Ho izon al accu acy assessmen o a no el algo i hm o
app oxima e a su ace o a DEM
Ba e a, D. a,b, Ibáñez, M.J. a, Edda gani, S. a, c, Rome o, R. d, A iza-López, F.J.e, Reinoso-Go do,
J.F. ,*
a Depa men o Applied Ma hema ics, Uni e si y o G anada, Spain – [email p o ec ed], [email p o ec ed],
[email p o ec ed]
b IMAG -- Ins i u e o Ma hema ics, G anada, Spain
c Uni e si y Hassan Fi s , MISI Labo a o y, Mo occo
d Depa men o Compu e Science and A i icial In elligence, Uni e si y o G anada, Spain – [email p o ec ed]
e Depa men o Ca og aphic, Geodesic and Pho og amme y Enginee ing – [email p o ec ed]
Depa men o A chi ec u al and Enginee ing G aphic Exp ession – [email p o ec ed]
* [email p o ec ed]
Abs ac :
This s udy e alua es he ho izon al posi ional accu acy o a new algo i hm ha de ines a su ace ha app oxima es DEM
da a by means o a spline unc ion. This algo i hm allows e alua ing he su ace a any poin in i s de ini ion domain and
allows analy ically es ima ing o he pa ame e s o in e es , such as slopes, o ien a ions, e c. To e alua e he accu acy
achie ed wi h he algo i hm, we use a e e ence DEM 2 m × 2 m (DEM e ) om which he de i ed DEMs a e ob ained a
4 m × 4 m, 8 m × 8 m and 16 m × 16 m (DEMde ). Fo each DEMde i s spline app oximan is calcula ed, which is e alua ed
a he same poin s occupied by he DEM e cells, ge ing a esampled DEM 2x2m (DEM em). The ho izon al accu acy is
ob ained by compu ing he a ea amongs he homologous con ou lines de i ed om DEM e and DEM em, espec i ely. I
has been obse ed ha he planime ic e o s o he p oposed algo i hm a e e y small, e en in la a eas, whe e you could
expec majo di e ences. The e o e, his algo i hm could be used when an e alua ion o he ho izon al posi ional accu acy
o a DEM p oduc a lowe esolu ion (DEMp o) and a di e en p oducing sou ce han he highe esolu ion DEM e is
wan ed.
Keywo ds: Spline, Be ns ein basis, con ol poin s, Bézie o dina es, enso p oduc , esample, ho izon al accu acy, DEM
1. In oduc ion
Ha ing a ma hema ical unc ion ha ep esen s he e ain
h oughou i s con inuous de ini ion domain has di e en
ad an ages, among o he s he ollowing: a) I is possible o
sample egula meshes o gene a e digi al ele a ion models
(DEM) o bo h highe and lowe esolu ion han he da a
om which he ma hema ical unc ion was ob ained; and
his is achie ed hanks o i s de ini ion domain is
con inuous, b) mo phological a iables o in e es can be
ob ained om he co esponding ma hema ical o mulas o
he su aces, such as slope, o ien a ion, cu a u e and
no mal di ec ion, c) You could in e sec wo su aces
co esponding o homologous DEMs om di e en da es
and calcula e he inc ease o dec ease in he e ain
olume. The p o ision o unc ions o his ype has
allowed esampling h ough bilinea (Maune, 2007) and
bicubic (Keys, 1981) in e pola ions ha ha e been used in
di e en applica ions bo h o ob ain DEMs o highe and
lowe esolu ion. In he i s case, ob aining highe
esolu ion has been used o, o example, imp o e u ban
lood zones in he absence o dense models (Shen and
Tan, 2020); in he second case, i s use has been equen
when i was in ended o compa e he al ime ic accu acy o
a lowe esolu ion p oduc model wi h ano he highe
esolu ion e e ence model (Gao, 1998, Mukhe jee e al.
2013, Wang e al. 2015), al hough, in mos cases, he e o
in oduced by he esampling om a highe esolu ion o a
lowe one was le unanalysed, as indica ed by Mesa and
A iza, 2020. O he s udies ha e add essed he in luence o
esampling echniques on p oduc s de i ed om DEMs
such as s eam lows. (Leong e al., 2015).
Al hough p ocedu es a e a ailable o ex ac in o ma ion
di ec ly om a DEM, we p opose a new algo i hm o build
a su ace wi h low compu a ional cos ha adjus s he
ele a ions in o de o ha e an explici exp ession
( unc ion) om which o ind he elemen s o in e es . As
he e ain has many i egula i ies, a su ace should no be
cons uc ed oo egula . C1 con inui y is su icien . The
egula s uc u e o he DEM allows o de ine a piecewise
su ace de ined on a quad angula pa i ion o he e ain.
Mo e p ecisely, we de ine a piecewise bicubic su ace by
p o iding simple ules ha gi e he Bézie o dina es (c .
Fa in 2001) o he su ace pa ches ela i e o he squa es
ha make up he pa i ion.
Mos s udies on DEM accu acy o DEM THT compa e a
p oduc wi h a e e ence ha e analyzed he al i ude
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componen , lea ing he ho izon al componen un ouched.
The eason o he lowe numbe o in es iga ions de o ed
o he ho izon al componen is due o he di icul y o
inding a sa is ac o y me hod. In his wo k we will s udy
he ho izon al accu acy achie ed by he new algo i hm ha
we ha e p oposed. We will use he au oma ic algo i hm
based on homologous con ou lines in oduced in Reinoso,
2010 and igo ously demons a ed in Reinoso, 2011 o
e alua ing he ho izon al accu acy.
2. Ma e ial and Me hodology
Ou s udy was ca ied ou on a 2x2 m esolu ion DEM e
p oduced by he Ins i u o Geog á ico Nacional o Spain o
he Na a a egion. The ollowing phases ha e been
ca ied ou :
1. S a ing om he 2x2 m esolu ion DEM e , a
esampling has been pe o med using he nea es
neighbo me hod a esolu ions o 16x16, 8x8 and
4x4 m (DEMde ) which we espec i ely deno e
DEM4x4, DEM8x8, DEM16x16. In his way, he
DEMde ha e he same alues as he DEM e a he
poin s in e pola ed by he nea es neighbo
me hod.
2. To e alua e he capaci y o ou app oxima ion
algo i hm (Aapx) o es ima e he DEM e alues,
he su ace o each DEM e se ha calculado la
supe icie de cada DEMde (Sde ) has been
calcula ed using Aapx. Sde is assessed o ob ain
DEMs wi h he same esolu ion as DEM e which
we call DEM emXxX (DEM em4x4, DEM em8x8,
DEM em16x16).
3. The ho izon al displacemen be ween DEM e and
each one o he DEM emXxX is calcula ed using he
Reinoso 2011 con ou me hod.
4. The esul s ob ained wi h Aapx a e compa ed wi h
he adi ional bicubic esampling algo i hm.
2.1 Ma e ial
The DEM e has a cell size o 2x2m and y occupies an a ea
o 4.8x4.8 Km. Figu e 1 shows he geog aphical
cha ac e is ics o he en i onmen , as well as he DEM e
ha con ains la a eas along wi h o he s eep slopes. The
coo dina es a e e e ed in he ETRS89 sys em 30N UTM
zone.
Figu e 1: DEM used as e e ence and i s geog aphical
en i onmen
2.2 The app oxima ion algo i hm
We p opose o cons uc a spline su ace by means o a
enso p oduc o 1D spline app oximan s, de ining he
su ace pa ches di ec ly in he Be ns ein basis. Suppose
ha o a eal unc ion 𝑓 he alues 𝑓(𝑣𝑖), 𝑖 ∈ℤ, a e known,
whe e 𝑣𝑖=𝑖 ℎ, wi h ℎ > 0 he size o he pa i ion 𝛥≔
{𝑣𝑖:𝑖 ∈ ℤ}. The 1D app oxima ing spline 𝐴𝑓 educes on
each in e al 𝐼𝑖≔[𝑣𝑖,𝑣𝑖+1] o a cubic polynomial, whose
con ol polygon is o med by ou con ol poin s wi h
Bézie abscissae {𝑣𝑖,𝑣𝑖+ℎ
3,𝑣𝑖+1 −ℎ
3,𝑣𝑖+1}. I 𝐷3 is he
union (wi hou epe i ions) o hese Bézie abscissae and
𝑢𝑖=𝑣𝑖−ℎ
3y 𝑤𝑖= 𝑣𝑖+ℎ
3, hen 𝐷3=⋃{𝑢𝑖,𝑣𝑖,𝑤𝑖}
𝑖∈ℤ .
Fo 𝑥 ∈𝐼𝑖, i holds
𝐴𝑓(𝑥)=𝑉𝑖𝐵0(𝑡)+𝑊𝑖𝐵1(𝑡)+𝑈𝑖+1𝐵2(𝑡)+𝑉𝑖+1𝐵3(𝑡),
wi h 𝑡 = 𝑥
ℎ−𝑖 and 𝐵𝑘(𝑡)=(3
𝑘)𝑡𝑘(1−𝑡)3−𝑘, 0≤𝑘 ≤3.
The Bézie o dina es 𝑉𝑖,𝑊𝑖,𝑈𝑖+1 and 𝑉𝑖+1 a e de ined as
linea combina ions o poin alues 𝑓(𝑣𝑖): 𝑉𝑖=𝑓(𝑣𝑖),
𝑈𝑖=∑𝛼𝑟𝑓(𝑣𝑖+𝑟)
1
𝑟=−1 , 𝑊𝑖=∑𝛽𝑟𝑓(𝑣𝑖+𝑟)
1
𝑟=−1 ,
whe e he masks 𝛼 =(𝛼−1,𝛼0,𝛼1)∈ℝ3 and 𝛽 =
(𝛽−1,𝛽0,𝛽1)∈ℝ3 a e de e mined o achie e 𝐶1
con inui y as well as he ep oduc ion o he quad a ic
polynomials.
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P oposi ion 1 The unique masks yielding 𝐶1 con inui y
and he ep oduc ion o quad a ic polynomials a e
𝛼=(1
6,1,−1
6) and 𝛽=(−1
6,1,1
6).
Fu he mo e, he uni o m no m o he co esponding
ope a o 𝐴 is equal o 4/3.
F om he exac ness o 𝐴, he ollowing esul ega ding
he app oxima ion e o holds.
P oposi ion 2 The e exis cons ans 𝐾𝑙,𝑙 = 0,1,
independen o 𝑓 and ℎ and 𝑖, such ha
‖(𝑓−𝐴𝑓)(𝑙)‖∞,𝐼𝑖≤𝐾𝑙ℎ3−𝑙‖𝑓(3)‖∞,𝛺,
whe e 𝛺 =[𝑣𝑖−1,𝑣𝑖+2].
Now, gi en a 2D unc ion 𝑓(𝑥,𝑦) a bi-cubic piecewise
su ace 𝑠 is de ined as a enso p oduc app oximan : he
ope a o 𝐴 is applied o 𝑓 as a unc ion depending on 𝑥
(o 𝑦), and hen 𝐴 is again applied o de esul ing
unc ion, i.e. 𝑠(𝑥,𝑦)= 𝐴𝑦𝐴𝑥𝑓(𝑥,𝑦).
On each squa e 𝐼𝑖×𝐼𝑗 his unc ion is a bi-cubic Bézie
su ace, so ha i can be ep esen ed in Be ns ein-Bézie
o m. I is a linea combina ion o unc ions
𝐵𝑚(𝑥
ℎ−𝑖)𝐵𝑛(𝑥
ℎ−𝑗),0 ≤𝑚,𝑛 ≤3,
whose coe icien s a e exp essed in e ms o alues
𝑓(𝑣𝑖+𝑘,𝑣𝑗+𝑙),−1≤𝑘,𝑙 ≤1.
Fo F anke unc ion (see F anke 1982) ( op), he esul s
p o ided by he me hod p oposed o ℎ =2−7 a e shown
in Fig. 2.
Figu e 2: Resul s p o ided by ou Aapx o h=2-7 acco ding o
he F anke uncion
Also o Nielson es unc ion (Nielson 1978) i p o ides
good esul s o he same s ep leng h, shown in Fig.3.
Figu e 3: Resul s om ou Aapx algo i hm acco ding o he
Nielson es unc ion
2.3 Con ou s-based algo i hm o measu e he
ho izon al displacemen
The he ho izon al displacemen compu a ion o he ases
wi h espec o he DEM e is ca ied ou in he ollowing
phases:
1. The con ou s o bo h DEMs a e calcula ed (Fig. 4
a and b espec i ely), and hei homologous
cu es a e au oma ically iden i ied, e.g. cu es
C4a and C4b in Fig. 4 a and b.
2. A e supe imposing he homologous con ou s
(Fig. 4 c), he a eas enclosed be ween hem a e
calcula ed (g ay a ea in Fig. 4 d). The ho izon al
displacemen (Hdi) compu ed by he i h pai o
homologous con ou s (Cia, Cib) is o mula ed as
he a ea enclosed by bo h cu es (Ai) di ided by
he mean leng h o hose con ou s (𝐿𝑚𝑖=
𝐿𝑖𝑎+𝐿𝑖𝑏
2)
𝐻𝑑𝑖=𝐴𝑖
𝐿𝑚𝑖
3. And he mean displacemen o he DEM emXxX
espec o DEM e (𝐻𝑑DEM emXxX) is compu ed as
he weigh ed a e age o he displacemen s o all
he homologous con ou s, he weigh ing ac o
being he a e age leng h o hose con ou s,
di ided by he o al leng h o he a e age con ou s
being he o al leng h 𝐿𝑇𝑜𝑡 =∑𝐿𝑚𝑖
𝑛
𝑖=1 :
𝐻𝑑DEM emXxX =1
𝐿𝑇𝑜𝑡 ∑𝐴𝑖∗
𝑛
𝑖=1 𝐿𝑚𝑖
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Figu e 4: Hologous con ou s and a ea be ween hem
3. Resul s and discussion
To calcula e he 𝐻𝑑DEM emXxX a 10 m in e al be ween
con ou lines has been used, ha in ou DEM p oduces a
o al o 22 le els, speci ically hei heigh s anging om
450 o 660 m. In Fig. 5 you can see he homologous
con ou d awn on a shadow map, as well as a de ail whe e
he a ea enclosed be ween hose homologous con ou s a e
highligh ed on g een colo .
Figu e 5: A ea be ween homologous con ou s co esponding o
he DEM e and he DEM em16x16
Table 1 shows he mean ho izon al displacemen s as well
as hei s anda d de ia ions calcula ed o 4x4, 8x8 and
16x16m esolu ions using ou new Aapx algo i hm and he
adi ional bicubic esampling me hod.
DEM emXxX
𝐻𝑑DEM emXxX
Aapx (m)
Bicubic
(m)
DEM em4x4
Mean
0.07
0.99
S d
0.03
0.06
DEM em8x8
Mean
0.28
1.05
S d
0.13
0.06
DEM em16x16
Mean
0.81
1.38
S d
0.38
0.27
Figu e 6: Ho izon al displacemen om he
𝐻𝑑𝐷𝐸𝑀𝑟𝑒𝑚𝑋𝑥𝑋 espec o he DEM e , compa ing ou algo i hm
and he adi ional bicubic.
Table 1 shows ha ou new algo i hm p oduces be e
esul s han he adi ional bicubic algo i hm ega dless o
he cell size used as DEMde . While ou algo i hm seems o
dec ease he e o due o ho izon al displacemen a a a e
o ¼ as he esolu ion inc eases a a a e o 2, in he bicubic
algo i hm he a e o dec ease is much lowe . Howe e , no
la ge di e ences a e obse ed in he alues o he s anda d
de ia ions i bo h algo i hms a e compa ed o each le el
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o esolu ion. Howe e , addi ional es s should be ca ied
ou wi h a g ea e numbe o DEMs, co e ing all ypes o
e ain ( la , undula ing and moun ainous), in o de o
s a is ically alida e he appa en ly be e esul s o ou
algo i hm espec he adi ional bicubic one.
Ano he ad an age o he new algo i hm wi h espec o
he adi ional bicubic one is ha i has a lowe
compu a ional cos , making i a candida e o be
implemen ed in ca og aphic p oduc ion so wa e
packages.
We belie e ha his new algo i hm can be used when you
wan o know he posi ional accu acy (ho izon al and
e ical) o a lowe esolu ion DEM coming om a sou ce
o he han he e e ence one o ha has been c ea ed wi h
a di e en me hod om he e e ence one.
On he o he hand, i would also be in e es ing o ha e an
algo i hm ha no only epo ed he ho izon al
displacemen wi h a scala alue, bu also included
in o ma ion abou di ec ion in each o he cells, which
would be possible by adap ing he con ou s me hod by
Reinoso 2011.
4. Conclusions
In his wo k a new algo i hm (Aapx) is p esen ed o
app oxima e a DEM by means o a piecewise de ined
su ace (Sde ). I p esen s some ad an ages linked o i s
de ini ion ype, such as being able o ob ain he al i ude o
a poin in he en i e de ini ion domain o ha su ace, as
well as mo phological a iables ha cha ac e ize he
e ain su ace: slope, o ien a ion, cu a u e o no mal
di ec ion in an analy ical way. An immedia e applica ion
would be he possibili y o esampling Sde o ob ain DEMs
(DEM emXxX) o highe o lowe esolu ion han hose used
o c ea e Sde . One consequence o Aapx esampling
capabili ies is being able o assess he accu acy o a
p oduc DEM (DEMp o) agains a highe accu acy
DEM e . This assessmen could be ca ied ou bo h in he
e ical componen and in he ho izon al componen ,
which is he one s udied in his wo k. DEMp o can come
om bo h a sou ce o a me hod o he han he sou ce o
me hod used o c ea e he DEM e .
Aapx has shown a lowe ho izon al displacemen han he
adi ional bicubic in e pola ion algo i hm, which can be
in e p e ed as a lowe e o when esampling DEMs o
lowe esolu ion o o he s o highe esolu ion; These
p ocesses a e necessa y when ying o compa e he
accu acies o a DEMp o agains a DEM e , and whene e
possible i will be necessa y o choose hose algo i hms
ha p oduce he leas e o (ho izon al displacemen ).
The Aapx compu a ional cos is lowe han o he
concep ually simila such as he adi ional bicubic one.
Finally, in he u u e an expe imen will ha e o be
designed wi h a su icien ly la ge numbe o DEMs on
which o es Aapx so ha he esul s ha appea in his i s
Aapx s udy can be e i ied.
5. Re e ences
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P oceedings o he In e na ional Ca og aphic Associa ion, 4, 2021.
30 h In e na ional Ca og aphic Con e ence (ICC 2021), 14–18 Decembe 2021, Flo ence, I aly. This con ibu ion unde wen
single-blind pee e iew based on submi ed abs ac s. h ps://doi.o g/10.5194/ica-p oc-4-11-2021 | © Au ho (s) 2021. CC BY 4.0 License.