Incorporating uncertainty in hydrological predictions for gauged and ungauged basins in southern Africa

Figures & data

Fig. 1 The model-independent model application and evaluation framework that can be used in southern Africa.

Table 1 List of the parameters of the Pitman model including those of the reservoir water balance model (Hughes et al. 2006)

Parameter	Units	Parameter description
RDF	-	Controls the distribution of total monthly rainfall over four model iterations
AI	Fraction	Impervious fraction of sub-basin
PI1 and PI2	mm	Interception storage for two vegetation types
AFOR	%	% area of sub-basin under vegetation type 2
FF	-	Ratio of potential evaporation rate for Veg2 relative to Veg1
PEVAP	mm	Annual sub-basin evaporation
ZMIN	mm month^-1	Minimum sub-basin absorption rate
ZAVE	mm month^-1	Mean sub-basin absorption rate
ZMAX	mm month^-1	Maximum sub-basin absorption rate
ST	mm	Maximum moisture storage capacity
SL	mm	Minimum moisture storage below which no GW recharge occurs
POW	-	Power of the moisture storage–runoff equation
FT	mm month^-1	Runoff from moisture storage at full capacity (ST)
GPOW	-	Power of the moisture storage–GW recharge equation
GW	mm month^-1	Maximum groundwater recharge at full capacity, ST
R	-	Evaporation–moisture storage relationship parameter
TL	months	Lag of surface and soil moisture runoff
CL	months	Channel routing coefficient
DDENS	-	Drainage density
T	m^2 d^-1	Groundwater transmissivity
S	-	Groundwater storativity
GWSlope	%	Initial ground water gradient
AIRR	km^2	Irrigation area
IWR	Fraction	Irrigation water return flow fraction
EffRf	Fraction	Effective rainfall fraction
NIrrDmd	ML year^-1	Non-irrigation demand from the river
MAXDAM	ML	Small dam storage capacity
DAREA	%	Percentage of sub-basin above dams
A, B	-	Parameters in nonlinear dam area–volume relationship
IrrAreaDmd	km^2	Irrigation area from small dams
CAP	hm^3	Reservoir capacity
DEAD	%	Dead storage
INIT	%	Initial storage
A, B	-	Parameters in nonlinear dam area–volume relationship
RES 1–5	%	Reserve supply levels (percentage of full capacity)
ABS	hm^3	Annual abstraction volume
COMP	hm^3	Annual compensation flow volume

Fig. 2  A flow diagram of the Pitman model, indicating the main components of the model including the parameters (given in brackets).

Table 2 The information typically included in the South Africa AGIS land type database (AGIS 2007)

Property	Information available in database
Area (ha)	Total area covered by each land type
Geology	Rock type and geological formations in each land type
Terrain units	Between one and five units used to describe each land type. These terrain units are representations of hilltops/crests, scarps or upper slopes, middle slopes, foot/lower slopes and the valley bottoms. The percentage of area occupied by each terrain unit within the land type is also given
Slope	Range of the percentage slope for each terrain unit
Shape of slope	Concave, convex and straight slope forms
Soil information	Area (%) covered by each soil series within each terrain; range of soil depth; topsoil and subsoil clay content; soil texture type; type of depth limiting material

Fig. 3 The incorporation of uncertainty in the a priori parameter estimation procedure of the Pitman model. A secondary variable is a basin-scale equivalent of a variable that is measured at point scale.

Table 3 Details of the regional Budyko type relationships based on simulated and naturalized observed data. Equations are of the form ln(Q/P) = a × ln(P/PE) + b

	Regions
	1	2	3	4	5
Simulated data
No. of basins	397	702	317	202	325
Area range (km^2)	59–8647	43–18108	72–10274	72–3913	89–8037
Slope, a	2.527	2.293	2.168	2.126	1.770
Intercept, b	−1.113	−0.687	−0.304	0.194	0.478
R ^2	0.927	0.968	0.984	0.990	0.866
Naturalized observed flow data
No. of basins	40	135	45	23	27
Area range (km^2)	86–1887	81–1668	106–1691	84–873	101–1889
Slope, a	2.322	2.154	2.171	2.406	1.351
Intercept, b	−1.079	−0.741	−0.338	0.475	0.173
R ^2	0.932	0.905	0.890	0.917	0.820

Fig. 4 Hydrological metrics used to constrain model output ensembles. The regional relationships between (a) Q/P and P/PE based on observed flows, and (b) the resultant regions. (c) Slope of FDC relationship with aridity and relief, including the 90% prediction limits. (d) Range of groundwater recharge estimates between the lowest and the middle values (grey) and between the middle and the highest values (black).

Table 4 Percentage frequency of occurrence of possible outcome types A to E in the comparison of the model output ensembles with the constraints

Region	No. of basins	Possible outcome types
		A	B	C	D	E
Description:		Low parameter uncertainty	Excessive uncertainty uncertainty	Parameters giving outputs above limit	Parameters giving outputs below limit	Parameters giving outputs completely outside limits
Runoff ratio constraint:
Region 1	7	62%	25%	13%	0%	0%
Region 2	16	44%	31%	25%	0%	0%
Region 3	8	63%	0%	25%	13%	0%
Region 4	8	63%	38%	0%	0%	0%
Region 5	7	86%	14%	0%	0%	0%
Total	46	59%	24%	15%	2%	0%
FDC slope constraint:
All regions	46	31%	2%	2%	63%	2%
GW Recharge constraint (before constraining the GW parameter):
All regions	46	4%	63%	17%	19%	0%

Fig. 5 Possible outcome types (A to E) from an analysis of uncertainty in the parameter estimation process against the uncertainty related to the constraints (* is the intersection between the parameter priors and the regional constraints).

Fig. 6 Results of the comparison of the uncertainty between the runoff ratio constraint and the parameter estimation for the basins in regions 1–5. The observed data points are plotted for comparison with range of simulations, while the upper and lower 90% bounds are 95th and 5th prediction limits of the regression equations of the constraints.

Fig. 7 Results of the comparison of the uncertainty between the FDC constraint and the parameter estimation for the test basins.

Fig. 8 Results of ensemble analysis based on the groundwater recharge constraint.

Fig. 9 Illustration of the sensitivity analysis of the ensembles using a priori parameter estimates for parameter ZMIN for sub-basin R20C, showing the calculation of the sensitivity class of the parameters based on either traditional objective functions or constraint metrics.

Fig. 10 Results of the sensitivity analysis for some of the selected sub-basins used in the study. Black indicates a critical (highly sensitive) parameter, grey is important (sensitive) and white is negligible (insensitive).

Fig. 11 The variation in the range of: (a) mean monthly runoff and (b) FDC slope, before and after the calibration of the GW parameter. Uncertainty range refers to the difference between the maximum and minimum mean monthly runoff and FDC slope values related to simulation ensembles.

AGIS, 2007. Agricultural Geo-Referenced Information System. http://www.agis.agric.za (http://www.agis.agric.za)

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