Calculation of Propagated Errors in Airborne LiDAR Data Matrix ( , ,) α 0 β 0 γ 0 R rotates the...

1

Calculation of Propagated Errors in Airborne LiDAR Data

Ajay Dashora, Bharat Lohani

Department of Civil Engineering,

Indian Institute of Technology Kanpur, Kanpur (India)

{ajayd, blohani}@iitk.ac.in

KanGAL Report Number 2013011

Abstract

A process of determining the planimetric (or horizontal) and altimetric (altimetric) errors

of airborne LiDAR data is demonstrated using law of error propagation for LiDAR

observation equation. Standard values of the precisions are referred from relevant

literature and specifications of the sensors. A set of complete and simplified expressions

of propagated errors are also presented.

Introduction: This KanGAL report is prepared by referring Hoften et al. (2000), Glennie

(2007), May (2008), and Schär (2010). Standard law of propagation of random errors is

referred from Mekhail (1976).

The LiDAR observation equation is written as:

( )sb

s

bm

b

mm

p XRlRXX ++= 0 … (1)

For the purpose of analysis of random errors by statistical methods, epoch or duration of

time is not participating as variable. Therefore, all variables are considered without time

and denoted without subscript t, which indicates an instant of a time or epoch. Expanding

the above equation with matrices and vectors gives:

−+

+

=

ηηγβακϕω

cos

sin

0

),,(),,(),(

0

0

0

000

r

r

l

l

l

Z

Y

X

Z

Y

X

z

y

x

refinc

p

p

p

RRR … (2)

2

Matrix ),,( 000 γβαR rotates the scanner coordinate system to body coordinate system with

positive values of bore-sight angles (May, 2008). However, the positive signs of roll (ω ),

pitch (ϕ ) and yaw (κ ) angles rotates the mapping coordinate system to body coordinate

system (May, 2008). Therefore, matrix ),,( κϕωR is constructed by roll, pitch and yaw

angles with their negative values. Matrices are written with their full expressions as:

−

=

100

001

010

),( refincR

−

+−+

++−

=

−

−

−

=

−−−=

ωϕωϕϕωϕκωκωϕκωκϕκωϕκωκωϕκωκϕκ

ωωωω

ϕϕ

ϕϕκκκκ

ωϕκκϕω

coscossincossin

cossinsinsincossinsinsincoscoscossin

cossincossinsinsinsincoscossincoscos

cossin0

sincos0

001

cos0sin

010

sin0cos

100

0cossin

0sincos

)()()(),,( xyz RRRR

−

+−−

−+

=

=

00000

000000000000

000000000000

000),,(

coscossincossin

cossinsinsincossinsinsincoscoscossin

cossincossinsinsinsincoscossincoscos

)()()(000

αβαββαβγαγαβγαγβγαβγαγαβγαγβγ

αβγγβα xyz RRRR

Random errors propagated in the 3D coordinates of a LiDAR point can be calculated by

standard law of propagation of error. 3D vector coordinates shown by equation (2) are

functions of 14 variables: navigation sensor coordinates in mapping frame ( ZYX ,, ),

attitude angles ( κϕω ,, ), lever-arm offset components ( zyx lll ,, ), bore-sight angles

( 000 ,, γβα ), scan angle (η ) and range ( r ). Standard deviation and variance are denoted

by σ and 2σ with the subscript indicating the variable. Standard law of propagation of

error (Mekhail, 1976) with all parameters considered independent gives the propagated

error in the 3D vector coordinates of a LiDAR point as:

TAAC Σ= … (3)

3

In equation (3), MatrixΣ is the covariance-variance matrix, which is a square and

diagonal matrix having variances of parameter as non-zero diagonal elements. Matrix A is

jacobian of m

pX with respect to the 14 parameters. The diagonal elements of the matrixC

represent the propagated variances (2

pXσ ,2

pYσ ,2

pZσ ). Indicating the column vector of 14

parameters by matrix p and individual elements by 1421 ,, ppp L gives:

14...,,2,1;][

][

][

1413121110987654321

000000

==

=

=

jp

pppppppppppppp

rlllZYX

Tj

T

Tzyx φγβακϕωp

… (4)

Writing the parameter jp and corresponding precision by jσ (where 14...,,2,1=j ), the

the diagonal elements of the matrix C (propagated variances) can be written as:

( ) ( )

( ) ( )

( ) ( )214

1

2

,3

2

3,3

214

1

2

,2

2

1,2

214

1

2

,1

2

1,1

j

j

jZ

j

j

jY

j

j

jX

AC

AC

AC

p

p

p

σσ

σσ

σσ

∑

∑

∑

=

=

=

==

==

==

… (5)

Matrix A contains 42 elements or partial derivatives of 3D vector coordinates with

respect to variables in 3 rows and 14 columns. Elements of matrix A (i.e. jiA , ) can be

calculated by evaluating the partial derivatives. In order to express matrix A , elements of

this matrix are grouped in five categories, namely, elements for 3D navigation sensor

coordinates ( NSCA ), elements for attitude angles ( AAA ), elements for bore-sight angles

( BSAA ), elements for lever arm offsets ( LAOA ), and elements for scanner angle and range

observations ( SARA ). Matrix A is expressed in five partitions as:

( ) ( ) ( ) ( ) ( )

=

×××××× 2333333333)143(SARBSALAOAANSC AAAAAA MMMM

4

All variables are not interblended in equation (2). Therefore, variables, which are

independent from the remaining variables, can be treated as constant in matrix

calculations. As a result, additive variables can be ignored and multiplicative variables

remain constant in matrix multiplication and thus not affecting the partial derivative

calculations.

Elements of NSCA matrix:

( )

=

∂

∂

∂

∂

∂

∂∂

∂

∂

∂

∂

∂∂

∂

∂

∂

∂

∂

=×

100

010

001

000

000

000

33

Z

Z

Y

Z

X

Z

Y

Y

Y

Y

Y

Y

Z

X

Y

X

X

X

ppp

ppp

ppp

NSCA

Elements of AAA matrix:

( )

∂

∂

∂

∂

∂

∂∂

∂

∂

∂

∂

∂∂

∂

∂

∂

∂

∂

=×

κϕω

κϕω

κϕω

ppp

ppp

ppp

AA

ZZZ

YYY

XXX

33

A

))sinsinsincos(cossin

)sinsincossin(coscos ()sinsincossin(cos

)sinsincos coscoscos )(sinsinsin coscos(

00000

00000

0000

γβαγαη

γβααγηκϕωωκ

ηαβηβακωϕκωω

−−

++−−

+++−=∂

∂

r

rl

rrlX

y

z

p

))sinsinsin coscos(sin

)sinsincossincos(cos(sinsincos

))sinsincos sincos(sin

)sincoscossin(sincos(sinsin

)sinsincoscoscoscos(sincoscos

00000

00000

00000

00000

0000

γβαγαη

γβααγηκωϕ

βαγγαη

βγαγαηκϕ

ηαβηβακωϕϕ

−−

+++

+−

−+−

++=∂

∂

r

rl

r

rl

rrlX

y

x

z

p

5

))sinsincossincos(sin

)sincoscossinsin(cos(coscos

))sinsinsincoscos(sin

)sinsincossincos(cos)(sinsincossin(cos

)sinsincoscoscoscos)(sincoscossin(sin

00000

00000

00000

00000

0000

βαγγαη

βγαγαηκϕ

γβαγαη

γβααγηωϕκκω

ηαβηβαϕκωκωκ

+−

−++

−−

++−−

+++=∂

∂

r

rl

r

rl

rrlX

x

y

z

p


)sinsincossin(coscos)(sincoscossin(sin

)sinsincoscoscoscos)(sinsincossin(cos(

00000

00000

0000

γβαγαη

γβααγηϕκωκω

ηαβηβαωϕκκωω

−−

++++

++−=∂

∂

r

rl

rrlY

y

z

p


)sinsincossin(coscos(sincoscos

))sinsincossin(cossin

)sincoscossin(sincos(sincos

)sinsincoscoscoscos(coscoscos

00000

00000

00000

00000

0000

γβαγαη

γβααγηωκϕ

βαγγαη

βγαγαηϕκ

ηαβηβακωϕϕ

−−

+++

+−

−+−

++=∂

∂

r

rl

r

rl

rrlY

y

x

z

p




)sinsincossin(coscos)(sinsinsincos(cos

)sinsincos

coscoscos)(sinsincossin(cos

00000

00000

00000

00000

00

00

βαγγαη

βγαγαηκϕ

γβαγαη

γβααγηκωϕκω

ηαβ

ηβακϕωωκκ

+−

−+−

−−

+++−

+

+−=∂

∂

r

rl

r

rl

r

rlY

x

y

z

p


)sinsincossin(coscos(coscos

)sinsincoscoscoscos(sincos

00000

00000

0000

γβαγαη

γβααγηωϕ

ηαβηβαωϕω

−−

++−

++=∂

∂

r

rl

rrlZ

y

z

p


)sinsincossin(coscos(sinsin

)sinsincoscoscoscos(sincos


)sincoscossin(sincos(cos

00000

00000

0000

00000

00000

γβαγαη

γβααγηωϕ

ηαβηβαϕω

βαγγαη

βγαγαηϕϕ

−−

+++

+++

+−

−++=∂

∂

r

rl

rrl

r

rlZ

y

z

x

p

6

0=∂

∂

κpZ

Elements of LAOA matrix:

( )

∂

∂

∂

∂

∂

∂∂

∂

∂

∂

∂

∂∂

∂

∂

∂

∂

∂

=×

z

p

y

p

x

p

z

p

y

p

x

p

z

p

y

p

x

p

LAO

l

Z

l

Z

l

Z

l

Y

l

Y

l

Y

l

X

l

X

l

X

33

A

κϕ sincos=∂

∂

x

p

l

X

κωϕκω sinsinsincoscos +=∂

∂

y

p

l

X

ωκκϕω sincossinsincos −=∂

∂

z

p

l

X

κϕ coscos=∂

∂

x

p

l

Y

κωωϕκ sincossinsincos −=∂

∂

y

p

l

Y

ϕκωκω sincoscossinsin +=∂

∂

z

p

l

Y

ϕsin=∂

∂

x

p

l

Z

ωϕ sincos−=∂

∂

y

p

l

Z

ωϕ coscos−=∂

∂

z

p

l

Z

7

Elements of BSAA matrix:

( )

∂

∂

∂

∂

∂

∂∂

∂

∂

∂

∂

∂∂

∂

∂

∂

∂

∂

=×

000

000

000

33

γβα

γβα

γβα

ppp

ppp

ppp

BSA

ZZZ

YYY

XXX

A

))sincoscossin(sinsin

)sinsincossin(coscos(sincos

)cossincossincoscos)(sinsincossin(cos

)sinsinsincos(cos

))sinsincossin(cossin)sinsinsincos(coscos(

00000

00000

0000

0000000000

0

βγαγαη

βαγγαηκϕ

ηαβηβακϕωωκκωϕκω

γβααγηγβαγαηα

−+

++

−−−

+

++−=∂

∂

r

r

rr

rrX p

)coscoscoscossinsincoscos(sincos

)sinsincossin(cos)sinsinsincossincos(

)sinsinsincossincoscoscos()sinsinsincos(cos

000000

0000

000000

0

ηγβαηαγβκϕ

κϕωωκηβαηβα

ηγαβγηβακωϕκωβ

rr

rr

rrX p

+−

−++

++=∂

∂

)sinsinsincos(cos


)sincoscossin(sincos(



00000

00000

00000

00000

0

κωϕκω

βαγγαη

βγαγαη

γβαγαη

γβααγηκϕγ

+

+−

−−

−−

+=∂

∂

r

r

r

rX p



)sinsincossin(cos

))sinsincossin(cossin)cossinsincos(coscos(

)cossincossincoscos()sincoscossin(sin

00000

00000

000000000

0000

0

βγαγαη

βαγγαηκϕωϕκκω

γβααγηηβαγαη

ηαβηβαϕκωκωα

−+

++

−

++−−

−+=∂

∂

r

r

rr

rrYp

)coscoscoscossinsincoscos(coscos

)sincoscossin(sin)sinsinsincossincos(

)sinsinsincossincoscoscos()sinsincossin(cos

000000

0000

000000

0

ηγβαηαγβκϕ

ϕκωκωηβαηβα

ηγαβγηβαωϕκκωβ

rr

rr

rrYp

+−

++−

+−=∂

∂

8



)sinsincossin(cos

))sinsincossin(cossin)sincoscossin(sincos(

00000

00000

0000000000

0

γβαγαη

γβααγηκϕωϕκκω

βαγγαηβγαγαηγ

−−

++

−

+−−=∂

∂

r

r

rrYp

)cossincossincoscos(coscos


)sinsinsincos(coscos(sincos


)sinsincossin(coscos(sin

0000

00000

00000

00000

00000

0

ηαβηβαωϕ

γβααγη

γβαγαηωϕ

βγαγαη

βαγγαηφα

rr

r

r

r

rZ p

−−

++

−−

−+

+=∂

∂

)sinsinsincossincoscoscos(sincos

)coscoscoscossinsincoscos(sin

)sinsinsincossincos(coscos

000000

000000

0000

0

ηγαβγηβαωϕ

ηγβαηαγβϕ

ηβαηβαωϕβ

rr

rr

rrZ p

+−

+−

+=∂

∂





00000

00000

00000

00000

0

βαγγαη

βγαγαηωϕ

γβαγαη

γβααγηϕγ

+−

−+

−−

+=∂

∂

r

r

r

rZ p

Elements of SARA matrix:

( )

∂

∂

∂

∂∂

∂

∂

∂∂

∂

∂

∂

=×

r

ZZr

YYr

XX

pp

pp

pp

SAR

η

η

η

23

A

9



)sinsinsincos(cos


)cossincossincoscos)(sinsincossin(cos

00000

00000

0000000000

0000

βγαγαη

βαγγαηκϕκωϕκω

γβααγηγβαγαη

ηαβηβακϕωωκη

−+

+−

+

++−−

−−=∂

∂

r

r

rr

rrX p


)sincoscossin(sin(cossincos

)sinsincossin(cos)sinsincoscoscos(cos

)sinsinsincos(cos

)sinsinsincos(cossin)sinsincossin(cos(cos

00000

00000

0000

0000000000

βαγγαη

βγαγαηκϕ

κϕωωκηαβηβακωϕκω

γβαγαηγβααγη

+−

−+

−+−

+

−−+=∂

∂

r

X p



)cossincossincoscos()sincoscossin(sin

)sinsincossin(cos


00000

00000

0000

0000000000

βγαγαη

βαγγαηκϕ

ηαβηβαϕκωκωωϕκκω

γβααγηγβαγαηη

−+

+−

−+−

−

++−=∂

∂

r

r

rr

rrYp


)sincoscossin(sin(coscoscos

)sinsincossin(cos

))sinsinsincos(cossin)sinsincossin(cos(cos

)sincoscossin(sin)sinsincoscoscos(cos

00000

00000

0000000000

0000

βαγγαη

βγαγαηκϕωϕκκω

γβαγαηγβααγη

ϕκωκωηαβηβα

+−

−+

−

−−+−

++=∂

∂

r

Yp

)cossincossincoscos(coscos



)sinsincossin(cossin

)sinsinsincos(coscos(sincos

0000

00000

00000

00000

00000

ηαβηβαωϕ

βγαγαη

βαγγαηϕ

γβααγη

γβαγαηωϕη

rr

r

r

r

rZ p

−+

−+

+−

++

−=∂

∂


)sinsincossin(cos(cossincos

)sinsincoscoscos(coscoscos


)sincoscossin(sin(cossin

00000

00000

0000

00000

00000

γβαγαη

γβααγηωϕ

ηαβηβαωϕ

βαγγαη

βγαγαηϕ

−−

+−

+−

+−

−=∂

∂

r

Z p

10

Typical examples of parameters and their precision values are presented in following

table (1).

Table 1. Typical Values of Precisions in Airborne LiDAR Data Acquisition

S.N. Variable Precision Values Values and Remarks

1 3D Coordinates of

navigation sensor cm

cm

XZ

YX

5.7)(5.1

5

00

00

==

==

σσ

σσ

ppmcm

ppmcm

Z

YX

22

;12

0

00

+=

+==

σ

σσ

(Baseline length = km30 )

2 Orientation angles IMU model: Applanix

510

o

o

008.0

;005.0

=

==

kσ

σσ ϕω

o0=== κϕω

3 Lever-arm

components m

zyx lll 02.0±=== σσσ mlll zyx 5.0−===

4 Bore-Sight Angles Values by least squares :

o

o

004.0

;001.0

0

00

=

==

γ

βα

σ

σσ

o2000 === γβα

5 Range and scan

angle o0044.0

;02.0

=

=

φσ

σ mr

φsecHr =

Using values of various variables ( 000 ,,,,,,,, γβακϕω zyx lll ) given in table (1), the

elements of the matrix A are written below in terms of range and scan angle:

( )

=

=

∂

∂

∂

∂

∂

∂∂

∂

∂

∂

∂

∂∂

∂

∂

∂

∂

∂

=×

100

010

001

3,32,31,3

3,22,21,2

3,12,11,1

000

000

000

33AAA

AAA

AAA

Z

Z

Y

Z

X

Z

Y

Y

Y

Y

Y

Y

Z

X

Y

X

X

X

ppp

ppp

ppp

NSCA

11

( )

=

∂

∂

∂

∂

∂

∂∂

∂

∂

∂

∂

∂∂

∂

∂

∂

∂

∂

=×

6,35,34,3

6,25,24,2

6,15,14,1

33

AAA

AAA

AAA

ZZZ

YYY

XXX

ppp

ppp

ppp

AA

κϕω

κϕω

κϕω

A

)cos9988.0sin0349.05.0(4,1 φφ rrA −−=

05,1 =A

)sin0361.0cos0336.05.0(6,1 φφ rrA ++−=

04,2 =A

)sin0349.0cos9988.05.0(5,2 φφ rrA ++−=

)sin9987.0cos0361.05.0(6,2 φφ rrA +−=

)sin9987.0cos0361.05.0(4,3 φφ rrA +−=

)sin0361.0cos0336.05.0(5,3 φφ rrA −−−=

06,3 =A

( )

−

=

=

∂

∂

∂

∂

∂

∂∂

∂

∂

∂

∂

∂∂

∂

∂

∂

∂

∂

=×

100

001

010

9,38,37,3

9,28,27,2

9,18,17,1

33AAA

AAA

AAA

l

Z

l

Z

l

Z

l

Y

l

Y

l

Y

l

X

l

X

l

X

z

p

y

p

x

p

z

p

y

p

x

p

z

p

y

p

x

p

LAOA

07,1 =A

18,1 =A

09,1 =A

17,2 =A

08,2 =A

09,2 =A

12

07,3 =A

08,3 =A

19,3 −=A

( )

=

∂

∂

∂

∂

∂

∂∂

∂

∂

∂

∂

∂∂

∂

∂

∂

∂

∂

=×

12,311,310,3

12,211,210,2

12,111,110,1

000

000

000

33AAA

AAA

AAA

ZZZ

YYY

XXX

ppp

ppp

ppp

BSA

γβα

γβα

γβα

A

)sin0361.0cos9987.0(10,1 φφ rrA +=

)sin0012.0cos0349.0(11,1 φφ rrA +=

)sin0361.0cos0336.0(12,1 φφ rrA +=

)sin0336.0cos0361.0(10,2 φφ rrA −=

)sin0349.0cos9982.0(11,2 φφ rrA −−=

)sin9987.0cos0361.0(12,2 φφ rrA −=

)sin9988.0cos0349.0(10,3 φφ rrA −=

)sin0012.0cos0349.0(11,3 φφ rrA +=

012,3 =A

( )

=

∂

∂

∂

∂∂

∂

∂

∂∂

∂

∂

∂

=×

14,313,3

14,213,2

14,113,1

23AA

AA

AA

r

ZZr

YYr

XX

pp

pp

pp

SAR

η

η

η

A

)sin0361.0cos9987.0(13,1 φφ rrA −−=

)sin9987.0cos0361.0(14,1 φφ −=A

)sin0336.0cos0361.0(13,2 φφ rrA +−=

13

)sin0361.0cos0336.0(14,2 φφ −−=A

)cos0349.0sin9988.0(13,3 φφ rrA −=

)sin0349.0cos9988.0(14,3 φφ −−=A

Conclusion

This report demonstrates the calculation of propagated error in 3D coordinates obtained

by airborne LiDAR by law of error propagation. The standard equation of physical model

that describes that how range and half scan angle data are transformed into error

expressions using GPS, IMU and spatial arrangements between these sensor units, is used

for determining the propagated error. For choosen precision values of the specific GPS,

IMU, and LiDAR sensors, simplified expressions of the propagated errors are presented.

References

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Calculation of Propagated Errors in Airborne LiDAR Data Matrix ( , ,) α 0 β 0 γ 0 R rotates the...

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