Lesson 02 Pend Inversi Geofisika

8/15/2019 Lesson 02 Pend Inversi Geofisika

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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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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.


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];


= - - -

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


. . . .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


>> 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


. - . - . - .-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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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:

[email protected]

• Kertas A4

• Dikumpulkan hari Senin sebelum jam 16.00

WIB

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Lesson 02 Pend Inversi Geofisika

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