“Double Fourier Series as a mapping tool in marine cartography : analyses

and use”

Lic. Ing. Vande Wiele Tom

Area

Modelling

Settings

Results

Sampling

Data Set

Economical Factor

Cartographic Factor

Relief

Pattern

Quality

Double Fourier Series Model

Introduction

Periodic function

kTxfxf

aT

a

Tdxxfdxxf

0

unambiguous, finite, continuous and periodic function

1

021 sincos

kkk kxbkxaaxf

T

Tk dxkxxfa0

2 cos

T

Tk dxkxxfb0

2 sin

converging, infinite, trigonometric series

The General Equation

1 1 21

1 1 21

1 1 21

1 1 21

2sin

2sin

2cos

2sin

2sin

2cos

2cos

2cos

n m

jinm

n m

jinm

n m

jinm

n m

jinmij

ymxn

ymxn

ymxn

ymxnZ

2

*

1

2

*

1

2sin

2sin

2cos

2cos

jm

in

jm

in

ymS

xnS

ymC

xnC

1 1

****

n mmnnmmnnmmnnmmnnm SSCSSCCCijZ

Adjustment to the Data

0

0

0

0

1

2

1

2

1

2

1

2

nm

s

tijt

nm

s

tijt

nm

s

tijt

nm

s

tijt

ZZ

ZZ

ZZ

ZZ

Interpretation

z f x yi T i i i ,

z f x y Li T i i i i ,

Model Comparison

Model Results

properties

Construction Set

Test Set

Test

Method

H1

H2

Test Method

YX

Z0

Zp

TestSet

Model

TestMethod

Study Area

The Data Sets

Construction Sets :

main patterns according to orientation :

North West - South East oriented tracks [t]

The Data Sets

Construction Sets :

main patterns according to orientation :

North West - South East oriented tracks [t]

South West - North East oriented tracks [l]

The Data Sets

Construction Sets :

main patterns according to orientation :

North West - South East oriented tracks [t]

South West - North East oriented tracks [l]

combination of previous tracks [lt]

The Data Sets

Construction Sets :

main patterns according to orientation :

North West - South East oriented tracks [t]

South West - North East oriented tracks [l]

combination of previous tracks [lt]

distances between tracks differentiation :

l1 l2 l3 l4 l5

The Data Sets

Construction Sets :

main patterns according to orientation :

North West - South East oriented tracks [t]

South West - North East oriented tracks [l]

combination of previous tracks [lt]

distances between tracks differentiation :

l1 l2 l3 l4 l5

The Data Sets

Construction Sets :

main patterns according to orientation :

North West - South East oriented tracks [t]

South West - North East oriented tracks [l]

combination of previous tracks [lt]

distances between tracks differentiation :

l1 l2 l3 l4 l5

The Data Sets

Construction Sets :

main patterns according to orientation :

North West - South East oriented tracks [t]

South West - North East oriented tracks [l]

combination of previous tracks [lt]

distances between tracks differentiation :

t1 t5

l1 l5

lt1 lt5

Test Set

Results - Analyses

Influence Pattern

0.990

0.991

0.992

0.993

0.994

0.995

0.996

0.997

0.998

2x2 3x3 4x4 5x5 6x6

(m x n)

r -

corr

elat

ion

r-l1

r-l2

r-l3

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

2x2 3x3 4x4 5x5 6x6 7x7 8x8 9x9 10x10

(m x n)

r -

corr

elat

ion

r-t1

r-t2

r-t3

r-l1

r-l2

r-l3

r-lt1

r-lt2

r-lt3

r-lt4

r-lt5

Influence Test Set

0.990

0.992

0.994

0.996

0.998

1.000

2x2 3x3 4x4 5x5 6x6 7x7 8x8 9x9 10x10

(m x n)

r -

corr

elat

ion

test set

constr.set

lt4 (mxn) D) D) Min(D) Max(D) MAE RMSE RMSEs RMSEu a b r d2x2 0.24 0.62 -3.68 2.55 0.5027 0.6614 0.2507 0.6120 -0.038 0.9826 0.9919 0.9953

test set 3x3 0.25 0.49 -3.02 2.29 0.4201 0.5448 0.2542 0.4819 0.040 0.9869 0.9950 0.99684x4 0.25 0.44 -2.52 1.90 0.4028 0.5084 0.2541 0.4403 0.076 0.9890 0.9958 0.99725x5 0.25 0.42 -2.21 1.90 0.3821 0.4836 0.2515 0.4131 0.085 0.9897 0.9963 0.99756x6 0.24 0.40 -1.95 1.51 0.3707 0.4643 0.2405 0.3972 0.110 0.9919 0.9966 0.99777x7 0.25 0.40 -1.79 1.84 0.3845 0.4729 0.2593 0.3954 0.086 0.9893 0.9966 0.99768x8 0.24 0.43 -1.92 1.85 0.3874 0.4924 0.2430 0.4283 0.114 0.9920 0.9961 0.99749x9 0.24 0.60 -2.51 2.75 0.4843 0.6471 0.2488 0.5973 0.067 0.9888 0.9923 0.9955

10x10 0.22 1.60 -10.05 9.11 0.8506 1.6192 0.2208 1.6040 0.097 0.9923 0.9487 0.97302x2 0.00 0.67 -4.23 3.15 0.4533 0.6706 0.0834 0.6654 -0.255 0.9845 0.9922 0.9961

constr. 3x3 0.00 0.48 -3.25 2.27 0.3194 0.4817 0.0430 0.4798 -0.131 0.9920 0.9960 0.99804x4 0.00 0.39 -2.47 2.09 0.2661 0.3912 0.0284 0.3902 -0.087 0.9947 0.9974 0.9987

set 5x5 0.00 0.33 -2.41 1.88 0.2222 0.3318 0.0204 0.3311 -0.062 0.9962 0.9981 0.99916x6 0.00 0.29 -1.92 1.16 0.1975 0.2940 0.0160 0.2936 -0.049 0.9970 0.9985 0.99937x7 0.00 0.27 -1.78 1.28 0.1809 0.2737 0.0139 0.2734 -0.042 0.9974 0.9987 0.99948x8 0.00 0.26 -1.89 1.40 0.1775 0.2616 0.0120 0.2614 -0.037 0.9978 0.9988 0.99949x9 0.00 0.24 -1.90 1.47 0.1616 0.2438 0.0108 0.2436 -0.033 0.9980 0.9990 0.9995

10x10 0.00 0.23 -1.87 1.20 0.1488 0.2269 0.0095 0.2267 -0.029 0.9982 0.9991 0.9996

475000 475500 476000 476500 477000 477500 478000

Easting

5680500

5681000

5681500

5682000

5682500

5683000

Nor

thin

g

L1 = 3500L2 = 3000m = 10n = 10Data Set = lt4

Differences in m eter.

-5 .0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

25 .0Influence Point Density

0.990

0.991

0.992

0.993

0.994

0.995

0.996

0.997

0.998

2x2 3x3 4x4 5x5 6x6 7x7 8x8 9x9 10x10

(m x n)

r -

corr

elat

ion

r-100

r-50

r-25

r-10

r-5

r-1

Results - Use of the Model

Direct Mapping

Mean D 0.20Variance D 0.36

Min D -1.80Max D 1.35MAE 0.3328

RMSE 0.4161RMSEs 0.2039RMSEu 0.3627

a 0.1611b 0.9973r 0.9972d 0.9982

475000 475500 476000 476500 477000 477500 478000

Easting

5680500

5681000

5681500

5682000

5682500

5683000

Nor

thin

g

-1 .00

-0 .80

-0 .60

-0 .40

-0 .20

0.00

0.20

0.40

0.60

0.80

1.00

2.00Differences in m eter.

L1 = 3500L2 = 3000m = 10n = 10Data Set = lt1

Mapping Tool

(Semivariogram original data set)(Semivariogram reduced data set)

Combination - Double Fourier Series & Kriging

DFS - model

476000 476500 477000 477500 478000

Easting (m )

5681500

5682000

5682500

5683000

No

rth

ing

(m

)

-1.5 m

-1.3 m

-1.1 m

-0.9 m

-0.7 m

-0.5 m

-0.3 m

-0.1 m

0.1 m

0.3 m

0.5 m

0.7 m

0.9 m

1.1 m

1.3 m

1.5 m

UTM zone 31(ED 50)

Kriging - model

0.00

0.05

0.10

0.15

0.20

0.25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

SV 14°

SV 104°

sferical model

hole effect model

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0 500 1000 1500 2000Distance along track (m)

Devia

tio

ns f

rom

DF

S m

od

el

(m)

Z 0

Z Kriging

-0.8-0.6-0.4

-0.20.00.20.4

0.60.8

0 500 1000 1500 2000Distance along track (m)

Dif

fere

nces b

etw

een

mo

del

an

d

ob

serv

ati

on

s (

m)

Model Performance

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

Mean D Variance D Min D Max D

DFS

Kriging-DFS

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

MAE RMSE RMSEs RMSEu

DFS

Kriging-DFS

0.950

0.955

0.960

0.965

0.970

0.975

0.980

0.985

0.990

0.995

1.000

b r d

DFS

Kriging-DFS

-20

-18

-16

-14

-12

-10

-8

0 500 1000 1500 2000

Z o

-20

-18

-16

-14

-12

-10

-8

0 500 1000 1500 2000

Z o

Z DFS

-20

-18

-16

-14

-12

-10

-8

0 500 1000 1500 2000

Z o

Z DFS

Z Kriging

Conclusions

Influence number of terms used

Influence validation/test set

Influence pattern

Analyses :

Use :

Direct mapping

Indirect mapping / tool