Mining Engineering Department Faculty of Engineerin Engineering g
Mulawarman University
GEOSTATISTIK
Shalaho Dina Devy
1
Mining Engineering Department Faculty of Engineerin Engineering g
Mulawarman University
(Course-4)
2
1
Mulawarman University
Mining Engineering Department Faculty of Engineerin Engineering g
Semivar variog iogra ram m Ek Ekss perimental • Semi semivar ivariogr iogram am teo teorit ritis is • sem
3
Mulawarman University
Mining Engineering Department Faculty of Engineerin Engineering g
4
2
Mulawarman University
Mining Engineering Department Faculty of Engineerin Engineering g
Vario ariogr gram am teori eoriti tiss memp mempun unya yaii bentu entuk k kurv kurva a yang yang paling paling mendek mendekat atii variog variogra ram m eksper eksperime iment ntal. al.
Untuk keperluan analisis lebih lanjut variogram ekperimental harus diganti dengan variogram teoritis.
Terdapat beberapa jenis variogram yang sering digu digunak nakan an,, yait yaitu u: 1. Model bola (spherical (spherical mode) mode) 2. Model Model ekponensia ekponensiall (Exponen (Exponential tial Model) Model) 3. Model Model gauss (gaussion (gaussion model) model) 5
Mulawarman University
Mining Engineering Department Faculty of Engineerin Engineering g
Bentuk Bentuk vario variogr gram am ini dirumu dirumusk skan an sebag sebagai ai berik berikut ut
Denga Dengan n:
• H adala adalah h jarak jarak lokasi lokasi anta antarr sampel sampel • Co+C adalah adalah sill , yaitu yaitu nilai nilai vario variogr gram am untuk untuk jarak jarak pada pada saat saat besar besarny nya a konsta onstan n (teta (tetap). p). Nilai Nilai ini sama sama denga dengan n nilai nilai varia variansi nsi data data • a adalah range, yaitu yaitu jarak jarak pada pada saat saat nilai nilai vario variogr gram am menca mencapai pai sill
6
3
Mulawarman University
Mining Engineering Department Faculty of Engineering
7
Mulawarman University
Mining Engineering Department Faculty of Engineering
Pada model eksponensial terjadi peningkatan dalam semivariogram yang sangat curam dan mencapai nilai sill secara asimtotik, dirumuskan sebagai berikut
8
4
Mulawarman University
Mining Engineering Department Faculty of Engineering
9
Mulawarman University
Mining Engineering Department Faculty of Engineering
Model gaus merupakan bentuk kuadrat dari eksponensial sehingga menghasilkan bentuk parabolic pada jarak yang dekat dan dirumuskan sebagai berikut
10
5
Mulawarman University
Mining Engineering Department Faculty of Engineering
11
Mulawarman University
Mining Engineering Department Faculty of Engineering
Ada 2 metode yang umumnya digunakan untuk menfit variogram ekperimental dengan variogram teoritis yaitu 1. metode visual dan 2. metode least square. Dengan metode visual (manual) biasanya sudah cukup memuaskan, dan banyak digunakan oleh para ahli geostatistik. Karena sense yang banyak berperan dalam melakukan fitting tersebut, maka dalam pekerjaan ini pengalaman akan sangat menentukan kualitas fitting.
12
6
Mulawarman University
Mining Engineering Department Faculty of Engineering
• Tujuan utama dari fitting ini adalah untuk mengetahui parameter geostatistik seperti a (range), C, Co.
Berikut beberapa pedoman penting dalam melakukan fitting • Variogram yang mempunyai pasangan conto yang sangat sedikit agar diabaikan • Nugget variance (Co) didapat dari perpotongan garis tangential dari beberapa titik pertama variogram dengan sumbu Y • Sill (Co+C) kira-kira sama dengan atau mendekati varians populasi. Garis tangensial diatas akan memotong garis sill pada jarak 2/3 a, Sehingga selanjutnya dapat dihitung harga a. • Interpretasi nugget variance untuk variogram dengan sudut toleransi 180o (variogram rata-rata) akan sangat membantu untuk memperkirakan besarnya nugget Variance • Nugget variance diambil dari multiple variogram (dalam berbagai arah). Dalam multiple Variogram, best spherical line sebaiknya lebih mendekati variogram yang mempunyai pasangan conto yang cukup
Mulawarman University
13
Mining Engineering Department Faculty of Engineering
1. Metode Kriging 2. Metode Invers Distance Square
14
7
Mulawarman University
Mining Engineering Department Faculty of Engineering
Optimal interpolation based on regression against observed z values of surrounding data points, weighted according to spatial covariance values.
15
Mulawarman University
Mining Engineering Department Faculty of Engineering
16
8
Mulawarman University
Mining Engineering Department Faculty of Engineering
Kriging merupakan teknik perhitungan untuk estimasi suatu variabel teregional yang menggunakan pendekatan bahwa data yang dianalisis dianggap sebagai suatu realisasi dari suatu vriabel acak, Keseluruhan variabel acak yang dianalisis tersebut akan membentuk suatu fungsi acak menggunakan strukturak variogram
17
Mulawarman University
Mining Engineering Department Faculty of Engineering
Kriging Secara umum merupakan metode yang digunakan untuk menganalisis data geostatistik, yaitu untuk menginterpolasi suatu nilai kandungan mineral berdasarkan data sampel Data sampel diambil dari lokasi yang tidak beraturan
18
9
Mulawarman University
Mining Engineering Department Faculty of Engineering
Kriging digunakan untuk menonjolkan metode khusus dalam rata-rata pergerakan terbobot (weighted moving average) yang meminimalkan variasi dari hasil estimasi (Matheron, 1960) Estimasi besaran nilai karakteristik Z’ pada titik tidak tersempel berdasarkan informasi dari karakteristik titik tersempel Z yang berada di sekitarnya dengan mempertimbangkan korelasi spasial yang ada dalam data tersebut
Mulawarman University
19
Mining Engineering Department Faculty of Engineering
Mencari penaksir tak-bias linier
terbaik Kriteria terbaik: variansi minimum. Memilih rata-rata berbobot dari nilai sampel yang memiliki variansi minimum. Interpolasi spasial. 20
10
Mining Engineering Department Faculty of Engineering
Mulawarman University
Kriging
Block
Point Simple Kriging
21
Mulawarman University
Mining Engineering Department Faculty of Engineering
Estimation of a variable at an unmeasured location from observed values at surrounding locations. For example, estimating porosity at u = (2000 m, 4700 m) based on porosity values at nearest six data points in our Zone A data:
22
11
Mulawarman University
Mining Engineering Department Faculty of Engineering
23
Mulawarman University
Mining Engineering Department Faculty of Engineering
Gives estimate of estimation error (kriging variance), along with estimate of the variable, Z , itself (but error map is basically a scaled version of a map of distance to nearest data point, so not that unique)
24
12
Mulawarman University
Mining Engineering Department Faculty of Engineering
25
Mulawarman University
Mining Engineering Department Faculty of Engineering
Salah satu metode kriging memiliki asumsi matematik khas untuk penerapan intrinsic stationary Sifat-sifat ordainary kriging: • Linear • Unbiased • Best/optimu 26
13
Mulawarman University
Mining Engineering Department Faculty of Engineering
27
Mulawarman University
Mining Engineering Department Faculty of Engineering
28
14
Mulawarman University
Mining Engineering Department Faculty of Engineering
29
Mulawarman University
Mining Engineering Department Faculty of Engineering
30
15
Mulawarman University
Mining Engineering Department Faculty of Engineering
Untuk Mendapatkan matriks unsur invers 3x3 kita perlu memahami matriks - matriks berikut : 1. 2. 3. 4.
Matriks Kofaktor Adjoin Nilai elemen rumus invers Matriks ordo 3 x 3
31
Mulawarman University
Mining Engineering Department Faculty of Engineering
Matriks Kofaktor & Adjoin
Matriks Kofaktor adalah matriks yang unsurnya diganti dengan nilai determinan yang unsurnya tidak sebaris dan tidak sekolom dengan unsur asal.
Adjoin adalah matriks kofaktor yang di Transposkan ( baris jadi kolom , kolom jadi baris )
32
16
Mulawarman University
Mining Engineering Department Faculty of Engineering
33
Mulawarman University
Mining Engineering Department Faculty of Engineering
Langkah pertama maka kita harus mencari kofaktor dari A
34
17
Mulawarman University
Mining Engineering Department Faculty of Engineering
Langkah kedua, Setelah hasil dari Kofaktor A ditemukan , maka kita mencari Adjoin nya
35
Mulawarman University
Mining Engineering Department Faculty of Engineering
Langkah ketiga , Mencari nilai determinan A
36
18
Mulawarman University
Mining Engineering Department Faculty of Engineering
Langkah terakhir adalah mencari invers matriks A dengan rumus Invers Matriks nxn = 1
/ nilai determinan X Matriks Adjoinnya
jadi matriks invers A
37
Mulawarman University
Mining Engineering Department Faculty of Engineering
1.
2.
38
19
Mulawarman University
Mining Engineering Department Faculty of Engineering
39
Mulawarman University
Mining Engineering Department Faculty of Engineering
40
20
Mulawarman University
Mining Engineering Department Faculty of Engineering
1.
41
Mulawarman University
Mining Engineering Department Faculty of Engineering
42
21
Mulawarman University
Mining Engineering Department Faculty of Engineering
2.
43
Mulawarman University
Mining Engineering Department Faculty of Engineering
44
22
Mulawarman University
Mining Engineering Department Faculty of Engineering
1.
45
Mulawarman University
Mining Engineering Department Faculty of Engineering
46
23
Mulawarman University
Mining Engineering Department Faculty of Engineering
47
Mulawarman University
Mining Engineering Department Faculty of Engineering
2.
48
24
Mulawarman University
Mining Engineering Department Faculty of Engineering
49
Mulawarman University
Mining Engineering Department Faculty of Engineering
50
25
Mulawarman University
Mining Engineering Department Faculty of Engineering
51
Mulawarman University
Mining Engineering Department Faculty of Engineering
52
26
Mulawarman University
Mining Engineering Department Faculty of Engineering
53
Mulawarman University
Mining Engineering Department Faculty of Engineering
54
27
Mulawarman University
Mining Engineering Department Faculty of Engineering
55
Mulawarman University
Mining Engineering Department Faculty of Engineering
56
28
Mulawarman University
Mining Engineering Department Faculty of Engineering
57
Mulawarman University
Mining Engineering Department Faculty of Engineering
58
29
Mulawarman University
Mining Engineering Department Faculty of Engineering
59
Mulawarman University
Mining Engineering Department Faculty of Engineering
60
30
