Sudan GCN — Scientific Methodology & Equations

Section 01

Processing Pipeline Overview

The application follows the GCN250 framework (Jaafar et al., 2019) enhanced with slope and vegetation corrections. The complete processing chain:

ESA WorldCover
10 m → 250 m

→

USDA Land
Use Classes

→

CN Lookup
Table

→

CN_avg
(ARC II)

→

5 CN
Products

SoilGrids
Clay & Sand %

→

HSG
A / B / C / D

↗

SRTM Slope
30 m

→

Slope
Correction

→

MODIS NDVI
Seasonal

Section 02

Hydrologic Soil Group Classification

Hydrologic Soil Groups (HSG) classify soils by their infiltration capacity. We derive HSG from OpenLandMap SoilGrids clay and sand fractions at the surface (0 cm depth):

HSG Decision Rules Eq. 1

$$\text{HSG} = \begin{cases} \textbf{A} & \text{if } \text{Sand} > 85\% \;\text{ and }\; \text{Clay} < 10\% \\[6pt] \textbf{B} & \text{if } 10\% \leq \text{Clay} < 20\% \;\text{ and }\; \text{Sand} \geq 50\% \\[6pt] \textbf{C} & \text{if } 20\% \leq \text{Clay} \leq 40\% \\[6pt] \textbf{D} & \text{if } \text{Clay} > 40\% \\[6pt] \textbf{C} & \text{otherwise (default)} \end{cases}$$

HSG	Soil Type	Infiltration Rate	Runoff Potential
A	Deep sand, loamy sand, sandy loam	High (> 7.6 mm/hr)	Low
B	Silt loam, loam	Moderate (3.8–7.6 mm/hr)	Moderate
C	Sandy clay loam	Slow (1.3–3.8 mm/hr)	Moderately High
D	Clay, silty clay, clay loam	Very slow (< 1.3 mm/hr)	High

ℹ️

Source: USDA-NRCS National Engineering Handbook, Part 630, Chapter 7 — Hydrologic Soil Groups. Soil texture data from ISRIC OpenLandMap SoilGrids at 250 m resolution.

Section 03

Curve Number Lookup Table

Each pixel receives a CN value based on its land cover class (from ESA WorldCover) crossed with its Hydrologic Soil Group. The lookup table follows USDA NEH-630 Chapter 9:

CN Assignment Eq. 2

$$CN_{\text{raw}}(x) = \text{LUT}\Big[\,\text{LandCover}(x),\;\text{HSG}(x)\,\Big]$$

WorldCover Class	Code	HSG-A	HSG-B	HSG-C	HSG-D
Tree Cover	10	33	58	72	78
Shrubland	20	42	62	75	81
Grassland	30	39	61	74	80
Cropland	40	57	68	75	78
Built-up	50	89	92	94	95
Bare / Sparse	60	77	86	91	94
Water Bodies	80	100	100	100	100
Wetlands	90	100	100	100	100
Mangroves	95	100	100	100	100
Moss / Lichen	100	49	69	79	84

Final Average CN (ARC II) Eq. 3

$$CN_{\text{avg}} = \sum_{g=A}^{D} \Big[\, CN_g \cdot \mathbb{1}\{\text{HSG}=g\} \,\Big], \quad CN_{\text{avg}} \in [0, 100]$$

Where $\mathbb{1}\{\cdot\}$ is the indicator function. Pixels with $CN_{\text{avg}} \leq 0$ default to Group C values, and remaining zeros default to $CN = 80$ (bare ground assumption).

Section 04

CN for Antecedent Runoff Conditions

The SCS method defines three Antecedent Runoff Conditions (ARC) based on 5-day antecedent rainfall and season:

Condition	Description	5-Day Antecedent Rainfall
ARC I (Dry)	Soils are dry, low runoff potential	< 36 mm (growing) / < 13 mm (dormant)
ARC II (Average)	Normal conditions, standard design	36–53 mm (growing) / 13–28 mm (dormant)
ARC III (Wet)	Saturated soils, high runoff potential	> 53 mm (growing) / > 28 mm (dormant)

4.1 — Dry Condition: CN(I)

Hawkins et al. (1985) — Dry CN Eq. 4

$$CN_{\text{dry}} = \frac{4.2 \cdot CN_{\text{avg}}}{10 - 0.058 \cdot CN_{\text{avg}}}$$

4.2 — Wet Condition: CN(III)

Hawkins et al. (1985) — Wet CN Eq. 5

$$CN_{\text{wet}} = \frac{23 \cdot CN_{\text{avg}}}{10 + 0.13 \cdot CN_{\text{avg}}}$$

⚠️

Clamping: All CN values are clamped to the physical range $[0, 100]$. A value of 100 indicates complete runoff (impervious surface); a value near 30 indicates high infiltration capacity.

Section 05

Slope Adjustment — Sharpley-Williams Equation

The standard SCS-CN method was developed for moderate slopes (≤ 5%). Sudan's terrain varies from flat alluvial plains to steep mountain slopes. The Sharpley-Williams (1990) correction adjusts CN based on local slope:

Slope-Adjusted CN Eq. 6

$$CN_{\text{slope}} = \big(CN_{\text{wet}} - CN_{\text{avg}}\big) \cdot \Big[1 - 2 \cdot e^{-13.86 \cdot \tan(\alpha)}\Big] + CN_{\text{avg}}$$

Variable	Description	Units	Source
$CN_{\text{slope}}$	Slope-adjusted Curve Number	—	Output
$CN_{\text{wet}}$	CN for ARC III (wet condition)	—	Eq. 5
$CN_{\text{avg}}$	CN for ARC II (average condition)	—	Eq. 3
$\alpha$	Surface slope angle	degrees	SRTM 30 m DEM
$\tan(\alpha)$	Slope gradient (rise/run)	m/m	Computed

📐

Behavior:
• At $\alpha = 0°$ (flat): $CN_{\text{slope}} \approx CN_{\text{avg}}$ (no correction)
• As $\alpha$ increases: $CN_{\text{slope}} \to CN_{\text{wet}}$ (steeper slopes → more runoff)
• The exponential term $e^{-13.86 \cdot \tan(\alpha)}$ decays rapidly, reaching ~0 at slopes > 15°

Slope Conversion Eq. 6a

$$\tan(\alpha) = \tan\!\left(\text{slope}_{\text{degrees}} \cdot \frac{\pi}{180}\right)$$

Section 06

Seasonal NDVI Adjustment

Vegetation density affects infiltration through canopy interception and root zone storage. We use MODIS MOD13A2 annual mean NDVI to apply a seasonal reduction factor:

Vegetation Factor Eq. 7

$$f_{\text{veg}} = 0.15 \cdot \frac{\text{NDVI} - 0.1}{0.4}, \quad f_{\text{veg}} \in [0,\; 0.15]$$

Seasonal CN Eq. 8

$$CN_{\text{seasonal}} = CN_{\text{avg}} \cdot (1 - f_{\text{veg}})$$

Variable	Description	Range
$\text{NDVI}$	Annual mean Normalized Difference Vegetation Index (2023)	$[-1, 1]$
$f_{\text{veg}}$	Vegetation reduction factor	$[0, 0.15]$
$0.1$	Minimum NDVI threshold (bare soil)	—
$0.5$	Maximum NDVI for full effect ($0.1 + 0.4$)	—
$0.15$	Maximum CN reduction (15%)	—

🌿

Interpretation: Dense vegetation ($\text{NDVI} \geq 0.5$) reduces CN by up to 15%, reflecting increased infiltration from root channels and organic matter. Bare desert ($\text{NDVI} < 0.1$) receives no reduction.

Section 07

SCS-CN Runoff Equation

The core of the SCS Curve Number method. Developed by the USDA Soil Conservation Service (now NRCS), this empirical equation estimates direct surface runoff from a rainfall event:

7.1 — Potential Maximum Retention

Maximum Soil Retention Eq. 9

$$S = \frac{25{,}400}{CN} - 254 \quad \text{(mm)}$$

$S$ represents the maximum depth of water the soil can retain after runoff begins. Units are millimeters. The constants 25,400 and 254 come from the original inch-based formula ($S = \frac{1000}{CN} - 10$ inches) converted to metric.

7.2 — Initial Abstraction

Initial Abstraction Eq. 10

$$I_a = \lambda \cdot S = 0.2 \cdot S \quad \text{(mm)}$$

$I_a$ accounts for water intercepted by vegetation, stored in surface depressions, and infiltrated before runoff begins. The standard $\lambda = 0.2$ is used (some recent studies suggest $\lambda = 0.05$).

7.3 — Direct Runoff Depth

SCS-CN Runoff Equation Eq. 11

$$Q = \begin{cases} \displaystyle\frac{(P - I_a)^2}{(P - I_a) + S} & \text{if } P > I_a \\[12pt] 0 & \text{if } P \leq I_a \end{cases}$$

Variable	Description	Units	Typical Range
$Q$	Direct surface runoff depth	mm	$0$ to $P$
$P$	Total rainfall depth for the event	mm	$0$ to $200+$
$S$	Potential maximum retention	mm	$0$ to $\infty$
$I_a$	Initial abstraction	mm	$0$ to $0.2S$
$CN$	Curve Number (any of 5 products)	—	$30$ to $100$
$\lambda$	Initial abstraction ratio	—	$0.2$ (standard)

7.4 — Runoff Coefficient

Runoff Ratio Eq. 12

$$C_r = \frac{Q}{P} \times 100\%$$

The runoff ratio indicates the fraction of rainfall that becomes surface runoff. Higher values mean less infiltration and higher flood risk.

7.5 — Behavior by CN Value

CN	$S$ (mm)	$I_a$ (mm)	$Q$ at $P=50$ mm	Runoff Ratio	Interpretation
40	381	76.2	0.0 mm	0%	High infiltration, no runoff at P=50
60	169	33.9	1.3 mm	2.6%	Low runoff
75	84.7	16.9	8.1 mm	16.2%	Moderate runoff
85	44.8	9.0	18.9 mm	37.8%	Significant runoff
95	13.4	2.7	35.3 mm	70.6%	Very high runoff (urban)

Section 08

NASA GPM Rainfall Processing

The app integrates NASA GPM IMERG V07 (Global Precipitation Measurement) for real satellite rainfall data:

Daily Rainfall Accumulation Eq. 13

$$P_{\text{daily}}(x) = \overline{R_{\text{half-hourly}}(x)} \times 24 \quad \text{(mm/day)}$$

Variable	Description	Units
$R_{\text{half-hourly}}$	GPM IMERG half-hourly precipitation rate	mm/hr
$\overline{R}$	Mean half-hourly rate over 24 hours (48 timesteps)	mm/hr
$P_{\text{daily}}$	Total daily rainfall accumulation	mm/day

🛰

Dataset: NASA GPM IMERG V07 Final Run, available from 2000–present at 0.1° (~11 km) resolution. Available in GEE as NASA/GPM_L3/IMERG_V07. The daily rainfall is then fed into Eq. 11 to compute spatially distributed runoff.

GPM-Based Runoff Eq. 14

$$Q_{\text{GPM}}(x) = \text{SCS-CN}\Big(\,P_{\text{daily}}(x),\; CN_{\text{avg}}(x)\,\Big)$$

Section 09

Flood Risk Composite Index

The Flood Risk Index combines runoff potential, terrain susceptibility, and population exposure into a normalized composite score:

Flood Risk Index Eq. 15

$$\text{FRI} = w_1 \cdot \hat{CN} + w_2 \cdot \hat{S}_{\text{inv}} + w_3 \cdot \hat{P}_{\text{pop}}$$

Component Weights Eq. 15a

$$w_1 = 0.4, \quad w_2 = 0.3, \quad w_3 = 0.3, \quad \sum w_i = 1.0$$

9.1 — Normalized Components

CN Component (Runoff Potential) Eq. 16

$$\hat{CN} = \frac{CN_{\text{avg}} - 30}{70}, \quad \hat{CN} \in [0, 1]$$

Slope Component (Terrain Susceptibility) Eq. 17

$$\hat{S}_{\text{inv}} = 1 - \frac{\min(\text{slope}, 30°)}{30°}, \quad \hat{S}_{\text{inv}} \in [0, 1]$$

Population Component (Exposure) Eq. 18

$$\hat{P}_{\text{pop}} = \frac{\log_{10}(\max(\text{Pop}, 0) + 1)}{4}, \quad \hat{P}_{\text{pop}} \in [0, 1]$$

Component	Weight	Logic	Data Source
$\hat{CN}$	40%	Higher CN → more runoff → higher risk	SCS-CN (this app)
$\hat{S}_{\text{inv}}$	30%	Flatter terrain → water accumulates → higher risk	USGS SRTM 30 m
$\hat{P}_{\text{pop}}$	30%	More people → more exposure → higher risk	WorldPop 2020

⚠️

Interpretation: FRI = 0.0 (low risk, infiltrating terrain, unpopulated). FRI = 1.0 (extreme risk, impervious surface, flat, densely populated). The inverse slope is used because flat areas accumulate water and are more flood-prone.

Section 10

Statistical Analysis & Validation

10.1 — AOI Statistics

Spatial Statistics Eq. 19

$$\bar{CN}_{\text{AOI}} = \frac{1}{N}\sum_{i=1}^{N} CN(x_i)$$ $$\sigma_{CN} = \sqrt{\frac{1}{N-1}\sum_{i=1}^{N}\big(CN(x_i) - \bar{CN}\big)^2}$$

Where $N$ is the number of pixels within the Area of Interest, computed at 5 km resolution with bestEffort: true to avoid timeouts for large areas.

10.2 — Point Query Data Stack

Clicking on the map samples 13 bands at a single point at 250 m resolution:

#	Band	Description	Units
1	`CN_average`	CN for ARC II	—
2	`CN_dry`	CN for ARC I	—
3	`CN_wet`	CN for ARC III	—
4	`CN_slope_adjusted`	Sharpley-Williams corrected	—
5	`CN_seasonal`	NDVI-adjusted	—
6	`HSG`	Hydrologic Soil Group (1–4)	A/B/C/D
7	`slope`	Terrain slope	degrees
8	`elev`	Elevation	meters
9	`NDVI`	Vegetation index	—
10	`flood_risk`	Composite FRI	0–1
11	`clay`	Clay fraction	%
12	`sand`	Sand fraction	%
13	`LC`	Land cover class	WorldCover code

10.3 — Reference

📚

Primary Reference:
Jaafar, H.H., Ahmad, F.A. & El Beyrouthy, N. (2019). GCN250, new global gridded curve numbers for hydrologic modeling and design. Scientific Data, 6, 145.
DOI: 10.1038/s41597-019-0155-x

SCS-CN Method:
USDA-NRCS (2004). National Engineering Handbook, Part 630 — Hydrology. Chapter 9: Hydrologic Soil-Cover Complexes. Chapter 10: Estimation of Direct Runoff from Storm Rainfall.

Slope Adjustment:
Sharpley, A.N. & Williams, J.R. (1990). EPIC—Erosion/Productivity Impact Calculator. USDA Technical Bulletin No. 1768.

Scientific Methodology & Equations

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