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International Journal of
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Relationship between
profile length and roughness
variables for natural surfaces
N. Baghdadi a , P. Paillou b , G. Grandjean a , P.
Dubois c & M. Davidson d
a BRGM
b France
c av. C. Guillemin, B.P. 6009, Orle ans cedex,
45060
d CESBIO, B.P. 2801, Toulouse, 31401, France
Version of record first published: 25 Nov 2010.
To cite this article: N. Baghdadi, P. Paillou, G. Grandjean, P. Dubois & M. Davidson
(2000): Relationship between profile length and roughness variables for natural
surfaces, International Journal of Remote Sensing, 21:17, 3375-3381
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int. j. remote sensing, 2000, vol. 21, no. 17, 3375–3381
Relationship between proŽ le length and roughness variables for natural
surfaces
N. BAGHDADI , P. PAILLOU , G. GRANDJEAN , P. DUBOIS ,
2
1
and M. DAVIDSON
1
3
4
1
2
3
4
BRGM, 3 av. C. Guillemin, B.P. 6009, 45060 Orleans cedex 2, France
´
Observatoire de Borbeaux, B.P. 89, 33270 Floirac, France
/
DGA CEGN, 16bis av. Prieur de la Cote d’Or, 94114 Arcueil, France
ˆ
CESBIO, B.P. 2801, 31401 Toulouse, France
(Received 23 September 1999; in Ž nal form 2 May 2000)
Abstract. Surface roughness variables such as the rms height and the correlation
length are commonly used as inputs for electromagnetic models to predict the
backscattering coe cient. The objective of this study is to investigate the in uence
of the roughness proŽ le length on the estimation of roughness variables. Surface
proŽ les of 25 m long obtained by a laser proŽ ler have been examined. A clear
dependence of exponential type was observed between the roughness variables
and the proŽ le length.
1. Introduction
A considerable eŒort has been devoted to the use of radar backscattering response
of natural surface to estimate geophysical variables, such as soil moisture and
roughness. Such variables are important for environmental studies in diŒerent Ž elds
such as hydrology and management. Concurrently with experimental studies, direct
models have been developed to predict the backscattering coe cient (e.g. Fung
1994). These models need input variables which describe the surface and the output
is a prediction for the backscattering coe cient. The value of this coe cient is
governed by the dielectric properties of the soil and the surface roughness. Surface
roughness is known to play a dominant role in the scattering of electromagnetic
waves for natural surfaces. However, the agreement between experimental measure-
ments and theoretical predictions using physical models has often been weak.
DiŒerences between the SAR data values and the ones estimated using theoretical
models (e.g. IEM, GOM) appear to be due to an inadequate description of surface
variables rather than a failure of the models themselves. For modelling purposes a
given surface is deŽ ned statistically by two variables, determined from the surface
height proŽ les: the standard deviation of surface height (S), which speciŽ es the
vertical scale of the roughness, and the correlation length (L ), representing the
horizontal scale.
Recent works have indicated that roughness variables estimated on the basis of
Ž eld measurement data are very sensitive to proŽ le length (e.g. Davidson et al.
1998). These investigations have shown that S and L increase with the proŽ le length.
But the relation between proŽ le length and roughness variables is not understood
International Journal of Remote Sensing
ISSN 0143-1161 print/ISSN 1366-5901 online © 2000 Taylor & Francis Ltd
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3376
N. Baghdadi et al.
su ciently. A major di culty in the investigation of the trends in S and L variables
have been the limited length of conventional proŽ lers (one or two metres long) which
lead to large uncertainties in the estimated roughness variables because of the limited
length of the series from which the statistical variables are extracted. In fact, on a
theoretical level Oh and Kay (1998) have demonstrated the need for long proŽ les.
In order to estimate the correlation length and the rms height with a precision of
Ô
10% of their mean values, they concluded that the proŽ le length should be at
least 200L and 40L , respectively.
The purpose of this letter is to investigate the eŒect of proŽ le length on the
estimation of surface roughness variables for geological surfaces of varying roughness.
This study is carried out from roughness proŽ les of 25 m long collected in the
Republic of Djibouti. The attainment of this objective should be viewed as a precursor
to the use of roughness information in application models for predicting watershed
runoΠand to correctly estimate the signal penetration depth in arid surfaces.
2. Study site and data description
A Ž eldwork campaign was carried out during February 1999 over a study site
located in the Republic of Djibouti (East Africa; lat. 11ß35 N and long. 42ß30 E).
¾
¾
The study area, the Assal rift, is composed of well-exposed volcanic (basalts) and
sedimentary rocks.
Surface roughness measurements using a laser proŽ ler developed jointly by
CESBIO and ESA (Davidson et al. 1998) were conducted over ten training areas.
Two surface categories have been chosen: sedimentary areas and lava plateaux
(Ž gure 1). At least one roughness proŽ le of 25 m long was collected for each training
site. A proŽ le is a juxtaposition of Ž ves proŽ les of 5 m long, with a spatial resolution
of 5 mm. Once an individual 5 m sub-proŽ le is acquired, the proŽ ler is displaced
exactly of 5 m and realigned using a theodolite. The height precision of measurements
is of the order of 1.5 mm. The two roughness variables, standard deviation of surface
height (S) and correlation length (L ), were extracted for each training site for proŽ le
sections of 1, 2, 3, 5, 7, 10, 15 and 25 m long, using the mean of the autocorre-
lation function. Gaussian and exponential curves were Ž tted to the computed
autocorrelation function for rough and smooth areas, respectively (Fung 1994).
3. Data analysis
3.1. Relationship between the correlation length and the rms surface height
Figure 2 shows the relationship between the correlation length and the rms
surface height. Each symbol represents the variation of two roughness variables
(L and S) as a function of the proŽ le length. Two remarks can be made:
1. The sedimentary sites present generally higher correlation length compared
to the volcanic ones: for rough lava sites, the maximum value of correlation
length is around 20 cm when the proŽ le length is 25 m, whereas for sedimentary
areas, the correlation length can reach 60 cm for this same proŽ le length.
2. Overall the roughness lava surfaces are characterised by higher rms height
values than the smoother sedimentary areas.
3.2. Relationship between the surface correlation length and the proŽ le length
The relationship between the correlation length (L ) and the proŽ le length (L )
has been studied from the large database obtained for proŽ le sections of 1, 2, 3, 5,
p

Remote Sensing L etters
3377
(a)
(b)
Figure 1. Photos of surfaces from which surface roughness measurements were made. (a)
Owed deposits, a very smooth surface, and (b) aphyric basaltic lava, a very rough surface.
7, 10, 15, and 25 m long. Figure 3 illustrates the exponential dependence of the
correlation length on proŽ le length. We observe that the correlation length increases
as a function of proŽ le length. The correlation length increases linearly with the
length of proŽ le until L saturates for L
10 m as shown in Ž gure 3. It is also shown
>
p

3378
N. Baghdadi et al.
Figure 2. Relationship between the correlation length (L ) and the rms surface height (S).
Each symbol type corresponds to one proŽ le with the corresponding eight segments
of 1, 2, 3, 5, 7, 10, 15 and 25 m. For each proŽ le, the lower value of L and S corresponds
to proŽ le segment length of 1 m whereas the higher value corresponds to proŽ le
segment length of 25 m.
Figure 3. Relationship between the correlation length and the proŽ le length. Rms surface
height values for a proŽ le length of 25 m are indicated.

Remote Sensing L etters
3379
that L increases more rapidly for sedimentary areas than for lava plateaux (generally
rougher than sedimentary areas).
The behaviour of L versus the proŽ le length L can be modelled as an exponential:
p
b
p
Õ
L
a(1
e
)
L
(1)
5
Õ
Where a and b are the regression coe cients determined through a Ž t with
proŽ le data. The coe cient a is equal to L when L tends towards inŽ nite (L
5
p
p
25 m). Likewise, we verify easily that L
0 for L
Firstly, the coe cients a and b have been calculated for each proŽ le. The relation-
0.
5
5
p
ship (1) has been established with a signiŽ cant coe cient of determination: R lies
2
between 0.86 and 0.99. Next, the behaviour of coe cient b has been studied for all
proŽ les: b can be written as a function of a as following:
b
1.2a
(2)
5
0.315
Õ
Using (2), equation (1) was obtained for each proŽ le with a coe cient of deter-
mination varying from 0.84 to 0.99.
3.3. Relationship between the surface rms height and the proŽ le length
The relationship between the standard deviation of height (S) and the proŽ le
length (L ) has also been examined. The analysis shows that the rms height increases
when the proŽ le length varies between 1 m and 5 m (Ž gure 4). For proŽ le lengths
superior to 5 m, S remains unchanged.
p
Figure 4. Relationship between the rms surface height and the proŽ le length. Correlation
length values for a proŽ le length of 25 m are indicated.

3380
N. Baghdadi et al.
The relationship between S and L can be expressed by a function equivalent
p
to (1):
g
p
Õ
S
c(1
e
)
L
(3)
5
Õ
Where the coe cient c is equal to S when L tend to inŽ nite (L
we verify that S 0 for L
25 m). Likewise,
5
p
p
0.
5
The coe cients c and g have been calculated for each proŽ le with R from 0.89
5
p
2
to 0.99. Next, the behaviour of coe cient g has been studied for all proŽ les. The
coe cient g depends on c by the following equation:
g
1.08c
(4)
5
0.23
Using (4), equation (3) illustrates the behaviour of each proŽ le with a coe cient
of determination R varying between 0.88 and 0.98.
2
Note that the two relationships (1) and (3) show that the two roughness variables
follow exponential laws with only one coe cient (a for L and c for S). These
coe cients are equal to the value of L and S respectively when the proŽ le length is
large ( 5 m for S and 10 m for L ).
>
>
3.4. Precision analysis
In a theoretical study based on simulated proŽ les, Oh and Kay (1998) demon-
strated that the errors in proŽ le-based estimates of rms height and correlation length
depend on the proŽ le length. Within the context of this study, the results from Oh
and Kay can be used to interpret the errors in the derived trends as a function of
proŽ le length. At 1 m proŽ le length, for instance, the correlation length estimates lie
within the range of 5 cm to 15 cm. The associated L /L ratios lie therefore between
20 and 7 which implies errors of the order of 40% for L estimates close to 5 cm
p
Ô
Ô
2 cm) and over 50% for L estimates close to 15 cm (over 7.5 cm). The associated
(
precision on the rms height varies between 14% for L 5 cm and 17% for L 15 cm.
5
5
If we interpret these errors in terms of the trends displayed in Ž gure 2 it is clear from
the above analysis that there is a considerable overlap between the estimates at this
proŽ le length and very poor separability between surface roughness types based on
their associated correlation lengths.
At 25 m proŽ le lengths on the other hand a better separability of the diŒerent
trends can be achieved. At this proŽ le length all correlation length estimates lie below
60 cm so that the critical L /L ratio (the one associated with the highest percentage
p
error) is 40. The associated error in the estimated L for such a ratio is approximately
Ô
25% or
15 cm for L 60 cm. This implies that, while not all individual proŽ le
5
trends are distinguishable, the overall trends associated with lava plateau and sedi-
mentary surfaces can be distinguished and discriminated. The precision on the rms
height varies between 4% for L 10 cm and 8% for L 60 cm.
5
5
For the surfaces and associated trends examined within this letter it appears that
the critical proŽ le length lies around 10 m. At this proŽ le length the uncertainties in
Ô
Ô
24 cm for L around 60 cm
L range from
(40%) implying that the overall trends in L can be distinguished at this proŽ le
length. The precision on the rms height measurements varies between 5% for L
1 cm (10%) for L around 10 cm to
5
10 cm and 13% for L 60 cm.
5

Remote Sensing L etters
3381
4. Conclusion
The purpose of this study was to investigate the possibility of estimating a
relationship between proŽ le length of natural surfaces and their roughness variables,
using a large data set acquired in Republic of Djibouti. A clear dependence of
exponential type was observed between the roughness variables (correlation length
and standard deviation of height) and the proŽ le length. Our results show in particu-
lar that proŽ le lengths of at least 10 m long should be considered in order to derive
valuable roughness variables. The precision on rms height will be between 5% and
13% and the one on correlation length between 10% and 40%.
Acknowledgments
This work has been Ž nanced by a joint project between CNRS/INSU, DGA/
CEGN and BRGM. The authors would like to thank the ESA for allowing the use
of the laser proŽ ler. The assistance of military personnel from the 13
Djibouti is greatly acknowledged.
DBLE in
eme
References
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Artech House).
Oh, Y., and Kay, Y., 1998, Condition for precise measurement of soil surface roughness. IEEE
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