lesson 02 pend inversi geofisika
TRANSCRIPT
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Pendahuluan Inversi Geofisika
Pelajaran 02
Dr. Sugeng Pribadi
2016
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Outline
• Metode Kuantitatif
• Teknik Pendukung
–Pemrograman Matlab
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Part 2
(or theories)
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A. Implicit Theory
relationships between the data and the model are known
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Example
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measure
mass,
size, , , ,
d=[ , , ,
density,m=[
f 1(d,m)=0
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note
No guarantee that
contains enough informationfor unique estimate m
determining whether or not there is enoughis part of the inverse problem
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B. Explicit Theory
the equation can be arranged so that is a function of
= N one equation per datum
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Example
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measure
C== d=[ ,
want to knowL=H=
m=[ ,
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C. Linear Explicit Theory
the function g(m) is a matrix times
N rows and M columns
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C. Linear Explicit Theory
the function g(m) is a matrix times
N rows and M columns“data kernel”
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Example
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D. Linear Implicit Theory
The relationships between the data are linear
rows
N+M columns
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in all these examples m is discrete
discrete inverse theory
one could have a continuous m(x) instead
continuous inverse theory
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as a discrete vector m
in this course we will usually approximate
a continuous m(x)
m = [m( ), m(2 , m(3 … m(M ]T
but we will spend some time later in
the course dealing with the continuous
problem directly
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Part 3
Some Examples
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0.5
1
d e g C )
A. Fitting a straight line to data
1965 1970 1975 1980 1985 1990 1995 2000 2005 2010-0.5
0
time, t (calendar years)
t e m p
e r a t u r e a n o m a l y , T i
(
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each data pointis predicted by a
straight line
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matrix formulation
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B. Fitting a parabola
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each data pointis predicted by a
strquadratic curve
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matrix formulation
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straight line parabola
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Soal (1)
1. Inversi linier, temperatur (Ti ) naik bertambah
kedalaman dengan proses (m1,m2),
m1 + m2 zi = Ti
NO DEP TEMP
1 5 35.4
2 16 50.1
a, b, m = variable model
T = variabel data (terikat)
z, G = variabel bebas
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3 25 77.3
4 40 92.3
5 50 137.6
6 60 147.07 70 180.8
8 80 182.7
9 90 188.5
10 100 223.2
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Soal (1)
1. Inversi linier, temperatur (Ti ) naik bertambah
kedalaman dengan proses (m1,m2),
m1 + m2 zi = Ti
NO DEP TEMP
1 5 35.4
2 16 50.1
a, b, m = variable model
T = variabel data (terikat)
z, G = variabel bebas
=
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3 25 77.3
4 40 92.3
5 50 137.6
6 60 147.07 70 180.8
8 80 182.7
9 90 188.5
10 100 223.2
m1 + m2 z1= T1
m1 + m2 z2= T2
………………….
m1 + m2 z1= T10
1 z1 t1
1 z2 t2
1 z3 t3
1 z4 m1 t4
1 z5 m2 t51 z6 t6
1 z7 t7
1 z8 t8
1 z9 t9
1 z10 t10
G.m = d
GT .G.m = GT .d
[GT .G]-1 . GT .G.m = [GT .G]-1 . GT .d
m = [GT .G]-1 . GT .d
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Soal (2)2. Inversi parabola, modifikasi data temperatur Vs
kedalaman, lebih rumit,
m1 + m2 zi + m3 zi 2 = Ti
NO DEP TEMP
1 5 20.8
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2 8 22.6
3 14 25.3
4 21 32.7
5 30 41.5
6 36 48.2
7 45 63.7
8 50 74.6
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Soal (2)2. Inversi parabola, modifikasi data temperatur Vs
kedalaman, lebih rumit,
m1 + m2 zi + m3 zi 2 = Ti
NO DEP TEMP
1 5 20.8
1 z1 z1^2 t1
1 z2 z2^2 t2
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2 8 22.6
3 14 25.3
4 21 32.7
5 30 41.5
6 36 48.2
7 45 63.7
8 50 74.6
1 z3 z3^2 t3
1 z4 z4^2 m1 t4
1 z5 z5^2 m2 t5
1 z6 z6^2 m3 t61 z7 z7^2 t7
1 z8 z8^2 t8
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Berdasarkan metode seismik refleksitunggal horisontal, carilah nilai
kecepatan (v) dan kedalaman (z) di
lapisan !
Diketahui kecepatan seismik konstan (V ),
jarak ( x ), waktu (t )
Rec Dist m Time s
1 60 0.5147
2 80 0.5151
3 100 0.5155
4 120 0.5161
Soal (3)
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z v +x v =
Model matematika
m1+m2x2=t2
m1= 4.z2/v2
m2= 1/v2
5 140 0.5167
6 160 0.5175
7 180 0.5183
8 200 0.5192
kecepatan = 2797 m/s ;
kedalaman = 719 m
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PEMROGRAMAN MATLAB
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• Andry Pujirianto. 2004. Cepat Mahir Matlab.
Kuliah berseri IlmuKomputer.com
•http://www.mathworks.com/
Acuan (referensi)
• http://www.math.ohiou.edu/
• http://www.miislita.com/information-
retrieval-tutorial/matrix-tutorial-2-matrix-operations.html
Operasi Matriks, LIB 33
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• Program untuk analisis dan komputasi numerik
yang merupakan bahasa pemrograman
matematika lanjutan dengan dasar pemikiran
menggunakan sifat dan bentuk matrik
MATrics LABoratoryMATrics LABoratory
• Awalnya merupakan interface untuk koleksi rutin-
rutin numerik LINPACK dan EISPACK yang
menggunakan FORTRAN
• Sekarang menjadi produk komersial Mathworks
Inc. yang menggunakan C++
PS Pend. Matematika UNEJ
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•
Gunakan untuk memasukkan variabel,menjalankan fungsi dan “m-file”.
Command Window
Ketik fungsi dan variabel pada “MATLAB
prompt”
MATLAB prompt
Tampilan hasil
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• Semua kaidah dan aturan operasi matriks berlaku.
• Mis. Matriks A dengan ukuran m x n (baris, kolom) dikalikan dengan matriks B berukuran sama p x q. AB
hanya dapat dikalikan jika n= p. . Vektor kolom dianggap sebagai matriks p x 1 dan vektor baris 1 x q.
• transpose dari vektor u, ukurannya 4 x 1
a i an engan u,
A*u
Karena A adalah 3 x 4 dan u adalah 4 x 1, maka valid dan hasilnya vektor 3 x 1.
Operasi Matriks, LIB 36
ans =
1234
Masih ingat perkalianmatriks ini??
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• Mis. Membuat matriks 5 elemen.
•
Gunakan tanda ; untuk memulai baris baru
Membuat matriks sederhana
A = [12 62 93 -8 22];
Operasi Matriks, LIB 37
= - - -
A =
12 62 93 -8 22
16 2 87 43 91-4 17 -72 95 6
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• Mis. Membuat matriks acak ukuran U ukuran4 x 4
Mengindeks matriks
>> U=rand(4,4)
U =
• Ambil komponen baris 3 kolom 4
Operasi Matriks, LIB 38
. . . .0.2311 0.7621 0.4447 0.73820.6068 0.4565 0.6154 0.17630.4860 0.0185 0.7919 0.4057
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•
Ambil semua komponen baris 3
>> U(3,:)
ans =
0.6068 0.4565 0.6154 0.1763
• Ambil semua komponen kolom 2
Operasi Matriks, LIB 39
>> U(:,2)
ans =
0.89130.76210.45650.0185
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• Invers matriks U
Contoh fungsi matematika
>> inv(U)
ans =
Syarat inv dan det ; ukuran harusbujur sangkar
•
Determinan matriks U
Operasi Matriks, LIB 40
. - . - . - .-0.7620 1.2122 1.7041 -1.2146-2.0408 1.4228 1.5538 1.37301.3075 -0.0183 -2.5483 0.6344
>> det(U)
ans =
0.1155
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Operator in MATLAB
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each data pointis predicted by a
strquadratic curve
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straight line parabola
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in MatLab
G= ones N 1 t t.^2
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in MatLab
G= ones N 1 t t.^2
for k = 1:n
G(k,1) = 1;
G(k,2) = z(k);
G(k,3) = z(k).^2;
end
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G.m = d
GT .G.m = GT .d
GT .G -1 . GT .G.m = GT .G -1 . GT .d
in MatLab
m = [GT .G]-1 . GT .d
In MATLAB m=inv(G’*G)*G’*d
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TUGAS INDIVIDU
• Kerjakan soal 1 - 3 menggunakan cara:
– Perhitungan manual (matrik) tulis tangan
– Pemrograman MATLAB
• r n -ou an r m so copy e:
• Kertas A4
• Dikumpulkan hari Senin sebelum jam 16.00
WIB
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