On the calculation method of Ce Map

SP/elevation gradient (Ce-gradient) calculation method proposed by Lénat (2007), which was used to produce Ce-gradient map of Nemrut volcano (Ulusoy et al., 2008) calculates the gradient in four directions (N-S, E-W, NW-SE and NE-SW).

Using an array-type operator could overcome the limitations of calculations with

four directional operators. For this purpose, a computer code using an array-based operator was designed and tested. The code uses a procedure that may be named as “Swirl procedure”. Swirl procedure depends on swirling two (in this case PS-PS or DEM-DEM) images (arrays) and applying a mathematical operation to overlapped pixels during swirling (Fig. 5.15a). Using PS image and DEM, this operation allows calculating ΔV or Δz between two points in image array (Fig 5.15a).

In swirl operation, the duplicate of the image array is shifted on the same image by predefined number of pixels. This predefined “number of pixels” can be called as

“swirl degree”. A duplicate (red) of a 5x5 array (yellow) is shifted over the same image by 2 pixels in Figure 5.15b; therefore swirl degree is “2”. Using this condition of the shifted images (Fig. 5.15b), a mathematical operation between points A and B (or C and D) can be made (Fig. 5.15c). When the duplicate image is swirled on the original image, depending on the swirl degree, a mathematical operation can be applied on certain angles. These specific angles according to swirl degree are listed in Table 5.2. With the change of swirl degree, the lateral difference between pixels (dx) also changes for every point calculated.

Figure 5.15. Principles of Swirl operation a) Swirl procedure using a duplicated Self-Potential (PS) image pair, to produce ΔV image. b) Image pairs shifted with a swirl degree of 2. c) Sample Pixel pairs (A-B and C-D) that the calculation will be based on with a swirl degree of 2.

Swirl degree 1 2 3 4 5 6 7 8 9 10

Operation angle 45.00 22.50 11.25 5.63 2.81 1.41 0.70 0.35 0.18 0.09 Table 5.2. Swirl degree and the corresponding angle between calculated pixels.

Two ways of calculation can be applied using swirl procedure: “Full-swirl” and

“limited-swirl”. In the full-swirl calculation, all the neighboring pixels up to swirl degree are taken into account and final value of center pixel is calculated upon these neighboring pixels. On the other hand, limited-swirl operator uses only the pixel values defined by the swirl degree for the calculation. The contribution of pixels to the operation in full-swirl procedure (Fig. 5.16a), and in limited-swirl procedure (Fig. 5.16b), when the swirl degree = 2 is shown in Figure 5.16. Red point symbolizes the calculated center value and the black points symbolize the contributing neighboring pixels. Desired mathematical operation could be applied for all the pixels in the image. If the swirl degree = 1 there is no difference between full-swirl and limited-swirl procedures (Fig. 5.16c).

Figure 5.16. Principles of Full-swirl and Limited-swirl procedures. Contributions of pixels in, a) Full- swirl procedure b) Limited-swirl procedure, for swirl degree = 2 and, c) for swirl-degree =1.

Ce gradient calculation

PS-altitude gradient (Ce) can be calculated using swirl procedure. A copy of PS image can be swirled on itself to calculate ΔV between two pixels (Fig. 5.17).

Similarly, when swirl procedure is applied to a DEM covering the same area and resolution with PS image, Δz for same points used in PS swirl procedure can be calculated (Fig. 5.17). Using ΔV and Δz values four different methods can be applied to calculate Ce values:

Method 1: Ce-mean calculation using full-swirl procedure, Method 2: Ce-mean calculation using limited-swirl procedure, Method 3: Ce-max calculation using full-swirl procedure, Method 4: Ce-max calculation using limited-swirl procedure.

Figure 5.17. Swirl procedures used to calculate Ce-gradient. Black and grey points are representing the contributing pixel values to calculate final (red) value. a) Calculation of ΔV and Δz in full-swirl procedure for swirl degree = 2, b) Calculation of ΔV and Δz in limited- swirl procedure for swirl degree = 2.

We wrote an IDL code to calculate the Ce-gradient map, which is using the following equations for the four methods described above:

Method 1 and 2:

( )

⎟⎟

⎟⎟

⎠

⎞

⎜⎜

⎜⎜

⎝

⎛

Δ +Δ Δ +

+Δ Δ +

+Δ Δ Δ

=

− n

z V z

V z

V z V mean

Ce

f ⁿ

... n

...

7 7 2

2 1 1

Method 3 and 4:

( )

⎠

⎜⎜ ⎞

⎝

⎛

Δ Δ Δ

Δ Δ

Δ Δ

= Δ

−

n n

z V z

V z

V z MAX V Ce

f max , ,..., ,...,

7 7 2

2 1

1

At the end of the calculation Ce_calc code can filter the final image using median filter of selected kernel size. Advantages of this method are: 1) array based code can make the calculations rapidly, 2) Ce gradient calculations can be made using wide range of neighboring values (and more then 4 directions) which increases the reliability of the result. Where Δz values are very low (i.e. <1m for a dataset with 25 m spatial resolution), resulting Ce-gradient is relatively more reliable than a calculation in 4-direction. Still, where Ce values are abnormally high due to low Δz values, it is possible to eliminate this effect by blinding low Δz values, using a NaN (not a number) definition in the code. Finally, when the code ran, due to the calculation method, a cropping related to the swirl degree is occurred on the final Ce-gradient image.

Artificial dataset test run

We have produced an artificial dataset (Fig. 5.18) which consists of PS and DEM sets (each set is an array of 90x58 pixels and have 16 m resolution). Using this dataset, Ce maps were calculated by our code with four methods described above (Fig. 5.19a,b,c,d). Calculation of Ce-mean, using Method 1 gave the most reliable output. Both positive and negative Ce-value ranges and the anomalies of Ce are consistent with the PS dataset. Ce-max calculators failed because: a) they don’t enclose the maximums in the negative range; b) maximum values are abnormally high. Code may be changed in an appropriate way to recalculate the negative maximum values (minimums), but the resulting positive and negative range of values will be still too high.

Figure 5.18. Artificial dataset to calculate Ce-gradient image. Artificial a) DEM and b) PS images.

Real dataset test run

The code was also used to produce Ce-gradient from Nemrut datasets and the output is given in Figure 5.20a. The old Ce map produced by code of Lénat is given in Figure 5.20b. Note that, identical color ramps were used for Ce-gradient images for a better comparison.

Figure 5.19. Ce-gradient maps produced with Swirl method, using artificial dataset. Swirl degree = 5, median filter size = 5x5. Final Ce-gradient image is overlapped on hillshade of artificial DEM. Contour lines show the artificial PS data. a) Ce-mean calculated with method 1, b) Ce-mean calculated with method 2, c) Ce-max calculated with method 3 and d) Ce-max calculated with method 4.

Figure 5.20. Ce-gradient maps of Nemrut caldera. Both gradient images were filtered with a 5x5 median filter. a) Ce-gradient (Ce-mean) calculated with swirl procedure (swirl degree = 5), method 1, b) Ce-gradient calculated using the method defined by Lénat (2007).

Swirl procedure could be used in various applications (e.g. ΔT calculations with Thermal satellite imagery, topographic corrections in geophysics). Our studies on various uses of swirl procedure continue; a short paper describing the procedure and its applications will soon be prepared for publishing.