THE VI INTERNATIONAL SCIENTIFIC CONFERENCE “SCIENTIFIC FOUNDATIONS FOR THE USE OF
INFORMATION TECHNOLOGIES OF A NEW LEVEL AND MODERN PROBLEMS OF AUTOMATION”,
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190
MODELING OF SEISMIC DATA PROCESSING USING
INTEGRAL GEOMETRY PROBLEMS ON A FAMILY OF
BROKEN LINES
A.K.Seidullea 1,2, S.S.Nizama dino a1
1Ka akalpak S a e Uni e si y, Nukus, Uzbekis an
2V. I. Romano skiy Ins i u e o Ma hema ics, Uzbekis an Academy o Sciences, Tashken ,
Uzbekis an
h ps://doi.o g/10.5281/zenodo.17740297
In eg al geome y p oblems p o ide a ma hema ical amewo k ex ensi ely applied in
seismological da a p ocessing. I in ol es he ans o ma ion o seismic wa e o ms in o al e na i e
domains o enhance analy ical ac abili y and in e p e a i e cla i y. This app oach is pa icula ly
e ec i e o de ec ing and cha ac e izing linea ea u es wi hin seismic da ase s, which a e
indica i e o cons an ay pa ame e s and play a c i ical ole in subsu ace imaging.
The Radon ans o m, commonly e e ed o as he slan s ack, is a ma hema ical echnique
employed o ep ojec seismic da a om he con en ional ime-o se domain in o he
p
−
domain, whe e
deno es he in e cep ime and
p
ep esen s he ay pa ame e [1]. This
ans o ma ion acili a es he alignmen o seismic aces along ajec o ies o cons an slowness,
he eby enhancing he in e p e abili y o wa e ield cha ac e is ics. I is pa icula ly ad an ageous
o analyzing high- esolu ion e lec ion and e ac ion da ase s, especially hose acqui ed om
linea sou ce geome ies in ho izon ally laye ed media. The me hod p o es e ec i e in isola ing
and in e p e ing nea - e ical e lec ion e en s, con ibu ing o imp o ed subsu ace imaging and
eloci y model es ima ion [2].
This s udy add esses he in e se p oblem o econs uc ing an unknown unc ion
( , )u x y
, which ep esen s he spa ial dis ibu ion o a wa e ield om obse ed seismic measu emen s
( , ) x y
. The p oblem is o mula ed wi hin he amewo k o in eg al geome y, speci ically o e
a amily o b oken-line ajec o ies. An analy ical ela ionship is de i ed ha connec s he in eg al
ans o ms o he unknown wa e ield unc ion
( , )u x y
wi h he measu ed da a
( , ) x y
.
Fu he mo e, i is demons a ed ha he solu ion can be exp essed h ough he applica ion o he
Laplace ope a o o he known unc ion
( , ) x y
, p o iding a pa hway o e icien compu a ional
eco e y.
We in es iga e a class o in eg al geome y p oblems cha ac e ized by he equa ion
( ) ( )
( )
( )
,
,,
xy
y u ds x y
−=
whe e
( )
,u
is he unknown unc ion ep esen ing he wa e ield dis ibu ion, and
( , ) x y
is a known unc ion de i ed om seismic measu emen s. The in eg a ion is pe o med
along he cu e
( , )xy
, de ined by he condi ion:
( )
, , ,0x y x y x R y H
= − = −
whe e
ds
deno es he di e en ial a c leng h along he cu e. This o mula ion models he
p opaga ion o seismic wa es along b oken-line ajec o ies and se es as a ounda ion o
THE VI INTERNATIONAL SCIENTIFIC CONFERENCE “SCIENTIFIC FOUNDATIONS FOR THE USE OF
INFORMATION TECHNOLOGIES OF A NEW LEVEL AND MODERN PROBLEMS OF AUTOMATION”,
NOVEMBER 20, 2025
191
econs uc ing he unde lying wa e ield om in eg al measu emen s.
Theo em. Le
( , ) x y
be a p esc ibed unc ion de ined on he domain
[0, )
H
LH=
, and le
( , )u x y
deno e an unknown unc ion ha is ini e on
H
L
and sa is ies he in eg al
equa ion
( , )
( ) ( , ) ( , ),
xy
y u ds x y
−=
whe e
( , )xy
ep esen s a speci ic cu e in he domain, and $ ds $ deno es he di e en ial
a c leng h along his cu e.
Unde he assump ion ha
2
0
( , ) ( )
H
u x y C L
he solu ion o his in e se p oblem is
unique and can be explici ly exp essed in e ms o he known unc ion
( , ) x y
ia he ela ion
2
22
22
2
( , ) ( , ),
4
u x y x y
xy
= −
whe e
22
22
xy
= +
deno es he Laplace ope a o . This esul p o ides a di ec
analy ical mechanism o eco e ing he wa e ield dis ibu ion om in eg al measu emen s along
b oken-line ajec o ies.
Nume ical expe ience. To alida e he heo em nume ically, we conside ed a
smoo h es unc ion
( )
,u x y
on a pe iodic domain
)
)
0, 0,
xy
LL
wi h
2
x
L=
and
1
y
L=
,
sampled on a
192 192
g id. The chosen unc ion was a combina ion o Fou ie modes:
( )
2 4 4 6
, sin sin 0.6cos sin
eal
x y x y
x y x y
u x y L L L L
=+
.
Using spec al di e en ia ion, he Laplacian in Fou ie space is:
( ) ( ) ( )
22
ˆ
,,
x y x y x y
u k k k k u k k = − +
,
whe e
22
,
xy
xy
mn
kk
LL
==
o in ege modes
,mn
. Using FFT-based spec al
di e en ia ion, we compu ed
u
and hen eco e ed
( )
, x y
om he iden i y
( )
2
22
xy
u c = −
whe e
2
4
c=
. In Fou ie space, he ope a o
( )
2
22
xy
−
becomes
( )
2
22
yx
kk−
. So:
( ) ( )
( )
2
22
,
ˆ,xy
xy
yx
u k k
k k c k k
=−
, o
( )
2
22
yx
kk
−
,
whe e
is a small cu o o a oid di ision by ze o. Nex , we econs uc ed
explici
u
by
applying he explici o mula:
Compu e
( )
( )
2
22
,xy
g x y c = −
, in Fou ie space
THE VI INTERNATIONAL SCIENTIFIC CONFERENCE “SCIENTIFIC FOUNDATIONS FOR THE USE OF
INFORMATION TECHNOLOGIES OF A NEW LEVEL AND MODERN PROBLEMS OF AUTOMATION”,
NOVEMBER 20, 2025
192
( ) ( ) ( )
2
22
ˆ
ˆ,,
x y y x x y
g k k c k k k k=−
.
Then we sol e
explici
ug=
, so
( ) ( )
( )
explici 22
ˆ,
ˆ,xy
xy
xy
g k k
u k k kk
=−+
,
se ing he ze o-mode o ze o (ze o-mean solu ion).
We obse e he nume ical isualiza ion o he a o emen ioned o mulas using nume ical
me hods as ollows:
Fig1. Hea map o
( )
eal ,u x y
(ze o mean)
Fig2. Hea map o
( )
explici ,u x y
Fig3. E o map
eal explici
uu−
Fig4. P o ile a
2
y
yL=
We sub ac he mean om bo h ields and compu e:
( )
2
eal explici
1
RMSE u u
N
=−
,
eal explici
22
eal 2
Rela i e uu
Lu
−
=
,
( )
eal explici
eal explici
,
Co ela ion Co u u
=
.
The nume ical expe imen con i ms he heo e ical esul wi h high accu acy. S a ing om
a p esc ibed smoo h es unc ion
( )
,u x y
, we compu ed he Laplacian
u
and
( )
, x y
.
THE VI INTERNATIONAL SCIENTIFIC CONFERENCE “SCIENTIFIC FOUNDATIONS FOR THE USE OF
INFORMATION TECHNOLOGIES OF A NEW LEVEL AND MODERN PROBLEMS OF AUTOMATION”,
NOVEMBER 20, 2025
193
We hen econs uc ed
explici
u
by applying he ope a o
( )
2
22
xy
c −
o
and sol ing he
Poisson equa ion
ug=
spec ally. The compa ison be ween he o iginal
eal
u
and he
econs uc ed
explici
u
yielded an RMSE o app oxima ely
13
1.4 10−
, a ela i e
2
L
e o o
13
2.4 10−
, and a co ela ion coe icien o nea ly 1.0, indica ing ag eemen o machine p ecision.
Visualiza ions o bo h ields and hei di e ence con i med ha he explici o mula p o ides an
accu a e and s able mechanism o eco e ing he unknown unc ion om in eg al measu emen s.
This demons a es ha he heo e ical app oach is nume ically easible and highly eliable when
implemen ed wi h spec al me hods on smoo h pe iodic da a.
REFERENCES
1. Munadi, S. (2022). The hype bolic adon ans o m and some o i s applica ion in seismic da a
p ocessing. Scien i ic Con ibu ions Oil and Gas, 15(1), 12–19.
h ps://doi.o g/10.29017/scog.15.1.1113
2. Chapman, C. H. (1981). Gene alized Radon ans o ms and slan s acks. Geophysical Jou nal
In e na ional, 66(2), 445–453. h ps://doi.o g/10.1111/j.1365-246x.1981. b05966.x
