. For
$\ell=1,2,3$ and $p_{l}=l/4$, and for any $A\in\mathcal{B}(0,1)$, the
quantities $f_{\ell}=f(p_{\ell})$ satisfy

$\displaystyle\textstyle\tau(A)$
$\displaystyle=\mathbb{E}\int_{A}\mathbbm{1}\\{F(Y)\leq
p\\}\,\mathrm{d}\nu_{F}(p)$ (52)
$\displaystyle=\int_{A}f(p)\,\mathrm{d}\mu(p)=f_{1}\,\frac{2t_{1}}{3}\,\delta_{1/4}(A)+f_{2}\,\frac{t}{3}\,\delta_{1/2}(A)+f_{3}\,\frac{2t_{2}}{3}\,\delta_{3/4}(A),$

where the expectation in (52) may be calculated by the law of total
expectation:

$\displaystyle\textstyle\mathbb{E}\int_{A}\mathbbm{1}\\{F(Y)\leq
p\\}\,\mathrm{d}\nu(p)$
$\displaystyle=\frac{1}{3}\sum_{i=1}^{3}\mathbb{E}\left(\int_{A}\mathbbm{1}\\{F(Y)\leq
p\\}\,\mathrm{d}\nu_{F}(p)\hskip 2.84526pt\big{|}\hskip
2.84526ptF=F_{i}\right)$
$\displaystyle=\frac{1}{3}\sum_{i=1}^{3}\int\int_{A}\mathbbm{1}\\{F_{i}(y)\leq
p\\}\,\mathrm{d}\nu_{i}(p)\,\mathrm{d}Q_{i}(y)$
$\displaystyle=\frac{t_{1}}{6}\,\delta_{1/4}(A)+\frac{t}{6}\,\delta_{1/2}(A)+\frac{t_{2}}{2}\,\delta_{3/4}(A).$

We conclude that $f_{\ell}=p_{\ell}$ for $\ell=1,2,3$, and hence ${\rm
MCB}_{\rm HB}=0$.


[a]Elisa Resconi

# The Pacific Ocean Neutrino Experiment

###### Abstract

Neutrino telescopes are unrivaled tools to explore the Universe at its most
extreme. The current generation of telescopes has shown that very high energy
neutrinos are produced in the cosmos, even with hints of their possible
origin, and that these neutrinos can be used to probe our understanding of
particle physics at otherwise inaccessible regimes. The fluxes, however, are
low, which means newer, larger telescopes are needed. Here we present the
Pacific Ocean Neutrino Experiment, a proposal to build a multi-cubic-kilometer
neutrino telescope off the coast of Canada. The idea builds on the experience
accumulated by previous sea-water missions, and the technical expertise of
Ocean Networks Canada that would facilitate deploying such a large
infrastructure. The design and physics potential of the first stage and a
full-scale P-ONE are discussed.

The intrinsic properties of 1) the high-energy universe beyond our Galaxy and
2) obscured astronomical objects can not be fully revealed using photon-based
astronomy only. Neutrinos interact only weakly; hence are ideal astrophysical
messengers but higher statistics is needed to access to the nature of active
galaxies, blazars, dark matter and explore the cosmos up to the highest energy
frontier. The observatory leading high energy neutrino astronomy research is
the IceCube Neutrino Observatory (IceCube) at the South Pole (see
https://icecube.wisc.edu). The milestones achieved with the neutrinos
collected by IceCube during a period of more than 10 years of data searching
for cosmic accelerators are:

  * •

The discovery of a diffuse astrophysical flux of neutrinos [1, 2, 3, 4, 5];

  * •

The hint of the association of neutrinos to blazars [6, 7, 8, 9, 10, 11, 12];

  * •

The evidence of the association of an alert event and a neutrino flare to the
$\gamma$-ray emitting blazar TXS 0506+056 [13, 14, 15];

  * •

The hint of the association of neutrinos to the Seyfert galaxy NGC 1068 [16,
17].

While this progress is exciting, it also reveals that with a telescope only of
the size of the order of a cubic kilometer, the neutrino sample collected from
the cosmos is still too limited to advance on this promising, rich path of
fundamental discoveries in astro and particle physics. To gain more statistics
and exposure to the entire sky, a global effort is mobilizing to establish
dramatic improvements in the integral exposure to astrophysical neutrinos. New
feasible technologies, methodologies, and observatories, ultimately linked
through a global network are now to be fostered [18].

Figure 1: Map of existing or under construction neutrino telescopes [18].

A network of telescopes, as the one in Fig. 1, will expand neutrino
observation beyond the limits of the northern sky covered by IceCube to
include also the southern sky.
Neutrino astronomers and oceanographers alike have envisioned exploiting deep
sea infrastructure to support environmental and fundamental research.
Thousands of kilometers of underwater telecommunications cables are seen as a
potentially ideal host for massive, cubic-km scale neutrino telescopes and as
an interface to study the secrets of ocean systems and global environmental
systems, however the neutrino astronomy community has missed the
infrastructure available to be used as a piggyback. We therefore conceived the
Pacific Ocean Neutrino Experiment (P-ONE) as the first neutrino telescope
hosted by an existing oceanographic infrastructure.

##  1 Visualize P-ONE

The Pacific Ocean Neutrino Experiment (P-ONE) [19] is a proposed multi-cubic
kilometer-scale neutrino telescope to be deployed off the coast of Vancouver
Island, Canada (see Sec. 2). The main goal of P-ONE is to significantly
advance the field of neutrino astronomy by extending the cosmic frontier at
the highest energies. P-ONE is a discovery-oriented mission aiming to reveal
previously unknown astronomical phenomena, test fundamental physics at the PeV
scale, and provide crucial information for multi-messenger follow-up
observations. P-ONE leverages the expertise in deep-sea operations from Ocean
Networks Canada (ONC). P-ONE will complement the existing network of neutrino
telescopes around the globe (Antares [20], KM3NeT [21], GVD Baikal [22],
IceCube [23]) and significantly contribute to a complete neutrino sky coverage
at the PeV scale [18]. P-ONE is currently envisioned as a segmented structure,
consisting of 7 clusters with ten mooring lines each. Each line will include
20 optical modules of two different types: a calibration module and an optical
receiver module, see Fig. 2.

Figure 2: P-ONE first vision [19]. The left side shows the complete telescope
configuration, consisting of 7 clusters. An individual cluster with 10 mooring
lines is depicted on the right. A full optimization of the telescope is
presently on-going.

P-ONE is designed to detect astrophysical neutrinos above the TeV energies
with excellent directional and energy reconstruction for neutrino events,
focusing on horizontal tracks where discovery potential is maximal. Another
critical aspect of the design is a modular structure that allows for
installation of varying sizes and easy scalability of the array in different
stages. A single P-ONE cluster will consist of 10 strings with 20 optical
modules each. Two pathfinder experiments are currently studying the optical
properties of the Cascadia Basin, Strings for Absorption Length in Water
(STRAW) and STRAW-b, deployed in 2018 and 2020 (see Sec. 3) [24, 25]. The next
milestone towards P-ONE is the development of the prototype instrumented line
and the construction of at least three for in-situ validation of performances
(see Sec. 4) [26]. The role of P-ONE in the global network of neutrino
telescopes has been addressed by the PLE$\nu$M team, summarized in Sec. 5
[18].

##  2 Ocean Networks Canada, NEPTUNE, and the Cascadia Basin Node

Ocean Networks Canada (ONC) Ocean Networks Canada
(ONC)111http://oceannetworks.ca/ is a multidisciplinary organization with an
exemplary track record in deploying large-scale, real-time sensor networks in
the deep ocean with increasing reliability over the last decade. The ONC has a
success rate of off-shore maintenance cruises of the order of 95%. Those rates
are the result of a set of strictly applied standard operating procedures that
put the emphasis on thorough pre-testing, and are supported by rigorously
observed work-flows. ONC is a leading global organization supporting research
and discovery in the oceans, including, since 2017, the P-ONE mission. It is
composed of many observatories, the largest being the North East Pacific Time-
series Underwater Networked Experiment (NEPTUNE)[27].

The NEPTUNE observatory (see Fig. 3) is operated by ONC since 2009. It is
comprised of an 800 km loop of telecommunication fibre-optic cables to power
and transfer data to a wide variety of sensors. The high-speed data link (up
to 4 Gbit/s) and high power (of the order of 8 kW/node) supports 5 nodes
acting as a local hub for core observations and experiments. A total of 17
primary junction boxes are cabled to the nodes and used to connect hundreds of
instruments. The connections rely on underwater-mateable connectors with field
proven reliability (a failure lower than 2% in deployed connector pairs over a
period of 10 years). The NEPTUNE observatory offers an ideal and available
oceanographic infrastructure and in this way the opportunity for the
construction of a large volume neutrino telescope. In 2017, we set up a new
collaboration with ONC for its use in the context of neutrino astronomy. Among
the various ONC-powered nodes, the Cascadia Basin at a depth of 2660 meters
has been selected, after two years of in-situ measurements, to host the
Pacific Ocean Neutrino Experiment (P-ONE, see http://www.pacific-
neutrino.org).

Cascadia Basin is a deep area lying between the Juan de Fuca Ridge and the
Pacific Northwest, which hosts one of the five instrumented nodes of NEPTUNE.
Already cabled and constantly powered by ONC, Cascadia Basin is a geological
and environmental body of water known to be a calm sedimental area with weak
currents (3-7 cm/s) and temperature around 2∘C [28]. It has been instrumented
with several monitoring systems, such as hydrophones, tiltmeters,
thermometers, accelerometers, and acoustic Doppler current profilers (ADCP),
providing an excellent starting point for the assessment of the environmental
conditions.

Figure 3: Map of the 800 km long NEPTUNE cable observatory operated by Ocean
Networks Canada (white loop) with an exploded view of the instrumentation
presently taking date at Cascadia Basin (black window) and a vision of P-ONE
telescope (white window).

##  3 Find the path to P-ONE

Fascinated by the ONC success rate, their world-wide uniqueness and their ease
of operation deep in the ocean, a collaboration between TU Munich, University
of Alberta, Simon Fraser University, Queen’s University, and ONC was formed in
late 2017. If a lot is already measured and understood about the environment
in Cascadia Basin from ONC, the determination of the optical properties has
required a dedicated effort. For this reason, the P-ONE collaboration has
launched in 2018 a first pathfinder mission, the STRings for Absorption length
in Water (STRAW), and in 2020 a second one (STRAW-b) [29, 25]. Both missions
resulted in installation of optical and calibration sensors.

The STRAW Pathfinder
As a first pathfinder mission for P-ONE, ONC top-down deployed two 145 m tall
instrumented mooring lines in Spring 2018 , see Fig. 4 [29]. The lines have
been designed, commissioned by the nascent P-ONE collaboration. Thanks to
strict quality control based on standardized testing and final qualification
tests in ONC’s external facilities, all instruments, three Precision Optical
CAlibration Modules (POCAMs), and five photosensors are reliably operating for
more than three years with a duty cycle of 98%. Data are flowing from the deep
Pacific Ocean to the P-ONE collaboration, and we achieved the preliminary
assessment of the attenuation length vs. wavelength, rate of bioluminescence,
and ${}^{40}K$ [30, 24].

Figure 4: The two STRAW mooring lines equipped with three calibration modules
(POCAM) and five photosensors (sDOM). The lines have been daisy chained and
connected to a mini Junction Box (mJB). The mJB is connected to a primary JB
already integrated in the ONC infrastructure. The geometry was optimized to
obtain various baselines between the light emitter (POCAM) and receiver
(sDOM). The lines were oriented face-on at the time of the submarine
connection, thanks to a rotatable anchor.

Figure 5: Quality control at the Marine Technology Center (MTC) of ONC. On the
left is the saltwater pool with an overhead crane. The crane is permanently
mounted to lift and immerse heavy infrastructure, including fully instrumented
mooring lines. On the right, one of the STRAW lines enters the pool for long-
term tests. The submerged instrumentation can be powered and monitored in its
operation during submersion tests. The pool can be covered to avoid damaging
the photosensors.

Results from the STRAW pathfinders are reported in [31]. Most notably, we
preliminary report the attenuation length measured at four wavelengths:
$\lambda_{365nm}\approx$ 11 m, $\lambda_{400nm}\approx$ 15 m,
$\lambda_{450nm}\approx$ 30 m, $\lambda_{585nm}\approx$ 8 m. Results are shown
in Fig. 6 where measurements from other sites are also show for comparison
[32, 33, 34]. Details on how the analysis has been performed and about the
comparisons are available in [31]. If the observed value of the attenuation
length at 450 nm is similar to other sites, the values at lower wavelengths
appear reduced. A possible explanation could be related to the fact that STRAW
is relatively close to the Cascadia Basin floor, where more significant
sediment dissolution is likely. We anticipate that the attenuation length is
dominated by absorption in the water and that scattering is subdominant.
Direct verification of both the attenuation length in the uppermost part of
the water column and the scattering length motivated the construction of a
second pathfinder, STRAW-b.

Figure 6: Attenuation length estimated from the STRAW data collected over four
different wavelength and comparisons with other sites [30, 24] (see text for
references and details).

Another environmental parameter of fundamental importance is the level of
locally observed bioluminescence and the temporal variation of this point
emission of light by organisms. Also covered in more details in [31], we
report in Fig. 7 (left), the hourly percentiles of the background rates over a
period of four days. Such behaviour is representative of the observed
behaviour recorded constantly (98% duty cycle).

Figure 7: (left) Bioluminescence rate observed at Cascadia Basin, hourly
percentiles over a representative four days of data taken. The time structure
visible in the distribution corresponds to the 12.5 hour tidal cycle. We note
that we report the percentiles instead of the mean rate since the percentiles
are unaffected by the DAQ saturation limit [30, 24]. (right) A comparison of
the data gathered by STRAW-b [25] and the simulation presented here. We
performed a fit on data, while minimizing the number of organisms. This
resulted in a population of three distinct species to model the data, denoted
G1, G2, and C1." Is the textblob in the paper (the new version) G1, G2 denote
organisms which emit light in the form of a Gamma distribution (for their
time-series) C1 emits light in the form of a Cauchy distribution [35].

In parallel to the data taken and bioluminescence empirical study, members of
the P-ONE collaboration have anticipated a modelling work predicting deep sea
bioluminescence simulating the water flow around the mooring lines of the
P-ONE telescope [35]. Two types of causes producing flashes have been
investigated: the encounter and the shear stress generated by the tides and
the telescope’s structure. Preliminary findings indicate that shear stress
play a far larger role at the low densities expected in the deep sea.
Moreover, investigation of unique bioluminescence signatures once compared
with the simulation might reveal insights in the species producing the
bioluminescence, topic of interest for the marine biologists in P-ONE. More
studies related to the spectrum of the bioluminescence are on-going in STRAW-b
pathfinder, see Fig. 7 (right).

While the primary background consists of stochastic bioluminescence spikes, it
is also expected that a continuous noise floor due to the decay of radioactive
isotopes occurring in sea salt as the potassium-40 (40K) is also present.
Although the measurement of 40K was not among the goals of STRAW because the
salinity is measured with high accuracy, the photons coincidence between the
top and bottom PMTs of an sDOM have been isolated revealing a pile up due to
the 40K. The level of 40K measured is in excellence agreement with the
salinity measured by ONC ($3.482\pm 0.001$% [36]), confirming a good
understanding of the STRAW data [30].

The STRAW experience in the context of the large-scale ONC operation is an
early demonstration that deep sea installation and operation of neutrino
sensors has the potential today to achieve the scale of what would be needed
for an extremely large-volume ocean-based neutrino telescope. STRAW was
financed through the German Research Foundation (DFG) grant SFB 1258
“Neutrinos and Dark Matter in Astro- and Particle Physics”, cluster of
excellence “Origin and Structure of the Universe”, the University of Alberta
and the Natural Science and Engineering Research Council of Canada (NSERC).

The STRAW-b Pathfinder
Having observed no significant challenges in the construction, installation,
and operation of STRAW’s two 145 meter long lines, and with the ambitious goal
of building P-ONE with one-kilometer instrumented lines, we decided to
challenge our capabilities by designing and building a 500-meter line named
STRAW-b [25]. STRAW-b is equipped with ten modules all housed in spherical
13′′ high-pressure resistant glass spheres. Three of the modules, named
Standard Modules, monitor the environmental conditions (pressure, temperature,
humidity) and host magnetic field sensors and accelerometers. The other
modules are two LiDARs (Light Detection And Ranging), a muon tracker, two PMT-
based spectrometers, a mini spectrometer, and a Wavelength shifting Optical
Module (WOM) prototype built by the University of Mainz, see Fig. 8. Single
cameras are also hosted in most of the modules to allow visual inspection.

Figure 8: STRAW-b pathfinder: a bundle of 10 hybrid cables are equipped with
modules. The modules are mechanically sustained by a backbone steel cable. As
for STRAW, a mini Junction Box connects the line to a main JB integrated in
the NEPTUNE system.

Completed in spring 2020 in Munich, the instrumented line was shipped to ONC
where the Canadian part of the P-ONE collaboration took care of the final
integration and tests. Deployment of STRAW-b occurred in Fall 2020 with less
than optimal conditions due to both the pandemic and ocean conditions. Despite
these additional challenges, the deployment with the anchor at the front of
the line was successful. However, we lost connection with one of the standard
modules right after the deployment. The most probable scenario is that one of
the connectors failed, producing such a loss. The STRAW-b modules have been
taking data continuously for several months and data are being integrated into
the ONC database, Oceans 2.0 (https://data.oceannetworks.ca).

STRAW-b was financed through the German Research Foundation (DFG) grant SFB
1258 “Neutrinos and Dark Matter in Astro- and Particle Physics”, cluster of
excellence “Origin and Structure of the Universe”, and the University of
Alberta.

##  4 Start up P-ONE: the Prototype Line Project

Encouraged by the experience with STRAW and STRAW-b, since early 2021, we have
started the design of a fully-instrumented P-ONE line [26]. In addition to the
knowledge gathered by P-ONE since 2017 and ONC over 15 years of experience, we
integrated in the design phase lessons learned from IceCube, GVD, Antares, and
KM3NeT. The P-ONE protoype line hardware will be optimized for minimal risks,
maximal longevity, and scientific return.

A simplified integration between the backbone cable and the optical and
calibration modules is guaranteed by a new connector-less cable design and
sensor mounting that resembles the fly’s eye anatomy, Fig. 9. Work is on-going
on the finalization of the P-ONE optical module. As anticipated by KM3NeT
work, to achieve the best possible performance within the high light-
background environment of the deep ocean (O(10) kHz per PMT), a multi photo-
multiplier tube (mPMT) configuration has been adopted. The photosensors in the
P-ONE optical modules will be PMTs with a diameter between 3-3.5 inches. At
the current development stage, candidate PMTs for the optical modules are
being evaluated at calibration setups at the University of Alberta and the
Technical University Munich.

Experts from Michigan State University and the Technical University of Munich
are developing the read-out electronics, time synchronization, and the data
acquisition system based on state-of-the-art FPGA (Field Programmable Gate
Array) and ML technologies to filter in-situ the extensive background, which
is central to meet the boundary conditions of the ONC’s infrastructure. For
the first installation of up to 10 lines, bandwidth (4 Gbit/s) and power
consumption (8 kW) available at the node will not be a limiting factor.

Figure 9: Fly’s eye anatomy of the P-ONE optical module. The integration of
the active components of the telescope with the P-ONE cable will take place
without the use of external connectors. The instrumented hemispheres will be
assembled through titanium rings to an internal junction box. The details of
these connections are still being finalised.

Looking ahead, the challenge is to reduce bandwidth usage, keeping power
consumption affordable. The time synchronization of the array will follow
follow the White Rabbit protocol[, as in the future IceCube-Gen2 and the under
construction KM3NeT telescopes [37, 38]. A new calibration strategy developed
by experts based a the Simon Fraser University, is also proposed for
monitoring the floating geometry without using an additional acoustic
triangulation system. This prototype phase aims to demonstrate the
scalability, robustness, and maintainability of neutrino telescopes from the
operation of ideally more than three prototype lines over several years.

##  5 Realize and Operate P-ONE in the Global Context of Multimessenger
Astronomy

IceCube has been foundational to neutrino astronomy. After 10 years of
operation of IceCube, new radical steps, have to be take to advance the field
beyond infancy. The field of neutrino astronomy must progress from the cubic-
km scale of IceCube to the multi-cubic km era, and drastically expand coverage
to open up a view of the entire sky, for the first time ever, through one
network of telescopes. As studied in [18], a complementary operation among the
multiple neutrino telescopes will be decisive and will boost the discovery
potential in the Southern Hemisphere, an unexplorable land for a single
telescope alone (Fig. 10), by up to three orders of magnitude. This will also
address the need for expanded sky coverage for real-time astronomical
observations with neutrinos all around the clock.

Figure 10: Evolution of the discovery potential for present (IceCube) and
future neutrino telescopes (KM3NeT, GVD, P-ONE = PLE$\nu$M) and for two
spectral indices: $\gamma=2.0$ (left) and $\gamma=3.0$ (right) [18]. A
synergetic operation among the neutrino telescopes will be decisive for the
future of the field. Figure 11: P-ONE prototype line under design. The line is
one kilometer tall instrumented with 15 optical modules and 5
optical/calibration modules. The modules are engaged with the electrical-
optical-mechanical cable through titanium rings which penetrates the module’s
head through a junction box that defends the vital elements of the cable from
infiltration. The optical sensor hemispheres (eyes) are laterally mounted via
a titanium flange which is glued to the glass hemisphere [26].

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## Full Authors List: P-ONE Collaboration

Nicolai Bailly1, Jeannette Bedard1, Michael Böhmer2, Jeff Bosma1, Dirk
Brussow1, Jonathan Cheng1, Ken Clark3, Beckey Croteau1, Matthias Danninger4,
Tyce DeYoung6, Fabio De Leo1, Nathan Deis1, Matthew Ens4, Christopher Fink2,
Rowan Fox1, Christian Fruck2, Andreas Gärtner5, Roman Gernhäuser2, Darren
Grant6, Dilraj Ghuman4, Christian Haack2, Rob Halliday6, Helen He1, Felix
Henningsen7, Kilian Holzapfel2, Ryan Hotte1, Reyna Jenkyns1, Hamish Johnson4,
Akanksha Katil5, Claudio Kopper6, Carsten B. Krauss5, Ian Kulin1, Klaus
Leismüller2, Sally Leys8, Tony Lin1, Paul Macoun1, Thomas McElroy5, Stephan
Meighen-Berger2, Jan Michel9, Roger Moore5, Mike Morley1, Laszlo Papp2, Benoit
Pirenne1, Tom Qiu1, Mark Rankin1, Immacolata Carmen Rea2, Elisa Resconi2,
Adrian Round1, Albert Ruskey1, Ryan Rutley1, Lisa Schumacher2, Christian
Spannfellner2, Jakub Stacho4, Ross Timmerman1, Meghan Tomlin1, Matt
Tradewell1, Michael Traxler10, Matt Uganecz1, Braeden Veenstra5, Seann
Wagner1, Eva Laura Winter2, Nathan Whitehorn6, Juan Pablo Yañez5, Yinsong
Zheng1

1Ocean Networks Canada, University of Victoria, Victoria, British Columbia,
Canada.
2Department of Physics, Technical University of Munich, Garching, Germany.
3Department of Physics, Engineering Physics and Astronomy, Queen’s University,
Kingston, Ontario, Canada.
4Department of Physics, Simon Fraser University, Burnaby, British Columbia,
Canada.
5Department of Physics, University of Alberta, Edmonton, Alberta, Canada.
6Department of Physics and Astronomy, Michigan State University, East Lansing,
MI, USA.
7Max-Planck-Institut für Physik, Munich, Germany.
8Department of Biological Sciences, University of Alberta, Edmonton, Alberta,
Canada.
9Institut für Kernphysik, Goethe Universität, Frankfurt, Germany.
10Gesellschaft für Schwerionenforschung, Darmstadt, Germany.