Frequency spectrum of the elevation of the topography

complicated integrals that can be found in standard tables, e.g., by Gradshteyn and Ryzhik. 7 1. l p 1. l → 0 implies that g ¼ ¹ m, which corresponds to homogeneous conductivity. Algebraic calculations yield: ˆ f y , z ¼ pRe{ e ¹ y þ iz 2 erfc z ¹ iy } 54 The steepest slope of ˆ f s y is for y ¼ 1=  2 p 0.707. This point is a critical point in the sense that a stream- line can only start at the free surface inside this point. At a point at the free surface outside the critical point, water flows out of the domain. The solution, eqn 54, and the corresponding cylindrical solution, have been studied by Rehbinder. 9,10 2. l p 1 l → ` implies that g ¼ ¹ 1l. This is not true if m . l , but for these m-values, the integrand in eqn 48 is dominated by e ¹ m 2 , which is negligible. Algebraic calculations yield the equation of a streamline: z ¼ 1 2l ln y y 55 A streamline starting at free surface inside the critical point can never reach the free surface again, which implies that the residence time is infinite. The first of the extremes is not realistic, but the second is because the horizontal length scale is more likely to far exceed the vertical one. The consequence is that the residence time of a fluid particle is long and that its point of return to the free surface is far from the ridge.

2.2 Frequency spectrum of the elevation of the topography

Possible periodicity of the topography can be identified using either of two methods, which are described below. The basis for all calculations are elevation data in GIS. As mentioned in the introduction, the elevation is given at the nodes of a quadratic mesh. The distance between the points is 500 m. The fact that the mesh is quadratic considerably simplifies the calculations. Details of the topics of this section can be found in the textbook on multidimen- sional signal processing by Dudgeon and Merserau. 5 All numerical calculations which are described below have been performed with the aid of the standard code MATLAB. 2.2.1 Fast Fourier transform The most straightforward and commonly used frequency analysis is the fast Fourier transform, in the sequel called FFT. Since the elevation yk 1 ,k 2 is a function of two variables, the 2D discrete Fourier transform: Y n 1 , n 2 ¼ X N 1 k 1 ¼ 1 X N 2 k 2 ¼ 1 y k 1 , k 2 e ¹ 2pi n 1 k 1 =N 1 þ n 2 k 2 =N 2 n 1 ¼ ¹ N 1 =2:::N 1 =2 ¹ 1 n 2 ¼ ¹ N 2 =2:::N 2 =2 ¹ 1 ð 56Þ applies. An estimate of the power spectrum is then: ˆ F n 1 , n 2 ¼ lY n 1 , n 2 l 2 57 The amplitude corresponding to a particular frequency is estimated from the FFT as a ¼ 2 lY n 1 , n 2 l N 1 N 2 58 Details on the derivation can be found in the textbook by Brigham. 1 Each Fourier coefficient corresponds to the continuous frequencies in each of the two coordinate directions. q 1 ¼ 2pn 1 =N 1 P 1 59 q 2 ¼ 2pn 2 =N 2 P 2 where P 1 and P 2 are the spatial sampling intervals. Eqn 8 yields the corresponding wavelengths. L 1 ¼ N 1 P 1 =n 1 60 L 2 ¼ N 2 P 2 =n 2 The FFT gives a spectrum defined only by discrete frequency points. If any of n i is small, i.e., L i ,N i P i , the resolution is poor. However, the resolution can be enhanced by the aid of so-called zero-padding. This method is described in the textbook by Brigham. 1 2.2.2 Autoregression Another common method, albeit somewhat more compli- cated, is autoregression called AR in the sequel. In contrast to the FFT, the AR gives the spectrum at any point. On the other hand, the AR does not give any information about the amplitude of the components. According to the AR method, the elevation yn 1 ,y 2 is written: y n 1 , n 2 ¼ ¹ X k 1 X k 2 a k 1 , k 2 y n 1 ¹ k 1 , n 2 ¹ k 2 þ e n 1 , n 2 61 where en 1 ,n 2 is white noise, the mean value and variance of which are equal to zero and m respectively. The two- dimensional z-transform applied to eqn 61 yields: Y z 1 , z 2 ¼ 1 A z 1 , z 2 E z 1 , z 2 62 where the filter polynomial Az 1 ,z 2 is given by: A z 1 , z 2 ¼ 1 þ X k 2 X k 2 a k 1 , k 2 z ¹ k 1 1 z ¹ k 2 2 63 516 G. Rehbinder A. Isaksson The summation, i.e., the region of support S mn , remains to be chosen. Here a non-symmetrical half-plane has been used. Hence, S mn ¼ {k 1 , k 2 lk 1 ¼ , 1 k 2 m , 1 k 1 n , ¹ m k 2 m} ð 64Þ where m,n is known as the model order. Estimates of m and a k 1 , k 2 can be obtained by applying the least squares method to the elevation data. An estimate of the spectrum in this case is: ˆ F q 1 , q 2 ¼ ˆ m l ˆA e iq 1 , e iq 2 l 2 65 Just for the FFT, the dominating length scales are: L 1 ¼ 2pP 1 =q 1 66 L 2 ¼ 2pP 2 =q 2 Details on two-dimensional AR-modelling have been reported by Isaksson. 8 3 DISCUSSION 3.1 Periodicity of the Swedish topography A number of areas in Sweden have been suggested for a future repository of radioactive waste. They are shown in Fig. 7. Forsmark, item 1, and Oskarshamn, item 2, lay at the coast of the Baltic and Ringhals, item 3, at the coast of the Skagerrak. Nuclear power plants are already located at these places. Mala˚, item 5, and Storuman, item 4 in the north, have also been suggested. Most parts of the south of Sweden are low-lying and flat and any possible periodicity of the topography has not been studied. The highland of Sma˚land, between Oskarshamn and Ringhals, is interesting from this point of view. The highland slopes radially and there are good reasons to believe that it has an azimuthal periodic structure. Such a periodicity, if it exists, cannot be identified because the elevation points are given in a rectan- gular coordinate system. Nonetheless, it does not seem too difficult to investigate azimuthal periodicity, if the GIS points are transformed to polar coordinates. The dominating hydraulic gradients at Oskarshamn and Ringhals are determined by the ratio between the altitude of the highland of Sma˚land and half the distance between the coasts of the Skagerrak and the Baltic. Analogously, the hydraulic gradient at Forsmark is determined by the ratio between the altitude of southern Norway and the distance between the Caledonian and the Baltic. In the north, viz., the west part of Lapland, the topography is hilly. Although the Caledonian dominates the topography, there are large valleys with rivers running eastwards to the Baltic. The direction of the rivers is approximately 1208. The elevation data of some areas in Lapland were analyzed by means of FFT and AR and one specific area, item 6, in Fig. 7, was studied thoroughly. The domain is rectangular, 0 k 1 N 1 , 0 k 2 N 2 . Since the dominating length scale of the Caledonian far exceeds the extent of the domain, the elevation points yk 1 ,k 2 were replaced by reduced points y r k 1 ,k 2 , which are equal to the elevation points minus the elevation of the average slope plane of the domain. In this way the dominating period that is equal to infinity is removed. A superficial visual inspection of a standard map, such as the one in Fig. 8, or a level map of the reduced elevation, as the one in Fig. 9, does not give any clear indication whether periodic variations exist within the domain. A preliminary analysis showed that the elevation data can be down-sampled by a factor of 10, without any loss of information. The mesh width of the grid is thus increased to 5000 m. The result of the FFT and AR analyses of y r k 1 ,k 2 are shown in Figs 10 and 11, respectively. The fastest frequency that can be represented is given by the Nyquist frequency, which is half the sampling fre- quency, here corresponding to a period of 10 km. The 2-D FFT applied to the domain for which N 1 ¼ 40 and N 2 ¼ 30, together with zero-padding to 1024 3 1024, yields the power spectrum which is shown in Fig. 10. It is important to note that n 1 and n 2 in Fig. 10 are running from ¹512 to 512. This implies that N 1 ¼ N 2 ¼ 1024 in eqn 60, whereas N 1 ¼ 40 and N 2 ¼ 30 in eqn 58. There are six peaks, positive and negative. They Fig. 7. A schematic map of Scandinavia: 1 Forsmark, 2 Oskar- shamn, 3 Ringhals, 4 Mala˚, 5 Storuman, 6 analyzed area. Large-scale flow of groundwater in Swedish bedrock. An analytical calculation 517 Fig. 8. A topographical map of Sarek in the northwest part of the rectangular area in Fig. 6. 518 G. Rehbinder A. Isaksson correspond to three sets of periodic topographical waves, viz., L 1 ¼ 74.2 km, L 2 ¼ 170.7 km, a ¼ 58.2 m, L 1 ¼ 146.3 km, L 2 ¼ 232.7 km, a ¼ 47.4 m and L 1 ¼ ¹ 255.7 km, L 2 ¼ 204.9 km, a ¼ 46.6 m. The last one corresponds to the longest wave, whereas the first one cor- responds to the most pronounced wave in the spectrum. Alternatively, every wave can be described as one length scale L ¼ L 1 sin v in the direction v ¼ arctan L 2 L 1 to the latitude. Hence the shortest, the intermediate and the longest waves, with indices s, i and l, respectively, are L s ¼ 68 km, v s ¼ 678, a s ¼ 60 m, L i ¼ 124 km, v i ¼ 588, a i ¼ 47 m and L l ¼ 160 km, v l ¼ ¹ 398, a l ¼ 47 m. The longest of the above topographical waves can be questioned, since their lengths are approximately the same as or even exceed the extent of the domain. The AR power spectrum, obtained by estimating a half plane of model order 7,4, is shown in Fig. 11. The peaks are for q 1 ,q 2 ¼ 0.49, 0.25, which correspond to L 1 ¼ 64 km and L 2 ¼ 126 km. As above these values correspond to one length scale L ¼ 57 km at an angle v ¼ 638 to the latitude. The longer waves can be discerned, but not evalu- ated. This is in agreement with the comment above, viz., that the number of periods of the long waves in the domain is insufficient. Despite the reservations, the two methods yield surpris- ingly consistent results, in that the magnitudes as well as the directions of the pronounced topographic waves agree. Although the shortest wave is dominant in comparison with the others in Fig. 10, the question is if it is dominant from a hydraulic point of view, i.e., is the condition eqn 47 sufficiently well violated? The answer is yes, and the reason is the following. Because a s a l L l L s 2 ¼ 7.1 . 1 and a s a i L i L s 2 ¼ 4.3 . 1, it follows that the influence on the hydraulic gradient of the shortest wave is most likely strong. The lateral length scales are so large, i.e., L l z . L i z . L s z q 1 and the location of a possible repository is so shallow, i.e., z r z , that the streamlines are very shallow. Consequently, the short but pronounced wave in Figs 10 and 11 determines the flow pattern and the residence time.

3.2 The residence time of a fluid particle