Initial run results

During the past 11 days I have completed 1064 simulations of the river, using the combination of parameters described previously. Runs are performed on 2 Apple Macintosh computers having 2.0 and 2.4 GHz Intel dual-core CPUs running OS X 10.5.5. When I am not doing anything else with the computers, 2 simultaneous simulations (as Unix processes) can be run, taking less combined time than 2 sequential simulations.

Because the runs are progressing well, I have decided to perform an additional 768 simulations using the same combinations of the 5 variables {width, depth, diam, flow, sdist}, but with an initial erosion rate of 1.8 ha/yr/km, so that 4 values of initial erosion will be used: {1.2, 1.8, 2.4, 3.6}. This is partly because many of the simulations decrease their erosion rate over time (so that some end with erate << 1.2), but also because after viewing the initial results I feel that an intermediate value between 1.2 and 2.4 is needed, whereas there is less final difference between runs started at 2.4 and 3.6 ha/yr/km. This will yield a new total of 3072 rather than 2304 distinct simulations. At the current rate of 96/day, this should take a total of 32 days, allowing all runs to be completed by the end of October, barring unforseen problems.

I have also written a simple database to collect, display, and manipulate the results, in both numerical and graphical form. This database has a programming and query syntax similar to some other common programs, so I have elected to call it 'RiverSquirrel', after 'my squirrel' (MySQL) or 'post grey squirrel' (PostgreSQL). A database creation program peruses all simulation files and accumulates several numerical results, creating a large table of values. The database query program can perform the following operations on the table:

• Print entries and fields in any order.
• Sort entries about any field.
• Select subsets of entries using numerical relationships and combinations of 'and' and 'or' clauses.
• Plot the river for any parameter combination.
• Accumulate and plot common coverage and average erosion as 2d arrays for subsets of entries.

Currently the query interface is a programming one (from Python), but it could be modified to either read text command files, or to take interactive input from the terminal. At present it is sufficient to perform rudimentary searches and display subsets of results as text and images. It is expected that this interface and its capabilities will evolve as necessary over the next few months as the data are analyzed.

Here are several fields from the table showing the 6 independent, discrete, input parameters (sorted by width->depth->diam->flow->sdist->erate) and 5 dependent, continuous, output values:

> ../RiverSquirrel.py runs.rsq 
Reading: runs.rsq...
1064 entries read
1064 entries:
                          file         avwidth0            depth             diam             flow            sdist           erate0          avwidth           length            cArea            eArea            erate
------------------------------ ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- ----------------
            iMac/test0000.mnrr       433.661987         2.750000         0.000500       800.000000         0.750000         1.199998       437.179789    148234.005405       214.250000     11188.422908         0.886913
           iMac2/test5000.mnrr       433.661987         2.750000         0.000500       800.000000         0.750000         1.800001       440.097217    174636.130130       265.500000     17384.646105         1.127514
            iMac/test0001.mnrr       433.661987         2.750000         0.000500       800.000000         0.750000         2.400007       441.617137    177842.965221       319.750000     23891.941898         1.492773
            iMac/test0002.mnrr       433.661987         2.750000         0.000500       800.000000         0.750000         3.599992       439.382579    167536.841460       395.000000     36242.357280         2.462988
            iMac/test0003.mnrr       433.661987         2.750000         0.000500       800.000000         1.000000         1.199999       438.327311    157784.485527       228.000000     12822.685652         0.946889
           iMac2/test5001.mnrr       433.661987         2.750000         0.000500       800.000000         1.000000         1.799999       440.352742    171500.594897       286.500000     19754.559753         1.347048
            iMac/test0004.mnrr       433.661987         2.750000         0.000500       800.000000         1.000000         2.400005       441.954097    181279.258680       342.750000     26948.249198         1.700486
            iMac/test0005.mnrr       433.661987         2.750000         0.000500       800.000000         1.000000         3.599994       435.376554    164998.048398       410.000000     39488.989870         2.714484
            iMac/test0006.mnrr       433.661987         2.750000         0.000500       800.000000         1.250000         1.199998       440.585635    172485.286508       249.500000     15000.749204         0.998835
           iMac2/test5002.mnrr       433.661987         2.750000         0.000500       800.000000         1.250000         1.800002       441.104491    185249.360552       315.500000     22951.506781         1.331328
...
         MacBook/test1657.mnrr       548.391695         3.500000         0.001000       800.000000         0.750000         2.399998       569.720380    154960.473825       300.250000     20249.570461         1.449807
         MacBook/test1658.mnrr       548.391695         3.500000         0.001000       800.000000         0.750000         3.599997       569.411951    164236.967117       365.750000     30577.516346         2.052708
         MacBook/test1659.mnrr       548.391695         3.500000         0.001000       800.000000         1.000000         1.200000       565.412960    142317.306038       236.000000     12126.988516         0.961104
         MacBook/test1660.mnrr       548.391695         3.500000         0.001000       800.000000         1.000000         2.399999       571.851009    166821.030775       337.250000     24538.043265         1.599996
         MacBook/test1661.mnrr       548.391695         3.500000         0.001000       800.000000         1.000000         3.600003       570.268799    165831.200249       391.000000     36116.163596         2.589623
         MacBook/test1662.mnrr       548.391695         3.500000         0.001000       800.000000         1.250000         1.200000       569.830171    144333.457406       255.500000     14002.956943         1.107369
         MacBook/test1663.mnrr       548.391695         3.500000         0.001000       800.000000         1.250000         2.400001       562.370133    143929.554365       353.750000     27202.992166         2.021015
         MacBook/test1664.mnrr       548.391695         3.500000         0.001000       800.000000         1.250000         3.599989       568.442438    166832.196873       411.750000     40639.004881         2.800044
         MacBook/test1665.mnrr       548.391695         3.500000         0.001000      1000.000000         0.750000         1.200001       558.115380    144798.917813       227.000000     11134.616033         0.928373
         MacBook/test1666.mnrr       548.391695         3.500000         0.001000      1000.000000         0.750000         2.400001       559.951468    157448.331464       328.750000     23593.496280         1.753849

Using the database, one can plot, say, the longest and shortest final river:

5 entries:
          length                           file         avwidth0            depth             diam             flow            sdist           erate0
---------------- ------------------------------ ---------------- ---------------- ---------------- ---------------- ---------------- ----------------
   245477.926265             iMac/test0143.mnrr       433.661987         2.750000         0.001250      1400.000000         1.250000         3.599963
   242306.678435             iMac/test0281.mnrr       433.661987         3.000000         0.001250      1400.000000         0.750000         3.600000
   236444.152114             iMac/test0404.mnrr       433.661987         3.250000         0.001250       800.000000         1.250000         3.600008
   235817.205567             iMac/test0413.mnrr       433.661987         3.250000         0.001250      1000.000000         1.250000         3.599998
   234978.637504             iMac/test0245.mnrr       433.661987         3.000000         0.001000      1400.000000         0.750000         3.599993
5 entries:
          length                           file         avwidth0            depth             diam             flow            sdist           erate0
---------------- ------------------------------ ---------------- ---------------- ---------------- ---------------- ---------------- ----------------
   132485.937320          MacBook/test1539.mnrr       548.391695         3.250000         0.001000      1400.000000         0.750000         1.199999
   132274.379441          MacBook/test1326.mnrr       548.391695         3.000000         0.000500      1400.000000         1.000000         1.200000
   132070.553290          MacBook/test1512.mnrr       548.391695         3.250000         0.001000       800.000000         0.750000         1.200000
   131320.759969          MacBook/test1260.mnrr       548.391695         2.750000         0.001250       800.000000         0.750000         1.200002
   130198.496148          MacBook/test1632.mnrr       548.391695         3.500000         0.000750      1000.000000         1.000000         1.200001

Or the ones with the most cutoffs (also the longest river) and with the longest average cutoffs:

5 entries:
nCutoffs                           file         avwidth0            depth             diam             flow            sdist           erate0
-------- ------------------------------ ---------------- ---------------- ---------------- ---------------- ---------------- ----------------
      60             iMac/test0143.mnrr       433.661987         2.750000         0.001250      1400.000000         1.250000         3.599963
      55             iMac/test0116.mnrr       433.661987         2.750000         0.001250       800.000000         1.250000         3.600012
      51             iMac/test0404.mnrr       433.661987         3.250000         0.001250       800.000000         1.250000         3.600008
      44             iMac/test0080.mnrr       433.661987         2.750000         0.001000       800.000000         1.250000         3.599996
      42             iMac/test0260.mnrr       433.661987         3.000000         0.001250       800.000000         1.250000         3.599987
5 entries:
       avCLength                           file         avwidth0            depth             diam             flow            sdist           erate0
---------------- ------------------------------ ---------------- ---------------- ---------------- ---------------- ---------------- ----------------
    21548.912916             iMac/test0142.mnrr       433.661987         2.750000         0.001250      1400.000000         1.250000         2.399992
    18132.580194          MacBook/test1289.mnrr       548.391695         2.750000         0.001250      1400.000000         0.750000         3.599988
    16023.012955          MacBook/test1532.mnrr       548.391695         3.250000         0.001000      1200.000000         0.750000         3.600005
    15947.509123             iMac/test0092.mnrr       433.661987         2.750000         0.001000      1200.000000         0.750000         3.600009
    15674.649523             iMac/test0101.mnrr       433.661987         2.750000         0.001000      1400.000000         0.750000         3.599998

Or the 'wildest' and 'tamest' rivers (by final erosion rate):

5 entries:
           erate                           file         avwidth0            depth             diam             flow            sdist           erate0
---------------- ------------------------------ ---------------- ---------------- ---------------- ---------------- ---------------- ----------------
       10.922526             iMac/test0143.mnrr       433.661987         2.750000         0.001250      1400.000000         1.250000         3.599963
        8.472486             iMac/test0134.mnrr       433.661987         2.750000         0.001250      1200.000000         1.250000         3.600004
        7.848543             iMac/test0137.mnrr       433.661987         2.750000         0.001250      1400.000000         0.750000         3.599998
        7.526362             iMac/test0287.mnrr       433.661987         3.000000         0.001250      1400.000000         1.250000         3.599978
        7.320121             iMac/test0140.mnrr       433.661987         2.750000         0.001250      1400.000000         1.000000         3.600050
5 entries:
           erate                           file         avwidth0            depth             diam             flow            sdist           erate0
---------------- ------------------------------ ---------------- ---------------- ---------------- ---------------- ---------------- ----------------
        0.741212          MacBook/test1593.mnrr       548.391695         3.500000         0.000500      1000.000000         0.750000         1.200000
        0.730534          MacBook/test1620.mnrr       548.391695         3.500000         0.000750       800.000000         0.750000         1.200000
        0.717399          MacBook/test1440.mnrr       548.391695         3.250000         0.000500       800.000000         0.750000         1.200000
        0.717024             iMac/test0432.mnrr       433.661987         3.500000         0.000500       800.000000         0.750000         1.200000
        0.666959          MacBook/test1584.mnrr       548.391695         3.500000         0.000500       800.000000         0.750000         1.200000

By looking at the combinations of input parameters in the previous printout, one can see that the 'wilder' rivers tend to be narrower, shallower, have more discharge, larger grain size, depend on more upstream geometry, and have higher initial erosion rates (pretty much as expected).

Here are some 'more nominal' results (for length < 195 Km and erate < 1.95):

10 entries:
                          file         avwidth0            depth             diam             flow            sdist           erate0          avwidth           length            cArea            eArea            erate
------------------------------ ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- ----------------
            iMac/test0025.mnrr       433.661987         2.750000         0.000500      1200.000000         1.250000         2.400003       439.860193    194729.247457       361.500000     30530.064193         1.765710
            iMac/test0308.mnrr       433.661987         3.250000         0.000500      1200.000000         0.750000         3.600006       442.100802    194720.333067       384.000000     33110.756979         1.855819
            iMac/test0337.mnrr       433.661987         3.250000         0.000750      1000.000000         1.000000         2.400002       441.005884    194194.613023       358.250000     29184.843497         1.805953
            iMac/test0190.mnrr       433.661987         3.000000         0.000750      1000.000000         0.750000         2.400000       441.125405    193950.637765       343.250000     26738.039039         1.783816
            iMac/test0334.mnrr       433.661987         3.250000         0.000750      1000.000000         0.750000         2.399995       441.602400    193822.916461       339.750000     25834.549258         1.661886
         MacBook/test1474.mnrr       548.391695         3.250000         0.000500      1400.000000         1.250000         2.399994       557.226743    193648.949332       352.750000     25410.387029         1.357874
            iMac/test0457.mnrr       433.661987         3.500000         0.000500      1200.000000         1.250000         2.399995       443.906511    192998.064211       346.750000     27170.359716         1.505281
            iMac/test0311.mnrr       433.661987         3.250000         0.000500      1200.000000         1.000000         3.600000       435.604436    192996.759572       404.750000     36491.492542         1.932142
            iMac/test0317.mnrr       433.661987         3.250000         0.000500      1400.000000         0.750000         3.600004       433.366096    192672.337384       385.250000     33417.286202         1.857648
            iMac/test0199.mnrr       433.661987         3.000000         0.000750      1200.000000         0.750000         2.400005       436.158985    192262.792834       344.500000     26798.066912         1.829786

Relationships between all combinations of the output values can also be plotted (in the following, the single-point outlier in many plots is run 143 shown above):

Although these plots show only about 35% of all runs (and only 2 initial widths so far), they do inspire a few observations/hypotheses:

1. There appears to be a downward trend in river length, coverage and eroded area, and final erosion rate with an increase in river width.
   
   
2. There appears to be a positive trend in both coverage and eroded area with respect to length, superimposed on a more random variation. Limiting these plots to specific subsets of the initial parameters might show where this transition occurs. However, although the relationship between river length and erosion rate is also positive, it appears that almost all lengths can be generated from fairly low values of erosion. This may be due to the increased number of cutoffs which are also associated with increased erosion rate, and which limit river length.
   
   
3. As expected, there are strong positive correlations between coverage and eroded area, and between both and the final erosion rate.

Further sorting and selecting of combinations of input variables (rather than all combinations lumped together as shown above) might show differences in these plots and allow one to see where specific trends form and change (i.e. for fixed subsets of the other variables, do variations in a single or pair of input variables lead to specific variations in output values?).

For this project, the most important result might be the cumulative common coverage of the river (i.e. showing the fraction of all simulations which pass through 2d points in the valley), and the average cumulative total erosion over 100 years (also at 2d points in the valley):

Notes:

• The common coverage shows the fraction of all simulated rivers that occupy points in the valley at least once over 100 years. This goes from red (points where all 1064 rivers pass through), to green (where 50% of all rivers go), to dark blue (where only one river goes), to black (where no river goes). These values could also be interpreted as probabilities of occupancy by at least one hypothetical river over the course of 100 years. The limit (contour) of some specific probability value might then be called the 'migration corridor'.
  
  
• Total erosion is plotted as hectares per (0.5 km)^2. Note that 0.25 km^2 is 25 ha, so that points showing more than 25 ha total erosion were, on average, eroded by the river more than once. Also note that this value is the total continuously integrated (each 0.25 year) erosion at a point, not the difference between final and initial land areas (as are reported in EJ06). Thus, the river could stay mostly in one place and 'wiggle' slightly, and it would continue to erode surface area, possibly beyond the capacity of the landscape to continue to provide sediment. In this analysis, it is assumed that surface runoff, or some other mechanism, will continue to provide sediment necessary to satisfy any cumulative erosion.

I will post more intermediate results as they progress next month.