HYDROGRAPH SEPARATION BASED ON INFORMATION FROM

RAINFALL AND RUNOFF RECORDS

OUTLINE

Hydrograph Separation

Vertical

Separation

Horizontal

Separation

The portion of flow that comes from groundwater storage or other

delayed source. It is relatively stable. It includes contributions from

groundwater and return flow.

Contributed from direct precipitation, overland flow (infiltration

excess and saturation excess), interflow (shallow subsurface flow),

and groundwater flow.

Terminology

Method Examples Advantages Disadvantages

Tracer-

based

Method

In situ experiments Most realistic 1) Laborious, expensive

2) Restricted to small number

of events and basins

Graphical

Method

Straight Line Method

Fixed-base Method

Variable Slope Method

1) Intuitive

2) Easy-to-program

1) Empirically based

2) Not applicable for long-

term records

Filtering

Method

UKIH, RUKIH, AdUKIH,

FUKIH, SARR

Applicable for long-

term records

Arbitrary parameter

selection

Recursive

Digital

Filter

(RDF)

Nathan-McMahon Filter

Chapman Filter

Chapman-Maxwell Filter

1) Applicable for

long-term records

2) Physically-based

to some extent

1) Needs in situ

measured BFImax

2) Results vary

significantly with BFImax

Analytical

Solution of

Recession

Constant-k Method

Sellinger Method

Unit Hydrograph Method

Physically-based

to a larger extent

Applicability for

long-term records

Summary of Baseflow Separation Methods

Method Advantages Disadvantages

Manual

Selection

Intuitive 1) Laborious

2) Highly arbitrary

3) Restricted to mass application

Merz et al.

(2006)

1) Objective

2) Has physical basis since it

involves

a lumped hydrologic model

1) Needs endeavor to calibrate a

hydrologic model beforehand

2) High data requirements

SARR 1) Objective

2) Low data requirements

3) Reliable

1) Designate to coarse

(daily) resolution data

2) Lack a physical basis

Khannal

(2004)

Has physical basis since it

applied the unit hydrograph

method in separation

1) Disregard Multi-peak event

2) Quasi-automatic and subjective

because it put on visual inspections

for further refining events

Summary of Event Identification Techniques

Study Area

Gauge

Codes

Area

(km2)

Elevation (m) Flow Length (km) Slope (°)

Mean Range Mean Max Mean Max

T1 431.5 148 88-225 29.4 55 2.7 21.3

T2 1125.4 128 56-225 51.0 99 3.2 28.3

T3 2031.1 106 27-225 99.6 180 2.9 28.3

T4 2406.4 99 17-225 99.8 193 2.7 28.3

S 428.9 98 41-163 46.4 86 2.8 25.5

F1 461.1 83 38-143 34.6 65 2.8 20.8

F2 1373.9 81 25-158 66.1 120 2.8 21.3

TSF 5847.8 76 3-225 125.0 258 2.2 28.3

Study area is the Tar River basin in North Carolina, USA. The

study area was divided into eight nested sub-basins namely T1, T2,

T3, T4, S, F1, F2 and TSF.

Data

Type Dimension Coverage Resolution

Rainfal

l

SpaceEntire Tar River

Basin4km×4km

Time 2002~2009 Hourly

DEM SpaceEntire Tar River

Basin29m×29m

Runoff Time 2002~2009 Hourly• Rainfall: aggregate into the areal weighted rainfall time series.

• DEM: used for basin delineation

Gauge

Codes

Mean Annual

Precipitation

(mm/year)

Mean Annual

Runoff

(mm/year)

Annual

Runoff

Ratio

Mean Maximum

Annual Flow

(m3/s)

T1 1000 302 0.30 162.7

T2 1015 298 0.29 166.6

T3 1027 300 0.29 174.1

T4 1038 311 0.30 206.2

S 1059 326 0.31 55.5

F1 1080 280 0.26 56.2

F2 1070 279 0.26 104.1

TSF 1074 310 0.29 347.0

comparatively dry (less than 1000mm of yearly cumulative precipitation);

• 2003 is the wettest (yearly cumulative runoff over 500mm);

• 2007 and 2008 have relatively low yearly cumulative runoff (less than 200mm)

• 2003 is the

wettest

(1200mm of

yearly

cumulative

precipitation);

• 2005 and

2007 are

Methodology

• Baseflow separation

Revised Constant k (RCK) Method

o Envelope Baseflow Rate, ben

Recursive Digital Filter (RDF)

• Event identification

Rainfall Events

Flow Events

Events Combination

Revised Constant k (RCK) Method

• We assumed steamflow is consisted by the baseflow and direct flow.

• We assumed the linear reservoir model:

• k can be expressed as:

𝑄 = 𝑄0𝑒−𝑘𝑡

Q: discharge at time t (mm/h)

Q0: discharge at the beginning of recession (mm/h)

k: recession coefficient (h-1)

𝑘 = −𝑑

𝑑𝑡ln𝑄 (2)

(1)

• Change rate of k, k* (h-2), is analytically defined as:

• Time steps with absolute |k*| smaller than a given constant were defined as

recession points (RePs).

𝑘𝑡∗ =

𝑘𝑡 − 𝑘𝑡−1∆𝑡

(3)

Steps•Determine k by linear regression over a time window, LRW=19h, compute k*;

•Select and record the time steps with positive k, |k*| bounded by a null-change

ratio, RNC=5×10-5 h-2, and r2 smaller than a minimum acceptable r2 (r2MA).

•Find the rising points (RiPs) satisfying the following condition:

•Exclude the RePs and RiPs with flow rate greater than an envelope baseflow rate

(ben). The remaining RePs and RiPs are turning points (TPs).

•Connect the TPs with straight lines to form the baseflow hydrograph, bRCK.

•Apply the following restriction to the results:

𝑟2 > 𝑟𝑀𝐴

2

𝑄𝑡−1 ≥ 𝑄𝑡 < 𝑄𝑡+1(4)

𝑄𝑏 = 𝑄𝑏 𝑄𝑏 < 𝑄𝑄 𝑄𝑏 ≥ 𝑄

(5)

Envelope Baseflow Hydrograph• A null-change ratio is determined for every potential TPs from

recession limbs; it leads to an issue, this ratio is not able to satisfy

every single recession limb since the heterogeneity of temporal flow

properties. The envelope baseflow time series are constructed to limit

the occurrences of turning points at the high flow rate locations.

Steps• Divide the flow hydrograph into numbers of segments according to a

searching period, LSP:

𝐿𝑆𝑃 = 19.848𝐴0.2 (6)

• Find the time steps with maximum flow rate of every searching periods;

these time steps are represented as (tM,i , QM,i).

• Select the time steps which the maximum flow rate are the largest

compared to their two outliers:

• For the selected time steps (tM,j , QM,j), fill the time period from tb,j to te,j

with QM,j and formed the envelope flow time series Qen:

𝑄𝑀,𝑖−1 ≤ 𝑄𝑀,𝑖 ≥ 𝑄𝑀,𝑖+1

𝑡𝑏,𝑗 =𝑡𝑀,𝑗−1 + 𝑡𝑀,𝑗

2𝑡𝑒,𝑗 =

𝑡𝑀,𝑗 + 𝑡𝑀,𝑗

2

(7)

(8) (9)

• Times Qen with BFI for every time steps to form the ben;

• Constrain ben with an upper and a lower limit of baseflow rate, bu and bl;

• bu and bl are determined according to the mean flow rate. Construct a CDF for

Q and find the corresponding quantile at mean flow rate;

• Find the upper and lower envelope baseflow quantiles by plus and subtract

the mean flow quantile by 8% and 2%;

• Determine the corresponding flow rate for the two quantiles on the CDF

𝑏𝑒𝑛 =

𝑏𝑢 𝑏𝑒𝑛 ≥ 𝑏𝑢𝐵𝐹𝐼 ∙ 𝑄𝑒𝑛 𝑏𝑙 < 𝑏𝑒𝑛 < 𝑏𝑢𝑏𝑙 𝑏𝑒𝑛 ≤ 𝑏𝑙

(10)

Gauge

Codes

Upper Envelope Lower Envelope Mean Flow

Quantile

(%)

Flow Rate

(×10-2 mm/h)

Quantile

(%)

Flow Rate

(×10-2 mm/h)

Quantile

(%)

Flow Rate

(×10-2 mm/h)

T1 87.0 5.01 77.5 3.14 79.5 3.45

T2 84.0 4.52 74.5 3.11 76.5 3.40

T3 80.7 4.66 71.2 3.19 73.1 3.42

T4 78.9 4.66 69.4 3.29 71.4 3.55

S 79.9 4.75 70.4 3.54 72.4 3.72

F1 80.4 4.02 70.9 3.03 72.9 3.20

F2 79.4 4.01 69.9 3.02 71.9 3.18

TSF 76.2 4.67 66.7 3.31 68.7 3.54

Recursive Digital Filter (RDF)

• RDF has the following form:

• BFI is the baseflow index defined as the ratio of baseflow volume in a

given time interval (T stands for the interval of the entire flow record):

bRDF: baseflow time series under RDF

(11)𝑏𝑅𝐷𝐹,𝑡 =𝑎 1 − 𝐵𝐹𝐼

1 − 𝑎𝐵𝐹𝐼𝑏𝑅𝐷𝐹,𝑡−1 +

1 − 𝑎 𝐵𝐹𝐼

1 − 𝑎𝐵𝐹𝐼𝑄𝑡

𝐵𝐹𝐼 = 𝑇 𝑄𝑏𝑑𝑡

𝑇 𝑄𝑑𝑡(12)

• a is relative to the recession coefficient in the following way:

𝑎 = 𝑒−𝐾 (13) 𝐾 = −1

𝑄

𝑑𝑄

𝑑𝑡(14)

Steps

•Plot –dQ/dt vs Q for –dQ/dt>0 and r2≥r2MA.

•Fit a straight line for –dQ/dt vs Q with zero-intersection, the slope is K;

•Calculate a for every basin;

•Initialize BFI by UKIH;

•Filter the bRCK time series forward once by RDF and obtain bFRCK;

•Implement again Eq.(5) to control bFRCK;

•Calculate BFI based on bFRCK;

•Rerun procedure 6), 7), 8), 13), 14), and 15) with the new BFI 6 times.

𝑏𝐹𝑅𝐶𝐾,𝑡 =𝑎 1 − 𝐵𝐹𝐼

1 − 𝑎𝐵𝐹𝐼𝑏𝑅𝐶𝐾,𝑡−1 +

1 − 𝑎 𝐵𝐹𝐼

1 − 𝑎𝐵𝐹𝐼𝑄𝑡

The differences are decreasing exponentially

with more and more iteration times.

Gauge

Codesr2

MA

K

(×10-2 h-1)a BFII

LSP

(h)

T1 0.9 5.99 0.942 0.28 67

T2 0.9 2.94 0.971 0.38 81

T3 0.9 1.74 0.983 0.49 91

T4 0.85 1.46 0.986 0.48 94

S 0.88 1.77 0.982 0.50 67

F1 0.9 2.44 0.976 0.44 68

F2 0.9 1.53 0.985 0.49 84

TSF 0.88 0.75 0.992 0.60 112

Flow Events• A flow event has three characteristic points, a beginning, a peak and

an end. If these points are known, an event is established. The event

beginning/end could be found within the RiPs/RePs.

Steps

•Replace the consecutive time steps with equivalent flow rate by one time step;

form the new hydrograph (Ts,x , Qs,x);

•Form the maxima flow hydrograph, (Tm,y , Qm,y), by finding the maxima time

steps of (Ts,x , Qs,x) as:

𝑄𝑠,𝑥−1 ≤ 𝑄𝑠,𝑥 ≥ 𝑄𝑠,𝑥+1 (15)

•Exclude the time step from (Tm,y , Qm,y) with low Qm,y:

•Assign the same integers to (Tm,y , Qm,y) if:

•Select the maximum flow rate from each class assigned by the same number. These

points are regarded as the peak flow points (PFPs).

•Find the closest RiP/ReP for each PFP as the event beginning/end. For PFPs share

the same RiPs or RePs or both, these PFPs are grouped together to form a multi-peak

event (MPE). PFP with unique RiP and ReP is termed as single peak event (SPE).

𝑄𝑚,𝑦 ∙ 𝐵𝐹𝐼 ≤ max1

𝑇 𝑇

𝑄𝑑𝑡 , 𝑏𝐹𝑅𝐶𝐾,𝑦

𝑇𝑚,𝑦 − 𝑇𝑚,𝑦−1 ≤ 𝐿𝑆𝑃 (17)

(16)

Rainfall Events

• Periods with above-threshold basin-average rainfall intermitted by

limited below-threshold rainfall periods.

Steps

• Assign same integers for every consecutive basin areal rainfall time steps with

rain rate greater than r0=0 and intermittent with number of time steps with rain

rate smaller than r0 less than LMG=1;

• Record the beginnings and ends of every rainfall events; calculate the event

centroids.

Gauge

Codes

Num. of

Events

Duration

(Dr, h)

Volume

(Vr, mm)

Intensity

(mm/h)

Max rain

rate (mm/h)

T1 2033 4.1 3.9 0.5 1.5

T2 2126 4.6 3.8 0.4 1.3

T3 2289 5.4 3.6 0.3 1.0

T4 2295 5.3 3.6 0.3 1.1

S 1986 4.4 4.3 0.5 1.6

F1 1899 4.5 4.5 0.6 1.8

F2 2060 5.0 4.2 0.4 1.4

TSF 2361 6.0 3.6 0.3 1.0

Events Combination• Assumption 1: if the difference between the beginning of a rainfall

event to its subsequent flow event is within a threshold, the rainfall

event is contributing to that flow event. This rainfall event is called

Group A Event (GAE).

• Assumption 2: rainfall occur before and very close to the flow event

centroid are counted as contributing rainfall events to that flow event.

This type of rainfall events are termed as Group B event (GBE).

• Assumption 3: rainfall events after the flow event centroid but

distance from the flow event ending are contributed to the flow event.

This type of rainfall event is named as Group C event (GCE).

• Find the rainfall events occur before the beginning of each flow event within a

searching radius, RS in h, as:

Steps

0 ≤ 𝑡𝑏,𝑛 − 𝜏𝑏,𝑛 ≤ 𝑅𝑆,𝑛

tb/te and τb/τe are the beginning/end hour of flow event and rainfall event;

n is the index of flow event;

N is the total number of events for the basin

𝑅𝑆,𝑛 = min 𝑡𝑏,𝑛 − 𝑡𝑒,𝑛−1 ,1

𝑁

𝑛=1

𝑁

𝑡𝑙𝑎𝑔,𝑛 , 𝐿𝑆𝑃

(18)

(19)

• Find the rainfall events with centroids in between the centroid of the closest

GAE and the centroid of the flow event. The concept of centroid is given as:

• Merge the GAE and GBE as a whole rainfall event and calculate the time lag, tl,

between this combined rainfall event to its matching flow event;

Tf stands for the period of flow event.

𝑡𝑐 = 𝑇𝑓

𝑡 ∙ 𝑄𝑑𝑡

𝑇𝑓𝑄 𝑑𝑡

(20)

• Find the rainfall events after the flow event centroid which satisfies:

• Merge each corresponding GAE, GBE and GCE as a holistic event;

• Calculate the event time lag (tlag, difference between rainfall and flow event

centroid) and runoff coefficient (RC):

𝜏𝑐 > 𝑡𝑐𝑡𝑒 − 𝜏𝑐 > 𝑡𝑙

𝑅𝐶𝑛 = 𝑇𝑓,𝑛

𝑄𝑡 − 𝑏𝐹𝑅𝐶𝐾,𝑡 𝑑𝑡

𝑇𝑟,𝑛𝑝0,𝑡 𝑑𝑡

(22)

(21)

• Exclude events with negative tlag or RC greater than 1;

• Average tlag with the rest events;

• Rerun from steps 9) to 15) until the mean of tlag remain the same;

• Events with fairly low flow volume and high initial and final flow rate are

excluded:

1

𝐷𝑓,𝑛 𝑇𝑓,𝑛

𝑄𝑑𝑡 <1

𝐷 𝑇

𝑄𝑑𝑡 max𝑄 𝑡𝑏

𝑄 𝑡𝑝,

𝑄 𝑡𝑒

𝑄 𝑡𝑝> 𝐵𝐹𝐼𝑒

D and Df stand for the duration of the T and Tf periods;

tp is the peak flow hour.

(23) (24)

Results• By using the filter, some

rapid droppings and risings

from RCK are eliminated

• Beginnings and ends of every event are associated with high

BFIs.

• The similarity between FRCK hydrograph to RCK and RDF

hydrographs are quantified by the mean of mean relative error

(MRE) and correlation coefficient (CC) on a long term and an event

basis:Gauge

Codes

N. of Events RCK v.s. FRCK RDF v.s. FRCK

SPEMP

EMRE CC MRE CC

T1 76 13 -5.6% 0.67 59% 0.48

T2 66 13 -2.7% 0.84 46% 0.56

T3 62 9 -1.3% 0.93 31% 0.62

T4 57 11 -0.8% 0.88 22% 0.63

S 55 15 -1.9% 0.94 46% 0.43

F1 69 12 -3.2% 0.81 60% 0.49

F2 59 14 -1.8% 0.90 52% 0.53

TSF 44 8 -0.4% 0.99 18% 0.60

Gauge

Codes

Annual Basis Monthly Basis

RCK vs.

FRCK

RDF vs.

FRCK

RCK vs.

FRCK

RDF vs.

FRCK

MRE CC MRE CC MRE CC MRE CC

T1 0.23% 0.967 -2.9% 0.626 2.08% 0.945 -21.1% 0.779

T2 0.07% 0.992 -3.0% 0.669 0.69% 0.980 -14.0% 0.754

T3 0.09% 0.996 -3.1% 0.650 0.36% 0.987 -10.4% 0.763

T4 0.13% 0.998 -3.4% 0.716 0.41% 0.992 -12.3% 0.701

S 0.12% 0.994 -3.5% 0.646 0.08% 0.962 -4.3% 0.736

F1 0.17% 0.993 -4.3% 0.667 0.01% 0.955 -5.7% 0.746

F2 0.09% 0.997 -3.7% 0.660 0.23% 0.979 -8.7% 0.723

TSF 0.08% 0.999 -4.3% 0.742 0.22% 0.992 -12.0% 0.723

• Both tables indicate few discrepancies between the RCK hydrographs to the

FRCK ones;

• Long-term-based MREs display smaller magnitude compared to the event MREs

• Long-term-based CCs are higher than event-based CCs;

• The mean of long term MREs between RCK/RDF and FRCK are always

positive/negative while the event MREs between them are negative/positive;

• The event-based MREs and CCs show scale dependencies for every drainage

systems.

Long-term Baseflow Properties

• Positive scale dependencies for every drainage systems;

• Different correlations between BFI and area for every drainage systems.

• Negative correlation between BFI and RC on both yearly and monthly basses;

Properties of Events (Timing)

• Values of the MPEs timing properties are in general larger;

• The degree of variability of the SPE is smaller;

• Positive scale dependency;

Properties of Events (Water Budget)

• Vr, Vb from MPEs are in general larger than those from SPEs;

• Higher degree of variability for the MPEs’ Vr, Vb and RC;

• Positive scale dependency for Vr, Vb and BFI in terms of values and for Vr and

Vb by means of variability;

• Negative scale dependency for RC.

Questions?