GLY 6932/6862 Numerical Methods in Earth Sciences Spring 2009

Transcription

1 T GLY 6932/6862 Numerical Methods in Earth Sciences Spring 2009 Lesson 7 Geospatial Analysis (and other 3-d data) (With acknowledgements to C. Connors, W&L Univ.) In Earth sciences, we often need to express how a dependent variable behaves as a function two independent variables. A common example of such a system is topography, where the surface elevation (z, the dependent variable) is a function of where you are on a map grid (latitude & longitude, or northing & easting the independent variables). A non-topographic example is the joint histogram of wave climate, where the number of waves (dependent variable) is a function of wave heights and directions (the independent variables). In this lesson, we will go over an example that involves topographic data and an example from chemistry that deals with behavior of an ideal gas. Viewing Spatial Data Nowadays, it is very common to display topography or elevation (in color) as a function of position with uniformly spaced, gridded data. This is the essence of a DEM (Digital Elevation Model). In this lesson, we'll develop techniques for viewing non-uniformly spaced data, and then we'll show how to overlay some directional data on top of that. And along the way, we'll cover a number of ways to view spatial data. To get started, load up the dmap.txt data file from our course website. This is a 3 column vector data set (x,y,z) containing elevation data for a fictitious surface: load dmap.txt You can quickly examine where these data are in x-y space: plot(dmap(:,1),dmap(:,2),'ro') 1

2 Next, try visualizing the file with the plot3 command p_h=plot3(dmap(:,1),dmap(:,2),dmap(:,3),'o') The 'o' parameter plots the data points as individual circles as opposed to a 3D-line. The p_h is called a handle. You could have just typed plot3 without the p_h but by adding this you have the ability to later alter (or delete) this graphic element. Delauney Triangles Because data are often not regularly sampled, as above, surfaces are often modeled as a set of interlocking triangles. There is a particularly compact way of making surfaces of irregularly 2

3 spaced data called a Delaunay triangulation. Delaunay triangulation is the set of triangles built by connecting each of the points of an irregularly spaced data set where the vertices of the triangles are the data points. Thus it is an exact representation of your known data and linear interpolation in between. If you'd like to switch back and view the original data as a series of Delauney triangles: x=dmap(:,1); y=dmap(:,2); z=dmap(:,3); tri=delaunay(x,y); plot(x,y,'ro'),hold on trisurf(tri,x,y,z); axis equal axis square Note that the vertex of each triangle is one of the data points. If you want to see the plot without the points type: delete(p_h) 3

4 This surface plot is efficient but it doesn t produce a very pretty picture (well at least for this small data set, for large data sets, triangulated surfaces are often preferable because they are more compact and there is enough data that the surface looks good). Gridding and Various Ways of Viewing the Data At this point, are the data regularly spaced? No they are not. You've got to make them so. We often want a regular spacing of the X and Y locations to make it smoother and not faceted. A regularly spaced data set is called a grid. It uses up more memory and thus is slower to manipulate because data is defined at every location. To convert the irregularly spaced data to regularly-spaced we need to grid it. This requires a couple of steps. First you need to define the spacing and extent of the grid by making two vectors of the X and Y upper and lower limits at a given spacing. This command will be much akin to interp2 because that's what we're doing, but in this case, we really are interpolating 3 matrices: xi, yi, & zi: rangey=floor(min(dmap(:,2))):.2:ceil(max(dmap(:,2))); rangex=floor(min(dmap(:,1))):.2:ceil(max(dmap(:,1))); rangez=floor(min(dmap(:,3))):10:ceil(max(dmap(:,3))) [xi,yi]=meshgrid(rangex,rangey); zi=griddata(x,y,z,xi,yi); 0.2 is the spacing I have chosen. I was a little obscure in choosing the floor(min syntax. Try min(dmap(:,2)) to see what that gives you and then try floor(min(dmap(:,2))) to see what that does. Hint: it is a way of controlling the rounding direction. Let's see what we've created. We can view it the traditional way, as a colormap: pcolor(xi,yi,zi) shading flat 4

5 Or as a contour map with labels: [C,h]=contour(xi,yi,zi,rangeZ) c_h=clabel(c,h,rangez(1:2:end)) Or as a surface: surf(xi,yi,zi),hold on set(gca,'view',[145 30]) grid on 5

6 Or as a surface with contours projected below: surfc(xi,yi,zi),hold on set(gca,'view',[145 30]) grid on Or as an illuminated surface, with contour lines superimposed: surfl(xi,yi,zi) shading interp colormap copper hold on 6

7 set(gca,'view',[145 30]) grid on contour3(xi,yi,zi,10) and the original data superimposed: plot3(x,y,z,'ko','linewidth',2) Ok, that's all for now on the various types of 3d plots you can build. Hunt around and find some more like voronoi polygons, for example. Computing Directional Information from Elevation Data 7

8 Let's do one more thing. I want you to get familiar with directional data, so here's an example with the gradient. First re-build the colormap: close all,clear all load dmap.dat plot(dmap(:,1),dmap(:,2),'ro') x=dmap(:,1); y=dmap(:,2); z=dmap(:,3); x(length(x)+1:length(x)+4)=[ ]; y(length(y)+1:length(y)+4)=[ ]; z(length(z)+1:length(z)+4)=[ ]; ti=[0:0.25:6.5]; [xi,yi,zi]=griddata(x,y,z,ti,ti'); pcolor(xi,yi,zi) shading flat axis equal axis square axis tight Now, you need to calculate the gradient. Everybody remember what the gradient is? It is a vector, whose orientation represents the direction of most rapid rate of change of the function. In other words, it is the straight path that a ball will roll down hill. [px,py]=gradient(zi); hold on quiver(xi,yi,px,py,5,'k') 8

9 But these vectors are all pointing uphill! We don't want that. Point them down hill, please. Think about the unit circle to do this. Surfaces of Non-Spatial Data But we are not limited to topography when making surfaces to examine how a variable behaves. Consider the ideal gas law (PV=nRT), which you ll recall from any chemistry you ve taken. This equation relates pressure (P), volume (V), and temperature (T), for a set number of moles (n), through the universal gas constant (R), which is equal to Joules/(mol*K). If we want to explore how the pressure of an ideal gas changes over a range of temperatures, we must first fix the volume and the number of moles. This is somewhat limiting. By expanding to a surface analysis we can examine how pressure changes over a range of temperatures and a range of volumes, simultaneously. Let s do it: close all,clear all % % Script to illustrate 3d "surface" analysis of ideal gas law, pv=nrt, for % use in GLY 6862 Lesson 7 (Geospatial and Surface Analysis) % % by pna at home, M/03/16/2009 % %% Example 1 - constant n, vary P &V, T is dep. var. R=8.314; % univ. gas constant in J/(mol*K)==(kg*m^2)/(mol*K*s^2) n=40; % number of moles of gas in our sample P_range=[0:10000:200000]; % pressure in N/(m^2) V_range=[0:0.2:5]; % volume in m^3 [P,V]=meshgrid(P_range,V_range); T=(P.*V)/(n*R); 9

10 ('units','normalized','position',[ ]) surf(p/1e3,v,t) xlabel('pressure (kpa or kn/m^2)') ylabel('volume (m^3)') zlabel('temperature (K)') disp('hit any key to show isotherms.'),pause ('units','normalized','position',[ ]) contourf(p,v,t,20) xlabel('pressure (kpa or kn/m^2)') ylabel('volume (m^3)') title('isotherms of an ideal gas (K)') colorbar 10