Introduction
Hydrodynamic models are useful tools for exploring how climate change, rising
sea levels, and hydrological regime alterations might affect the interaction
between tides, rivers, and coastlines , as well as urban coastal
flooding . Similarly, such models
are vital in analysis of river hydrodynamics and floodplain inundation that
might be affected by changing climate patterns
. Unfortunately, modeling annual to
decadal timescales for management and climate change analyses typically
requires hydrodynamic model grid scales that might not adequately represent
narrow blocking features. Herein we develop new methods for upscaling a
digital elevation model of topography to ensure hydrodynamic blocking
features are retained.
The working hypothesis of this paper is that at any sufficiently coarse-grid
scale (ΔC) there might be topographic features of width scale
ℓW < ΔC and length scale ℓL ≥ ΔC that can be represented as
“edge” or “face” features of the grid cell. These features, if given the
correct continuity across multiple grid cells, can represent hydrodynamic
blocking that is lost when subgrid features are represented as topographic
roughness. We will call this an edge-blocking technique.
By way of background, the present state of the art for processed lidar data
can readily provide a ∼ 1 m × 1 m digital terrain model (DTM) for use
in high-resolution hydrodynamic modeling .
Unfortunately, modeling at such fine spatial resolution
invariably drives the model time step down to 1 s or lower, depending
on the numerical model scheme. At such scales, even a
small river delta of 104 ha will require 108 grid cells and
approximately 107 time steps per year of simulation – pushing even a
small system into supercomputer territory for multi-decadal and/or Monte
Carlo simulations for sensitivity analyses, which are both desirable for
adaptive coastal management. However, by coarsening to a 20 m × 20 m grid
resolution, 104 ha requires only 2.5 × 105 grid cells and can
typically be run at time steps of 10 to 30 s in a semi-implicit
model, i.e., ∼ 106 time steps per year of simulation. The
reduced memory requirement allows multiple model instances to be
simultaneously run on a standard multi-core desktop workstation. The larger
allowable time step allows faster simulations over longer timescales without
requiring extensive high-end computational resources. Indeed, it is likely
that, as computers get more powerful, our desire to decrease grid resolution
will be countered by our desire to run larger areas and longer timescale
simulations. Thus, the need for grid coarsening of lidar data for
hydrodynamics is unlikely to disappear, and the challenge we face is in
upscaling the topography while capturing the hydrodynamic effects of
unresolved features.
As an alternative to grid coarsening, grid nesting (from fine to coarse
grids) can be applied to somewhat reduce computational costs
e.g.,, but such methods inherently require an
expert modeler to make a judgement as to where model resolution might be
coarsened. Artificial porosity approaches seem appropriate for urban areas
with multiple pathways around unresolved objects ,
but it is not clear that they are necessarily useful in broader natural
settings or where narrow objects are blocking over multiple cells. Quadtree
subgrid nesting for hydrodynamic models appears to be one approach for
upscaling effects of fine-resolution topography without resorting to edge
features , but such methods are still in
the early stages of development and cannot be considered a definitive
solution. Indeed, it is not clear that simple application of quadtree methods
would necessarily preserve contiguous blocking edges, so combining quadtree
with the present edge-blocking technique might be necessary.
A 75 ha section of the Rincon Bayou in the Nueces River delta
shown in a Google Earth satellite photo (a), which is centered at
27∘53′20′′ N, 97∘34′11′′ W. Comparing the 1 m × 1 m resolution lidar
DTM (b) courtesy of J. Gibeaut, Texas A & M Corpus Christi, and the arithmetic
mean of the data computed on a 20 m × 20 m grid (c) illustrates the loss
of hydrodynamic blocking height and continuity when a simple mean is used for
coarse-grid elevations (results for a 20 m × 20 m median, not shown, are
almost indistinguishable). Frames (d) through (f) show applications of median
filtering (see Methods) with varying filter scales, retaining the original
1 m × 1 m resolution but eliminating unresolvable features from the data
set. All elevations are relative to mean sea level (m.s.l.).
Upscaling of lidar data to a coarser grid presents hydrodynamic modeling
challenges . Prior to lidar, our topographic data
were generally coarser than model grid scales , so
translating topography to a model grid was a matter of simple interpolation
from a sparse data set and calibrating roughness for effects of unknown
features. With the advent of lidar, as noted by , “we
need methods to identify and connect linear topographic features … given
their significant hydraulic impact.” Indeed, in associated work of
, Bates' research team pioneered the use of
image-processing techniques (skeletonization) for linear features that were either
incorporated in a highly resolved mesh or represented as roughness.
Nevertheless, despite more than 150 citations of these pioneering works, the
general challenge of automatically identifying and connecting linear topographic features has not been previously addressed, and thus provides
the fundamental motivation for the present work. Herein we address this
challenge with a new automated method for identifying, extracting, and
representing topographic features that extend over multiple coarse-grid
cells but are so narrow that their hydrodynamic blocking effects would be
lost in common upscaling techniques.
As a simple example of the challenge, consider the satellite photograph and
corresponding lidar data in Fig. a and b, which show a portion of a
railroad dike that cuts across the Nueces River delta in Texas, USA, just
outside the City of Corpus Christi. The dike is approximately 5 m across at
the top and 15 m across at the base. If the lidar data are rasterized to a
20 m × 20 m grid using a simple arithmetic mean of the finer-scale data
(Fig. c), the dike loses both its overall blocking height and
continuity. A hydrodynamic model using this coarser grid would have flow
paths through the dike at 1.5 to 2 m a.s.l. (above sea level) rather than being
contiguously blocked at a 3 m elevation.
Where dikes, embankments, and complex topography are narrower than a
practical hydrodynamic grid, the traditional solution has been use of
unstructured grids designed with cell boundaries coincident to narrow
blocking features e.g.,.
Unfortunately, unstructured grids usually require significant expertise and
“hands-on” artistry to develop an acceptable balance between hydrodynamic
modeling practicality and fidelity to the physical topography
. As another approach,
modeled the 7500 ha of the Nueces River delta
with a structured Cartesian grid, where the narrow railroad dike across the
delta (Fig. ) was represented as an elevated edge feature in a
2-D hydrodynamic model. Identifying this contiguous edge feature on the raster
grid was a labor-intensive manual task, but it was critical to obtaining the
correct hydraulic blocking effects.
It can be stated without reservation that hydrodynamic modeling is simpler if
a single size and shape of grid element is used across an entire domain. Such
a grid might be called an “arbitrary” grid as, by definition, it is not
tuned to match any particular topographic feature. The problem with arbitrary
grids is that at coarse resolution they will not have the proper connectivity
of either narrow flow paths or narrow blocking structures. Thus, the
simplicity of arbitrary gridding does not necessarily imply a better model.
However, if narrow blocking features can be upscaled to the edges of the
arbitrary grid cells and the correct continuity preserved over multiple
cells, then the feature's effect on flow blocking can be retained, albeit at
some distortion of the shape and location of the object. However, such
distortions are inherently at the model grid scale and hence should be
acceptable for a coarse-grid model. That is, both the landscape and an
associated hydrodynamic model are subject to distortions scaling on the grid
size (e.g., a single grid cell has a uniform elevation and roughness and a
single velocity representing the flow); thus, the distortion caused by moving
a subgrid blocking feature from the grid cell center to its edge is
consistent with approximations associated with the selection of the coarse-grid scale.
For an arbitrary gridding approach to actually simplify modeling, the
definition of the coarse-grid edge blocking must be automatic – otherwise we are
back to treating grid definition as a labor-intensive art form. The
“hands-off” approach introduced herein captures both the obvious
large-scale features (such as the railroad dike) and smaller features
that are more difficult to identify. The new approach uses Cartesian grids
that are relatively easy to create, modify, and hydrodynamically model.
Although unstructured grids have been popular over the past two decades as a
way to direct computational power at specifically desired scales, one can
make the argument that continually increasing computer power will eventually
lead to a return to structured grid modeling that provides simpler
automation, requires less expertise in model and mesh development, and allows
for easier communication between models.
This paper provides a set of methods to represent fine-scale topographic data
to allow hydrodynamic modeling of blocking features at coarser scales on a
Cartesian grid. Full implementation and testing of these ideas require a
hydrodynamic model, whose characteristics will influence the resulting
behaviors and would require detailed description and investigation. To keep
the focus of this paper on the topographic techniques, analysis of the
hydrodynamic solution is reserved for future investigations and will not be
addressed herein.
Methods
The goal of our lidar processing is to produce a coarse Cartesian grid at
some scale, ΔC, that retains valuable hydrodynamic characteristics
associated with contiguous blockage features visible at a finer resolution,
ΔF (e.g., the railroad dike in Fig. ). In a conventional
Cartesian raster grid, each cell has a single piece of data: the landscape
elevation. However, for the coarse grid we will store a representative value
for the elevation over the bulk of the grid cell and separate blocking
elevation values on each cell face, which is typically allowed in Cartesian
grid models e.g., and in some unstructured
grid models . For convenience in data processing, we will
limit our focus to systems with integer values of the coarse-to-fine raster
ratio, defined as
RΔ=ΔCΔF.
For sufficiently fine resolution lidar data, requiring integer values for
RΔ is not a significant limitation. It follows that the coarse-grid
raster is of size ncx × ncy such that
ncx=nfxRΔ,ncy=nfxRΔ,
where nfx and nfy are the number of fine-grid cells in the data set
(this approach generally requires truncating some fine-grid data at edges to
ensure integer values for ncx and ncy). For simplicity in
exposition, we will confine ourselves to the case where the x and
y directions are resolved with the same RΔ, although the method is
readily extensible to rectangular vice square grid cells. Further extension
to unstructured grids is theoretically possible, but the methods developed
herein would require significant modification for a non-Cartesian mesh.
The general procedure for data processing is as follows:
create a fine-grid background topography (Fig. d),
create a coarse-grid representation of background topography
(Fig. a),
compute the difference between fine and coarse topography
(Fig. b),
identify contiguous objects that occur in the difference set
(Fig. ),
identify blocking objects and assign elevations to grid cell faces (Figs. and ).
Fine-scale background topography – the first step is to separate
unresolvable topographic features (at coarse-scale ΔC) from a
background topography, i.e., estimating what the fine-scale landscape would
look like with coarse-scale unresolvable features removed. Herein we apply a
median filter, which was originally designed for image noise removal
but has proven more widely useful, e.g., removing the signature of large
woody debris from bathymetric data . A median
filter replaces the value at position (x, y) with the median of the values
in some neighborhood around the point; the neighborhood size is defined as
the filter size, ΔM. The filter operation is accomplished on a moving
window over the fine-scale grid to produce a smooth rendition of the
background elevations that are resolvable at a coarser resolution; that is,
the original resolution of the data set is maintained (e.g., unlike the
averaging in Fig. c), while the unresolvable features are
removed. For example, the 1 m × 1 m lidar data are processed with
different-size median filters as shown in Fig. d–f, providing
smooth, high-resolution background elevations.
Clearly, the filter scale for defining background topography should be equal
to or greater than the desired coarse-grid scale, i.e., ΔM ≥ ΔC.
If the median filter is smaller than the coarse grid, then objects that
cannot be resolved will remain in the filtered data set and hence not in the
difference data set (discussed below). Indeed, it seems prudent to generally
apply a median filter of ΔM ≈ 2ΔC to ensure that objects
only slightly larger than a grid cell are readily identified. That is, if a
20 m × 20 m coarse grid is desired and a 20 m × 20 m median filter
is applied, there can be features slightly larger than 20 m that will appear
across two coarse-grid cells and hence will not really be hydrodynamically
resolved. This effect is clear in Fig. f, where the 20 m × 20 m
filter shows a partial signature of the railroad dike that would be
lost in upscaling to the coarse grid as in Fig. c. Sensitivity
of blocking identification to the choice of the ΔM parameter is
provided in the Results section, below.
Coarse background and difference data set – the median filtered
fine-scale data, Fig. d, are used to produce a coarse-grid
approximation of the landscape elevation, Fig. a. This step
can be accomplished using either the simple arithmetic mean or median of
elevations inside the coarse-grid cell (herein the median is used). This
coarse-grid representation is pushed back to the fine grid (i.e., using
identical values for all the fine cells within a single coarse-grid cell) so
that a fine-scale difference map can be created (Fig. b).
Note that it is also possible to directly use the median filtered fine-scale
data for the difference map (i.e., without pushing back to the coarse grid),
but this approach affects the interpretation of the difference value relative
to the coarse-grid cell elevation.
(a) The 20 m × 20 m coarse-grid background topography based on
the 40 m × 40 m filter of Fig. d; (b) the difference
between the 1 m × 1 m lidar data of Fig. b and the
coarse-grid background topography.
Identification of objects – the difference map
(Fig. b) contains both negative objects (unresolved
depressions) and positive objects (unresolved blockages). The present work
focusses on the positive objects that are
relatively easy to handle in a hydrodynamic model that includes cell edge
elevations. To identify blocking positive objects, a cutoff height
(Δh) is specified above which an object is deemed a hydrodynamic blockage
rather than topographic roughness. The number and size of objects will be a
function of this cutoff. The object size, i.e., the number of fine grid cells
in a contiguous blocking object, is denoted as N with subscripts to
distinguish the x and y faces of a coarse-grid cell. For the present
demonstration, the cutoff height is Δh = 0.2 m. Sensitivity of
blocking results to the choice of Δh is provided in the Results
section, below.
A binary data set can be defined as {0, 1} based on whether fine-grid cells
are respectively below or above Δh, as shown in
Fig. a. Our goal is to identify discrete objects, where an
object is defined as a contiguous set of cells with the value of 1.
Algorithms to identify connected cells are relatively easy to write but are
difficult to make efficient for large data sets. Fortunately, binary object
identification is a standard image-processing task and efficient algorithms
are available .
However, even efficient algorithms have computational costs scaling on the
number of objects, so it is useful to first remove single cells, i.e., where
N = 1, which cannot cause hydraulic blocking. In image processing, single
pixels in a binary data set are considered “noise” and addressed with
noise-removal techniques. Again, one can readily write an algorithm to check the
neighbors for each cell with a value of 1 and eliminate all those whose
neighbors are all 0, but it is simpler to rely on the expertise of the
image-processing community for this task. In the present work the Matlab
Image Processing Toolbox bwmorph function with the “clean” option was
used to remove single-cell objects. Identification of multi-cell objects was
accomplished with the Moore neighbor tracing algorithm
as implemented in the bwboundaries function
in Matlab. The resulting objects are shown in Fig. b.
Binary image (a) of difference data set from
Fig. b, which can be used to identify separate objects,
shown in colors in (b). Note that objects smaller than 20 fine-grid cells
in (a) have been eliminated in (b).
It is also useful to remove objects that are too small to block a coarse-grid
cell; i.e., for a 20 m × 20 m coarse grid based on a 1 m × 1 m data
set (RΔ = 20), any object in Fig. a that consists of
fewer than 20 fine-grid cells cannot hydraulically block a coarse-grid cell
and can be excised from the object data set. To allow some flexibility, it is
useful to define a small object removal criterion based on N blocking cells as
N≤RΔ-δ,
where δ ∈ {0, 1, 2, …} is a user-defined parameter
that allows nearly blocking objects (δ > 0) to be retained in the
object data set. Herein δ = 1 is used.
Blocking caused by a small object. The red × and blue +
represent fine-grid blocking cells (gray □) snapped to the
coarse-grid faces Gy and Gx; black lines are the resulting blocked
coarse-grid faces. Here and throughout this paper δ = 1 is used for
defining blocking. The object cells in (b) are slightly shifted from (a) as
discussed in the text to better align with coarse grid.
Snap-grid object blocking – each object can be processed separately
to provide hydraulic blocking conditions on both row and column faces of a
coarse-grid cell. A typical object and its elevations (Fig. a)
has a binary representation at the fine-grid level, as shown in
Fig. b. Let fo(xc, yc) be the set of fine-grid cells of the
object, where xc and yc are coordinates measured relative to the coarse
grid. The coarse-grid demarcation lines are, by definition, integer values in
our coarse-grid numbering scheme. This is perhaps slightly unconventional for
hydrodynamic modelers as the coarse-grid cell centers are therefore at
non-integer values; i.e., the location
xc,yc=(i-1/2,j-1/2)
defines a coarse-grid cell in the raster set for
i=1,2,3,…ncxj=1,2,3,…ncy
with faces at coarse-grid indexes
xc,yc∈(i,j-1/2),(i-1,j-1/2),…(i-1/2,j),(i-1/2,j-1).
Using this indexing, we can define
Gy(i-1/2,j)=foxc,roundyc
as a set of fine-grid cells of varying xc that are “snapped” to an
integer yc face (the red × markers in Fig. c). Similarly
Gx(i,j-1/2)=foroundxc,yc
is the set of blocking cells of varying yc snapped to an integer xc
face (the blue + markers in Fig. c). For the purposes of
determining whether or not blocking occurs, the cells sets Gx and Gy
are true mathematical sets that do not include duplicate values. However,
data sets with duplicates (denoted as Gyy and Gxx) are retained for
computation of blocking height (discussed below).
To determine snap-grid blocking of coarse-grid faces
(Fig. d), the sizes of set Gy(i-1/2,j) and Gx(i,j-1/2)
are defined as Ny(i-1/2,j) and Nx(i,j-1/2), respectively. These are
the number of unique values of xc on a round (yc) face (red ×
markers on a column face) and the number of unique yc values on a
round(xc) face (blue + markers on a row face). Snap-grid blocking occurs
along coarse-grid column faces that have a number of blocking fine-grid cells
exceeding the removal criterion of Eq. (), that is,
Ny(i-1/2,j)≥RΔ-δ,
and coarse-grid row faces that satisfy
Nx(i,j-1/2)≥RΔ-δ.
Note that this approach allows the fine cells to serve as blocking in
the both x and y directions simultaneously, which is necessary to represent
the hydraulic blocking of objects at an angle to the coarse grid. After a
face is blocked, e.g., as shown in Fig. d, the corresponding
Gy(i-1/2,j) and Gx(i,j-1/2) are set to 0 so that the fine-grid cells
used to define a snap-grid block are not used in computing cross-cell blocking
(discussed below).
Small object shift – the snap-grid blocking approach will necessarily
depend on the spatial relationship between the objects and the coarse grid.
For small objects, a slight shift of the object position can change whether
or not the object is judged to be blocking. For example, the lower column
face blocking in Fig. d would not have been identified as blocking
in the original object position, shown in Fig. a, because some
of the fine-grid cells would have shown up in an adjacent column such that
Eq. () would not have been satisfied for either face. For small
objects (overall length less than 1.5ΔC), it is convenient to
simply shift the object (as in Fig. b) to maximize the number
of fine-grid blocking cells within a single coarse-grid cell. As long as the
shift is fewer than ΔC/4 cells, it does not significantly affect the
coarse-grid physical relationships. Object shifting can be accomplished with
an automated algorithm that is based on the total extent of a small object
and the overhang of the object into adjacent cells.
Blocking caused by a large object similar to Fig. for
frames (a) through (c). In (d) the remaining Gy and Gx are transformed
to cell-center Hy and Hx blocking shown with red ∘ and blue
△, and different possible blocking paths are illustrated with
dashed lines; (e) shows final blocking paths that are contiguous.
Cross-cell object blocking – depending on an object's topology, the
blocked faces determined by the snap-grid approach (described above) might
not provide a contiguous blocked path. Consider the larger object in
Fig. a, where snap-grid blocking provides the results in
Fig. b and c. It is clear that some hydraulic blocking has not been
captured: the Gx (the blue + markers) and Gy (red × markers)
in the second coarse-grid cell from the top do not satisfy Eqs. ()
or () as they are split between different faces. Furthermore in the
lowermost coarse-grid cell, the red × markers on either face are
insufficient for blocking. These effects arise because the snap-grid approach
uses rounding, which can split the blocking fine-grid cells to the upper and
lower faces of the coarse-grid cell. To address this issue, we define a
cross-cell blocking approach for column and row faces. Cross-cell blocking
is conducted after snap-grid blocking and only uses the fine-grid cells that
were not applied in snap-grid blocking, e.g., the remaining red ×
and blue + in Fig. c.
For the coarse-grid cell centered at (i-1/2, j-1/2), we define sets of
unique fine-grid blocking cells across the cell center as
Hy(i-1/2,j-1/2)=Gy(i-1/2,j)∪Gy(i-1/2,j-1),
Hx(i-1/2,j-1/2)=Gx(i,j-1/2)∪Gx(i-1,j-1/2),
which are illustrated in Fig. d. Blocking conditions are
defined similar to Eqs. () and (), using Ny and Nx
as the size of the unique Hy and Hx cell sets. For determining object
blocking height (discussed below), we also define sets that retain non-unique
elements:
Hyy(i-1/2,j-1/2)=Gyy(i-1/2,j)∪Gyy(i-1/2,j-1),
Hxx(i-1/2,j-1/2)=Gxx(i,j-1/2)∪Gxx(i-1,j-1/2).
Histograms of the number of fine-grid blocking cells at different
heights for five randomly selected coarse-grid cells in the data set. Dashed
line represents the blocking height for the coarse-grid face selected as the
75th-percentile blocking height. Note that the maximum number of fine-grid
cells in a single coarse-grid cell is 400.
As the Hy and Hx fine-cell sets cross the coarse-cell center, face
blocking could be at either of the grid cell faces, as shown by dashed
blocking lines in Fig. d. Indeed, there can be more than one
set of blocking faces that provides a reasonable representation of cross-cell
blocking. The critical issue is ensuring that cross-cell blocking is
contiguous; i.e., there are choices for cross-cell blocking faces in
Fig. d that would not provide contiguous blocking. The
simplest algorithm for selecting blocking is to process column faces (red ∘)
and row faces (blue △) sequentially. If a coarse cell
contains only a single blocked end point (e.g., the lowermost complete cell
in Fig. d), then the cross-cell blocked face (either row or
column) must connect to that blocked end. If a coarse cell contains two
blocked end points, then the algorithm must distinguish between
diagonally blocked points (e.g., the second coarse cell from the top in
Fig. d) and co-linear blocked end points along a face (either
row or column). Where two blocked end points are along a single column face
and cell-center blocking for a column face exists (red ∘), the
cell-center blocking logically must be along the face that connects the
blocked end points. Similarly, where two blocked end points are along a row
face and cell-center blocking for a row face is indicated (blue △),
the blocking is necessarily along the row face connecting the blocked
end points. However, where two blocked end points are co-linear along a
column face and the cell-centered blocking is indicated for a row face (or
vice versa), the choice of which face to block may be taken arbitrarily.
Similarly, when two blocked end points are diagonally opposed, the selection
of the blocking face is arbitrary. Note that these arbitrary choices will
necessarily set up a condition where three end points are blocked. Because
rows and columns are processed sequentially, diagonal blocking of two end
points solved using columns (first cycle) sets up a cell with three blocked
end points for solving using rows (second cycle). If three end points are
blocked, then the cross-cell blocking must connect the two blocked end points
that are co-linear along the column or row face (as appropriate), which
ensures continuity of the feature.
In the present data set, all the objects achieved contiguous edge-blocking
representations using the snap-grid and cross-cell object blocking algorithms
outlined above. However, one can imagine a feature that is longer and
narrower than shown in Fig. for which the procedure might
fail. For a long narrow feature, it is possible that multiple iterations of
the cross-cell algorithm would be required to define a blocking condition.
That is, the cross-cell algorithm described above combines (for example) the
Gx blocking cells of two opposite column faces into a single cell-centered
Hx that is tested for blocking. If Hx < RΔ - δ, there is no
blocking, and one can loop the algorithm to look for larger-scale blocking by
combining the Hx of two adjacent cells into an Ix that is evaluated on
the cell face for blocking. This iterative approach can be continued for
Jx, Kx, etc., until blocking is achieved or the coarse-grid length of
the object is reached.
Object blocking height – the snap-grid and cross-cell blocking methods
determine which faces are blocked by an object. Unfortunately, establishing
an exact fine-grid blocking height for a face requires analysis of the lowest
contiguous path through the blocking cells. Although such an algorithm is
theoretically possible, it has not been attempted. Given the uncertainties at
the fine-grid level within typical lidar data, there are questions as to
whether such a complicated analysis would be worthwhile; thus, a simple
statistical approach is adopted herein.
A blocking face is a single height value that represents a distribution of
the fine-grid cell heights, which are retained in the Gxx, Gyy,
Hxx, and Hyy sets described above. Typical distributions are shown
in Fig. . A reasonable estimation of the blocking height for
a Gxx set must be bounded by the maximum and minimum values in the set.
It is convenient to denote the number of cells in the Gxx as Nxx,
with similar definitions for other cell sets. We then define the relative
“thickness” of the blocking cell set for the xx face as
Txx≡NxxRΔ.
This thickness can be thought of as the number of rows of fine-grid cells
that would be blocked if the fine grid cells were rearranged against the
coarse-grid face. It can be argued that larger a Txx should lead to
higher blocking heights, i.e., making it less likely that a low-level path is
available through the object. In the present work, we take an ad hoc
approach: if Txx < 2, we use the median of Gxx as the blocking
height. Where Txx ≥ 2, we take the median of the largest half of the
data set, i.e., the height of the 75th percentile. Similar arguments are used
for, e.g., Gyy.
The snap-grid and cell-center blocking methods applied to the
median-filtered background topography from Fig. .
In (a) the locations of all blocked faces are shown; in (b), the blocked faces are
given colors corresponding to their blocking height.
Discussion
The present work is a first attempt at automating the identification of
linear features for hydrodynamic blocking; as such there remain a number of
areas where further analysis and improvement are needed. One of the
less-satisfying aspects in the present work is the ad hoc selection of the
75th-percentile height as the blocking height for a face. There are a wide variety
of statistical approaches that could be used for estimating the blocking
height, and the present approach merely tries to account for the greater
likelihood of higher blocking from a denser set of blocking fine-grid cells.
The effect of this estimation method remains an open question, as analysis
would be best accomplished in conjunction with hydrodynamic modeling.
A section of the Nueces River delta (2.2 km × 5 km of a larger
data set) centered at 27∘52′25′′ N 97∘32′21′′ W. Blocking
faces at bridges and culverts have been manually removed. For clarity in (a),
where data are missing from the original lidar DTM (principally at water
surfaces), the data have been replaced by -0.2 m. For the 20 m × 20 m
grid, these data have been replaced by survey data.
As an exercise in practical application to a larger system, the edge-blocking
methods have been used for the full 104 ha of the Nueces River
delta, with an extract shown in Fig. for 1100 ha of
marshes that are southeast of the railroad bridge of Fig. . The
blockages were computed using the automated techniques with manual removal of
blocking faces for known bridges and culverts. This figure illustrates a key
future challenge: identifying preferential flow paths that are unresolved at
the coarse-grid scale (i.e., the negative objects resulting from
unresolved depressions). Within the lidar data are several important narrow
channels that are not in the 20 m × 20 m grid. This effect can be seen
in greater detail in Fig. a, where we can clearly see a narrow
stream channel on one side of the railroad dike. This channel is entirely
absent in the final topography of Fig. . Such channels
could be readily identified by using the snap-grid and cell-center blocking
techniques as “path” techniques for negative objects (i.e., objects
determined similar to Fig. a, but using a negative Δh
for discrimination). However, it is not clear how such objects could be used
in most hydrodynamic models. Definition of preferential narrow flow paths
that are unresolved within coarse-grid topography remains an area with no
clear solution . However, there are
interesting possibilities in 1-D–2-D models, such as that of ,
which might provide a good starting point. In any case, inversion of the techniques
developed above provides a basis for defining preferential flow paths along
coarse-grid cell edges, which can be seen as a precursor for new hydrodynamic
modeling techniques.
A potential area where the edge-blocking method might be expanded is in the
estimation of topographic roughness, which has been a subject of extensive
prior research e.g.,. By defining
edge features, a portion of the difference between the grid cell elevation
and subgrid features can be removed from the roughness estimation; i.e., we
could use, for example, Gxx, Gyy, Hxx, and Hyy to remove pixels
that have been resolved in to edge features and only consider the remaining
pixels in a coarse-grid cell as contributing to roughness.
The methods developed above presume the lidar DTM is
a reasonable representation of the underlying topography. Naturally, any DTM
has noise, elevation inaccuracies, and the possible inclusion of permeable
features (e.g., vegetation) as impermeable landscape. For noise and permeable
features smaller than the coarse-grid scale, the present approach maintains
the typical advantages of a median filter in removing smaller objects while
retaining sharp edges. However, in larger forested areas or farmlands with
thick hedgerows, simple automated application of the present techniques is
likely to create undesirable blocking features. There is an open issue as to
whether the signature of such features could be a priori identified and
used to create permeable cell faces with increased roughness in a hydrodynamic model.
Finally, as pointed out by P. Bates (personal communication, 2015), we lack a
quantitative metric for evaluating how well the blocking faces model the
underlying topography. Unfortunately, there is no obvious approach for
comparing the results (Fig. b) to the original data
(Fig. b) without resorting to image-processing techniques, which
would simply provide a circular argument. It seems likely that a quantitative
metric will require using a hydrodynamic model at both the original DTM and
with the new coarse-grid topography to evaluate blockage effectiveness at
different water elevations.