#  A note on Hayman’s conjecture

Ta Thi Hoai An  Institute of Mathematics, Vietnam Academy of Science and
Technology
18 Hoang Quoc Viet Road, Cau Giay District
10307 Hanoi, Vietnam  and: Institute of Mathematics and Applied Sciences
(TIMAS)
Thang Long University
Hanoi, Vietnam<EMAIL_ADDRESS>and  Nguyen Viet Phuong  Thai Nguyen
University of Economics and Business Administration, Vietnam
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we will give suitable conditions on differential polynomials
$Q(f)$ such that they take every finite non-zero value infinitely often, where
$f$ is a meromorphic function in complex plane. These results are related to
Problem 1.19 and Problem 1.20 in a book of Hayman and Lingham [9]. As
consequences, we give a new proof of the Hayman conjecture. Moreover, our
results allow differential polynomials $Q(f)$ to have some terms of any degree
of $f$ and also the hypothesis $n>k$ in [1, Theorem 2] is replaced by $n\geq
2$ in our result.

Key words: Meromorphic Functions, Entire Functions, Nevanlinna theory, Value
Distribution, Differential Polynomial.

The authors are supported by Vietnam’s National Foundation for Science and
Technology Development (NAFOSTED), under grant number 101.04-2017.320

##  1\.  Introduction and main results

In 1940, Milloux obtained the first result involving the value distribution of
meromorphic functions together with their derivatives and called usually
Milloux’s inequality. This result has led to a series of important research
questions, for example Hayman [7] (or Problem 1.19 in [9]) proved that if $f$
is a meromorphic function in the plane omitting a finite value $a$, and if its
$k$-th derivative $f^{(k)}$, for some $k\geq 1$, omits a finite nonzero value
$b$, then $f$ is constant. This result is known as Hayman’s alternative.
Closely linked to Milloux’s inequality and Hayman’s alternative is Hayman’s
conjecture on the value distribution of certain differential monomials, if $f$
is a transcendental meromorphic function and $n\in\mathbb{N}$, then Hayman
conjectured that $(f^{n})^{\prime}$ takes every finite nonzero value
infinitely often for all $n\geq 2$. Hayman [7] himself proved this conjecture
for $n\geq 4,$ Mues [13] proved the case of $n=3$. In 1995, Bergweiler and
Eremenko [1], Chen and Fang [3], and Zalcman [15] independently proved the
conjecture for $n=2.$ In [2, 5, 10, 13, 15], the authors generalized Hayman’s
conjecture to $f^{(k)}$. The effective result in this direction was given in
[1, Theorem 2]: if $f$ is a transcendental meromorphic function in the plane
and $m>k\geq 1$ then $(f^{m})^{(k)}$ assumes every finite non-zero value
infinitely often. A closed connection was given in [9, Problem 1.20], which
was proven for $n\geq 5$ in [7], $n=4$ in [13] and $n=3$ in [2]: If $f$ is
non-constant and meromorphic in the plane and $n\geq 3$, then
$f^{\prime}-f^{n}$ assumes all finite complex values.

A question arising in connection with Hayman’s alternative was given by
Eremenko and Langley [6]: whether $(f^{m})^{(k)}$ can be replaced by a more
general term, such as a linear differential polynomial

$F=L[f]=f^{(k)}+a_{k-1}f^{(k-1)}+\dots+a_{0}f,$

where the coefficients $a_{j}$ are small functions of $f.$ The following
counterexample gives a negative answer to above question. Let
$f(z)=1+e^{-z^{2}}$ and
$F=L[f]=f^{\prime\prime}+2zf^{\prime}+3f=3+e^{-z^{2}}.$ Then $f(z)\neq 1$ and
$F(z)\neq 3$, but $f$ is not constant. Until now, most results related to
Hayman’s conjecture and Hayman’s inequality were considered for non-constant
differential polynomials $Q$ in $f$, where all of whose terms have degree at
least 2 in $f$ and its derivatives (see [4], [11]).

In this paper, we will generalize Hayman’s conjecture to differential
polynomials. Here, terms of the polynomials are not necessary all to have
degree at least 2.

Let $P(z)$ and $Q(z)$ be polynomials in $\mathbb{C}[z]$ of degree $p$ and $q$
respectively. We write

$\displaystyle
Q(z)=b(z-\alpha_{1})^{m_{1}}(z-\alpha_{2})^{m_{2}}\dots(z-\alpha_{l})^{m_{l}},\quad\text{and}$
$\displaystyle
P(z)=c(z-\beta_{1})^{n_{1}}(z-\beta_{2})^{n_{2}}\dots(z-\beta_{h})^{n_{h}}.$

where $b,c\in\mathbb{C}^{*}.$

Our results are as follows.

######  Theorem 1.

Let $k$ be an positive integer. If $f$ is a transcendental meromorphic
function and $q\geq l+1,$ then $[Q(f)]^{(k)}$ takes every finite non-zero
value infinitely often.

As a consequence, when we consider $k=1$ and $Q(z)=z^{n}$, Hayman conjecture
is obtained:

######  Corollary 1 (Hayman conjecture [1, 7, 8, 13] ).

If $f$ is a transcendental meromorphic function, then $f^{n}f^{\prime}$ takes
every finite non-zero complex value infinitely often, for any integer $n\geq
1$.

More generally, for any $n\geq 2$:

######  Corollary 2.

[1, Theorem 2] If $f$ is a transcendental meromorphic function, and if $n\geq
2$ and $k$ are positive integers, then $(f^{n})^{(k)}$ takes every finite non-
zero complex value infinitely often.

Note that in [1, Theorem 2], the authors need $n>k$. However, in Corollary 2,
we only need $n\geq 2$.

######  Corollary 3.

Let $k,l$ be positive integers and $a_{1},\dots,a_{l}$ be complex numbers. If
$f$ is a transcendental meromorphic function then
$[(f-a_{1})^{2}(f-a_{2})\dots(f-a_{l})]^{(k)}$ takes every finite non-zero
value infinitely often.

######  Theorem 2.

Let $P$ and $Q$ be polynomials of degree $p$ and $q$ respectively, and $k\geq
1$ be a positive integer. Let $f$ be a transcendental meromorphic function. If
$q\geq(k+1)p+l+2$ then $Q(f)+P(f^{(k)})$ has infinitely many zeros.

In the special case that $P(z)=z$ and $Q(z)=-az^{n}+b$ where $a\neq 0,b$ are
constants, we recover many known results, for example the results in [7], as
special cases of our result.

######  Corollary 4.

[7, Theorem 9] If $f$ is a transcendental meromorphic function, $n\geq 5$ and
$a\neq 0$, then $f^{\prime}(z)-af(z)^{n}$ takes every finite complex value
infinitely often.

######  Theorem 3.

Let $P$ and $Q$ be polynomials of degree $p$ and $q$ respectively. Let
$\alpha\neq 0$ be a small function with respect to $f$. If $f$ is a
transcendental meromorphic function of finite order and $p-q-h-2l-2>0$, then
$P(f)Q(f(z+c_{1})+c_{2}f(z))-\alpha$ has infinitely many zeros, for any
$c_{1},c_{2}\in\mathbb{C}$.

In particular, if $f$ is a transcendental entire function of finite order and
$p-h-l>0$, then $P(f)Q(f(z+c_{1})+c_{2}f(z))-\alpha$ has infinitely many
zeros, for any $c_{1},c_{2}\in\mathbb{C}$.

######  Corollary 5.

[12, Theorem 1.2] Suppose that $f$ is a transcendental meromorphic function,
$\alpha$ is a small function with respect to $f$ and $n,s$ are integers. If
$n\geq s+6$, then $f(z)^{n}(f(z+c)-f(z))^{s}-\alpha$ takes every finite non-
zero complex value infinitely often.

Acknowledgments. A part of this article was written while the first name
author was visiting Vietnam Institute for Advanced Study in Mathematics
(VIASM). She would like to thank the institute for warm hospitality and
partial support. The first name author also would like to thank the
International Centre for Research and Postgraduate Training in Mathematics
(ICRTM, grant number ICRTM01-2020.05) for a part of support.

##  2\. Proof of theorems

Let $f$ be a meromorphic function on the complex plane $\mathbb{C}.$ We use
standard notations, definitions and results of Nevanlinna theory in [8, 14].
We denote by $S(r,f)$ any function satisfying $S(r,f)=o(T(r,f))$ as
$r\rightarrow+\infty$ outside of a possible exceptional set with finite
measure. We first recall the following lemmas

######  Lemma 1.

[3, Theorem 2.1] Let $f(z)$ be a transcendental meromorphic function of finite
order. Then, for any $c\in\mathbb{C}$,

  * (i)

$m\Big{(}r,\frac{f(z+c)}{f(z)}\Big{)}=S(r,f)$ for all $r$ outside of a set of
finite logarithmic measure.

  * (ii)

T(r,f(z+c))=T(r,f(z))+S(r,f).

######  Lemma 2.

(Milloux, see Hayman [8, Theorem 3.2]) For $k\geq 1,$

$T(r,f)\leq\overline{N}(r,f)+N\Big{(}r,\dfrac{1}{f}\Big{)}+N\Big{(}r,\dfrac{1}{f^{(k)}-1}\Big{)}-N\Big{(}r,\dfrac{1}{f^{(k+1)}}\Big{)}+S(r,f).$

Recently, Yamanoi [16] proved the following result.

######  Lemma 3.

[16] Let $f$ be a transcendental meromorphic function in the complex plane,
$k\geq 1$ be an integer, and $\epsilon>0;$ let $A\subset\mathbb{C}$ be a
finite set of complex numbers. Then we have

$k\overline{N}(r,f)+\sum_{a\in A}N_{1}\Big{(}r,\dfrac{1}{f-a}\Big{)}\leq
N\Big{(}r,\dfrac{1}{f^{(k+1)}}\Big{)}+\epsilon T(r,f),$

for all $r>e$ outside a set $E\subset(e,\infty)$ of logarithmic density $0.$
Here, $E$ depends on $f,k,\epsilon$ and $A,$ and

$N_{1}\Big{(}r,\dfrac{1}{f-a}\Big{)}=N\Big{(}r,\dfrac{1}{f-a}\Big{)}-\overline{N}\Big{(}r,\dfrac{1}{f-a}\Big{)}.$

###### Proof of Theorem 1.

We will prove $[Q(f)]^{(k)}$ takes a non-zero constant $a$ infinitely often.
Without loss of generality, we may assume $a=1$. Applying Lemma 3 to the
transcendental meromorphic function $Q(f)$ and $k\geq 1,$ set
$A=\\{0\\}\subset\mathbb{C}$ and $\epsilon=\dfrac{1}{2q},$ we have

$\displaystyle
k\overline{N}(r,f)+N\Big{(}r,\dfrac{1}{Q(f)}\Big{)}-\overline{N}\Big{(}r,\dfrac{1}{Q(f)}\Big{)}$
$\displaystyle\leq
N\Big{(}r,\dfrac{1}{Q(f)^{(k+1)}}\Big{)}+\dfrac{1}{2q}T(r,Q(f))$
$\displaystyle\leq
N\Big{(}r,\dfrac{1}{Q(f)^{(k+1)}}\Big{)}+\dfrac{1}{2}T(r,f)+O(1)$

Together with Lemma 2, we get

$\displaystyle T(r,Q(f))$ $\displaystyle\leq
k\overline{N}(r,f)+N\Big{(}r,\dfrac{1}{Q(f)}\Big{)}-\overline{N}\Big{(}r,\dfrac{1}{Q(f)}\Big{)}$
$\displaystyle\quad-(k-1)\overline{N}(r,f)+\overline{N}\Big{(}r,\dfrac{1}{Q(f)}\Big{)}+N\Big{(}r,\dfrac{1}{(Q(f))^{(k)}-1}\Big{)}$
$\displaystyle\quad-N\Big{(}r,\dfrac{1}{(Q(f))^{(k+1)}}\Big{)}+S(r,f)$
$\displaystyle\leq\overline{N}\Big{(}r,\dfrac{1}{Q(f)}\Big{)}+N\Big{(}r,\dfrac{1}{(Q(f))^{(k)}-1}\Big{)}+\dfrac{1}{2}T(r,f)+S(r,f)$
$\displaystyle\leq\sum_{i=1}^{l}\overline{N}\Big{(}r,\dfrac{1}{f-\alpha_{i}}\Big{)}+N\Big{(}r,\dfrac{1}{(Q(f))^{(k)}-1}\Big{)}+\dfrac{1}{2}T(r,f)+S(r,f)$
$\displaystyle\leq\Big{(}l+\dfrac{1}{2}\Big{)}T(r,f)+N\Big{(}r,\dfrac{1}{(Q(f))^{(k)}-1}\Big{)}+S(r,f).$

Hence,

$\Big{(}q-l-\dfrac{1}{2}\Big{)}T(r,f)\leq
N\Big{(}r,\dfrac{1}{(Q(f))^{(k)}-1}\Big{)}+S(r,f).$

Thus $(Q(f))^{(k)}=1$ has infinitely many roots when $q\geq l+1.$ It follows
that $(Q(f))^{(k)}$ assumes every finite non-zero value infinitely often. The
proof is complete. ∎

###### Proof of Theorem 2.

Set

$\displaystyle F=Q(f)+P(f^{(k)}),$ (2.1) $\displaystyle
R(f)=\frac{[Q(f)]^{\prime}}{Q(f)}-\frac{F^{\prime}}{F},\quad
H(f)=P(f^{(k)})\Big{(}\frac{F^{\prime}}{F}-\frac{[P(f^{(k)})]^{\prime}}{P(f^{(k)})}\Big{)}.$
(2.2)

We have

$\displaystyle m(r,R(f))=S(r,f),$ (2.3) $\displaystyle Q(f)R(f)=H(f),$ (2.4)

and

$\displaystyle T(r,F)$ $\displaystyle\leq
T(r,Q(f))+T(r,P(f^{(k)})+S(r,f)=qT(r,f)+pT(r,f^{(k)})+S(r,f)$
$\displaystyle\leq(q+p(k+1))T(r,f)+S(r,f).$

Hence $S(r,F)=S(r,f)$.

If $R(f)\equiv 0,$ then $H(f)\equiv 0$. This means

$Q(f)=(c-1)P(f^{(k)}),$ (2.5)

where $c\neq 0,1$ is a constant. Since $q\geq(k+1)p+l+2$, Equation (2.5)
implies that $f$ cannot have poles. On the other hand, we have

$\displaystyle qm(r,f)=m(r,Q(f))$ $\displaystyle\leq m(r,P(f^{(k)})+O(1)\leq
pm(r,f^{(k)})+O(1)$ $\displaystyle\leq
pm(r,f)+pm\big{(}r,\frac{f^{(k)}}{f}\big{)}+O(1)=pm(r,f)+S(r,f).$

Therefore

$m(r,f)=S(r,f),$ (2.6)

and

$T(r,f)=N(r,f)+m(r,f)=S(r,f),$

which is a contradiction and then $R(f)\not\equiv 0.$

It is easy to see that

$\displaystyle m(r,H(f))$ $\displaystyle\leq
m(r,P(f^{(k)}))+m\Big{(}r,\frac{F^{\prime}}{F}-\frac{[P(f^{(k)})]^{\prime}}{P(f^{(k)})}\Big{)}+O(1)$
$\displaystyle\leq pm(r,f^{(k)})+S(r,f)\leq
p\Big{(}m(r,f)+m(r,\frac{f^{(k)}}{f})\Big{)}+S(r,f)$
$\displaystyle=pm(r,f)+S(r,f).$ (2.7)

We have

$\displaystyle qm(r,f)$
$\displaystyle=m(r,Q(f))+S(r,f)=m\Big{(}r,Q(f)R(f)\frac{1}{R(f)}\Big{)}+S(r,f)$
$\displaystyle\leq m(r,Q(f)R(f))+m\Big{(}r,\frac{1}{R(f)}\Big{)}+S(r,f)$
$\displaystyle=m(r,H(f))+m\Big{(}r,\frac{1}{R(f)}\Big{)}+S(r,f)$
$\displaystyle\leq pm(r,f)+m\Big{(}r,\frac{1}{R(f)}\Big{)}+S(r,f),$

hence

$(q-p)m(r,f)\leq m\Big{(}r,\frac{1}{R(f)}\Big{)}+S(r,f).$ (2.8)

By the first main theorem and (2.3), we have

$m(r,\frac{1}{R(f)})=T(r,R(f))-N(r,\frac{1}{R(f)})+O(1)=N(r,R(f))-N(r,\frac{1}{R(f)})+S(r,f).$
(2.9)

From the definition of $R(f)$, we see immediately that the possible poles of
$R(f)$ occur only at the poles of $f$ and the zeros of $Q(f)$ and $F.$ Note
that $R(f)$ can have only simple poles. Now, suppose that $z_{0}$ is a pole of
$f$ of order $s.$ Then $z_{0}$ is a pole of $Q(f)$ of order $qs$ and $H(f)$ of
order at most $(s+k)p+1.$ Since $q\geq(k+1)p+l+2,$ we have

$qs-(s+k)p-1=(q-p)s-kp-1>0.$

Thus, by (2.4), we deduce $z_{0}$ must be a zero of $R(f)$ of order at least

$qs-((s+k)p+1)=(q-p)s-kp-1.$

Hence, we get

$\displaystyle N(r,R(f))$
$\displaystyle\leq\overline{N}\big{(}r,\frac{1}{F}\big{)}+\overline{N}\big{(}r,\frac{1}{Q(f)}\big{)}\leq\overline{N}\big{(}r,\frac{1}{F}\big{)}+\sum_{i=1}^{l}\overline{N}\big{(}r,\frac{1}{f-\alpha_{i}}\big{)}$
(2.10)
$\displaystyle\leq\overline{N}\big{(}r,\frac{1}{F}\big{)}+lT(r,f)+O(1),$
(2.11)

and

$\displaystyle
N\big{(}r,\frac{1}{R(f)}\big{)}\geq(q-p)N(r,f)-(kp+1)\overline{N}(r,f).$
(2.12)

Combining (2.8), (2.9), (2.11), (2.12) and by the first main theorem, we get

$\displaystyle(q-p)T(r,f)$ $\displaystyle=(q-p)N(r,f)+(q-p)m(r,f)$
$\displaystyle\leq\overline{N}\big{(}r,\frac{1}{F}\big{)}+lT(r,f)+(kp+1)\overline{N}(r,f)+S(r,f)$
$\displaystyle\leq\overline{N}\big{(}r,\frac{1}{F}\big{)}+(l+kp+1)T(r,f)+S(r,f).$

Hence,

$\big{(}q-(k+1)p-l-1\big{)}T(r,f)\leq\overline{N}\big{(}r,\frac{1}{F}\big{)}+S(r,f).$

Thus $F=Q(f)+P(f^{(k)})$ assumes every finite value infinitely often when
$q\geq(k+1)p+l+2.$

∎

###### Proof of Theorem 3.

Denote by $G(z)=P(f(z))Q(f(z+c_{1})+c_{2}f(z)).$ Applying the Second Main
Theorem for the meromorphic function $G$ and $0,\infty$, $\alpha$, we have

$\displaystyle T(r,G)$
$\displaystyle\leq\overline{N}(r,G)+\overline{N}(r,\frac{1}{G})+\overline{N}(r,\frac{1}{G-\alpha})+S(r,f)$
(2.13) $\displaystyle\leq
2{T}(r,f)+\sum_{i=1}^{h}\overline{N}(r,\frac{1}{f-\beta_{i}})+\sum_{i=1}^{l}\overline{N}(r,\frac{1}{f(z+c_{1})+c_{2}f(z))-\alpha_{i}})$
(2.14) $\displaystyle\quad+\overline{N}(r,\frac{1}{G-\alpha})+S(r,f)$ (2.15)
$\displaystyle\leq(h+2l+2){T}(r,f)+\overline{N}(r,\frac{1}{G-\alpha})+S(r,f).$
(2.16)

On the other hand, we have

$\displaystyle\frac{1}{f^{q}P(f)}$
$\displaystyle=\frac{1}{G}\frac{Q(f(z+c_{1})+c_{2}f(z))}{f^{q}}$
$\displaystyle=\frac{1}{G}\prod_{i=1}^{l}\Big{[}\frac{f(z+c_{1})+c_{2}f(z)-\alpha_{i}}{f(z)}\Big{]}^{m_{i}}.$
(2.17)

Therefore,

$\displaystyle(p+q)T(r,f)$ $\displaystyle=T(r,f^{q}P(f))+O(1)$
$\displaystyle\leq
T(G)+\sum_{i=1}^{l}m_{i}T\Big{(}r,\frac{f(z+c_{1})+c_{2}f(z)-\alpha_{i}}{f(z)}\Big{)}+O(1)$
$\displaystyle\leq
T(G)+\sum_{i=1}^{l}m_{i}T\Big{(}r,\frac{f(z+c_{1})-\alpha_{i}}{f(z)}\Big{)}+O(1)$
$\displaystyle\leq T(G)+2\sum_{i=1}^{l}m_{i}T(r,f)+O(1)$ $\displaystyle\leq
T(G)+2qT(r,f)+O(1),$

which implies

$\displaystyle T(r,G)\geq(p-q)T(r,f)+O(1).$ (2.18)

Combining (2.16) and (2.18), we have

$\displaystyle(p-q-h-2l-2)T(r,f)\leq\overline{N}(r,\frac{1}{G-\alpha})+S(r,f).$

Thus, $P(f)Q(f(z+c_{1})+c_{2}f(z))-\alpha$ has infinitely many zeros if $f$ is
a transcendental meromorphic function of finite order and $p-q-h-2l-2>0$.

In particular, if $f$ is an entire function then Lemma 1 (i) implies

$\displaystyle T(r,f(z+c_{1})+c_{2}f(z))-\alpha_{i})$
$\displaystyle=T(r,f(z+c_{1})+c_{2}f(z))+O(1)$
$\displaystyle=m(r,f(z+c_{1})+c_{2}f(z))+O(1)$
$\displaystyle=m\Big{(}r,f(z)\big{(}\frac{f(z+c_{1})}{f(z)}+c_{2}\big{)}\Big{)}+O(1)$
$\displaystyle\leq m(r,f)+m\Big{(}r,\frac{f(z+c_{1})}{f(z)}+c_{2}\Big{)}+O(1)$
$\displaystyle\leq m(r,f)+m\Big{(}r,\frac{f(z+c_{1})}{f(z)}\Big{)}+O(1)$
$\displaystyle\leq T(r,f)+S(r,f).$

Together with (2.16), we have

$\displaystyle T(r,G)$
$\displaystyle\leq\overline{N}(r,\frac{1}{G})+\overline{N}(r,\frac{1}{G-\alpha})+S(r,f)$
(2.19)
$\displaystyle\leq\sum_{i=1}^{h}\overline{N}(r,\frac{1}{f-\beta_{i}})+\sum_{i=1}^{l}\overline{N}(r,\frac{1}{f(z+c_{1})+c_{2}f(z)-\alpha_{i}})$
(2.20) $\displaystyle\quad+\overline{N}(r,\frac{1}{G-\alpha})+S(r,f)$ (2.21)
$\displaystyle\leq\sum_{i=1}^{h}\overline{N}(r,\frac{1}{f-\beta_{i}})+\sum_{i=1}^{l}T(r,f(z+c_{1})+c_{2}f(z)-\alpha_{i})$
(2.22) $\displaystyle\quad+\overline{N}(r,\frac{1}{G-\alpha})+S(r,f)$ (2.23)
$\displaystyle\leq(h+l){T}(r,f)+\overline{N}(r,\frac{1}{G-\alpha})+S(r,f).$
(2.24)

On the other hand, by (2.17), we obtain

$\displaystyle(p+q)T(r,f)$ $\displaystyle=T(r,f^{q}P(f))+O(1)$
$\displaystyle\leq
T(G)+\sum_{i=1}^{l}m_{i}T\Big{(}r,\frac{f(z+c_{1})+c_{2}f(z)-\alpha_{i}}{f(z)}\Big{)}+O(1)$
$\displaystyle\leq
T(G)+\sum_{i=1}^{l}m_{i}T\Big{(}r,\frac{f(z+c_{1})-\alpha_{i}}{f(z)}\Big{)}+O(1)$
$\displaystyle\leq T(G)+\sum_{i=1}^{l}m_{i}T(r,f)+O(1)$ $\displaystyle\leq
T(G)+qT(r,f)+O(1),$

where the fourth inequality follows from

$\displaystyle T\Big{(}r,\frac{f(z+c_{1})-\alpha_{i}}{f(z)}\Big{)}$
$\displaystyle=m\Big{(}r,\frac{f(z+c_{1})-\alpha_{i}}{f(z)}\Big{)}+N\Big{(}r,\frac{f(z+c_{1})-\alpha_{i}}{f(z)}\Big{)}$
$\displaystyle\leq
m\Big{(}r,\frac{f(z+c_{1})}{f(z)}\Big{)}+m\Big{(}r,\frac{1}{f(z)}\Big{)}+N(r,f(z+c_{1})-\alpha_{i})$
$\displaystyle\quad+N\Big{(}r,\frac{1}{f(z)}\Big{)}+O(1)$ $\displaystyle\leq
T(r,f)+S(r,f).$

Hence, we get

$\displaystyle T(r,G)\geq pT(r,f)+S(r,f).$ (2.25)

Combining (2.24) and (2.25), we have

$(p-h-l)T(r,G)\leq\bar{N}(r,\frac{1}{G-\alpha})+S(r,f).$

Hence, $P(f)Q(f(z+c_{1})+c_{2}f(z))-\alpha$ has infinitely many zeros when
$p-h-l>0$ if $f$ is a transcendental entire function of finite order.

∎

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scheme

Sixiu Liu, Claire Birnie, Tariq Alkhalifah
King Abdullah University of Science and Technology,
{sixiu.liu, claire.birnie<EMAIL_ADDRESS>

###### Abstract

Coherent noise regularly plagues seismic recordings, causing artefacts and
uncertainties in products derived from down-the-line processing and imaging
tasks. The outstanding capabilities of deep learning in denoising of natural
and medical images have recently spur a number of applications of neural
networks in the context of seismic data denoising. A limitation of the
majority of such methods is that the deep learning procedure is supervised and
requires clean (noise-free) data as a target for training the network.
Blindspot networks were recently proposed to overcome this requirement,
allowing training to be performed directly on the noisy field data as a
powerful suppressor of random noise. A careful adaptation of the blind-spot
methodology allows for an extension to coherent noise suppression. In this
work, we expand the methodology of blind-spot networks to create a blind-trace
network that successfully removes trace-wise coherent noise. Through an
extensive synthetic analysis, we illustrate the denoising procedure’s
robustness to varying noise levels, as well as varying numbers of noisy traces
within shot gathers. It is shown that the network can accurately learn to
suppress the noise when up to 60% of the original traces are noisy.
Furthermore, the proposed procedure is implemented on the Stratton 3D field
dataset and is shown to restore the previously corrupted direct arrivals. Our
adaptation of the blind-spot network for self-supervised, trace-wise noise
suppression could lead to other use-cases such as the suppression of coherent
noise arising from wellsite activity, passing vessels or nearby industrial
activity.

##  1 Introduction

In seismic surveys, field data are often contaminated by coherent noise that
exhibit coherency in time and/or offset, which makes seismic processing and
imaging difficult. Often, the existence of such noise may mask the useful
reflection signal [12, 19], which can lead to unreliable subsequent inversion
results [40], and thus should be removed early in seismic processing to
improve data quality. Coherent noise, such as trace-wise noise caused by
poorly coupled receivers, is one such cause of coherent noise in seismic
records.

Typically noise attenuation approaches are subdivided into four general
categories. The first category includes prediction-based methods that utilize
both the predictable property of useful signals and the unpredictable property
of noise to design prediction filters. For example, the polynomial fitting
(PF) method has been used to suppress noise by estimating local linear
coherent components in seismic data [29, 27]. The second category of
suppression procedures include sparse transform-based approaches that separate
signals and noise in a transformed domain. Signals and noise are sparsely
represented, separated by thresholding coefficients in the transformed domain
and transformed back to the original domain. These methods include the Fourier
transform [40], the radial trace (RT) transform [12, 19, 20] and the wavelet
transform (WT) [13, 16], among others. Similarly, mathematical morphological
filtering and median filtering attenuate linear noise based on the difference
in the shape of seismic waves in the transformed coordinate system [21] and
the energy difference between noise pixels and the median value of the
neighborhood [39, 14, 10], respectively. In some cases, differences between
useful signals and noise in a transformed domain or a coordinate system are
too small to be distinguished. Therefore, the applications of the second
category are limited. The third category, decomposition-based approaches,
decompose noisy data into different components and then recreate the useful
signals based on the principal components. For example, singular value
decomposition (SVD) [23, 7, 33] and empirical mode decomposition (EMD) [2, 15]
have both been illustrated as successful suppressors of coherent noise. The
last category, rank-reduction-based approaches, assume the seismic data to be
of low-rank after some data rearrangement. For example, multichannel singular
spectrum analysis (MSSA) was employed to denoise a 3D cross-spread data set
that was severely contaminated with ground roll [11].

Recently, deep learning has been widely used in the field of geophysics and
has made breakthroughs in seismic denoising. Among these are Convolutional
Neural Networks (CNNs)-based models for random noise removal, such as standard
CNNs [31, 43, 37], CNNs with residual learning [45, 22, 38], CNNs with
transfer learning [41] and denoising convolutional neural network (DnCNN) [26,
25]. CNNs-based models have also proven successful at coherent noise
attenuation within seismic data. For example, standard CNNs has been utilized
for the likes of deblending [1, 36] and linear noise removal [30]. The DnCNN
architecture has also been employed for the suppression of linear noise within
seismic data [46, 42].

Conventional deep learning noise suppression procedures typically take noise-
free seismic sections as labels for training the neural network in a
supervised way to attenuate noise in seismic data. However, for field data,
noise-free records are not available. Different from supervised learning
techniques that require labeled data for training, AutoEncoders (AEs) have the
ability to automatically learn and store features of the unlabeled, noisy data
in an unsupervised learning manner. Working similarly to traditional
decomposition-based procedures, the denoising ability of AEs are exploited for
both random [9, 34] and coherent noise suppression [17]. However, the learned
features by AEs are not sufficient to fully represent seismic data [44]. The
latent space in AEs is typically small, making the AEs unable to reconstruct
all the seismic features accurately. This motivates the investigation of other
self-supervised procedures, which only use the noisy data as the training
samples. Recently, a self-supervised denoising method, Noise2Void (N2V) was
proved to effectively attenuate random noise in seismic data in a self-
supervised manner by training directly on the data to be denoised [4, 5]. This
is achieved by masking the center pixel (blind spot) of the receptive field of
the network [24]. N2V can be considered as a prediction-based approach due to
it’s ability to accurately recreate the coherent signal and it’s inability to
predict the noise component due to the noise’s random nature. A similar
masking strategy was employed by [32] to train a CNN over a J-invariant
function for seismic random noise suppression in a self-supervised manner.
Within the field of microscopy, [6] proposed an extension of N2V, termed
StructN2V, where the blind-spot is extended beyond a single pixel to create a
mask that conceals the coherency of noise. This masking scheme was
successfully shown to suppress linear fluorescence noise. An initial
investigation into the use of such networks for processing seismic data was
proposed by [28].

In this work, we first introduce our adaptation of the StructN2V approach to
effectively remove trace-wise correlated noise in seismic data using only
noisy images as both the input and target for the blind-trace networks. The
performance of the proposed algorithm is evaluated on synthetic and field data
examples. We further investigate the proposed algorithm from the perspective
of differing noise levels and varying levels of noisy-traces within the
seismic data.

##  2 Theory

In conventional CNN procedures for denoising, a pixel value within an image is
predicted based on the values of pixels within a neighboring square (receptive
field) centralised on the predicted pixel. Hereon in, we shall refer to this
central pixel as the active pixel. As the receptive field contains the active
pixel, the active pixel’s value in the input is utilised to predict the active
pixel’s value in the target image. [24] proposed the use of a blind-spot
scheme, to be applied on an active pixel, to avoid its value influencing the
network’s prediction of the pixel’s value. Under the assumption of random
noise and coherent signal, employing such a scheme prohibits the network from
learning the noise component of the pixel’s value and therefore, only the
signal component can be predicted.

A limitation of the blind-spot implementation of [24] is that it requires
noise to be independent between pixels. To overcome this, [6] proposed the
extension of the blind-spot to create a ‘blind-patch’, which we will refer to
as noise mask throughout this study. The mask is designed prior to training to
hide the coherent noise information in neighbouring pixels from one specific
direction, which is calculated by cross correlation of the pure noise.
However, accurately designing the noise mask in relation to the noise is
challenging. As signals are closely correlated between pixels and vital for
predicting signal components of pixels values, improper selection of the
mask’s orientation has the potential to conceal the signals as opposed to the
desired noise concealment. The size of the mask also needs careful
consideration. A larger mask size will potentially hide the signals,
information that is required to predict the signal value of an active pixel. A
smaller mask may not be big enough to hide all the noise information, and
therefore might cause noise leakage into the value of a predicted pixel. In
our case, the noise in seismic data we are targeting is trace-wise dependent
along the whole trace. As such, a vertical mask across the whole trace of
seismic shot records is used.

The noise mask is implemented as a pre-processing step on the raw, noisy, shot
gathers prior to being passed to the neural network. In this masking scheme,
first some pixels are selected as the active pixels for training. These active
pixels may or may not fall on traces contaminated with trace-wise noise. Then,
all pixels belonging to the same trace location of the active pixels are
replaced (i.e., masked) by random values drawn from a uniform distribution, as
illustrated in Figure 1 by red arrows. The purpose of the noise mask is to
make the noise within the masked locations unpredictable prior to being passed
into the network. After training, a pixel’s signal value can still be
predicted from its neighbouring pixels (outside the mask) due to its spatial
predictability along the offset, but the noise component cannot be predicted
due due to its temporal unpredictability along the time axis.

As illustrated in Figure 1, a standard U-Net architecture is utilized
following other blind-spot denoising applications [6, 5, 28]. The corrupted
versions of raw noisy shot gathers created through the mask scheme and the
original noisy shot gathers are regarded as the inputs and the labels
(targets) to the network during training, respectively. In the process of
training, the network can learn to recreate the seismic arrivals due to their
spatial properties however cannot recreate the coherent noise. Therefore,
whilst the label is the noisy data, the output should be the denoised result,
as illustrated in Figure 1. Instead of computing all the pixels for loss as
done in classical deep learning methods, the loss function here is computed
only on the corrupted pixels. As part of our experiments, we have considered
the loss on both the active pixels and across the full masked trace. It should
be noted that the denoising loss is not desired to be zero, which is the case
in traditional supervised learning methods, otherwise the identity of the
noisy image is learned and the network will only output the original noisy
image. The loss $L$ is computed as follows,

$\begin{split}\mathcal{L}&=\sum_{j}\sum_{i}\mathcal{L}_{d}\left(f\left(\tilde{\boldsymbol{x}}_{\Omega}^{ji};\boldsymbol{\theta}\right),\boldsymbol{x}^{ji}\right),\end{split}$
(1)

where $i$ and $j$ are the number of active pixels within an image and the
number of the training samples, respectively. The trace that is randomized by
the blind-trace operation is represented by $\Omega$. The value of the $i$th
active pixel value within the $j$th training label (i.e., raw patch) and the
value of the $i$th active pixel value within the $j$th training input (i.e.,
corrupted patch) are represented by $\boldsymbol{x}^{ji}$ and
$\tilde{\boldsymbol{x}}_{\Omega}^{ji}$, respectively. The trained network
architecture and network parameters are denoted by $f$ and $\theta$,
respectively. The loss function applied to either only active pixels (field
case) or the full masked trace (synthetic case), and the total loss normalised
by Mean Absolute Error (MAE) are represented by $\mathcal{L}_{d}$ and
$\mathcal{L}$, respectively. After a specified number of iterations in the
optimization process, the network should hopefully have learnt how to recreate
the desired signal without recreating the targeted noise in the noisy data.

Figure 1: Schematic diagram of the training workflow using StructN2V to
suppress coherent noise. Red arrows in input, output and label illustrate
three corrupted traces, three denoised traces and the original three noisy
traces, respectively.

##  3 Numerical Examples

We evaluate the approach on two different datasets: synthetic seismic data
with different levels and different numbers of noisy traces, and a field
seismic dataset. In order to quantitatively compare the quality of the
denoised results with corresponding noise-free data, two performance metrics
are used: the Image Peak Signal-to-Noise Ratio (PSNR) and the Structural
Similarity Index Measure (SSIM). PSNR is a classic seismic metric whilst SSIM
is the preferred metric from the deep learning community for denoising
applications.

PSNR is expressed as

$PSNR=10\log_{10}\frac{\max\|D\|_{2}^{2}}{\|D-d\|_{2}^{2}},$ (2)

where $D$ and $d$ stand for the clean data and the compared data (either noisy
or denoised). The unit of PSNR is decibel (dB). The greater the PSNR value,
the better the denoised result.

SSIM measures the similarity between two images ($D$ and $d$) and it is
expressed as

$SSIM(D,d)=\frac{\left(2\mu_{D}\mu_{d}+c_{1}\right)\left(2\sigma_{Dd}+c_{2}\right)}{\left(\mu_{D}^{2}+\mu_{d}^{2}+c_{1}\right)\left(\sigma_{D}^{2}+\sigma_{d}^{2}+c_{2}\right)},$
(3)

where the average of $D$ and the average of $d$ are denoted by $\mu_{D}$ and
$\mu_{d}$, respectively. The variance of $D$ and the variance of $d$ are
represented by $\sigma_{D}^{2}$ and $\sigma_{d}^{2}$, respectively. The
covariance of $D$ and $d$ is $\sigma_{Dd}$. Two variables to stabilize the
division with weak denominator are $c_{1}=(k_{1}L)^{2}$ and
$c_{2}=(k_{2}L)^{2}$. The dynamic range of the pixel value is $L$. By default,
$k_{1}=0.01$ and $k_{2}=0.03$. The structural similarity ranges from 0 to 1.
The greater the SSIM value, the better the denoised result. When the two
images are identical, the value of SSIM is equal to 1.

###  3.1 Synthetic example

The proposed Self-Supervised Learning (SSL) procedure is first validated on
synthetic seismic records constructed via acoustic, finite-difference
modelling of the SEG Hess VTI model, shown as the “original patch” in Figure
2. Figure 2 shows how the training data are prepared using data augmentation
procedures to increase the training data size, where $n$ represents the number
of the training patches generated at the different stages of the pipeline.
Initially, 101 shots are generated and are then increased four-fold by the
data augmentation techniques of mirroring and polarity reversal. In order to
generate noisy patches, ten different noise realisations are created for each
shot in the last stage by randomly replacing ten traces with a random value
between [-0.8, 0.8] to represent the raw, noisy data used as the target
(label) for the deep learning network. The blind-trace denoising scheme is
then employed on each of the noisy patches, in which the masked traces, shown
by red vertical lines in the mask, will be corrupted by random values from a
uniform distribution over [-0.8, 0.8] to generate the input data, referred to
corrupted patch. The corrupted noise is preferred to have a similar magnitude
as the “recorded” noise to avoid changing the raw, noisy image too much while
employing the masking scheme. This corruption scheme is implemented per patch
and per epoch, therefore changing the input to the network on each training
iteration. A ReLU activation function and batch normalisation are used in the
convolutional layers, and a linear activation function is used on the output
layer. The training parameters were optimised via hyper-parameter tuning
following a grid-search procedure, with the optimal parameters detailed in
Table 1.

Figure 2: Data preparation workflow. Table 1: Training parameters for the two
data sets

|  Synthetic |  Field
---|---|---
Patch size |  512x256 |  56x56
Train:Val |  3200:840 |  7000:1140
# Active Traces |  3 |  3
Noise level |  [-0.8, 0.8] |  [-1.5, 1.5]
UNet depth |  4 |  3
Batch size |  64 |  16
# Initial filters |  32 |  32
Kernel |  3 |  3
LR |  0.004 |  0.004
Loss |  MAE |  MAE
# Epochs |  128 |  72
Loss coverage |  trace |  active

The results from the denoising scheme applied to the Hess VTI synthetic
seismic dataset trained with the blind-trace procedure are shown in Figure 3,
alongside the observed PSNR and SSIM. It can be seen that some of the signal
in the raw, noisy data (Figure 3(b)) is hidden by the strong trace-wise noise.
Compared to the noisy data shown in Figure 3(b) with a PSNR of 41.33 dB and a
SSIM of 0.948, the denoised result shown in Figure 3(c) has a higher PSNR of
44.39 dB and a higher SSIM of 0.995. The denoised result illustrates almost no
remaining coherent noise. Figure 3(d) and (e) highlight the noise removed by
the method and the difference between the noise-free and denoised datasets,
respectively, both of which show that the proposed method can accurately
suppress the trace-wise noise. However, these figures also show the existence
of a slight difference in the amplitudes of the first arrival, indicating
slight signal leakage for the high amplitude first arrival.

Figure 3: Trained SSL model applied to a synthetic dataset with coherent
noise. (a) The noise-free synthetic, (b) the noisy synthetic given as input to
the model,and (c) the result of the SSL denoising procedure. (d) and (e) are
the differences between the noisy and denoised datasets and between the noise-
free and denoised datasets, respectively. Red arrow in (b), (c) and (d)
illustrate the noisy traces, the denoised traces and the removed noise,
respectively.

#### Effect of noise level

To further evaluate the denoising performance, we apply different levels of
trace-wise noise contamination for synthetic seismic shot gathers. After
obtaining the 404 clean created for each shot by randomly replacing 10 traces
among the total 256 traces with random noise level ranging from 0.1 to 0.9,
with step of 0.1, to represent the 4040 raw, noisy data with different noise
levels as the target (label) for the deep learning network. For comparitive
purposes, the value ranges for the first arrival and the reflections are
[-4.4, 8.2] and [-0.6, 1], respectively. For post-training analysis, 100
additional noisy shot gathers are generated, again by randomly replacing 10
traces with random values over the pre-described range. All the training
parameters are as detailed in Table 1 and are the same for the nine different
cases, except for the noise levels to corrupt traces. The PSNR and SSIM are
both used to evaluate the denoising performance and are averaged across 100
test images.

Figure 4 illustrates the relationship between data quality (PSNR and SSIM)
values and percent changes over noise contamination level (the amplitude
values of noisy traces in a patch). As shown in Figure 4(a), as the noise
level of noisy traces increases, the denoised PSNR value and change (between
the noisy input data and the denoised results) both increase with a drop
between 0.4-0.8. All the changes for noise levels larger than 0.2 are above
zero, which means the denoised result has a higher PSNR than the noisy image,
indicating a higher tolerance with higher noise level. The fluctuation in PSNR
is possibly due to the random elements introduced during the training -
corruption values and locations. In Figure 4(b), the SSIM change increases
gradually while the denoised SSIM value increases slowly as the contamination
level increases. Similar to the PSNR, the denoised result has a higher SSIM
value than the noisy image as the changes are always above zero. Both the PSNR
and the SSIM of the noisy data decrease gradually due to the introduction of
higher noise contamination level. The proposed procedure can be considered as
a stable and powerful denoiser for different levels of trace-wise noise.

(a) PSNR Change

(b) SSIM Change

Figure 4: PSNR value and PSNR change over noise levels (a), and SSIM value and
SSIM change over noise levels (b). The noisy data has a consistent standard
deviation of 0.1 and 0.002 for PSNR and SSIM, respectively.

#### Effect on number of noisy traces

To conclude the synthetic analysis, we investigate the effect of including
varying numbers of noisy traces within the ‘raw’ shot gathers. After obtaining
the 404 clean patches in Figure 2, 10 different noise realisations are
generated for each shot by randomly replacing 10-90% of the traces among the
total 256 traces with a random value between [-0.8, 0.8]. These represent the
4040 raw, noisy data samples as the target (label) for the deep learning
network. As before, 100 new test images are generated following the same
procedure as the training data generation. All the training parameters are as
detailed in Table 1 and are the same for the nine different cases. Again, the
PSNR and SSIM are both used to evaluate the denoising performance and are
averaged across the 100 test images.

(a) PSNR Change

(b) SSIM Change

Figure 5: PSNR value and PSNR change over numbers of noisy traces (a), and
SSIM value and SSIM change over numbers of noisy traces (b). The noisy data
has a consistent standard deviation of 0.1 and 0.02 for PSNR and SSIM,
respectively.

Figure 5 illustrates the relationship between the data quality (PSNR and SSIM)
values and percent changes over the percent of noisy traces. As shown in
Figure 5(a), as the percentage of noisy traces increases, both of the denoised
values and changes of PSNR increase before 30% and then drop, due to the
introduction of more trace-wise noise. In Figure 5(b), the denoised SSIM value
stays just below $1.0$ and the SSIM percent change increases almost linearly
before 60%. At around 60%, both the denoised SSIM value and the percent change
start to drop, highlighting that the denoising network is no longer working
under its optimal conditions. Both the PSNR and the SSIM of the noisy data
decrease gradually due to the introduction of higher number of noisy traces.
Both plots in Figure 5 show a positive change in SSIM and PSNR, indicating the
network can still successfully suppress a portion of the noise when up to 90%
of the raw shot is contaminted by noise. Therefore, the method can denoise
trace-wise noise contamination percentage of 90% traces in seismic data. The
method is shown to be optimal where less than 60% of the raw traces are noisy.

###  3.2 Field data application

Shot gathers from the Stratton land 3D dataset from South Texas are used to
verify the practical effectiveness of the proposed method. For this field data
example, a data patching technique is applied on the original noisy images of
size $1024\times 56$ to generate 8140 noisy patches of size $56\times 56$, as
shown in Figure 6. In other words, the data are patched into time windows
containing 56 samples, 112 ms, covering the full receiver array. Patching is
utilised to ensure adequate data is available for the network training. We
take 7000 noisy patches with $56\times 56$ pixels as our training dataset and
1140 noisy patches of the same size as the test dataset.

Figure 6: Schematic of the data patching workflow for Stratton 3D land
dataset. On the left is the full shot gather whilst on the right are three
extracted patches.

Figure 7(b) illustrates the result of the trained SSL model applied to the
field dataset. For comparative purposes, Figure 7(c) illustrates the denoising
results of a median filter (MF) applied on the raw data, with filter
parameters of $3\times 3$ footprint. The red arrows in Figure 7 denote the
traces 18, 19 and 25 - three noticably noisy traces that exist within the
field dataset and that need to be suppressed. Figure 7(b) and Figure 7(c) show
that the trained SSL network and the MF result in more consistent reflections
across the full shot, than observed in the raw shot. The SSL result is notably
more consistent than the MF result in both earlier and later arrivals,
indicating that the SSL procedure offers a better denoising performance than
the MF. The differences between the original noisy image, the SSL denoised
result and the MF result show that the majority of the coherent noise has been
removed by both methods with a notably larger noise removal provided by the
SSL procedure. The MF causes larger signal leakage at far offsets while the
SSL network causes larger signal leakage on later arrivals. For field data,
PSNR and SSIM values cannot be computed as we do not have the corresponding
noise-free data. Therefore, instead of evaulating the denoised results of
field data quantitatively, we judge the result in a zoomed-in comparison for
three areas of interest, as depicted in Figure 8. Figure 8(b) shows that the
SSL denoised result can fill the trace gaps that are present in the noisy
section better and can remove more noise than the MF (Figure 8(c)). Most of
the coherent noise has been removed by the two methods, however, both of them
have caused a small amplitude reduction to some of the seismic arrivals.

Figure 7: Comparison of the trained SSL model and MF applied to the field
dataset with coherent noise. (a) The original noisy data, (b) the result of
the SSL denoising procedure and difference between the noisy and denoised
result, and (c) the result of the MF and difference between the noisy and
denoised result. Red arrows illustrate the original noisy traces in raw data,
the corresponding denoised traces and the removed coherent noise,
respectively. Figure 8:  A close-up of the original data (a), SSL denoised
result (b), and MF denoised result (c).

##  4 Discussion

We have proven blind-trace networks to be efficient suppressors of trace-wise
noise in seismic data; and, only requiring noisy images to train the network
in a self-supervised way. Adapting the methodology of [6], a vertical noise
mask is designed in an attempt to suppress trace-wise noise for seismic data.
The network was successfully applied on field data and shows a better
denoising performance than a conventional, non-machine learning method-the
median filter. As there are some cases that seismic data is severely
contaminated by noise, synthetic tests were performed with different noise
levels and different numbers of noisy traces. Results show that the network
can satisfactorily handle the scenarios of both large noise amplitudes and
high numbers of noisy traces in the raw data. The potential of the blind-trace
network to suppress other trace-wise noise, such as deblending in seismic
data, can be explored based on this paper.

Unlike interpolating gathers to recover the missing information within a trace
(of value zero) across the receivers, the SSL network with the blind-trace
scheme learns to remove the existing trace-wise noise. In this sense, the SSL
model is not easily adaptable for regular interpolation tasks due to the
computation of the loss being against the original noisy, data. Despite this,
we analyse it’s ability as a tool for regular interpolation. Similar to the
other synthetic experiments, noisy realisations are generated for each shot in
the 404 clean patches in Figure 2 by regularly replacing every 60, 50, 40, 30,
20, 10 and 5 traces among the total 256 traces with a constant value of 0 to
represent the 404 raw, noisy data as the target (label) for the deep learning
network. The test images are noisy realisations generated from the clean data
by regularly replacing every 60, 50, 40, 30, 20, 10 and 5 traces with zero
values. All the training parameters are as detailed in Table 1 and are the
same for the seven different cases. PSNR and SSIM are both used to evaluate
the denoising performance, computed against the corresponding noise-free
synthetic data. Figure 9 illustrates the denoised results of different trace-
wise noise patterns of contamination for synthetic seismic shot gathers. For
all seven cases, the differences between the noisy and denoised datasets
demonstrate some amount of noise are removed by the trained network but it
struggles to recover the noisy traces completely. The difference between the
noise-free and denoised datasets demonstrates that the majority of the noise
still remains.

Figure 9: Trained SSL model applied to a synthetic dataset with different
patterns of coherent noise. (a)-(e) The noise-free synthetic, the noisy
synthetic given as input to the model, the result of the SSL denoising
procedure, the difference between the noisy and denoised datasets and the
difference between the noise-free and denoised datasets, respectively, for the
seven different noise patterns of contamination.

One limitation of the blind scheme is the design of noise mask. In field data,
coherent noise often exhibits a relationship in both time and space, which
makes it difficult to design a precise shape of noise mask. However, this
paper can still provide inspirations to some other different noise types. For
example, industrial noise from nearby power stations for carbon dioxide
($\mathrm{CO_{2}}$) storage applications, as observed by [35], traffic from
nearby roads, as observed by [3] and [8], and tube waves within borehole
monitoring [18], to mention a few possibilities.

##  5 Conclusions

Blind-trace networks allow a self-supervised denoising scheme requiring only
the noisy seismic data, as opposed to the common requirement of clean data as
training labels. Contrary to previous applications where blind-spot networks
are shown to be good random denoisers assuming the unpredictability of the
targeted noise based on neighbouring values; in this paper, we have proposed
an adaptation of the blind-spot strategy to remove temporally-dependent noise.
By designing a vertical noise mask, the trace-wise, targeted noise can be
suppressed in a self-supervised manner. The proposed method creates training
and validation sets based on only the raw noisy shot gathers; to be specific,
the raw noisy shot gathers become the network’s labels and corrupted versions
of the raw shot gathers, created by the blind-trace strategy, become the
network’s inputs. Such adaptations make the proposed method capable of
removing coherent noise through a well-trained CNN model. Experiments on
synthetic and field seismic shot gathers have shown that the proposed method
can effectively suppress trace-wise coherent noise in seismic data with
minimal damage to the signals. Furthermore, we illustrate that the denoising
scheme performs well with different levels and different numbers of trace-wise
noise contamination, highlighting its robust nature to varying noise levels.

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