Research Progress

English Engineering Education Program (e3)

What the Box-Counting Method Does

It measures how complicated a river network is by checking how many grid boxes are needed to cover it at different scales.

The procedure (as described on the page):

Overlay a grid of square boxes of size 𝜀 on the river map.
Count how many boxes \(𝑁 ( 𝜀 )\) contain any part of the river.
Repeat with smaller and smaller boxes.
Plot:

horizontal axis: \(log ( 1 / 𝜀 )\)
vertical axis: \(log 𝑁 ( 𝜀 )\)

Compute the slope of the line. That slope is the fractal dimension \(𝐷\) .

The formula shown on the page:

Box‑Counting Formula In the box‑counting method, the formula shown on your page is:

\[ D = \lim_{\varepsilon \to 0} \frac{\log N(\varepsilon)}{\log(1/\varepsilon)} \tag{1}\]

As shown in Equation Equation 1, the box-counting dimension…

means:

This is the standard mathematical definition of box-counting domension.

What \(N\) Represent in the Box-Counting Formula

In the box-counting method, the formular shown in Equation 1. The variable \(N(\epsilon)\) means: The number of grid boxes (of size \(\epsilon\)) that contain any part of the river network. If the river network is more complex or space-filling, it will occupy more boxes at each scale, giving a higher \(N(\epsilon)\) and therefore a higher fractal dimension.

Box size \(\epsilon\)	What happens	Meaning of \(N(\epsilon)\)
Large boxes	River fits in fewer boxes	Small \(N(\epsilon)\)
Smaller boxes	River spans more boxes	Larger \(N(\epsilon)\)
Very small boxes	You capture fine wiggles and branches	\(N(\epsilon)\) grows faster

An example for calculation

Simple “river” on grid Imagine a binary image (or raster) that is 4x4 cells, each cell of size 1. The “river” passes through these 6 cells:

Row 1: columns 2, 3
Row 2: column 3
Row 3: column 3
Row 4: columns 2, 3 So there are 6 river cells in total.

Choose box sizes \(\epsilon\)

\(\epsilon_1=2\) (big boxes)
\(\epsilon_2=1\) (original cellsize)

For \(\epsilon_1=2\)

We overlay a 2x2 grid of big boxes (each box is 2x2 cells):

- Box (rows 1-2, cols 1-2): contains rivers? Yes (row 1, col 2; row 2, col 2 is empty but box still counts).
- Box (rows 1–2, cols 3–4): contains river? Yes (row 1, col 3; row 2, col 3).
- Box (rows 3–4, cols 1–2): contains river? Yes (row 4, col 2).
- Box (rows 3–4, cols 3–4): contains river? Yes (row 3, col 3; row 4, col 3).

So all 4 boxes contain at least one river cell:

\[ N(\epsilon_1=2)=4 \tag{2}\]

For \(\epsilon_2=1\) Now each original cell in a box. We count how many cells contain river: We already know: 6 cells. SO:

\[ D\approx \frac{logN(\epsilon_2)-logN(\epsilon_1)}{log(1/\epsilon_2)-log(1/\epsilon_1)} \tag{3}\]

Substituting to Equation 3:

\(N(\epsilon_1)=4, \epsilon_1=2\)
\(N(\epsilon_2)=6, \epsilon_2=1\)

So:

\[ D\approx \frac{log6-log4}{log(1/1)-log(1/2)} \tag{4}\]

So:

\[ D\approx \frac{1.7918-1.3863}{0-(-0.6931)}=\frac{0.4055}{0.6931}\approx0.585 \tag{5}\]

Nonlinear Shallow Water Equations (SWE)

Explained in the context of your river-analysis

It is called nonlinear because the velocity and water depth interact in multiplicative ways, producing complex flow behavior (exactly the kind of complexity that motivates fractal analysis of river networks).

What SWE represent

Water depth changes over time
Flow velocity evolves in space
Momentum is transported downstream
Gravity, inertia, and bed slope shape the flow

These equations are the backbone of hydrodynamic models used to simulate:

River discharge
Channel branching behavior
Flow accumulation patterns

All of these processes influence the geometry and fractal structure of river networks

What Occupancy: \(\phi(\Delta)\)

The fractal of area that is “active” (covered by the river network) inside a square of size \(\Delta \times \Delta\). So:

\[ \phi(\Delta)=\frac{\text{active area in }\Delta^2}{\text{total area in }\Delta^2} \tag{6}\]

when:

\(\Delta\) is the obdervation scale (the size of the box you use to measure how much of the river network is inside it).

As you zoom out \(\text{(larger }\Delta)\), the river network occupies a smaller fraction of the box.

As you zoom in \(\text{(smaller }\Delta)\), the river network occupies a larger fraction of the box. This scale-dependence is the hallmark of fractal geometry.

Part I: Workflow Optimization — Identifying the Optimal Pipeline

Goal: Compare all 48 workflow combinations (3 Ground Filters × 4 Interpolations × 2 Sink-Fills × 2 Flow Directions) at each resolution using pairwise Intersection over Union (IoU). The workflow achieving the highest Median IoU is carried forward to the fractal analysis in Part II.


====================================================
  IoU Optimisation: 1m
====================================================
  Files found: 48 
  Sample filename → component mapping:
    csf_idw_area1_1m_20251024_Hororo_area1_fillpd_d8.tif  →  CSF_IDW_PLAN_D8
    csf_idw_area1_1m_20251024_Hororo_area1_fillpd_dinf.tif  →  CSF_IDW_PLAN_Dinf
    csf_idw_area1_1m_20251024_Hororo_area1_fillwl_d8.tif  →  CSF_IDW_WANG_D8
    csf_idw_area1_1m_20251024_Hororo_area1_fillwl_dinf.tif  →  CSF_IDW_WANG_Dinf
  Loaded: 48 rasters
  Computing 1128 pairwise comparisons...

  ★ BEST [1m]: CSF_KRG_WANG_D8   (Median IoU = 0.9650)

====================================================
  IoU Optimisation: 5m
====================================================
  Files found: 48 
  Sample filename → component mapping:
    csf_idw_area1_5m_20251024_Hororo_area1_fillpd_d8.tif  →  CSF_IDW_PLAN_D8
    csf_idw_area1_5m_20251024_Hororo_area1_fillpd_dinf.tif  →  CSF_IDW_PLAN_Dinf
    csf_idw_area1_5m_20251024_Hororo_area1_fillwl_d8.tif  →  CSF_IDW_WANG_D8
    csf_idw_area1_5m_20251024_Hororo_area1_fillwl_dinf.tif  →  CSF_IDW_WANG_Dinf
  Loaded: 48 rasters
  Computing 1128 pairwise comparisons...

  ★ BEST [5m]: PMF_TIN_WANG_D8   (Median IoU = 0.9344)

====================================================
  IoU Optimisation: 10m
====================================================
  Files found: 48 
  Sample filename → component mapping:
    csf_idw_area1_10m_20251024_Hororo_area1_fillpd_d8.tif  →  CSF_IDW_PLAN_D8
    csf_idw_area1_10m_20251024_Hororo_area1_fillpd_dinf.tif  →  CSF_IDW_PLAN_Dinf
    csf_idw_area1_10m_20251024_Hororo_area1_fillwl_d8.tif  →  CSF_IDW_WANG_D8
    csf_idw_area1_10m_20251024_Hororo_area1_fillwl_dinf.tif  →  CSF_IDW_WANG_Dinf
  Loaded: 48 rasters
  Computing 1128 pairwise comparisons...

  ★ BEST [10m]: PMF_TIN_PLAN_D8   (Median IoU = 0.9220)

==========================================================

  OPTIMAL WORKFLOWS IDENTIFIED PER RESOLUTION

==========================================================

  1m     |  CSF_KRG_WANG_D8               |  Median IoU = 0.9650
  5m     |  PMF_TIN_WANG_D8               |  Median IoU = 0.9344
  10m    |  PMF_TIN_PLAN_D8               |  Median IoU = 0.9220

==========================================================

1 m Resolution — Optimization Results

Summary Statistics by Component

IoU Statistics by Workflow Component – 1 m Resolution
Component	Option	Median IoU	Mean IoU	SD	Q25	Q75	N
Ground Filter	CSF	0.9544	0.9563	0.0154	0.9459	0.9666	632
Ground Filter	MCC	0.9534	0.9538	0.0136	0.9451	0.9612	376
Ground Filter	PMF	0.9672	0.9662	0.0157	0.9543	0.9739	120
Interpolation	IDW	0.9599	0.9590	0.0140	0.9487	0.9688	354
Interpolation	KRG	0.9560	0.9602	0.0158	0.9497	0.9710	306
Interpolation	MBA	0.9469	0.9472	0.0135	0.9391	0.9546	258
Interpolation	TIN	0.9567	0.9585	0.0140	0.9491	0.9651	210
Sink-Fill	PLAN	0.9557	0.9586	0.0150	0.9479	0.9689	588
Sink-Fill	WANG	0.9545	0.9542	0.0152	0.9442	0.9644	540
Flow Direction	D8	0.9574	0.9597	0.0148	0.9500	0.9689	576
Flow Direction	Dinf	0.9517	0.9532	0.0151	0.9433	0.9635	552

Component IoU Distribution

Top 10 Workflows

Top 10 Workflows by Median IoU – 1 m Resolution
Workflow	Median IoU	Mean IoU	SD	Min IoU	Max IoU	N Comparisons
CSF_KRG_WANG_D8	0.9650	0.9640	0.0166	0.9299	1.0000	47
PMF_KRG_WANG_D8	0.9650	0.9640	0.0166	0.9299	1.0000	47
PMF_IDW_WANG_D8	0.9645	0.9630	0.0138	0.9361	0.9924	47
PMF_TIN_WANG_D8	0.9644	0.9635	0.0158	0.9172	0.9909	47
PMF_IDW_PLAN_D8	0.9640	0.9658	0.0118	0.9471	0.9985	47
CSF_TIN_PLAN_D8	0.9635	0.9638	0.0118	0.9400	0.9973	47
CSF_IDW_WANG_D8	0.9630	0.9600	0.0147	0.9332	0.9930	47
PMF_IDW_PLAN_Dinf	0.9629	0.9643	0.0119	0.9453	0.9985	47
CSF_TIN_PLAN_Dinf	0.9619	0.9601	0.0131	0.9369	0.9973	47
CSF_TIN_WANG_D8	0.9617	0.9605	0.0131	0.9304	0.9903	47

Best Option per Component

Consensus – Best Option per Component – 1 m Resolution
Workflow Component	Best Option	Median IoU	Improvement vs. Worst
Ground Filter	PMF	0.9672	1.4%
Interpolation	IDW	0.9599	1.4%
Sink-Fill	PLAN	0.9557	0.1%
Flow Direction	D8	0.9574	0.6%

Kruskal-Wallis Significance Tests

Kruskal-Wallis Tests – 1 m (*** p<0.001, ** p<0.01, * p<0.05)
Component	χ²	df	p-value	Sig.
Ground_Filter	57.35	2	0.00000	***
Interpolation	131.47	3	0.00000	***
Sink_Fill	14.34	1	0.00015	***
Flow_Direction	50.30	1	0.00000	***

Correlation Heatmap

5 m Resolution — Optimization Results

Summary Statistics by Component

IoU Statistics by Workflow Component – 5 m Resolution
Component	Option	Median IoU	Mean IoU	SD	Q25	Q75	N
Ground Filter	CSF	0.9152	0.9193	0.0270	0.8983	0.9389	632
Ground Filter	MCC	0.9046	0.9069	0.0275	0.8889	0.9185	376
Ground Filter	PMF	0.9423	0.9331	0.0347	0.9012	0.9588	120
Interpolation	IDW	0.9208	0.9221	0.0277	0.9039	0.9412	354
Interpolation	KRG	0.9069	0.9174	0.0331	0.8927	0.9393	306
Interpolation	MBA	0.9015	0.9069	0.0236	0.8915	0.9118	258
Interpolation	TIN	0.9159	0.9184	0.0290	0.9023	0.9317	210
Sink-Fill	PLAN	0.9121	0.9183	0.0309	0.8945	0.9378	588
Sink-Fill	WANG	0.9100	0.9149	0.0271	0.8965	0.9337	540
Flow Direction	D8	0.9118	0.9184	0.0307	0.8962	0.9371	576
Flow Direction	Dinf	0.9103	0.9148	0.0274	0.8947	0.9349	552

Component IoU Distribution

Top 10 Workflows

Top 10 Workflows by Median IoU – 5 m Resolution
Workflow	Median IoU	Mean IoU	SD	Min IoU	Max IoU	N Comparisons
PMF_TIN_WANG_D8	0.9344	0.9292	0.0217	0.8878	0.9929	47
CSF_TIN_PLAN_Dinf	0.9302	0.9266	0.0254	0.8672	0.9970	47
CSF_TIN_WANG_Dinf	0.9284	0.9260	0.0252	0.8765	0.9961	47
CSF_IDW_PLAN_D8	0.9282	0.9253	0.0239	0.8771	0.9960	47
PMF_TIN_WANG_Dinf	0.9282	0.9260	0.0243	0.8812	0.9929	47
PMF_IDW_WANG_D8	0.9282	0.9276	0.0280	0.8833	0.9980	47
CSF_TIN_PLAN_D8	0.9265	0.9231	0.0260	0.8652	0.9970	47
PMF_TIN_PLAN_Dinf	0.9262	0.9262	0.0258	0.8727	0.9898	47
PMF_IDW_WANG_Dinf	0.9259	0.9253	0.0284	0.8736	0.9980	47
CSF_IDW_WANG_Dinf	0.9257	0.9311	0.0225	0.8902	0.9970	47

Best Option per Component

Consensus – Best Option per Component – 5 m Resolution
Workflow Component	Best Option	Median IoU	Improvement vs. Worst
Ground Filter	PMF	0.9423	4.2%
Interpolation	IDW	0.9208	2.1%
Sink-Fill	PLAN	0.9121	0.2%
Flow Direction	D8	0.9118	0.2%

Kruskal-Wallis Significance Tests

Kruskal-Wallis Tests – 5 m (*** p<0.001, ** p<0.01, * p<0.05)
Component	χ²	df	p-value	Sig.
Ground_Filter	86.81	2	0.00000	***
Interpolation	63.90	3	0.00000	***
Sink_Fill	1.77	1	0.18305	ns
Flow_Direction	1.66	1	0.19714	ns

Correlation Heatmap

10 m Resolution — Optimization Results

Summary Statistics by Component

IoU Statistics by Workflow Component – 10 m Resolution
Component	Option	Median IoU	Mean IoU	SD	Q25	Q75	N
Ground Filter	CSF	0.9012	0.9063	0.0308	0.8840	0.9241	632
Ground Filter	MCC	0.9014	0.9063	0.0316	0.8860	0.9213	376
Ground Filter	PMF	0.9255	0.9274	0.0344	0.8998	0.9415	120
Interpolation	IDW	0.9100	0.9105	0.0291	0.8869	0.9286	354
Interpolation	KRG	0.9064	0.9095	0.0362	0.8858	0.9247	306
Interpolation	MBA	0.9033	0.9095	0.0268	0.8930	0.9213	258
Interpolation	TIN	0.8953	0.9028	0.0358	0.8786	0.9212	210
Sink-Fill	PLAN	0.9063	0.9113	0.0357	0.8861	0.9271	588
Sink-Fill	WANG	0.9012	0.9055	0.0273	0.8859	0.9235	540
Flow Direction	D8	0.9065	0.9105	0.0335	0.8866	0.9268	576
Flow Direction	Dinf	0.9012	0.9065	0.0304	0.8859	0.9232	552

Component IoU Distribution

Top 10 Workflows

Top 10 Workflows by Median IoU – 10 m Resolution
Workflow	Median IoU	Mean IoU	SD	Min IoU	Max IoU	N Comparisons
PMF_TIN_PLAN_D8	0.9220	0.9163	0.0338	0.8355	1.0000	47
PMF_TIN_WANG_D8	0.9214	0.9178	0.0318	0.8411	1.0000	47
CSF_KRG_WANG_D8	0.9213	0.9221	0.0340	0.8657	1.0000	47
PMF_KRG_WANG_D8	0.9213	0.9221	0.0340	0.8657	1.0000	47
PMF_TIN_PLAN_Dinf	0.9213	0.9136	0.0330	0.8446	0.9975	47
PMF_TIN_WANG_Dinf	0.9198	0.9136	0.0321	0.8435	0.9975	47
CSF_KRG_PLAN_D8	0.9196	0.9207	0.0370	0.8591	1.0000	47
PMF_KRG_PLAN_D8	0.9196	0.9207	0.0370	0.8591	1.0000	47
PMF_IDW_WANG_Dinf	0.9154	0.9166	0.0268	0.8774	0.9901	47
CSF_KRG_PLAN_Dinf	0.9141	0.9159	0.0392	0.8545	1.0000	47

Best Option per Component

Consensus – Best Option per Component – 10 m Resolution
Workflow Component	Best Option	Median IoU	Improvement vs. Worst
Ground Filter	PMF	0.9255	2.7%
Interpolation	IDW	0.9100	1.6%
Sink-Fill	PLAN	0.9063	0.6%
Flow Direction	D8	0.9065	0.6%

Kruskal-Wallis Significance Tests

Kruskal-Wallis Tests – 10 m (*** p<0.001, ** p<0.01, * p<0.05)
Component	χ²	df	p-value	Sig.
Ground_Filter	42.36	2	0.00000	***
Interpolation	16.96	3	0.00072	***
Sink_Fill	3.89	1	0.04856	*
Flow_Direction	2.82	1	0.09303	ns

Correlation Heatmap

Cross-Resolution Optimization Summary

==============================================================

  CROSS-RESOLUTION OPTIMAL WORKFLOW SUMMARY

==============================================================

 Resolution   Best_Workflow Median_IoU  Mean_IoU     SD_IoU N_Rasters N_Pairs
     <char>          <char>      <num>     <num>      <num>     <int>   <int>
         1m CSF_KRG_WANG_D8  0.9650456 0.9639831 0.01663718        48    1128
         5m PMF_TIN_WANG_D8  0.9344262 0.9292265 0.02170846        48    1128
        10m PMF_TIN_PLAN_D8  0.9219858 0.9162808 0.03379022        48    1128

==============================================================

Optimal Workflow per Resolution – used in Part II fractal analysis
Resolution	Optimal Workflow	Median IoU	Mean IoU	SD	# Rasters	# Pairs
1m	CSF_KRG_WANG_D8	0.9650	0.9640	0.0166	48	1128
5m	PMF_TIN_WANG_D8	0.9344	0.9292	0.0217	48	1128
10m	PMF_TIN_PLAN_D8	0.9220	0.9163	0.0338	48	1128

**Figure X.** Horizontal bar chart of the top-5 workflows ranked by median Intersection over Union (IoU) at each spatial resolution — 1 m (green), 5 m (orange), and 10 m (blue) — derived from pairwise workflow comparisons across the Kushiro Wetland study area (EPSG:6681). At 1 m resolution, IDW- and KRG-based interpolation workflows with WANG filling and D8 routing achieved the highest median IoU (0.964–0.965), indicating near-identical channel network delineation across top-performing configurations. At 5 m, TIN-interpolated workflows dominate (IoU: 0.928–0.934), while at 10 m all top-5 workflows converge narrowly between 0.921 and 0.922, reflecting reduced discriminability at coarser resolution. PMF_TIN_WANG_D8 is the only workflow appearing in the top-5 across all three resolutions, reinforcing its robustness as an optimal processing chain regardless of input DEM resolution.

Part II: Multi-Resolution Binary Channel Mask Extraction & Box-Counting Fractal Dimension Estimation

For each input resolution, the single optimal workflow raster selected in Part I is converted to a binary channel mask (channel pixel = 1, non-channel = 0). Box-counting fractal analysis is performed in R using physical box sizes in metres as the log–log axis scale, so that the resulting fractal dimension D is directly comparable across the 1 m, 5 m, and 10 m resolutions.

------------------------------------------------------------
  Fractal: 1m  |  File: pmf_tin_area1_1m_20251024_Hororo_area1_fillwl_d8.tif
------------------------------------------------------------
  Water coverage: 1.28%
  Binary raster saved

  Binary map plot saved
  Box-counting [pmf_tin_area1_1m_20251024_Hororo_area1_fillwl_d8]  native res = 1.00 m
       2 px |     2.00 m | N = 4589
       4 px |     4.00 m | N = 2401
       8 px |     8.00 m | N = 1169
      16 px |    16.00 m | N = 528
      32 px |    32.00 m | N = 229
      64 px |    64.00 m | N = 88
     128 px |   128.00 m | N = 27
     256 px |   256.00 m | N = 9
  → Fractal D = 1.2825  (R² = 0.9893)

  Log-log plot saved
------------------------------------------------------------
  Fractal: 5m  |  File: pmf_tin_area1_5m_20251024_Hororo_area1_fillwl_d8.tif
------------------------------------------------------------
  Water coverage: 4.26%
  Binary raster saved

  Binary map plot saved
  Box-counting [pmf_tin_area1_5m_20251024_Hororo_area1_fillwl_d8]  native res = 5.00 m
       2 px |    10.00 m | N = 650
       4 px |    20.00 m | N = 344
       8 px |    40.00 m | N = 157
      16 px |    80.00 m | N = 55
      32 px |   160.00 m | N = 18
      64 px |   320.00 m | N = 8
     128 px |   640.00 m | N = 2
     256 px |  1280.00 m | N = 1
  → Fractal D = 1.3933  (R² = 0.9936)

  Log-log plot saved
------------------------------------------------------------
  Fractal: 10m  |  File: pmf_tin_area1_10m_20251024_Hororo_area1_fillpd_dinf.tif
------------------------------------------------------------
  Water coverage: 6.87%
  Binary raster saved

  Binary map plot saved
  Box-counting [pmf_tin_area1_10m_20251024_Hororo_area1_fillpd_dinf]  native res = 10.00 m
       2 px |    20.00 m | N = 257
       4 px |    40.00 m | N = 127
       8 px |    80.00 m | N = 49
      16 px |   160.00 m | N = 17
      32 px |   320.00 m | N = 8
      64 px |   640.00 m | N = 2
     128 px |  1280.00 m | N = 1
     256 px |  2560.00 m | N = 1
  → Fractal D = 1.2609  (R² = 0.9787)

  Log-log plot saved


==========================================================

  FRACTAL DIMENSION SUMMARY

==========================================================

   Resolution Fractal_D        R2 Water_pct N_scales
       <char>     <num>     <num>     <num>    <int>
1:         1m  1.282512 0.9893326      1.28        8
2:         5m  1.393294 0.9936281      4.26        8
3:        10m  1.260886 0.9786732      6.87        8

==========================================================

1 m — Binary Water / Non-Water Map (Optimal Workflow)

**Figure X.** Binary channel network map extracted from the optimal 1 m resolution workflow (PMF–TIN–WANG–D8), Kushiro Wetland, Japan. Blue pixels represent classified channel (water = 1) and light grey pixels represent non-channel areas (non-water = 0). Total water coverage is 1.28% of the study area extent. Axes show UTM coordinates in metres; CRS: JGD2011 / UTM Zone 54N (EPSG:6681).

1 m — Box-Counting Log–Log (Fractal Dimension)

**Figure X.** Box-counting log–log plot of the binary channel network at 1 m resolution (optimal workflow: PMF–TIN–WANG–D8). Each blue point corresponds to one box size in the doubling sequence (2–256 pixels; 2–256 m physical size). The slope of the OLS regression line (red) yields a fractal dimension **D = 1.2825** (R² = 0.9893), reflecting strong self-similar channel network structure across scales. Kushiro Wetland, Japan; CRS: EPSG:6681.

5 m — Binary Water / Non-Water Map (Optimal Workflow)

**Figure X.** Binary channel network map extracted from the optimal 5 m resolution workflow (PMF–TIN–WANG–D8), Kushiro Wetland, Japan. Blue pixels represent classified channel (water = 1) and light grey pixels represent non-channel areas (non-water = 0). The coarser 5 m pixel size results in broader channel representations compared to the 1 m map, with reduced fine-scale detail. Axes show UTM coordinates in metres; CRS: JGD2011 / UTM Zone 54N (EPSG:6681).

5 m — Box-Counting Log–Log (Fractal Dimension)

**Figure X.** Box-counting log–log plot of the binary channel network at 5 m resolution (optimal workflow: PMF–TIN–WANG–D8). Each blue point corresponds to one box size in the doubling sequence (2–128 pixels; 10–640 m physical size). The slope of the OLS regression line (red) yields a fractal dimension **D = 1.3933** (R² = 0.9936), reflecting strong self-similar channel network structure across scales. Kushiro Wetland, Japan; CRS: EPSG:6681.

10 m — Binary Water / Non-Water Map (Optimal Workflow)

**Figure X.** Binary channel network map extracted from the optimal 10 m resolution workflow (PMF–TIN–PLAN–Dinf), Kushiro Wetland, Japan. Blue pixels represent classified channel (water = 1) and light grey pixels represent non-channel areas (non-water = 0). At 10 m resolution, narrow tributaries and fine-scale channel features visible at 1 m and 5 m are progressively lost due to spatial generalisation. Axes show UTM coordinates in metres; CRS: JGD2011 / UTM Zone 54N (EPSG:6681).

10 m — Box-Counting Log–Log (Fractal Dimension)

**Figure X.** Box-counting log–log plot of the binary channel network at 10 m resolution (optimal workflow: PMF–TIN–PLAN–Dinf). Each blue point corresponds to one box size in the doubling sequence (2–64 pixels; 20–640 m physical size). The slope of the OLS regression line (red) yields a fractal dimension **D = 1.2609** (R² = 0.9787), reflecting strong self-similar channel network structure across scales. Kushiro Wetland, Japan; CRS: EPSG:6681.

Cross-Resolution Fractal Comparison

Cross-Resolution Summary: Optimal Workflow → Binary Extraction → Fractal Dimension
Resolution	Optimal Workflow	Median IoU	Water Coverage (%)	Fractal D	R²
1m	pmf_tin_area1_1m_20251024_Hororo_area1_fillwl_d8	0.9650	1.28	1.2825	0.9893
5m	pmf_tin_area1_5m_20251024_Hororo_area1_fillwl_d8	0.9344	4.26	1.3933	0.9936
10m	pmf_tin_area1_10m_20251024_Hororo_area1_fillpd_dinf	0.9220	6.87	1.2609	0.9787

**Figure X.** Fractal dimension D estimated via box-counting on the binary channel mask for each optimal workflow per resolution, Kushiro Wetland, Japan. Box sizes are expressed as log(1/box_m) in metres; CRS: EPSG:6681. The 5 m resolution workflow (PMF–TIN–WANG–D8) yields the highest fractal dimension (D = 1.3933), indicating greater self-similar complexity in the extracted channel network compared to 1 m (D = 1.2825; PMF–TIN–WANG–D8) and 10 m (D = 1.2609; PMF–TIN–PLAN–Dinf). The higher D at 5 m likely reflects an optimal balance between channel detail and spatial generalisation, whereas the 1 m and 10 m networks capture either excessive fine-scale noise or insufficient structural detail, respectively.

**Figure X.** Combined box-counting log–log plot comparing fractal dimension D across all three optimal workflows at 1 m (red; PMF–TIN–WANG–D8), 5 m (blue; PMF–TIN–WANG–D8), and 10 m (green; PMF–TIN–PLAN–Dinf) resolutions, Kushiro Wetland, Japan. The x-axis represents log(1/box size) in metres, ensuring box sizes are physically comparable across resolutions, and the y-axis represents log(N occupied boxes). The slope of each OLS regression line directly yields the fractal dimension D: 1 m (D = 1.2825), 5 m (D = 1.3933), and 10 m (D = 1.2609). The steeper slope at 5 m indicates greater self-similar structural complexity in the extracted channel network relative to 1 m and 10 m, where fine-scale noise and spatial generalisation, respectively, reduce the apparent network complexity. CRS: JGD2011 / UTM Zone 54N (EPSG:6681).

Export Final Results


=== All Exports Complete ===

Files written to: ./output/data

  optimal_workflows_per_resolution.csv

  fractal_dimensions_optimal_workflows.csv

  all_pairwise_iou_all_resolutions.csv

Research Progress

What the Box-Counting Method Does

The procedure (as described on the page):

What \(N\) Represent in the Box-Counting Formula

An example for calculation

Nonlinear Shallow Water Equations (SWE)

Explained in the context of your river-analysis

What SWE represent

What Occupancy: \(\phi(\Delta)\)

Part I: Workflow Optimization — Identifying the Optimal Pipeline

1 m Resolution — Optimization Results

Summary Statistics by Component

Component IoU Distribution

Top 10 Workflows

Best Option per Component

Kruskal-Wallis Significance Tests

Correlation Heatmap

5 m Resolution — Optimization Results

Summary Statistics by Component

Component IoU Distribution

Top 10 Workflows

Best Option per Component

Kruskal-Wallis Significance Tests

Correlation Heatmap

10 m Resolution — Optimization Results

Summary Statistics by Component

Component IoU Distribution

Top 10 Workflows

Best Option per Component

Kruskal-Wallis Significance Tests

Correlation Heatmap

Cross-Resolution Optimization Summary

Part II: Multi-Resolution Binary Channel Mask Extraction & Box-Counting Fractal Dimension Estimation

1 m — Binary Water / Non-Water Map (Optimal Workflow)

1 m — Box-Counting Log–Log (Fractal Dimension)

5 m — Binary Water / Non-Water Map (Optimal Workflow)

5 m — Box-Counting Log–Log (Fractal Dimension)

10 m — Binary Water / Non-Water Map (Optimal Workflow)

10 m — Box-Counting Log–Log (Fractal Dimension)

Cross-Resolution Fractal Comparison

Export Final Results

Discussion

Key Findings

Methodological Advantages

Limitations and Considerations

Conclusions

Applications

Future Work

Data Availability

Satellite Data

Code Repository

References