Calculating centerline and width

During the past week I have worked on calculating the river centerline and width from the digitized banks, and on reconstructing new synthetic banks from that centerline and width. I have also smoothed these curves in preparation for input to the JP and other meandering routines. The result is an 'idealized' river geometry that closely follows the actual river, but which can be described by a single smoothly varying curve and width at every downstream point.

The centerline calculation is computationally intensive (< 5 min for two 1000 point banks), but seems to work well for all parts of the river, including areas where the banks change direction and/or are highly curved. The basis of this calculation is the following observation/axiom/criterion, which attempts to quantitatively answer the question "what exacty is the 'width' of the river at a given downstream location'? My answer is that the 'width' is only well-defined for those corresponding points {A(i), B(i)} along the two banks where:

• For a given point A on bank 1, the closest point on bank 2 is B, and for that point B on bank 2, the closest point on bank 1 is also A. The width of the river at {A, B} is then the distance between the two points, and the centerline passes halfway between them.

Application of this constraint yields surprisingly few mutually corresponding points along both banks, as in general the closest point B' to point A has as its closest point a different point A' ≠ A. However, this criterion can be applied iteratively, beginning with the corresponding points which are closest to one other, and then adding additional pairs of points in order of greater global distance between them such that:

1. New {A, B} combinations are selected as above from remaining points after all previous {A(i), B(i)} pairs are excluded, and
   
   
2. A new pair {A, B} is only added if the line segment connecting A and B does not cross any previous line segments connecting already existing A(i) and B(i).

Note that this process is not necessarily one-to-one, as a point on one bank may be connected to several points on the opposite bank, as long as the two specified conditions hold. And, some points may not be connected at all, if they fail the conditions. In general, however, nearly all points on both banks will be connected to at least one other point, so that the resulting centerline is well represented. Nevertheless, some post smoothing and reinterpolation of the centerline is useful.

Once a centerline and widths have been calculated, new synthetic banks can be generated on both sides of the centerline. Although these new banks are not required as input for the meandering routines, it is useful to verify that they correspond well to the original river banks, indicating that the centerline and widths have been calculated properly. Note that because each new bank is placed symmetrically around the centerline, this will erase details which occur in one original bank but not in the other, and will tend to act as a low-pass filter on the banks. Again, both smoothing and reinterpolation of the new banks is appropriate.

Here is an example of calculating centerline, widths, and new banks for one segment of the MNRR. First, the background map and digitized banks (at 100 m):

Next, the centerline is calculated, smoothed, and interpolated (again to 100 m):

Here, the background has been removed to better see the curves:

The centerline coordinates and widths can be printed, to be used as input for the meandering simulations (the banks and any other curve can also be printed. Output is {lon, lat, length, width}):

2 landmarks
     -97.458333333000    42.916666666000
     -96.583333333000    42.541666666000
3 curves
328 points
32319.24 meters

Centerline
111 points
10915.72 meters
  -96.975836169755    42.757693048201        0.000000      649.421425
  -96.974645255967    42.757495855266       99.919934      756.330894
  -96.973552118951    42.757099734514      199.639667      857.427623
  -96.972588082563    42.756547603693      299.589686      961.934324
  -96.971594080531    42.756025408675      399.521826     1055.633468
  -96.970604459558    42.755498413781      499.474071     1110.555963
  -96.969735336323    42.754867170403      599.372091     1140.379598
  -96.968970955009    42.754165158200      699.359214     1178.276840
  -96.968257485001    42.753434582184      799.350393     1234.188110
  -96.967660295512    42.752650392268      899.245759     1292.987857
...

Next, new banks are generated using the centerline and widths (with orientation provided by the locations of triangulated circles tangent to the centerline), and are also smoothed and interpolated:

Finally, the centerline and new banks alone:

Here is another example showing the same sequence of operations:

This process can be applied to the entire digitized river. At top are the digitized banks and centerline, at center are all curves, and at bottom are the centerline and new banks alone:

(Click for larger image)

Here are the centerline coordinates and widths.

Next month I will use the geometry generated this month to initialize some meandering simulations, including the JP89 method from April. I will still need access to depth, flow, and bed composition data. Robb Jacobson has provided several references to hydraulic data, and additional historical figures appear in the 2006 USGS report. Initially, depth can be set to a constant, although Robb has also mentioned that he has access to bathymetric data and models. Ideally, some cross-stream depth profiles would be useful, as it is apparent from the maps and satellite photos that the MNRR is definitely not an idealized trapezoidal channel. Furthermore, in many places the calculated width is likely to be considerably in excess of the actual flow width, due to the presence of bars and other shallow shoals, and the thalweg may be far from the calculated centerline. It may be possible to partly compensate for this based on the bar and vegetated bar percentage estimates in the 2006 report. In any case, these are all variables with which to experiment next month.

Finally, some estimate is needed of any of the following equivalent information:

• The bed particle size and density distribution, or
• The various Shields stress values (possibly from surveys of similar rivers), or
• Empirical measurements of several bed slopes wrt centerline curvature, or
• Some other way to estimate the 'scour factor' used in the equations for u_1b.

In conclusion, I would predict that the JP89 method will not provide a very realistic simulation for this river. It is clear from the maps and photos that the MNRR violates both the requirements and the basic principle of the JP method (i.e. that the flow is faster on the outside radius of curves, and that material is eroded there and deposited on the inside radius). The MNRR is generally very wide, has very unsymmetric banks, a very non-linear bed depth profile, and often a shallower bed depth at the outside radius of curves, where material has accumulated rather than eroded. This may follow partly from the hypothesis that the primary flow is confined to a narrow off-center region which is not well delineated by the actual banks, especially in wide segments of the river. Ideally, I would like to develop a meandering model which can take these issues into consideration, by both (a) changing the geometries (erosion/accumulation) of each bank independently of one another, rather than as a single idealized centerline/width/flow, and (b) modeling the cross-stream depth and flow profiles as other than linear gradients so as to better represent both shallow, slow-moving flows near the outer banks and faster, deeper flows away from the bars.

We'll see how this works out next month!