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represents the conditional simulation that reproduces the distribution of conditional to the Y-data.
Conditioning to the actual samples is done as a post-process through the method of kriging residuals. Let
be a non-conditional simulation, the random field defined by:
where are lines randomly distributed over the noitcnuf ecnairavoc htiw eulav modnar a si , erehps
and is the location where we want to simulate. In taht esoporp srohtua lareves ,esac lanoisnemid-eerht eht
is a number of lines su cient to match the expected statistics of the random function.
The final simulated value is obtained as the projection of values from many lines:
For every location on the three dimensional simulated grid, the simulated values generated over the line are projected orthogonally into the simulated location.
Simulated values are generated over lines randomly oriented over the unit sphere, with covariance function .
4
3
2
1
conditioning
simulation
Turning Bands simulation can be used to generate multiple stochastic scenarios of such spatial distribution.
TURNING BANDS METHOD
GPUN-2 GPUN-1GPU1
GLOBAL MEMORY
K K + 1 K + 1 K + 1
M
2K + 1
SHARED MEMORY
2K + 1
K + 1
t = 0t = 0
t = 1
M + (K + 1)
(K + 1) - Kt =
K + 1K
SORTED
M + (K + 1) (K + 1)
- Kafter t =
Knn
SHARED MEMORY
for K timesrepeat
d4
d3
d2
d1
d2k+1
2K + 1
thread 1 thread kthread 2
d1
d2
d3
if > :d1
d2
swap(d , d )1 2
if > :
swap(d , d )
d2
32
d3
THREAD
block N
...
block 1
block 0
GPU GRID
K threads
Nblocks
thread K-1
...
thread 1
thread 0
K
K
parallel threadexecution
syncrhonizedby column
L ( thread id, j ) = U ( j, thread id ) (3)
UL
K
K
Usecuential
threadexecution
block N
...
block 1
block 0
GPU GRID
K threads
Nblocks
thread K-1
...
thread 1
thread 0
Weights
=x
STEP 3 RESULT
block N
...
block 1
block 0
GPU GRID
K threads
Nblocks
thread K-1
...
thread 1
thread 0
K
K
secuentialthread
execution
LY
Cov (point, knn )6()
=x
knn
LU Decomposition
Linear System Solver
conditioning
thread row K-1
...
thread row z
...
thread row 1
thread row 0
M
threads
block N
...
...
block 1
block 0
GPU GRID
M
threads
Npoints
K threads
K
SHARED MEMORY
thread
M-1
...
thread 1
thread 0
M threads
Y (x j ) =1
√N l
N l
i=1
ψi (< x j , U i > ) (9)
REDUCE
M threads
simulation
GPU0
malloc
save results
pseudorandom number generation
CPU
0data chunk
0results
GPUN
Ndata chunk
Nresults
timeline
METALS
COPPER
GOLD
SILVER
MOLYBDENUM
32%
2%
5%
14%
SHARE IN GLOBALPRODUCTION
1º
14º
8º
3º
RANKING
28%
8%
14%
21%
SHARE IN GLOBAL ORE RESERVES
MINERAL RESERVES AND PRODUCTION IN CHILE - 2012
(*) Chilean Mining Council, "Minería en Cifras" 2013
The mining industry drives the chilean economy, representying more than 22% of chilean GDP and about 55% of its exports, being its main industry.
In 2013 Chile broke the world record in copper production, delivering more than 5,700,000 metric tonnes. It is expected that chilean mining will grow by 5% during 2014 and the copper production will rise up to 6 million metric tonnes.
Besides copper, chilean mining production also includes other minerals as shown in the following table (*):
CHILE AND THE MINING INDUSTRY
PROCESS &STORAGE
OPERATIONDEVELOPMENTPLANNINGEXPLORATION &MODELLING
SIMULATIONESTIMATIONREALITY
Mineral reserves are quantified using simulation methods to characterize the distribution of metal concentrations over space, from a limited number of drillhole samples available at few locations.
MINING PRODUCTION PROCESS
Time results obtained using:
Conditioning data : 20.000 samples
- 2 GPU : Tesla 2050 and Tesla 2075- 1 CPU : Intel Xeon 3.3 Ghz x 1 Core (16 GB Ram)
GPU PARALLELIZATION OF GEOSTATISTICAL SIMULATION FOR MINERAL RESERVES QUANTIFICATIONDaniel Baeza, Oscar Peredo, Felipe Navarro, Julián OrtizALGES Laboratory, Advanced Mining Technology Center (AMTC), University of Chile http://alges.cl
1MM
2MM
4MM
6MM
8MM
10MM
Grid
siz
e
time [minutes]
0 160 320 480 640 800
767
530
395
230
128
59
CPU simulation
CPU simulation + conditioning
GPU simulation
GPU simulation + conditioning
58x
1 GPU
84x
2 GPU
Speedups average
10x
1 GPU
18x
2 GPU
Speedups average
RESULTS AND SPEEDUPS
REFERENCES- G. Matheron, The intrinsic random functions and their applications, Advances in Applied Probability 5 (3) (1973) 439–468.- A. Journel, Geostatistics for conditional simulation of ore bodies, Economic Geology 69 (5) (1974) 673–687.- X. Emery, A turning bands program for conditional co-simulation of cross-correlated gaussian random fields, Comput. Geosci. 34 (12) (2008) 1850– 1862.- X. Emery, C. Lantuéjoul, Tbsim: A computer program for condi- tional simulation of three-dimensional gaussian random fields via the turning bands method, Comput. Geosci. 32 (10) (2006) 1615–1628.
contact name
Daniel Baeza: [email protected]
P4248
category: Climate, Weather, OCean mODeling - CW02
