Ó Birkha¨user Verlag, Basel, 2004
Pure appl. geophys. 161 (2004) 907–930 0033 – 4553/04/040907 – 24 DOI 10.1007/s00024-003-2478-x
Pure and Applied Geophysics
Application of Wavelet Transform to Magnetic Data Due to Ruins of the Hittite, Civilization in Turkey1 A. MUHITTIN ALBORA2 , Z. MU¨MTAZ HISARLI2 , and OSMAN N. UCAN3
Abstract — In the recent years, geophysical methods have been applied successfully in archaeological studies. In this article we have studied the application of wavelet transform to magnetic data in order to estimate boundaries of various synthetic examples and real data. Enhanced Horizontal Derivative (EHD) method is also applied as an alternative method for boundary estimation. The performance of wavelet transform and the EHD method are evaluated using magnetic data of the Sarissa-Kusakli archaeological site. All boundary results are mutually compared. Based on these comparisons, we conclude that the wavelet transform provides reasonable results. Key words: Archaeology, magnetic data, wavelet transform, EHD boundary analysis, boundary estimation.
Introduction The most significant culture in Anatolian history was created by Hittites (2000– 1200 B.C) who, probably coming from the northern part of Europe established themselves around the Kızılırmak River ( Fig. 1) in Anatolia at the third millennium B.C. The first town settled by the Hittites was Nesa, near present-day Kayseri, Turkey. Shortly after 1800 B.C. they conquered the town of Hattusas, near the site of present-day Bogazko¨y. Kusaklı-Sarissa was also another town settled by Hittites (Fig. 1). In Kusaklı, the fourth Hittite archive of cuneiform tablets was discovered after the famous findings from Bogazko¨y-Hattusa. The site has been investigated since 1992. During the excavations, two new buildings (A and B in Fig. 2) are found on the west side of the Acropolis (KARPE, 1998,1999). Magnetic resistivity and georadar methods have been applied by STUMPEL (1996, 1997, 1998) at the Kusakli site.
1
This work was supported by the TUBITAK under Project YDABCAG-100Y021. Istanbul University, Engineering Faculty, Geophysics Department, 34850, Avcilar, Istanbul, Turkey. E-mail:
[email protected],
[email protected] 3 Istanbul University, Engineering Faculty, Electrical and Electronic Department, 34850, Avcilar, Istanbul, Turkey. E-mail:
[email protected] 2
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Figure 1 General information about Hittite Imperial.
Wavelets are mathematical functions which split data into different frequency components and then each component was studied with a resolution matched to its scale. Wavelet transforms have advantages to traditional Fourier methods in analyzing physical situations where the signal contains discontinued and sharp spikes. Wavelets were developed independently in the fields of mathematics, physics, electrical engineering, and geophysics. Interchanges between these fields during the last ten years have led to many new wavelet application such as image compression, turbulence, human vision, radar, earthquake prediction, and separation of the gravity and magnetic anomalies. The wavelets, first mentioned by HAAR in 1909, had compact support which means vanish outside of a finite interval. Haar wavelets are not continuously differentiable. In the 1930s, representation of functions using scale-varying basis functions that can vary in scale and conserve energy has been researched by several researchers. GROSSMAN and MARLET (1985) defined wavelets within the context of physics. MALLAT (1989) elevated to digital processing by discovering pyramidal algorithms, and orthonormal wavelet basis. DAUBECHIES (1990) applied Mallat’s work to construct a set of wavelet orthonormal basis functions that are the cornerstone of wavelet application today. Since wavelet is an interesting tool for improving geophysical data, during the last decade it has been applied to geophysical data. Some of the recent applications in
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Figure 2 Topographic map including excavation results.
geophysics are processing of potential data (DAVIS et al, 1994; LI and OLDENBURG, 1997; FEDI and QUARTA, 1998; RIDSDILL-SMITH and DENTITH, 1999) and processing of seismic data (CHAKRABORTY and OKAYA, 1995; GRUBB and WALDEN, 1997). In this article, we have applied wavelet transform to magnetic data to estimate the boundary of the archaeological site. The anomalies on the horizontal, vertical and diagonal detail coefficients are condensed towards the horizontal, vertical and diagonal sides of the body, respectively. The boundaries of the estimated body are obtained when the centers of the anomalies are interconnected. The causative
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archaeological structures are obtained in Sarissa-Kusakli archaeological site when the wavelet transform is applied to the magnetic data of the sites. The results are also confirmed by Enhanced Horizontal Derivative Method(EHD)(FEDI and FLORIO, 2001). EHD is used to estimate the horizontal location of potential field source boundaries. In conclusion, we can impart that the wavelet transform is a suitable tool to estimate the boundaries of the possible causative bodies.
Wavelet Transform We consider the space L2 ðRÞ of measurable functions f(x), defined on the real line R, that satisfy 2
f ðxÞ 2 L ðRÞ )
Z1
jf ðxÞj2 dx < 1:
ð1Þ
1
The continuous wavelet transform ðW~ Þ of f(x) is given as, 1 W~ ða; bÞ ¼ pffiffiffi a
Z1
f ðxÞw
x a dx; a
ð2Þ
1
where w is called the mother wavelet. The variables a and b are integers that scale and dilate the mother wavelet (w). The scale index a indicates the wavelet’s width, and the location index b gives its position. Setting a ¼ 2j and b ¼ a:k in Eq.(2), the discrete wavelet transform (DWT) is written as, XX Wj;k ða; bÞ ¼ f ðxÞ2j=2 wð2j x kÞ: ð3Þ The dilations and translations are chosen based on power two, so-called dyadic scales and positions, which make the analysis efficient and accurate. Multi Resolution Analysis (MRA) MALLAT (1989) introduced an efficient algorithm to the DWT known as the Multi-Resolution Analysis (MRA). Multi-scale representation of signal f(x) may be achieved in different scales of the frequency domain by means of an orthogonal family of functions /ðxÞ. MRA is described further in the Appendix. Now, we illustrate how to compute the function in Aj (Fig. 3). The projection of the signal f ðxÞ 2 A0 on Aj defined by Aj f ðxÞ is given by X Aj f ðxÞ ¼ cj;k /j;k ðxÞ: ð4Þ k
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LOW RESOLUTION (AJf(x))
A0 A1 A2 A3
D1 D2
D3 Figure 3 The wavelet decomposition tree.
Aj f ðxÞ is called a discrete approximation of f(x). Here, pffiffiffi X hj2k Cj;k : cj;k ¼ 2
ð5Þ
j
Similarly, the projection of the function f(x) on the subspaces Dj is also defined by Dj f ðxÞ ¼
X
dj;k wj;k ðxÞ:
ð6Þ
k
Dj f ðxÞ is called discrete detail signal, where, pffiffiffi X dj;k ¼ 2 gjk cj;k :
ð7Þ
j
Because of Aj ¼ Aj1 Dj1 , the original function f ðxÞ 2 A0 can be written as X X f ðxÞ ¼ cj;k /ðxÞ þ dj;k wj;k ðxÞ: ð8Þ k
k
The transform coefficients cj;k and dj;k are given by, pffiffiffi X cj1;k ¼ 2 hj2k cj;k ;
ð9Þ
j
and dj;k ¼
pffiffiffi X 2 gj2k cj;k : j
ð10Þ
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2-D Multi Resolution Analysis (MRA) 2-D separable scaling function is, /ðx; yÞ ¼ /ðxÞ /ðyÞ; LL
ð11Þ
then 2-D separable wavelet functions are, wðx; yÞ ¼ /ðxÞ wðyÞ; LH
ð12Þ
wðx; yÞ ¼ wðxÞ /ðyÞ; HL
ð13Þ
wðx; yÞ ¼ wðxÞ wðyÞ; HH
ð14Þ
/ðxÞ and wðxÞ are known as, a perfect low-pass and a perfect high-pass filter, respectively. LL corresponds to lower horizontal and vertical frequencies of the original image. HL gives the vertical high frequencies and horizontal low frequencies (horizontal edges), LH the horizontal high frequencies and vertical low frequencies (vertical edges) and HH the high frequencies in both direction (the corners). Figure 4 shows how 2D-DWT is performed. Because of the orthogonal property of the DWT, it can perform 1D-DWT for each row, then 1D-DWT for each column, as shown in Figure 4. As analyzed before, due to data becoming very small after passing through high-pass filter, it can be compressed or ignored. The LL part in Figure 4 is the major part of the transformed image, thus we can prepare it recursively for two small LL parts. By doing so we can leave most of the image information concentrated on the
Orginal Image
DWT for each row
L
H
Low
High
Pass
Pass Data each column
Data
LL2 HL2
DWT for
LL1
HL1
LH1
HH1
HL1 LH2 HH2
LH1
HH1
Figure 4 2-D-DWT.
Figure 5 Block diagram for 2-D-DWT. H: Highpass Filter, L: Lowpass Filter, 2 #: Down sampling 2.
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a
b Figure 6 a) Black and white image. b) Wavelet decomposition at level 2.
top left corner of the final transformed data. The block diagram for the recursive 2DDWT is shown in Figure 5. Figure 6 shows an example of wavelet transform on the two resolution levels. The input data are chosen as black and white images (Fig. 6a). Wavelet transform is illustrated by the decomposition of white squares on a black background explained by Figure 6b. The black, gray and white pixels correspond to negative, zero and positive coefficients, respectively. As expected, the detail coefficients have high and low amplitudes on the horizontal edges, the vertical edges and the corners of the square.
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Table 1 Parameters of the synthetic structure in Figure 7. Parameters
Dip
Center (m)
Depth (m)
Intensity (nT)
Datum (nT)
Width (m)
Bottom (m)
Prism
90
(25, 25)
1
100
0
10
5
40 30 20 10
a 0
0
10
20
30
40
40
30
30
20
20
10
b 10
20
30
40
40
10
c 10
20
30
40
Figure 7 a) Synthetic total magnetic anomaly generated by prismatic source (Table 1). b) EHD signal c) EHD maxima.
The Wavelet Application on Synthetic Magnetic Data To test the ability of the wavelet transform for detection of borders, we have used synthetic magnetic anomalies given in Table 1 (Fig. 7a). The results of the wavelet transform are also confirmed for the EHD method used to estimate the horizontal location of potential fields source bodies. EHD signal is computed starting from the magnetic field up to the sixth vertical derivative. EHD maxima are shown as
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Figure 8 Pseudo-gravity anomaly and wavelet evaluation of synthetic example (Table 1). Here, A0 is pseudo-gravity anomaly of the synthetic model. A1 ,..,A4 are approximation coefficients, H1 ,..,H4 horizontal, V1 ,..,V4 vertical and D1 ,..,D4 diagonal detail coefficients for 1, 2, 3, 4 levels, respectively.
diamond form dots in Figure 7c. In this figure, solid lines indicate the borders of the synthetic model. There is a close match between EHD maxima and the borders of the synthetic model. To find the borders of the synthetic model, wavelet transform using the Daubechies 1.1 wavelet coefficient for four levels is applied to pseudo-gravity of the synthetic magnetic anomaly (Fig. 8). In this study, the pseudo-gravity anomaly (A0 ) is used instead of the total magnetic anomaly, because the source magnetization direction is generally unknown. In this and the entire following text, A1 indicates the
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approximation coefficient of the wavelet transform in Eq. (11) as LL for level 1. In the same manner, H1 , V1 and D1 show horizontal, vertical and diagonal details coefficient given by Eqs. (12), (13) and (14) as HL, LH and HH, respectively. As can be seen in Figure 8, A0 can be obtained as a summation of various wavelet coefficients for each level as: For the first level, A0 ¼ A1 þ H 1 þ V 1 þ D 1 :
ð15Þ
A0 ¼ A2 þ H2 þ V2 þ D2 þ H1 þ V1 þ D1 :
ð16Þ
A0 ¼ A3 þ H3 þ V3 þ D3 þ H2 þ V2 þ D2 þ H1 þ V1 þ D1 :
ð17Þ
For the second level,
For the third level,
For the fourth level, A0 ¼ A4 þ H4 þ D4 þ V4 H3 þ V3 þ D3 þ H2 þ V2 þ D2 þ H1 þ V1 þ D1
ð18Þ
Since the original input data is not noisy, the approximation coefficient distribution is distorted as levels are increased. A similar activity can be done for H ; V and D detail coefficients. There are either negative or positive contours for horizontal detail coefficient, H . The horizontal detail coefficients can have high frequency data for vertical direction and low frequency data for horizontal direction. As in Figure 6, detail coefficients of horizontal wavelet procedure information about horizontal borders. The longer axis of negative and positive contours correspond to horizontal borders of the structure. In the same manner, vertical detail coefficients have higher frequency property in the horizontal direction and lower frequency property in the vertical direction. Here distribution is focused on vertical borders. Since diagonal detail coefficients have high frequencies in both directions, it carries the information on diagonal corners as in Figure 8. When negative-positive contours are connected in dashed lines, the borders of the synthetic model are well matched. In Figure 8, as levels are increased, these cause distortion in approximation and in detail coefficients. Thus, the best separation is achieved at the first level. To evaluate the de-noising property of wavelet transform, we add 3% ratio of Gaussian noise for A0 to Figure 8 as to Figure 9. The results of the wavelet transform using Daubechies 1.2 coefficient for three levels are given in Figure 9. As shown in this figure, the best solution is not obtained in the first level. After de-noising the input data at the first level, at level 2 the horizontal, vertical and diagonal values match the borders of the synthetic example. As shown in synthetic examples, EHD method and wavelet transform have found the borders matching the original data. If data is noisy, wavelet transform first denoises the input, then detects borders. This is the important property of wavelet.
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Figure 9 Pseudo-gravity anomaly and wavelet evaluation of the synthetic model with 3% Gaussian noise. Here, A0 is pseudo-gravity anomaly of the synthetic model. A1 ,..,A4 are approximation coefficients, H1 ,..,H4 horizontal, V1 ,..,V4 vertical and D1 ,..,D4 diagonal detail coefficients for 1, 2, 3, 4 levels, respectively.
The Wavelet Application Archaeological Magnetic Data A topographic map of the Kusakli-Sarissa archaeological site is shown in Figure 2. The magnetic measurements have been collected by STUMPEL (1996, 1997, 1998) since 1993. A flux-gate magnetometer is used in magnetic measurements. The magnetic registration system consists of eight-channel data with a 12 V power supply and trigger input for distance marker. The area is bordered by from 81 E to 147 E
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longitude and from 260 N latitudes (Fig. 2), the western part of the Kusakli-Sarissa archaeological site is selected for this work as Kusakli-Sarissa archaeological site covers a very large area. The magnetic anomaly map of the area is shown in 0.2 m sample intervals in both forms with contouring (Fig. 10a) and imaging (Fig 10b). Approximately 15000 data points are used for digital processing. The important magnetic anomalies called A, B andC (Fig 10a) are chosen as subjects for this work and the wavelet transform is used to display more clearly the magnetic anomalies. Application of the wavelet transform to each of three anomaly regions will be discussed in detail in the following the section. Region A Region A covers 30 35 m2 . In magnetic data, to eliminate the effects of shallow mass, wavelet transform is applied to real data in region A, with Daubechies 1.1 wavelet coefficients with four levels. In Figures 11b, c, d and e, approximation coefficients are drawn for various levels. As explained previously, these coefficients have low frequency components of horizontal and vertical directions. As levels are increased, short wavelength differences disappear. The best separation is achieved at level 4 as in Figure 11e, since the effects of the structures close to the surface are well separated. In order to model the causative body, three profiles designated as I I 0 ; II II 0 and III III 0 are taken (Fig. 11e). These profiles are modeled by using a least-square inverse solution algorithm which is proposed by MARROBHE (1989)
Figure 10 Sivas-Kusakli Hittite archeological ruins. a) Magnetic contour anomaly map. b) Image map.
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Figure 11 Approximation coefficients of Region A for Daubechies of 1.1 wavelet coefficient. a) Total magnetic anomaly of region, b) approximation coefficient at level 1, c) at level 2, d) at level 3, e) at level 4.
(Figs. 12a,b,c). The model-parameters obtained for selected profiles are listed in Figure 12. These obtained models and their anomalies are shown in Figures 12a, b and c.
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Figure 12 Cross-sections obtained from the anomaly map in Figure 11e. The anomalies and model parameters from a) I I 0 cross section, b) II II 0 cross section, c) III III 0 cross-section.
The wavelet transform has been applied to detect the boundary of these buried archaeological structures. At the first step a pseudo-gravity anomaly map has been prepared (Fig. 13, A0 ). Next, the pseudo-gravity anomaly is evaluated by Daubechies 1.1 wavelet coefficients with five levels. In Figure 13, H1 shows a level one horizontal detail coefficient, as in the synthetic case, long axes of the anomalies are connected
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A0
A1
H1
V1
D1
A2
H2
V2
D2
V3
D3
A3
H3
A4
H4
V4
D4
A5
H5
V5
D5
Figure 13 Approximation coefficients of Region A for Daubechies of 1.1 wavelet coefficient, A0 . Here, A0 is pseudogravity anomaly of the synthetic model, A1 ,..,A4 are approximation coefficients, H1 ,..,H4 horizontal, V1 ,..,V4 vertical and D1 ,..,D4 diagonal detail coefficients for 1, 2, 3, 4 levels, respectively.
with a dashed line. These lines correspond to horizontal sides for the possible causative bodies. The vertical detail coefficient (V1 ) shows various patterns compared to the horizontal wavelet transform. To describe the vertical side of possible causative bodies, as in the synthetic case, long axes of anomalies are connected with dashed lines. These dashed lines show the horizontal sides of the bodies. In level one diagonal detail coefficients (D1 ), the center of the concentric anomalies, indicates the corners of the possible causative bodies. These corners are also, connected with
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Figure 14 a) EHD signal of Region A and b) location of EHD maxima. Dashed lines show boundary estimated from wavelet transform.
dashed lines. During connecting anomaly counters, the concentric anomaly polarity of being positive or negative should be considered. Otherwise, some errors can occur in border detection. The borders of three structures are well detected by the dashed lines of the diagonal detail coefficients in Region A. As in the synthetic case, EHD analysis (FEDI and FLORIA, 2001) is also applied to Region A in Figure 14a to compare with wavelet boundary results. The EHD signal is computed starting from the magnetic field extending to the sixth vertical derivative. The EHD maxima are shown as diamond dots in Figure 14b. The borders detected by wavelet approach are drawn as dashed lines in Figure 14. In general, the results of both methods have similar results. The borders of wavelet output involve EHD borders, thus wavelet approach gives a generalized border of buried objects. If the input data is noisy in wavelet approach, the regional anomaly will be separated since details disappear as levels are increased. Thus border detection is better in wavelet transform in noisy input data. Here, we have observed that the anomalies studied in this paper are composed of three structures. Region B Region B covers 10 10 m2 . This region is very close to the antic city wall as can be seen from Figure 2. Observed maximum magnetic anomaly values in
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a
IV'
IV
b
c
d
e Scale 0
1
2
3
4 m
Figure 15 Approximation coefficients of Region B for Daubechies of 1.1 wavelet coefficient. a) Total magnetic anomaly of region, b) approximation coefficient at level 1, c) at level 2, d) at level 3, e) at level 4.
Region B (Figure 15) are around 8500 nT. The identical way is followed here as with Region A. To decrease the effects of the masses closer to the surface, Daubechies 1.1 wavelet coefficients are used with four levels. In Figure 15a,
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nT 8000 7000 6000 5000 4000 Parameters of Model Dip 112.6
3000 2000 1000 0 -1000 -2000 0
1
2 meter
1
meter
Figure 16 Model structure of Region B and model parameters.
approximation coefficients with various levels are obtained. The noise effect is eliminated at the first level. We have inversely modeled this region by taking a cross section of IV IV 0 and using the MARROBHE (1989) approach as in Figure 15. The model outputs are given in Figure 16. Taking the pseudo-gravity of the total magnetic anomaly of Region B and applying wavelet transform, we have detected the possible borders as in Figure 17 and then drawn the output as dashed lines. At the same time the EHD method is applied to the total magnetic anomaly of Region B. The EHD signal is computed starting from the magnetic field up to the sixth vertical derivative (Fig. 18b). As in Region A, the results of two methods are given in Figure 18b. Results obtained with both methods match each other. We believe that there is a highly concentrated furnace closer to the surface, resulting in magnetization and thus there is high magnitude anomaly in this region.
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Figure 17 Approximation coefficients of Region B for Daubechies of 1.1 wavelet coefficient, A0 . Here, A0 is pseudogravity anomaly of synthetic model, A1 ,..,A4 are approximation coefficients, H1 ,..,H4 horizontal, V1 ,..,V4 vertical and D1 ,..,D4 diagonal detail coefficients for 1, 2, 3, 4 levels, respectively.
Region C This region which is situated at the left corner of the research area (Fig. 10a) is close to the wall of the antic city (Fig. 2). The magnetic anomaly map of Region C has many small anomalies. As in other regions, pseudo-gravity of the total magnetic anomaly is used to detect the borders, Daubechies 1.1 wavelet coefficients with four various levels are studied in Figure 19. There is a some
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327 327 326
326
325
325
324
324
323
323
82
83
84
85
81
a
82
Scale 0
1
83 b
84
85
2 meter
Figure 18 EHD outputs of Region B. a) EHD function. b) Maximum of EHD functions.
distortion after the first level of detailed coefficients. Thus as we focus on the first level, we have also shown the borders in dashed lines. In Region C, the EHD method is applied as in Figure 20a and EHD maxima are plot as diamond dots in Figure 20b. Here also, the dashed lines show borders of the wavelet transform. Similar results are obtained in both methods. In general there seems to be a circular structure surrounded by small bodies. The borders estimated from the wavelet transform are better defined. The same borders are also detected by EHD. The methods have different outputs unlike previous sections, since the structures are very small.
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Figure 19 Approximation coefficients of Region C for Daubechies of 1.1 wavelet coefficient, A0 . Here, A0 is Pseudogravity anomaly of the synthetic model, A1 ,..,A4 are approximation coefficients, H1 ,..,H4 horizontal, V1 ,..,V4 vertical and D1 ,..,D4 diagonal detail coefficients for 1, 2, 3, 4 levels, respectively.
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Figure 20 EHD outputs of Region C. a) EHD function. b) Maximum of EHD functions.
Conclusion The wavelet transform has been used for regional-residual anomaly separation in the potential fields by many researchers. In this study, wavelet transform is used to estimate boundaries of bodies. First, wavelet transform is applied to synthetic magnetic data. It is seen that the anomalies on the horizontal, vertical and diagonal detail coefficients are condensed towards the horizontal, vertical and diagonal sides of the body, respectively. The boundaries of the body are obtained when the centers of the anomalies are connected to each other with dashed lines. The synthetic examples show that boundaries of bodies can be obtained by wavelet transform. Boundaries of bodies are also confirmed by the EHD method. Then wavelet transform is applied to real magnetic data of Regions A, B and C of the Kusakli-Sarissa archaeological site. The wavelet transform and EHD methods have similar results. Although the wavelet transform provides a general solution as EHD approach, nonetheless it is more successful for noisy input data. Three prismatic structures which we have found in the Kusakli-Sarissa site may be a tomb in Region A. The prismatic body in Region B may be a furnace. The structures in Region C may be surroundings of water collection.
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Acknowledgement We thank Professor Dr. M. Fedi for valuable comments which aided us in improving the final version of this paper. We thank Professors Dr. Herald Stumpel and Dr. Filiz Bilgili who have supplied the archeological data for the Hittite civilization.
Appendix In MRA, L2 ðR) is nested subspaces Aj . . . A2 A1 A0 A1 A2 . . .
ðA1Þ
such that the closure of their union is L2 (R) 1 [
Aj ¼ L2 ðRÞ;
ðA2Þ
j1
and their intersection contains only the zero function 1 \
Aj ¼ f0g:
ðA3Þ
j¼1
In the dyadic case, i.e., when each subspace Aj is twice as large as Ajþ1 ; a function f(x) that belongs to one of the subspaces Aj has the following properties: f ðxÞ member of Aj , dilation f ð2xÞ member of Ajþ1
ðA4Þ
f ðxÞ member of A0 , translation f ðx þ 1Þ member of A0
ðA5Þ
if we can find a function /ðxÞ 2 A0 such that the set of functions consisting of /ðxÞ and its integer translates f/ðx kÞgk member of Z
ðA6Þ
form a basis for the space A0 , we call it a scaling function. For the other subspaces Aj (with j 6¼ 0) we define /ðxÞ ¼ 2j=2 /ð2j x kÞ;
j; k member of Z:
ðA7Þ
We can express /ðxÞ and wðxÞ in terms of basis functions of A1 /ðxÞ ¼ 2
N 1 X
hk /ð2x kÞ;
ðA8Þ
gk /ð2x kÞ:
ðA9Þ
k¼0
wðxÞ ¼ 2
N 1 X k
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Due to the multi-resolution analysis, these relations are also valid between Ajþ1; Aj and Dj for arbitrary j. hk and gk are called filter coefficients that uniquely define the scaling function /ðxÞ and the wavelet wðxÞ:
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