Statistical properties of the sea bottom in the Ormen Lange area

Transcription

1 Technical Report No. Computational Mathematics Unit Statistical properties of the sea bottom in the Ormen Lange area Joakim Hove Jarle Berntsen December, 3 The Bergen Center for Computational Science A section of UNIFOB - University of Bergen

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3 Summary We have used a large high resolution dataset from the Ormen Lange area to estimate subgrid parameters. In particular we investigate how subgrid parameters vary with cell size x, and the short length h used for finite difference operations, this dependence varies greatly between the different subgrid parameters. We find that for small values of x / h much of the statistical behavior is universal from between different areas. The slope, and the surface standard deviation are the most important subgrid parameters. For these parameters we find that the distribution can be nicely parameterized with the Log Normal distribution, furthermore we find that the variation of the average surface standard deviation with x can be quite well described with a power law with a universal exponent H.

4 Contents Introduction Overview of the topography. Simple griding Subgrid properties 6 3. Subgrid parameters Subgridding Estimators for subgrid parameters Small estimator test case Alternative estimators for γ, σ and θ Two length scales in the estimation Results / figures 4 4. Discussion of figures 7 and Discussion Detailed discussion 3 5. Standard deviation Fractal properties Standard deviation from plane Distribution of Z Slope σ Anisotropy γ Principal axis θ Suggestion for recipe for oceanographic simulation Concluding remarks 4 A Detailed results 4 B Transformation to rectangular coordinates 5 List of Figures An approximate view of the resolution of the data, the color shading indicates the average distance in meters between measurements

5 This figure schematically shows the complete sampling area Γ, along with a grid area Ω which has been divided in 3 3 cells The left figure shows the complete sampling area. The box in the lower right corner of this figure shows the bounding box of the right figure, which is a more detailed view of the area containing the pipeline. The two areas shown have a size of 9km 3km and 9km 7km respectively These two figures show highlights of the topography at the start and the end of the pipe respectively. Both of the two areas shown have size km km Illustration of how the topographic variations on subgrid scale lead to dissipation. 8 6 This figure shows the griding process schematically, with a micro-grid in the lower left grid cell. Observe the two different length scales x and h, ideally we should have results which are independent of the short scale h, which only the represents the (arbitrary) micro-griding Illustrating the algorithm used to form micro-grids, only the depth value closest to the cell center, i.e. the value within the circle, is used. The grid shown in this figure, corresponds to the shaded micro grid in the lower left corner of Fig A stream of random numbers x i N(, ) is generated, and the averages Σ x (n) are formed. This is repeated many times, and the plots show histograms of Σ x (n) with n =, and respectively, observe how the variance σ (n) decreases with n Demonstration of the XY estimators. The left column is from a random, uncorrelated topography, and the right column is from a plane topography with similar some random noise added. The second row shows the vector field m(x) from Eq. 9. We observe that to the left the arrows have random length and direction, whereas to the right there is a well defined length and direction. The third row shows the resulting vector M from Eq., and the phase θ XY (in the left figure M has been elongated, a faithful representation would have left M invisible). The dashed circle is M =, which corresponds to perfect ordering, i.e. all arrows in row two of equal length and direction. The slope σ XY is given by the average length (completely ignoring coherence) of the vectors in the second row, i.e. we can see that σ XY is small to left and larger to the right This figure shows the various areas used for the analysis of scale dependence, the topography of the different areas is shown in figures The topography of the area Large The topography of the area Small The topography of the area EdgeN, this and following topography in figure 4 are included to investigate the effect of the edge The topography of the area Ekes The topography of the area HighResN, this area is from around the pipeline and sampled with higher spatial resolution, here the resolution is about 5m The topography of the area HighResS, higher spatial resolution like

6 7 The area EdgeN, with cells, and h = m. The eight plots show: Topography (56 56 cells), µ, σ KLM, σ XY, γ KLM, γ XY, θ and z. The white spots indicate cells with insufficient data The same as figure 7, but with 6 6 cells, and h = 6m The black dots indicate the available depth measurements, within each cell random values of µ and σ have been drawn from the appropriate distributions. The hatched point indicates a missing data point, this has been sampled from the distribution of z The topography is measured in the dotted points, with spatial resolution x. The dashed boxes indicate the new grid cells with spatial resolution ξ = 4 x, by using x analogously to h it is possible to estimate subgrid properties at the coarse scale xi Histograms of σ KLM in: Large. Fit of these curves to the Log Normal distribution is found in table Histograms of σ KLM in: Small. Fit of these curves to the Log Normal distribution is found in table Histograms of σ KLM in: EdgeN. Fit of these curves to the Log Normal distribution is found in table Histograms of σ KLM in: EdgeS. Fit of these curves to the Log Normal distribution is found in table Histograms of σ KLM in: HighResN. Fit of these curves to the Log Normal distribution is found in table Histograms of σ KLM in: HighResS. Fit of these curves to the Log Normal distribution is found in table Histograms of σ XY in: Large. Fit of these curves to the Log Normal distribution is found in table Histograms of σ XY in: Small. Fit of these curves to the Log Normal distribution is found in table Histograms of σ XY in: EdgeN. Fit of these curves to the Log Normal distribution is found in table Histograms of σ XY in: EdgeS. Fit of these curves to the Log Normal distribution is found in table Histograms of σ XY in: HighResN. Fit of these curves to the Log Normal distribution is found in table Histograms of σ XY in: HighResS. Fit of these curves to the Log Normal distribution is found in table Histograms of γ KLM in: Large Histograms of γ KLM in: Small Histograms of γ KLM in: EdgeN Histograms of γ KLM in: EdgeS Histograms of γ KLM in: HighResN

7 38 Histograms of γ KLM in: HighResS Histograms of γ XY in: Large Histograms of γ XY in: Small Histograms of γ XY in: EdgeN Histograms of γ XY in: EdgeS Histograms of γ XY in: HighResN Histograms of γ XY in: HighResS Histograms of θ in: Large Histograms of θ in: Small Histograms of θ in: EdgeN Histograms of θ in: EdgeS Histograms of θ in: HighResN Histograms of θ in: HighResS Histograms of µ in: Large. Fit of these curves to the Log Normal distribution is found in table Histograms of µ in: Small. Fit of these curves to the Log Normal distribution is found in table Histograms of µ in: EdgeN. Fit of these curves to the Log Normal distribution is found in table Histograms of µ in: EdgeS. Fit of these curves to the Log Normal distribution is found in table Histograms of µ in: HighResN. Fit of these curves to the Log Normal distribution is found in table Histograms of µ in: HighResS. Fit of these curves to the Log Normal distribution is found in table Histograms of µ PR in: Large. Fit of these curves to the Log Normal distribution is found in table Histograms of µ PR in: Small. Fit of these curves to the Log Normal distribution is found in table Histograms of µ PR in: EdgeN. Fit of these curves to the Log Normal distribution is found in table Histograms of µ PR in: EdgeS. Fit of these curves to the Log Normal distribution is found in table Histograms of µ PR in: HighResN. Fit of these curves to the Log Normal distribution is found in table Histograms of µ PR in: HighResS. Fit of these curves to the Log Normal distribution is found in table The h limit of σ KLM in Large The h limit of σ KLM in Small

8 65 The h limit of σ KLM in EdgeN The h limit of σ KLM in EdgeS The h limit of σ KLM in HighResN The h limit of σ KLM in HighResS The h limit of σ XY in Large The h limit of σ XY in Small The h limit of σ XY in EdgeN The h limit of σ XY in EdgeS The h limit of σ XY in HighResN The h limit of σ XY in HighResS The h limit of γ KLM in Large The h limit of γ KLM in Small The h limit of γ KLM in EdgeN The h limit of γ KLM in EdgeS The h limit of γ KLM in HighResN The h limit of γ KLM in HighResS The h limit of γ XY in Large The h limit of γ XY in Small The h limit of γ XY in EdgeN The h limit of γ XY in EdgeS The h limit of γ XY in HighResN The h limit of γ XY in HighResS The h limit of θ in Large The h limit of θ in Small The h limit of θ in EdgeN The h limit of θ in EdgeS The h limit of θ in HighResN The h limit of θ in HighResS Scaling plot of: µ Scaling plot of: µ PR Histograms of z in: Large. See table 3 for fits to a Lorenzian Histograms of z in: Small. See table 3 for fits to a Lorenzian Histograms of z in: EdgeN. See table 3 for fits to a Lorenzian Histograms of z in: EdgeS. See table 33 for fits to a Lorenzian Histograms of z in: HighResN. See table 34 for fits to a Lorenzian Histograms of z in: HighResS. See table 35 for fits to a Lorenzian Scaling plot of: z Numerical fits in table

9 Scaling plot of: z List of Tables Table of symbols used Size of the different areas studied Overview of figures/tables in this appendix Overview of figures/tables in this appendix Overview of figures/tables in this appendix Results of curve fitting the average value of µ with block size, according to Eq. 7. The left columns (C and α ) are based on unweighted curve fits (i.e. dominated ny large x), whereas the right columns are based on weighted fits (i.e. dominated by small x) Analogous to table 4 for the deviation from a plane, i.e. µ PR Fit of distribution of µ in Large to the Log Normal distribution. Rawdata is shown in Fig Fit of distribution of µ in Small to the Log Normal distribution. Rawdata is shown in Fig Fit of distribution of µ in EdgeN to the Log Normal distribution. Rawdata is shown in Fig Fit of distribution of µ in EdgeS to the Log Normal distribution. Rawdata is shown in Fig Fit of distribution of µ in HighResN to the Log Normal distribution. Rawdata is shown in Fig Fit of distribution of µ in HighResS to the Log Normal distribution. Rawdata is shown in Fig Fit of distribution of µ in HighResS to the Log Normal distribution. Rawdata is shown in Fig Fit of distribution of µ PR in Large to the Log Normal distribution. Rawdata is shown in Fig Fit of distribution of µ PR in Small to the Log Normal distribution. Rawdata is shown in Fig Fit of distribution of µ PR in EdgeN to the Log Normal distribution. Rawdata is shown in Fig Fit of distribution of µ PR in EdgeS to the Log Normal distribution. Rawdata is shown in Fig Fit of distribution of µ PR in HighResN to the Log Normal distribution. Rawdata is shown in Fig Fit of distribution of µ PR in HighResS to the Log Normal distribution. Rawdata is shown in Fig

10 8 Fit of distribution of σ XY in Large to the Log Normal distribution. Rawdata is shown in Fig Fit of distribution of σ XY in Small to the Log Normal distribution. Rawdata is shown in Fig Fit of distribution of σ XY in EdgeN to the Log Normal distribution. Rawdata is shown in Fig Fit of distribution of σ XY in EdgeS to the Log Normal distribution. Rawdata is shown in Fig Fit of distribution of σ XY in HighResN to the Log Normal distribution. Rawdata is shown in Fig Fit of distribution of σ XY in HighResS to the Log Normal distribution. Rawdata is shown in Fig Fit of distribution of σ KLM in Large to the Log Normal distribution. Rawdata is shown in Fig Fit of distribution of σ KLM in Small to the Log Normal distribution. Rawdata is shown in Fig Fit of distribution of σ KLM in EdgeN to the Log Normal distribution. Rawdata is shown in Fig Fit of distribution of σ KLM in EdgeS to the Log Normal distribution. Rawdata is shown in Fig Fit of distribution of σ KLM in HighResN to the Log Normal distribution. Rawdata is shown in Fig Fit of distribution of σ KLM in HighResS to the Log Normal distribution. Rawdata is shown in Fig Fit of z distribution to Lorenzian in: Large. Rawdata is shown in Fig Fit of z distribution to Lorenzian in: Small. Rawdata is shown in Fig Fit of z distribution to Lorenzian in: EdgeN. Rawdata is shown in Fig Fit of z distribution to Lorenzian in: EdgeS. Rawdata is shown in Fig Fit of z distribution to Lorenzian in: HighResN. Rawdata is shown in Fig Fit of z distribution to Lorenzian in: HighResS. Rawdata is shown in Fig Fit of w z of z histograms. These fits are shown in Fig Fit of w z of z histograms. These fits are shown in Fig

11 Table : Table of symbols used. Symbol Explanation page z(x, y) Depth at position x, y h Short length used in finite difference approximations: 7,3 x z(x, y) (z(x + h, y) z(x, y))/h. z The distribution of z(x + h, y) z(x, y) and z(x, y + h) z(x, y). 6 Observe that the two directions are just joined together; this is probably not optimal (should rotate??) µ Subgrid parameter: standard deviation of bottom topography 6 µ PR Standard deviation of residues R(x, y) = z(x, y) (αx + βy + γ), 6 where α, β and γ come from a least squares fit to a plane. σ Subgrid paramater: slope 7 σ KLM The original definition of slope from Lott and Miller [] 7 σ XY The slope definition from the XY model. γ Subgrid parameter: anisotropy 7 γ KLM The original definition of anisotropy from Lott and Miller [] 7 γ XY The definition of anisotropy from the XY model θ Subgrid parameter: principal axis 7 x Size of grid cell N N Number of grid cells, this is more used than x to denote the size of grid cells. x can then be found by looking up the total size of the actual area in table. n n The number of grid cells in a microgrid within one conventional 9 grid cell. x, n and h are related through x = nh. Area This formatting is used to indicate one of the six areas Small, Large, 4 EdgeN, EdgeS, HighResN and HighResS. See figure and table. C µ Amplitude in the power law fit between µ and cell size x. This 33 amplitude is nonuniversal, and will vary depending on large scale features of the topography, bottom material e.t.c. C µ + Like C µ, but estimated without weights ( u), i.e. with main 33 emphasize on large values of x. Cµ Like C µ, but estimated with weights ( u), i.e. with main emphasize 33 on small values of x. H Hurst exponent, relating x and µ. The value of this exponent 33 should be universal, i.e. independent of details like bottom material. When this applies we have (statistical) self-similarity. H + Hurst exponent, estimated without weights ( µ). 33 H Hurst exponent, estimated with weights ( µ). 33 S, M Parameters in the Log Normal distribution. Used for the distribution 3. of µ, µ PR and σ, without any further indexing. e z Mean value of Lorenz fit of z distribution. 34 w z Width of Lorenz fit of z distribution. 34 9

12 Introduction Numerical simulations in oceanography are necessarily performed with a finite spatial resolution of the topography, values of x O( km) is quite typical. With such large grid cells there can clearly be significant topographic variations within one cell, so called subgrid variations. Lott and Miller [] have devised a method to calculate an effective drag coefficient from the subgrid topography. This drag is calculated based on a few subgrid parameters. Norsk Hydro will build a pipeline from the Norwegian coast, and out into the North Sea. To find an optimal route for the pipeline, and further to precisely calculate current induced loads, Norsk Hydro have collected depth measurements with high spatial resolution in a large area. With the use of this dataset we have been able to calculate subgrid properties for a wide range of x values, and in different areas. The remaining part of this report is organized as follows: In section we give a short introduction to the dataset, and the topography. In section 3 we introduce the subgrid properties and subgrid parameters. The results, in the form of figures and tables, are presented in section 4, and the results are discussed in section 5. Some details can be found in the appendices.

13 Overview of the topography In this section we will show some simple figures of the sea-bottom to become familiar with the topography. The depth varies from more than 6m in the north-west corner, to about m on the continental shelf along the eastern edge of the area. The shelf running approximately north-south along the eastern edge of the topography is a prominent feature, but there is an overall decline to-wards north west in the whole area. In the area around the pipeline the depth is sampled with a resolution of 5m 5m, whereas the remaining parts are sampled with 5m 5m, Fig. shows the resolution schematically Resolution Latitude Longitude 4 Figure : An approximate view of the resolution of the data, the color shading indicates the average distance in meters between measurements.. Simple griding The original raw data is a long list of triplets: (latitude,longitude,depth). The data is not regularly sampled, the spatial resolution varies over the region, see Fig., and the total amount of data (a total of. 8 triplets) can not be loaded in computer memory simultaneously. It has therefore been essential to form various numerical grids of the data. The Figures 3 and 4 have been made with the following simple algorithm. A rectangular region Ω is chosen, and this is subdivided in N N cells.

14 . Average depth is calculated for each cell. This algorithm is applied to get an overview of the topography, it is illustrated in Fig.. Ω Γ Figure : This figure schematically shows the complete sampling area Γ, along with a grid area Ω which has been divided in 3 3 cells.

15 Measurement area Detailed map of pipeline Latitude Latitude Longitude Longitude Figure 3: The left figure shows the complete sampling area. The box in the lower right corner of this figure shows the bounding box of the right figure, which is a more detailed view of the area containing the pipeline. The two areas shown have a size of 9 km 3 km and 9km 7km respectively

16 Detailed map around station TP8 Detailed map around station TP Latitude Latitude Longitude Longitude Figure 4: These two figures show highlights of the topography at the start and the end of the pipe respectively. Both of the two areas shown have size km km

17 The Figures 3 and 4 show the topography in a geographic coordinate system, i.e. with latitude and longitude. For the detailed calculations of subgrid parameters we have used rectangular coordinates, the transformation from geographic to rectangular coordinates is described in appendix B. 5

18 3 Subgrid properties For a typical oceanographic simulation the spatial resolution is quite poor, with grid cells at least in the km range. This means that there can be considerable topographic variations within one grid cell. These variations clearly affect the near bottom water flow, however the discretized equations governing the ocean dynamics are completely oblivious to topographic variations on the subgrid scale. Lott and Miller [] have introduced an effective near-bottom drag coefficient which is a function of a few subgrid parameters which characterize the topographic variations within one grid cell. The current dataset has a unique resolution, hence we can use it to calculate subgrid parameters, and in particular we can study (i) the scale dependence, i.e. how the subgrid parameters change with the size of the grid cell, and (ii) the characteristics of spatial distribution of subgrid parameters. There is clearly large variation in the subgrid parameters between different locations, however it is our belief that in particular the two properties listed will show some universal behavior. The remaining part of this section is organized as follows: In section 3. we introduce, and discuss the various subgrid parameters, in section 3. we describe the selection of data to use for calculation of subgrid parameters. In section 3.3 we introduce, and discuss, the statistical estimators used, and finally section 3.4 is devoted to a discussion of the short length scale h. 3. Subgrid parameters The properties we have estimated are. The variance of the height z(x, y) within each cell µ = n and the variance of the residues µ R = n 3 (z i z), z = z i, () n i i (z i (a + bx i + cy i )), () i where a, b and c are determined by fitting a plane through the cell in a least squares sense.. The average vertical distance of neighboring points, z = z(x) z(x + hˆx). (3) 3. The matrix H = ( z ) z x x z y z x z y ( z y ) = The over-bars in Eq. 4 denote spatial averages. [ K + L M M K L ]. (4) 6

19 Initially we also calculated the median of the depth within each cell, however it turned out that the difference between the median and the standard arithmetic mean was insignificant. The matrix Eq. 4 measures the rate of change of the topography, as measured by the mean square gradient. If we diagonalize this matrix, we will find two eigenvalues λ and λ along with the corresponding eigenvectors x and x λ = K ± L + M, x = { M L+ L +M }, x = { M L L +M }. (5) In Eq. 5 λ / x measure the maximum rate of change of z, and the corresponding direction, as measured by the mean square gradient. Instead of using the eigenvalues/eigenvectors from Eq. 5 directly, it is customary to consider the subgrid parameters θ, γ and σ defined by γ = λ ( ) ˆx, σ x = λ and θ = acos = acos L + M / + L M + L λ x L + M + L (6) M + L Here γ is an (an)isotropy parameter, in a topography with γ all directions are equivalent, whereas topographies with a well defined direction will have γ. The angle between the direction of most rapid variation, and the x axis is given by θ and the slope parameter σ gives the average of the local square gradient. In the case of σ it is important to emphasize that it is the local square gradient which is averaged, this means that smooth topographies with a finite overall slope might give a small value for σ, whereas a rough topography which is on average flat, will give a large value for σ. Observe that θ, γ and σ are mathematically well defined for any topography. However it is essential to consider all three simultaneously to get a qualitative understanding of the properties of the topography. Some cases to consider are:. We will always get a value for θ, but if the same topography gives γ, i.e. all directions are essentially equivalent, this means that the θ is really irrelevant.. γ, i.e. there is a distinct direction in the topography. However the same topography may give σ. This would mean that yes - there is a singled out direction in the topography, but it is nearly flat anyway. Finally we mention that only for σ and γ is it possible to infer information from a numerical value alone, for a phase like θ the concept of large and small makes no sense. An illustration of the subgrid parameters γ and µ, along with the important blocking height Z b is given in Fig. 5 D b (z) = C d max ( r ), ρ σ µ 3. Subgridding ( ) Zb z / max(cos ψ, γ sin ψ) U U z + µ. (7) Practically the subgrid parameters, for one grid cell, are calculated by forming a new dataseries consisting of only the points falling into this particular cell, and then using this limited data set to estimate the subgrid parameters for this particular cell. 7

20 Incident flow with velocity U Z B σ σ σ σ µ Figure 5: Illustration of how the topographic variations on subgrid scale lead to dissipation. Observe that the actual calculation of the parameters γ, σ and θ requires spatially oriented data, i.e. an internal representation which allows for the computation of finite difference approximations, i.e. x z(x, y) (z(x + a, y) z(x, y))/a. To achieve this we form a micro grid, consisting of only the data-points within the particular cell. This is illustrated in Fig. 3. These points are simply averaged Micro grid, only the point nearest to the center is used. h x Figure 6: This figure shows the griding process schematically, with a micro-grid in the lower left grid cell. Observe the two different length scales x and h, ideally we should have results which are independent of the short scale h, which only the represents the (arbitrary) micro-griding. When we want to study statistical properties, and in particular the scale dependence of these statistical properties, the simple algorithm described in section. is no longer suitable. This is because. The distribution of raw-data is not uniform, consequently the averaging in the second point at page will alter the statistical properties in a complicated way.. The averaging will introduce an additional length scale a given by the average distance between depth measurements within this particular grid cell. This makes the study of scale dependence less controlled. 8

21 The micro grid has therefore been made considering only the depth measurement closest to the grid-cell center. This is illustrated in Fig. 7. h x Figure 7: Illustrating the algorithm used to form micro-grids, only the depth value closest to the cell center, i.e. the value within the circle, is used. The grid shown in this figure, corresponds to the shaded micro grid in the lower left corner of Fig. 6. For the rest of this report we will mainly use the notation N N to denote the size of subgrid cells. The functional dependence of the subgrid parameters is on the length x, and not N. However the use of N N is more convenient notation, see Table for conversions. However, tables 4 and 5 and figures 93 and 94 use the length x. Table : Size of the different areas studied. Area Horizontal extent Vertical extent Small 45 km 45 km Large km 5 km EdgeN 4 km 45 km EdgeS 4 km 45 km HighResN km 45 km HighResS 8.5 km 3 km Observe that not all combinations of N N and h are possible for all regions. The micro-grid within one grid cell of size x y has n n cells with, the most important limitation is 9

22 that the cells x y must be sufficiently large to embed a micro-grid with of resolution h within, i.e. we must have x = nh with n sufficiently large. We have used n = 4. In addition very large cells / small values of N, means that we get few estimates of subgrid parameters and correspondingly poor statistics, see for instance figure Estimators for subgrid parameters From equations 3-6 it seems quite simple how to estimate the subgrid parameters, however we have found that in for particular for situations where ( x z) ( y z) it is difficult to find consistent estimates. By constructing synthetic topographies with no preferred direction, i.e. γ we found that the estimated value of γ was to small, and the deviation from unity decreased systematically with sample size (n). To get an understanding of the problem we consider a small test problem in the next section Small estimator test case We have a series of n random numbers x i with unknown mean and variance, and want to estimate x. The mean is in principle unknown, but let us assume that the true mean is zero, that is the most problematic case. We estimate x in the following manner :. Form the new variable Σ x (n) Σ x (n) = n. We then use Σ x (n) as an estimator for x. n x i. Discussion: The new variable Σ x (n) is itself a random variable, probably normally distributed with zero mean and variance σ (n). Given random variable x (n) N(, σ ) we can calculate x (n) analytically [ y (n) = dyye y σ (n) dyye y ] σ (n) = σ(n). (8) πσ(n) π The variance σ (n) is clearly a decreasing function of n, we will typically have σ (n) /n. In Fig. 8 we show the the distribution of Σ x (n) for three different values of n. From Eq. 8 we see that the final answer depends on the variance σ (n), which again depends on the sample size n. That the variance of an estimator depends on the sample size is normal, but the expectation value of the estimator should not depend on the sample size. i= 3.3. Alternative estimators for γ, σ and θ Due to the problems of estimating γ, σ and θ described in the previous section, we have employed some alternative estimators of these quantities. These estimators are heavily in- This is not the way one would naturally estimate x, in fact it is basically an incorrect way to do it. However it closely mimics the natural/naive way to estimate L. Since Σ x(n) is the sum of many random variables the central limit theorem says it should be normally distributed, at least in the limit n.

23 N = N = N = Figure 8: A stream of random numbers x i N(, ) is generated, and the averages Σ x (n) are formed. This is repeated many times, and the plots show histograms of Σ x (n) with n =, and respectively, observe how the variance σ (n) decreases with n. spired by the XY model in statistical physics [], and these estimators will be designated with γ XY, σ XY and θ XY. The XY estimators are constructed in the following manner:. Form the vector field m(x), where ( h(x + ˆxax ) h(x) m(x) =, h(y + ŷa ) y) h(y), (9) a x a y i.e. m(x) is a finite difference approximation to h(x).. Form the vector sum M = N where N is the number of lattice points. x m(x) m(x), () 3. Then the estimators are given by γ XY = M, σ XY = N m(x) and θ XY = atan x ( My M x ). ()

24 Finally, the chapter on statistical estimators should be concluded with the following remark: The various estimates are found by spatial averaging, and not by an ensemble averaging, i.e. averaging over many realizations of the same random process. This is correct if the scale of spatial variations of the underlying random process is much larger than the averaging region. The most important advantage of these modified estimators, is that they are more consistent than the original estimators, i.e. the expectation value varies less with the sample size N. 3.4 Two length scales in the estimation From Fig. 3 we see that there are two length scales involved, x which is the cell size in the original grid, and h which is the lattice constant of the subgrid. We are interested in the variation of subgrid parameters with x, but to actually calculate these parameters we must estimate x z(x, y) by using the finite difference scheme x z(x, y) z(x + h, y) z(x, y), this induces an unwanted dependence on the arbitrary scale h. The length x represents a physical length which will enter in subsequent oceanographic simulations, whereas h only represents an arbitrary length used to calculate the subgrid parameters, ultimately we wish to characterize the topographic variations on the scale x independently of the scale h. In practice this is of course impossible, but we have repeated all simulations with h = {5m, 5m, m, m, 4m, 8m, 6m} to investigate how the subgrid parameters vary with this short length scale. This variable implicitly affects the results, in particular a value of h which is much larger than the characteristic size of hills and valleys will lead to an unrealistic smoothing of the topography. An important exception is the quantity z from Eq. 3, which is only a function of the short length scale h. The main motivation to include this probability distribution for z is to be able to construct artificial topographies with at least this distribution correct.

25 Figure 9: Demonstration of the XY estimators. The left column is from a random, uncorrelated topography, and the right column is from a plane topography with similar some random noise added. The second row shows the vector field m(x) from Eq. 9. We observe that to the left the arrows have random length and direction, whereas to the right there is a well defined length and direction. The third row shows the resulting vector M from Eq., and the phase θ XY (in the left figure M has been elongated, a faithful representation would have left M invisible). The dashed circle is M =, which corresponds to perfect ordering, i.e. all arrows in row two of equal length and direction. The slope σ XY is given by the average length (completely ignoring coherence) of the vectors in the second row, i.e. we can see that σ XY is small to left and larger to the right. 3

26 4 Results / figures Resolution Large area Edge north 6 Small area 4 5 Edge south Y (km) High resolution north 8 5 High resolution south X (km) Figure : This figure shows the various areas used for the analysis of scale dependence, the topography of the different areas is shown in figures

27 9 8 7 Y (km) X (km) Figure : The topography of the area Large Y (km) X (km) 7 Figure : The topography of the area Small. 5

28 95 9 Y (km) X (km) Figure 3: The topography of the area EdgeN, this and following topography in figure 4 are included to investigate the effect of the edge. Y (km) X (km) 3 Figure 4: The topography of the area Ekes. 6

29 Y (km) X (km) Figure 5: The topography of the area HighResN, this area is from around the pipeline and sampled with higher spatial resolution, here the resolution is about 5 m Y (km) X (km) Figure 6: The topography of the area HighResS, higher spatial resolution like 5. 7

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31 Y (km) Y (km) X (km) X (km) Y (km) Y (km) X (km) X (km) X (km) Y (km) Y (km) X (km) Y (km) Y (km) X (km) X (km) Figure 7: The area EdgeN, with cells, and h = m. The eight plots show: Topography (56 56 cells), µ, σklm, σxy, γklm, γxy, θ and z. The white spots indicate cells with insufficient data. 9

32 Y (km) 75 8 Y (km) X (km) X (km) Y (km) Y (km) X (km) X (km) Y (km) Y (km) X (km) X (km) Y (km) 8 75 Y (km) X (km) X (km) Figure 8: The same as figure 7, but with 6 6 cells, and h = 6m. 3

33 4. Discussion of figures 7 and 8 There are several things we would like to emphasize regarding figures 7 and 8:. We see that the subgrid parameters are spatially correlated. We see however that the correlations are much less obvious in figure 8, i.e. the cell size x and h in this figure are approaching a characteristic length of the topography.. We see that the estimators σ XY and σ KLM give very similar results, whereas the estimators γ XY and σ XY differ more, in particular in figure There is significant difference in the spatial distribution of the subgrid parameters between the two figures, indicating that these things change with scale. 4. Discussion In this section the results are presented, in the form of figures fig. 63 to and tables 4 to 36. Figures to 6 show histograms over the subgrid parameters for the different regions and spatial scales x and h. These histograms constitute the raw-data. In the case of σ XY, σ KLM, µ, µ PR the results of fitting these histograms to a Log Normal distribution is shown in tables 6 to 9. The figures 63 to 94 are based on averages, with estimated errors, from the raw histograms. Figures 63 to 9 show the h limit of the different subgrid parameters, for the various areas. Figures 93 to 94 show the scaling of µ and µ PR with x, the results of fitting these curves to a power law is given in the tables 4 to 5. The figures 95 to show histograms of z for the various regions and h values. The results of fitting these histograms to a Lorenz distribution is shown in tables 3 to 35. Finally the results of fitting the parameters from the Lorenz distribution is shown in figures to and table. 5 Detailed discussion The amount of data is overwhelming, and it is important to extract the essentials. In this section we will try to extract the general properties from the different estimators. Spatial correlations: We have presented a large number of histograms of various subgrid parameters, sampled from different regions and at various resolutions. For instance h(µ )dµ is (an estimate of) the probability to find µ in the interval [µ, µ + dµ] at a randomly chosen point in the support region of h(µ). Clearly the various subgrid parameters are spatially correlated, i.e. Cov(x, y) = µ(x) µ(y). () The existence of such correlations can be clearly seen from e.g. figures 7-8. These correlations are obviously important to get a correct quantitative description of the topography, however we have chosen to ignore them for the time being. 3

34 5. Standard deviation The distribution of µ, shown in figures 5-56 show the following universal features. It is unimodal, with a quite sharp increase from zero, and a longer tail, i.e. negative skewness. Empirically it is found that the distribution function can be quite nicely fitted to the Log Normal distribution P (µ) = Properties of the Log Normal distribution: Sµ (ln µ M) π e S µ = e M+S /. (3) (a) The position of the maximum, i.e. most probable value, is given by x = e M S. (4) (b) S and M determine the shape of the distribution function. Roughly e M determines translation along the x axis, and S determines the width of the distribution. (c) It is not symmetric, the skewness is only a function of S ( γ = e S + e S). For large values of S the distribution is highly skewed, with a long tail. The expectation value µ = e M+S / is much larger than the most probable value, e M S. For small values of S the Log Normal distribution converges to a Gaussian with mean e M and variance S e M. (d) Given the two first moments M = x and M = x of Log Normally distributed variable the parameters S and M can be reasonably 3 well determined from M = ln M ln M (5) S = ln M ln M. (6). The shape of the distribution function seems to be quite universal, i.e. independent of the area studied. In particular the overall large-scale features of the topography are not prominent in the distribution function, see however the discussion of the areas EdgeN and EdgeS further down. 5.. Fractal properties During the last twenty years concepts and ideas from the mathematics of fractals have successfully been employed to describe a wide range of natural phenomena, in particular it has been shown that the topography of mountain areas is self affine[3]. 3 The Gaussian distribution is entirely determined from the two first moments, this is not the case for the Log Normal distribution. 3

35 From the the theory of self-affine surfaces it can be shown that the surface standard deviation, µ, should scale with linear extent of the surface, raised to a power H, where H is called the Hurst exponent. In our case this means that µ should behave as µ C x H. (7) Fig. 93 shows the spatial average of µ as a function of x, along with least squares fits to Eq. 7. The numerical results are given in table 4, and shown as straight lines in figure 93. The individual data points in figure 93 have error bars µ. These error bars are statistical errors estimated by re-sampling. The size of the errors systematically increases with increasing x. The fitting to the power law Eq. 7 has been done with and without the weights ( µ). Due to the systematic variation of µ with x this is equivalent to emphasizing small and large values of x respectively. From Table 4 we see that the amplitudes C i vary greatly from between different regions, whereas the exponent H is fairly constant, i.e. H.7 ±.7 and H +.83 ±.3 for weighted and unweighted estimates respectively. In particular the two areas called EdgeN and EdgeS (see Figures 3 and 4) have an overall feature, to a very low order approximation these surfaces (in particular EdgeS) can be described as planes z = m +. x. (8) From this very simple topography we can actually calculate how µ varies with block size, and find µ.5 x. (9) This is quite close to the C /H fits to these areas, hence for long distances these surfaces can be reasonably well described as planes with added fluctuations. For short distances, i.e. C /H, there are much larger deviations from Eq Standard deviation from plane In an attempt to reduce the dependence on the smooth properties of the topography we have also estimated the standard deviation of the residues from a least squares plane fitted through the area. The distribution of P (µ FP ) is also well described by the Log Normal distribution Eq. 3. Clearly we must have µ FP < µ, hence the amplitudes C FP but apart from that the scaling properties of µ and µ FP are very similar, i.e. compare tables 4 and 5. One notable difference is that the fits dominated by long length scales (C /α ) show less variability then the fits dominated by short length scales (C /α ). This is the opposite behavior of µ. Furthermore we see that for the two edge areas EdgeN and EdgeS the α exponent is much smaller than in the µ case, this is to be expected since the dominating linear plane behavior is subtracted in the FP method. Apart from these quite minor differences µ and µ FP show very similar behavior, and in the future one should probably only calculate the standard µ. 33

36 5.3 Distribution of Z The figures 95 - show distributions of µ z = z(x + hˆµ) z(x). Observe that for these histograms there is concept of grid blocking, only the short distance scale h enters. These distributions are not subgrid parameters in the same way as µ, σ and γ, but they give an impression of the how much the surface goes up and down. The general characteristics of these distributions are:. For small h the distributions are symmetric and highly peaked around the value z =.. The spatial distribution of z has similar shape in the different areas, but the width of the distribution varies from region to region. 3. For h exceeding about m the distributions are no longer universal, and obtain a non-universal shape specific to the particular area. We have not done systematic curve fitting of these distributions, but fits to a Lorenzian P ( z) = γ π (e z ), () + w z give reasonable results. The important point is that P ( z) is a distribution with a rather long tail. Knowledge of the distribution P ( z) can be used to synthetically increase the resolution of a sparsely sampled topography. We have calculated the mean and standard deviation of z distribution from the various regions. Qualitatively what we find can be summarized as:. The standard deviation ( width of the distribution) scales as h ν where ν varies from.46 to.67, the results are summarized in table 36 and figure. This is not 4 the same as γ in the tables 3-35, but it is close.. The mean value of z should scale linearly with h, we have not studied this quantitatively, but figure at least indicates that it is correct. For one-dimensional surface it is easy to see that the mean value of z must scale linearly with h: Given a a one dimensional topography of length L and total height difference H, if we subdivide this topography in n pieces each of length h = L/n, and with individual height differences z i we have: i.e. z scales linearly with h. z i = H n z = H z = h H L, () i For the remaining quantities, i.e. γ, σ and θ there are two length scales involved, and it is nontrivial to ascribe certain behavior to one particular of the two lengths. In many ways the effect of the short length scale h is easiest to understand, it works as the cut off in a low 4 The Lorenz distribution does not have well defined moments. 34