ISSN 10693513, Izvestiya, Physics of the Solid Earth, 2014, Vol. 50, No. 4, pp. 508–513. © Pleiades Publishing, Ltd., 2014. Original Russian Text © P.S. Martyshko, N.V. Fedorova, E.N. Akimova, D.V. Gemaidinov, 2014, published in Fizika Zemli, 2014, No. 4, pp. 50–55.
Studying the Structural Features of the Lithospheric Magnetic and Gravity Fields with the Use of Parallel Algorithms P. S. Martyshkoa, c, N. V. Fedorovaa, E. N. Akimovab, c, and D. V. Gemaidinova, c a
Institute of Geophysics, Ural Branch, Russian Academy of Sciences, ul. Amundsena 100, Yekaterinburg, 620016 Russia email:
[email protected],
[email protected],
[email protected] b Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, ul. Sof’i Kovalevskoi 16, Ekaterinburg, 620990 Russia c Ural Federal University, ul. Mira 19, Ekaterinburg, 620002 Russia email:
[email protected] Received March 26, 2013
Abstract—We describe the parallel algorithms for studying the structural features of the anomalies in the gravity and magnetic fields of the lithosphere, which are based on the height transformations of the data. The algorithms are numerically implemented on the Uran supercomputer. The suggested computer technology is used for constructing the maps of the regional and local anomalies of the magnetic and gravity fields for the northeastern sector of Europe within an area confined between 48°–62° E and 60°–68° N. DOI: 10.1134/S1069351314040090
INTRODUCTION In the geophysical practice, the problems of sepa rating potential fields into their components are often solved by the methods of downward and upward con tinuation (recalculation of a field into the upper and lower halfspace). By reducing the obtained results to the initial height level one can separate the low and highfrequency components of the amplitude spec trum of the anomalies generated by the lithosphere. Martyshko and Prutkin (2003) and Martyshko et al., (2010) suggested the following procedure of field transformation for separating the sources of gravity along the height. The initial stage involves solving the problem of separating the effects of the sources located between the surface and a certain depth H. The field is continued upwards to level H, due to which the effects of the local nearsurface sources (up to a depth H) are significantly suppressed if not eliminated altogether. In order to get rid of the influence of the local sources located in the horizontal layer between the surface and depth H, the field, which was calculated to the height level above the initial surface, is then continued down wards to depth H. Since this problem refers to the class of illposed problems, its solution requires the use of the regularization methods. At the final step of the procedure, the field is again continued upwards to the initial surface. The resulting field can be treated as the field of the sources that are located below level H. Sub traction of this field from the observed one gives the field that is generated by the layer. By repeating this
procedure for different heights and depths, we can separate the fields generated by the layers within the corresponding boundaries. This technique of separating the gravity field was used for constructing the density model of the Earth’s crust for the Ural region (Martyshko et al., 2010; 2011). The studies of vast territories require handling extensive data, which makes the required calculations on the singleprocessor machines highly timecon suming. By applying the parallel algorithms for multi processor computing systems, the time of the calcula tions can be significantly reduced. In this work, we describe the mathematical appara tus and algorithms of parallel computations that are used for designing the computer technology and present the results of applying this technology for the study of the structural features in the magnetic and gravity fields of the lithosphere in the northeastern sector of Europe.
MATHEMATICAL APPARATUS OF THE TRANSFORMATIONS We introduce the rectangular Cartesian coordi nates with axis Z pointing downwards and plane XOY coinciding with the surface of the observations. The upward continuation of the field U (x, y, z) z =0 mea
{
}
sured on area D = (x, y) ∈ R 2: a ≤ x ≤ b, c ≤ y ≤ d of
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the Earth’s surface to level z = –H is calculated by Poisson formula: +∞ +∞
U (x, y, −H ) = 1 2π
∫ ∫ ⎡(x − x')
−∞ −∞ ⎣
H + (y − y')2 + H 2 ⎤⎦
2
32
+∞ +∞
Ku ≡ 1 2π
∫ ∫ ⎡(x − x')
−∞ −∞ ⎣
H 2
+ (y − y')2 + H 2 ⎤⎦
32
(2)
× u(x', y',0)dx' dy' = U (x, y, H ). Solving the integral equation of the first kind (2) is an illposed problem, which requires application of the regularization methods. The operator of Eq. (2) is positive definite and selfadjoint, therefore we can apply the scheme of M.M. Lavrent’ev (1962). After the grid discretization of Eq. (2) and approx imation of the integral operator by the quadrature for mulas, the problem is reduced to solving the system of linear algebraic equations (SLAEs) with the symmet ric matrix K. In the regularized form, the system is (K + α I )u = U ,
(3) where I is the identity matrix and α is the regulariza tion parameter. For solving the SLAEs, we can apply the gradient type iterative methods, which include the following methods (Vasin and Eremin, 2005): (1) The iteratively regularized simple iteration method (4) u k +1 = u k − 1 ⎡⎣( K + αI ) u k − U ⎤⎦ , λ max where λmax is the maximal eigenvalue of matrix K + α I (symmetric case); (2) the minimal residual method
u
k +1
=u
k
( K ( Ku −
k
)
− U , Ku k − U
(
K Ku − U k
)
2
) Ku (
k
)
− U ; (5)
(3) the method of minimal error
Ku − U k
u
k +1
=u − k
K
T
(
2
Ku − U k
)
2
K
T
( Ku
k
)
−U ;
(6)
)
(7)
(4) the method of steepest descent
u
k +1
=u − k
K (Ku − U ) T
k
T
( Ku
KK
k
−U
2
)
2
K
T
( Ku
k
−U .
IZVESTIYA, PHYSICS OF THE SOLID EARTH
We note that for methods (5)–(7) in the regularized form, matrix K is replaced by K + α I. The stopping criterion in the iterative processes (4)– (7) is the fulfillment of the condition
(1)
× U (x', y',0)dx' dy'. For recalculating the filed downwards to depth z = H and finding the values U ( x, y, −H ), we solve Fredholm equation of the first kind:
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Ku k − U U
1), T1 is the execution time of a sequential algorithm on a single processor, and Tm is the sum of the pure time of computations and the time of the interprocessor exchanges Tm = Tc + To. Due to the memory limitations, efficient solution of the problem on a single Uran processor is impossi ble. With the technology of data storing to the hard drive and retrieving the resulting solution vector uk from the hard drive at each iteration, the time of com putations increases by more than a factor of 1000 and measures 62 h for a 300 × 300 grid. Since in our case the efficiency of the solution can not be estimated by formula (8), we suggest estimating it by the notion of scaling the parallel algorithm. The scaling factor of a parallel algorithm is under stood as the possibility to speed up the computations by a factor of M, proportional to the number of the used processors m, considering the fact that the vol ume of the data fits the memory limitations for each processor: (9) M m = Tmin Tm, where Tmin is the execution time of the parallel algo rithm on the minimum possible number of processors with allowance for the volume of the allocated mem ory. The execution time of one height transformation of the field (the successive solution of problems (1), (2), and again (1) in accordance with the described proce dure of the method) on the Uran cluster with the use of PAMV and PMSI parallel algorithms (Parallel
Algorithm of Matrix multiplication by the Vector and Parallel Method of Simple Iteration, respectively) and the values of the scaling factor as a function of number m of the used processors are presented in the table. The computational experiments show that the par allel algorithms have a high degree of scaling, since, as the number of the processes increases by a factor of M, the time of the computations also increases by a factor of M, and the speedup increases linearly. THE RESULTS OF THE COMPUTATIONS FOR THE NORTHEASTERN SECTOR OF EUROPE We applied the constructed parallel algorithms for studying the structural features of the gravity and mag netic anomalies over a large part of northeastern Europe, which is confined between 48°–62° E and 60°–68° N. Tectonically, this area comprises the northeastern part of the Russian (East European) Platform (RP), the Timan–Pechora Plate (TPP), and the northern segment of the Ural folded system adja cent to it in the east (Fig. 1). The amplitudes of the magnetic anomalies ΔТа do not exceed 3000 nT, which means that we can apply the methods developed for the harmonic functions (Magnitorazvedka..., 1990). In our calculations, we used the numerical models of the magnetic and gravity fields that are based on the car tographic data (Ovcharenko, 1996). The computations on the Uran cluster were carried out for the heights H = 5, 10, 20, 50, and 100 km. For each transformation, the field was recalculated upwards to height H by formula (1), then continued downwards to a depth 2H by formula (2), and again calculated upwards to height H by formula (1). The size of the computation area is 780 × 900 km2. The grid steps are Δx = 2.6 km and Δy = 3 km. After the grid discretization of Eq. (2) and approximation of the integral operator by the quadrature formula, the prob lem of the downward continuation of the field was reduced to solving the SLAEs with matrix K of size 3002 × 3002. After conducting a series of the computa tional experiments, it was decided to solve SLAE (3) by the method of simple iteration (4), which is the least laborious, while still as accurate as the other methods. For selecting the regularization parameter α, we used the results of the interpretation of the gravity and magnetic anomalies by the twodimensional method (Tsirul’skii et al., 1980). The deep structure of the Ural region was studied along the geotraverses and DSS (deep seismic sounding) profiles (Fedorova et al., 2010; Martyshko et al., 2011), part of which fall in the considered territory (Fig. 1b). As a result of the approximation of the gravity and magnetic fields, we obtained the distribution of the anomalous sources in the crustal layers. This allowed us to compare the anomalies from the model sources with the computed transformations of the gravity and magnetic fields for
IZVESTIYA, PHYSICS OF THE SOLID EARTH
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STUDYING THE STRUCTURAL FEATURES (a)
68°
(b)
68°
Agat
511
Malo zemel’skii Swell
TPP
nT
Varandei– Adz’va Zone
Khoreiver Pechora– Depression
nT
Kolva
Izhma
1000
Depression
68°
62°
Komi– Perm' Swell
0
(c)
Uplift
Western Uralian Zone
–200 Uralia n
Tr ou gh
–100 Eastern Uralian Uplift
60° 48°
im Tro ugh
Sysol'sk Swell
Ko zh
60° 48°
–600
Cis–Uralian Trough
Vy ch eg da
–400 ii insk olen n s a Kr
0
RP
–200
Ru bin –1
100
Tagil Sync linorium
0
200
Urals
Central
200
lift Up an Tim
Mezen'’S yneclise
400 us lob G
300
Zone
Pechora
Omroq– Luza Saddle
Kvarts
–300
500 km 62° (d)
68° nT
60° 48°
nT
200
200
100
20
0
0
–100
–20
–200
–200 60° 48°
62°
62°
Fig. 1. The maps of the magnetic field over the northeastern sector of Europe: (a) anomalous field; (b) lowfrequency component; (c, d) local anomalies calculated for (c) H = 20 km and (d) H = 5 km. The lines depict (a) DSS geotraverses and (b) the boundaries of the main tectonic structures of the region.
the set of the values of the regularization parameter and select the value of α for the further computations by the criterion of the best fit of the graphs of the fields. The anomalous magnetic field within the north eastern segment of Europe is shown in Fig. 1a. The neighboring Fig. 1b presents the map of the regional component obtained by the upward continuation of the field to a height of 20 km and its subsequent recal culation back to the zero level. This map is superim posed on the tectonic scheme of the region. In the geophysical practice, the local anomalies in the gravity and magnetic fields within the unexposed territories are efficiently used for determining the depths of the crystalline basement. By analyzing the distributions of local anomalies, one can reveal the faults in the crust, identify the magmatic intru sions, etc. IZVESTIYA, PHYSICS OF THE SOLID EARTH
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We used the computed data on the gravity and mag netic fields at H = 5 km and H = 20 km for construct ing the maps of local anomalies in the upper litho sphere of the northeastern European sector. The map (Fig. 1c) shows the magnetic anomalies whose sources are located in the upper layer of the Earth’s crust up to a depth of 20 km. The highestfrequency part of the anomalous magnetic field is illustrated in Fig. 1d. These maps are constructed by calculating the differ ence between the initial anomalous field ΔТа and the transforms ΔТа20 and ΔТа5 obtained by the continua tion of the field ΔТа to the depths H = 20 km and H = 5 km, respectively. The maps of the anomalous gravity field for the northeastern segment of Europe and the field trans forms are shown in Fig. 2. It can be seen that the maps of local anomalies (Figs. 1c, 1d and 2c, 2d) most dis tinctly reflect the belts of ultrabasic and basic intru No. 4
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MARTYSHKO et al. (a)
(b)
68°
68°
Malo zemel’skii Swell
TPP
mGal
Varandei– Adz’va Zone
Khorei ver Pechora– Depression
mGal
Kolva
Izhma Zone
Pechora
50
50
(c)
Uplift
Western Uralian Zone
Uralia n
0
Eastern Uralian Uplift
Komi– Perm' Swell
Tagil Sync linorium
Tro ugh
Sysol'sk Swell
60° 48°
62°
Cis–Uralian Trough
Vy ch eg da Tr ou gh
0
Central
60° 48°
RP
Ko zhi m
–50
Urals Omroq– Lu Saddle za
0
lift Up an Tim
Mezen'’S yneclise
Depression
–50
500 km 62° (d)
68°
68°
60° 48°
mGal
mGal
20
20
0
0
–20
–20
62°
60° 48°
62°
Fig. 2. The maps of the gravity field over the northeastern sector of Europe: (a) anomalous field; (b) lowfrequency component; (c, d) local anomalies calculated for (c) H = 20 km and (d) H = 5 km. The lines depict (b) the boundaries of the main tectonic structures of the region.
sions and other signs of magmatic and tectonic activity in the region. In the future, the results of field separa tion can be used for constructing the volumetric mod els of the Earth’s crust for the northeastern part of Europe. CONCLUSIONS
With the use of the computer technology devel oped, the maps of the regional and local anomalies of the gravity and magnetic fields are constructed for the northeastern segment of Europe. It is shown that for large data volumes, paralleling the algorithms signifi cantly reduces the time of the computations. ACKNOWLEDGMENTS
Parallel algorithms are developed for solving the problems of the height transformation of gravity and the magnetic fields to different levels, which makes it possible to separate the anomalies in the different intervals of the wavelengths. The algorithms are numerically implemented on the Uran cluster with highly efficient paralleling and scaling.
The work was supported by the Ural Branch of the Russian Academy of Sciences. The parallel algorithms and program codes are designed in basic research pro gram no. 18 of the Presidium of the Russian Academy of Sciences (project no. 12P152019). The studies of the deep structure of the Timan–Pechora region are carried out under project no. 2I52067.
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REFERENCES Akimova, E.N. and Gemaidinov, D.V., Parallel algorithms for solving the gravimetry problem of reconstructing the density in the layer, Tr. Inst. Mat. Mekh. Ural. Otd. Rus. Akad. Nauk, 2007, vol. 13, no. 3, pp. 3–21. Akimova, E.N., Parallel algorithms for solving the inverse problems of gravimetry and magnetometry on MVS1000, Vestnik Nizhegorodskogo Univ., 2009, no. 4, pp. 181–189. Fedorova, N.V., Kolmogorova, V.V., and Ladovskii, I.V., Geophysical models of crustal structure along the Krasno leninskii profile, Ural’skii Geofiz. Vestnik, 2010, no. 1, pp. 59–68. Lavrentiev, M.M., O nekotorykh nekorrektnykh zadachakh matematicheskoi geofiziki (On Some IllPosed Problems of Mathematical Geophysics), Novosibirsk: SO RAN SSSR, 1962. Magnitorazvedka: Spravochnik geofizika (Magnetic Pros pecting: Handbook for Geophysicist), Moscow: Nedra, 1990. Martyshko, P.S. and Prutkin, I.L., Technology for separat ing the gravity sources by the depth, Geofiz. Zh., 2003, vol. 25, no. 3, pp. 159–168. Martyshko, P.S., Ladovskii, I.V., and Tsidaev, A.G., Con struction of regional geophysical models based on the joint
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interpretation of gravity and seismic data, Izv., Phys. Solid Earth, 2010, vol. 46, no. 11, pp. 931–942. Martyshko, P.S., Druzhinin, V.S., Nachapkin, N.I., Osipov, V.Yu., Fedorova, N.V., Ladovskii, I.V., Kolmogor ova, V.V., and Tsidaev, A.G., Method and results of con structing the volumetric geologicalgeophysical model for the upper part of the lithosphere within the northern and middle segments of Ural region, in Dinamika fizicheskikh polei Zemli (Dynamics of the Physical Fields of the Earth), Moscow: IFZ RAN, 2011, pp. 9–30. Ovcharenko, A.V., Computer database of the geofields for Urals: informational framework for the new stage in the study of the region, in Glubinnoe stroenie i razvitie Urala (Deep Structure and Evolution of Urals), Yekaterinburg: Nauka, 1996. Tsirul’skii, A.V., Nikonova, F.I., and Fedorova, N.V., Metod interpretatsii gravitatsionnykh i magnitnykh anomalii s pos troeniem ekvivalentnykh semeistv reshenii (Method for Interpretation of Magnetic and Gravity Anomalies with Constructing Equivalent Families of the Solutions), Sverd lovsk: UNTs AN SSSR, 1980. Vasin, V.V. and Eremin, I.I., Operatory i iteratsionnye pro tsessy feierovskogo tipa. Teoriya i prilozheniya (Feyer Type Operators and Iterative Processes: Theory and Applica tions), Yekaterinburg: UrO RAN, 2005.
Translated by M. Nazarenko
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