Category Archives: Hydrology

Time of concentration: Pilgrim McDermott formula

There are many formulas for the time of concentration.  A previous post discussed the Bransby Williams approach. Here I look at the Pilgrim McDermott formula, which is another method commonly used in Australia and relates time of concentration to catchment area (A):

t_c = 0.76A^{0.38}    (hours)                                                     (equation 1)

where A is measured in km2.

This formula is a component of the Probabilistic Rational Method as discussed in Australian Rainfall and Runoff 1987 (ARR1987) Book IV and is recommended for use in:

  • Eastern New South Wales
  • Victoria (as developed by Adams, 1987)
  • Western Australia – wheatbelt region

McDermott and Pilgrim (1982) needed a formula for the time of concentration to develop their probabilistic rational method approach which was ultimately adopted in ARR1987.  They make the point that, for their statistical method, it is not necessary that the time of concentration closely matches the time for water to traverse a catchment, rather a characteristic time is required for a catchment to determine the duration of the design rainfall.  This characteristic time must be able to be determined directly by designers and lead to consistent values of the runoff coefficient and design flood values.

The basic formula for the probabilistic rational method is:

Q_y = C_y I_{(y,t_c)} A                                                                  (equation 2)

Where:

  • Q_y is the flood of y years average recurrence interval.
  • C_y is the runoff coefficient for a particular average recurrence interval.
  • I is the rainfall intensity which is a function of t_c (time of concentration) and y.
  • A is the catchment area.

For a catchment with a stream gauge, where flood frequency analysis can be undertaken, this will provide the Q_y values on the left hand side of equation 2. We also know the catchment area (A). If t_c can be estimated via a time of concentration formula, then the rainfall intensity can be looked up in an IFD table for the location and the only unknown is C_y.

C_y = \frac{Q_y}{I_{(y,t_c)} A}                                                              (equation 3)

This was the approach used in ARR1987. A large number of gauges were selected and C_y values calculated. Ultimately C_{10} values were mapped in Volume 2 of Australian Rainfall and Runoff.   For floods other than those with a 10 year average recurrence interval, frequency factors were provided to calculate the required runoff coefficient values.  This meant design floods could be estimated for ungauged catchments given information on design rainfall intensity which is available everywhere in Australia.

For this approach to work, some relationship is required between t_c and catchment characteristics i.e. we need a time of concentration formula. McDermott and Pilgrim (1982) began their development of such a formula by testing the Bransby Williams approach because that had been shown to be the best of 8 methods examined by French et al. (1974). McDermott and Pilgrim found that Bransby Williams wasn’t suitable for their purposes because it often resulted in runoff coefficients greater than 1 and they thought the use of such large values would be resisted by practising engineers. Equation 2 doesn’t preclude runoff coefficient values greater than 1 but the intuitive definition of C as being “the proportion of rainfall that runs off” requires it.

An alternative time of concentration formula was developed by considering the ‘minimum time of rise of the flood hydrograph’ which McDermott and Pilgrim collected or collated for 96 catchments. This is the time from when storm rainfall starts until stream discharge begins to increase. McDermott and Pilgrim adopted this as their definition of the time of concentration.

The measured times of concentration were regressed against catchment characteristics that included:

  • Catchment area
  • Main stream length
  • Main stream equal area slope
  • Main stream average slope
  • Catchment shape factor
  • Stream slope non-uniformity index
  • Vegetation cover
  • Median annual rainfall
  • Soil type.

Three formulas provided a similar fit to the data with the simple relationship with catchment area ultimately adopted (equation 1).

One of the important implications of the probabilistic rational method approach is that the time of concentration used for design must be calculated using the same formula that was used in the derivation of the runoff coefficients (equation 3).   So, in Victoria (and Eastern NSW and the Wheatbelt of WA), when using the probabilistic rational method to estimate floods in ungauged catchments, it is important to adopt the Pilgrim McDermott formula for the time of concentration and not use any of the many other approaches.

References

Adams, C. A. (1987) Design flood estimation for ungauged rural catchments in Victoria.  Road Construction Authority, Victoria. (link)

French, R., Pilgrim, D. H. and Laurenson, E. M. (1974) Experimental examination of the rational method for small rural catchments. Civil Engineering Transactions CE16: 95-102.

McDermott, G. E. and Pilgrim, D. H. (1982) Design flood estimation for small catchments in New South Wales.  Department of National Development and Energy.  Australian Water Resources Council Technical Paper No. 73, pp. 233. (link)

Pilgrim, D. H. and McDermott, G. E. (1982) Design floods for small rural catchments in eastern New South Wales. Civil Engineering Transactions.  Institution of Engineers CE24:226-234.

Modelling impervious surfaces in RORB – II

This blog builds on the previous post; looking at the runoff coefficient approach to modelling losses and the implications for representing impervious surfaces in the RORB model.

In addition to the IL/CL model discussed in the previous post, RORB can be run using an initial loss / runoff coefficient model, where the runoff coefficient specifies the proportion of rainfall lost in each time step after the initial loss is satisfied.  This reason these different loss models are of interest is that the new version of Australian Rainfall and Runoff is recommending that the IL/CL model is used in place of the runoff coefficient model (Book 5, Section 3.3.1).  In some areas, modelling approaches will need to change and this will have implications for flood estimates.

The runoff coefficient loss model is selected as shown in Figure 1.

RORB_RoC

Figure 1: A runoff coefficient loss model can be selected in RORB

The user inputs the runoff coefficient, C, for a pervious surface.  For an impervious surface, there is no opportunity to specify the runoff coefficient which is hard-wired in RORB as 0.9.  For mixed sub-areas, the runoff coefficient is scaled, the equations from the RORB manual are:

C_i = F_iC_{imp} +(1-F_i)C_{perv}, \qquad C_{perv} \le C_{imp} \qquad \mathrm{Equation  \;3.5}
C_i = C_{imp}, \qquad C_{perv} > C_{imp}\qquad\qquad \mathrm{Equation \; 3.6}

Where Ci is the runoff coefficient for the ith sub-area.

Example: For a fraction impervious, F_i = 0.6 and C_{perv} = 0.5
C_i = 0.6 \times 0.9 +(1-0.6) \times 0.5 = 0.74

The initial loss is calculated as as a weighted average of the pervious and impervious initial losses as shown in the previous post.  The impervious initial loss is always set to zero in RORB.

Let’s do the calculations for a 100% impervious surface.  RORB will set I\!L = 0 and C = 0.9.  Using the 6 hour, 1% rainfall as before, the rainfall excess hyetograph is shown in Figure 2.

RoC_impervious

Figure 2: Rainfall excess hyetograph for an impervious surface using the runoff coefficient model.  RORB sets IL to zero and the the runoff coefficient to 0.9 so 10% of rain is lost at each time step

Example calculation:

As explained in the previous post, the rainfall between 1.5 hour and 2 hour is 19.4 mm.  With a runoff coefficient of 0.9, the rainfall excess will be: 0.9 x 19.4 = 17.5 mm.

The rainfall excess hydrograph from a 10 km2 impervious sub-area can be calculated from the rainfall excess hyetograph using the method described in the previous post. The peak flow corresponding to the 17.5 mm rainfall peak is 97.2 m3s-1 (see the previous post for sample calculations).

The key point is that we have changed the peak flow from an impervious surface, just by changing the loss model.  With the IL/CL model, both initial and continuing loss for a 100% impervious surface are hard-wired to zero. The peak runoff was 107.8 m3s-1. For the runoff coefficient model, initial loss is hard-wired to zero, but the runoff coefficient is hard-wired to 0.9, i.e. we have some loss from the impervious surface. This changes the hydrograph as shown in Figure 3.

Hydro-Comp

Figure 3: Comparison of rainfall excess hydrographs from a 100% impervious surface; same rainfall, different loss model

The value of the runoff coefficient for an impervious surface is noted in the RORB manual:

The impervious area runoff coefficient Cimp is set by the program to 0.9, reflecting the fact that losses occur even on nominally impervious surfaces in urban areas.

This is reasonable, but inconsistent with the treatment of the continuing loss when the IL/CL loss model is used.  In this case, CL is hard-wired to zero so there are no losses from impervious surfaces; a feature of RORB for modellers to be aware of.

Also note Equation 3.6 above.  This suggests that if the user inputs a runoff coefficient larger than the impervious coefficient (i.e. larger than 0.9) then a value of 0.9 will be used.  This isn’t actually implemented.  If a runoff coefficient of 1 is input, there is a direct conversion of rainfall to runoff i.e. there is no loss.  It is even possible to input runoff coefficients greater than 1.

Equation 3.6 may just be the result of a typo.  Some experimenting suggests the behaviour in the model is represented the combination of equation 3.5, above and the following in place of equation 3.6:

C_i = C_{perv}, \qquad C_{perv} > C_{imp}

That is, the runoff coefficient for an impervious surface is 0.9 unless the runoff coefficient input by the user is larger than 0.9.

Calculations are available via a gist.

Modelling impervious surfaces in RORB

The previous post looked at rainfall excess hydrographs; here I explore how these hydrographs change when modelling impervious surfaces in RORB.  This post focusses on the initial loss/continuing loss modelling approach.

Usually, losses are reduced for impervious compared to pervious surfaces and RORB sets both initial and continuing loss to zero if a surface is 100% impervious.

As an example, consider the 6 hour 1% rainfall for Melbourne, which is 83.4 mm.  If we use the ARR1987 temporal pattern (see the previous post), the hyetograph is as shown in Figure 1.

TP_Melbourne_6h_1pc

Figure 1: The 6 hour 1% rainfall (83.4 mm) multiplied by the ARR1987 temporal pattern.  For an impervious surface, RORB sets both initial and continuing loss to zero

Example calculation:

In the ARR1987 temporal pattern, the time period between 1.5 and 2 hours has 23.3 percent of the rain.  The total rainfall is 83.4 mm so the rain in this period is 83.4 x 23.3% = 19.43 mm which is consistent with Figure 1.

The corresponding rainfall excess hydrograph, for an area of 10 km2, which is 100% impervious, is shown in Figure 2 (Note that Areal Reduction Factors have not been used).

RE_hydro

Figure 2: rainfall excess hydrograph for an area of 10 km2

Example calculation:

The instantaneous flow at a 2 hours will be

\frac{1}{3.6} \times \frac{1}{0.5} \times 19.4 \times 10 = 107.8 \mathrm{m^3 s^{-1}}

To explain factors at the start of the equation, 1/3.6 is for unit conversion, 1/0.5 is because the temporal pattern has a 0.5 hour time step.   The RORB output matches the calculations (Figure 3).

By default, RORB will show the rainfall excess hyetograph above the calculated hydrograph but this is  based on the initial and continuing loss as provided by the user.  In this case, I’ve specified IL = 10 mm and CL = 2 mm/h for the pervious areas.   These losses, and the hyetograph, are misleading where a sub-catchment has some impervious component.  In this case, for a 100% impervious sub-area, both IL and CL are set to zero by the program.  It would be best not to display the misleading hyetograph, which can be turned off as shown in Figure 4.

RORB.png

FInure 3: RORB output

RORB2.png

Figure 4: The hyetograph can be toggled off using the button outlined in pink

If a sub-area is a combination of both impervious and pervious surfaces, this must be specified to RORB as a Fraction Impervious (Fi).  The initial and continuing losses are scaled based on this fraction.

IL_i = (1 - F_i) I\!L_{perv}
CL_i = (1 - F_i) C\!L_{perv}

Where I\!L_{perv} and C\!L_{perv} are the initial and continuing losses for pervious areas as input by the user.

For example, if the pervious value of IL is set to 10 mm and CL to 2 mm/h, then for a sub-area with a Fraction Impervious value of 60%, the initial and continuing losses will be:

I\!L = (1 - 0.6) \times 10 = 4 \; \mathrm{mm}
C\!L = (1 - 0.6) \times 2 = 0.8 \; \mathrm{mm/h}

The continuing loss is 0.8 mm/h which is 0.4 mm per 30 min time step.

Running the model with these parameters results in a rainfall excess hydrograph as shown in Figure 5.  Note that the start of the rise of the hydrograph is delayed because of the initial loss.  The peak is reduced by a small amount (from  108 cumec to 106 cumec because of the continuing loss).

Example calculation, flow peak:

\frac{1}{3.6} \times \frac{1}{0.5} \times (19.4 - 0.4) \times 10 = 105.6 \mathrm{m^3 s^{-1}}

Rain_excess.jpeg

Figure 5: Rainfall excess hydrograph

For a real catchment with a 60% fraction impervious, we would expect some early runoff from the impervious surfaces that would provide flow directly into the urban drainage system.  RORB doesn’t model this process, which may not matter, depending on the application, but as modellers we need to be aware of this limitation.

Calculations are available as a gist.

Rainfall excess hydrograph

Ever wondered what a ‘rainfall excess’ hydrograph is and how they are calculated?  Then read on.

‘Rainfall excess’ is the rainfall left over after the initial and continuing loss are removed.  Rainfall excess hydrographs are used in the runoff-routing program RORB.  The RORB manual (Section 3.3.4) describes them as follows:

In catchment studies, the program calculates hyetographs for all sub-areas.  After deducting losses, it converts the hyetograph ordinates to ‘hydrographs’ of rainfall-excess on the sub-areas, in m3/s, and interprets the average ‘discharge’ during a time increments as an instantaneous discharge at the end of the time increment.

Lets look at an example.  I’m using the methods from the 1987 version of Australian Rainfall and Runoff so I can compare results with calculations in RORB.

1. Choose a design rainfall depth

I’m working on a catchment in Gippsland where the 1% AEP 6 hour rainfall is of interest.  Rainfall IFD data is available from the Bureau of Meteorology via this  link.

For the site of interest, the 1% (100-year), 6 hour rainfall depth is 90.9 mm.

2. Select a temporal pattern

Temporal patterns are available in Australian Rainfall and Runoff Volume 2, Table 3.2.  Gippsland is in zone 1 and ARI is > 30 years so we need the bottom row from the table below.   This shows the percentage of the rainfall depth in each 30 min time period

TemporalPatter_gt30y6h

Applying the temporal pattern to the design rainfall depth results in the following hyetograph.

Hyetograph

Figure 1: Design rainfall hyetograph

3. Remove the losses

Calculate the rainfall excess hyetograph by removing the initial loss and continuing loss.  For this example,

  • IL = 10 mm and
  • CL = 2 mm/h.

Note that the continuing loss is 2 mm/h and the time step of the hyetograph is 0.5 h so 1 mm is lost per time step.

The rainfall excess hyetograph is shown in Figure 2.

Hyeto_ILCL

Figure 2: Rainfall excess hyetograph

4. Convert to a hydrograph

The procedure to convert a rainfall excess hyetograph to a rainfall excess hydrograph is explained in the quote at the start of the blog.  We need to:

  • Multiply the rainfall excess by the catchment area (converts rainfall to a volume)
  • Divide by the time step (to calculate volume per unit time)
  • Ensure flow is allocated to the correct time step – the rainfall during a time step produces the instantaneous flow at the end of the time step
  • Ensure the units are correct – calculated flow is is m3/s, rainfall is in mm and catchment area is in km2.

There is also a discussion of this in ARR2016 Book 5, Chapter 6.4.3.1.

Example calculation: in this case, the sub-catchment area is 78.7 km2.  The rainfall in the 3rd time step,  between 1 hour and 1.5 hour, is 8.9 mm so the flow at the end of this time step will be:

Q = \frac{1}{3600} \times  10^{-3} \times 10^6 \times \frac{8.999}{0.5} \times 78.7 = 393.46 \; \mathrm{m^3s^{-1}}

The rainfall excess hydrograph is shown in Figure 3.

Rainfall_excess_hydro

Figure 3: Rainfall excess hydrograph

4. Comparison with RORB

Figure 4 shows the rainfall excess hydrograph as calculated by RORB.  The answers look close and I’ve confirmed this by looking at the calculated values.

Rainfall_excess_hydro_RORB

Figure 4: Rainfall excess hydrograph as calculated by RORB

Calculations are available as a gist.

Blue Books

The ‘Blue Books‘ are a key resource for Victorian Hydrologists – containing information about stream gauges and flow records.  There are on-line repositories of flow data but the ‘Blue Books’ always seem to have that bit of extra information that is really useful.  They have also been difficult to find.  Now, someone has kindly scanned them.  The four volumes are at:

 

BlueBooks

For Victorian flow data on-line see sites provided by the Victorian Government, or the Bureau of Meteorology:

On the calculation of equal area slope

As noted in the previous post, the equal area slope was adopted for use with the Bransby Williams time of concentration formula in the 1987 version of Australian Rainfall and Runoff to: “…give a better indication of flow response times, especially where there are large variations of slope within a catchment” (ARR 1987 Boo IV, Section 1.3.2(d)).  The equal area slope is also used as part of flood estimation in New Zealand (NZ, 1980; Auckland Regional Council, 1999), in the Papua New Guinea Flood Estimation Manual (SMEC, 1990) – where it is used in the estimation of overland flow times and runoff coefficients for use in the rational method – and is discussed in the Handbook of Hydrology (Pilgrim and Cordery, 1993).

The equal area slope is the slope of a straight line drawn on a profile of a stream such that the line passes through the outlet and has the same area under and above the stream profile.

An alternative to the equal area slope, as used by Bransby Williams, is the average slope.  The differences between the equal area and average slopes are highlighted in the Figure 1 below.

equal-area-slope-ave-slope

Figure 1: Average slope and equal area slope (McDermott and Pilgrim, 1982, page 28)

I haven’t been able to find the history of the equal area slope.  I’m guessing that the average slope was found to be too steep for use in hydrologic calculations.  The equal area slope may have been considered more representative and was easy to calculate in pre-computer times.  The procedure, perhaps to be undertaken by a draftsman, is  specified in NZ (1980) (see Figure 2).

The method involves the calculation of the slope of the hypothetical line AC, which is so positioned that the enclosed areas above and below it, i.e. areas X and Y, are equal. The procedure is to planimeter the total area under the longitudinal profile.  This area Ad, equals the area of the triangle ABC.

equal-area-slope

Figure 2: Definition diagram for the calculation of the equal area slope (NZ, 1980)

A_d = \frac{1}{2}AB \times BC

A_d = \frac{1}{2}L \times h

\therefore h= \frac{2A_d}{L}

(Point C is known at the ‘equal area slope ordinate’).

Hence the equal area slope S_a is given by

S_{ea} = \frac{h}{L} = \frac{2A_d}{L^2}

When the units for the elevation and length in the diagram above are used:

S_{ea} = \frac{2A_d}{1000L^2} \;\; \mathrm{m/m}

Notice that nothing iterative is required, we are just calculating the area under the profile and then working out the triangular area to match.

This simple calculation procedure seems to have been forgotten, as some modern approaches suggest an iterative procedure is necessary, for example, in this excel equal area slope tool.

An R function to calculate the equal area slope is available as a gist.

References

Auckland Regional Council (1999) Guidelines for stormwater runoff modelling in the Auckland Region. (link)

McDermott, G. E. and Pilgrim, D. H. (1982) Design flood estimation for small catchments in New South Wales.  Department of National Development and Energy.  Australian Water Resources Council.  Research Project No. 78/104. Australian Government Publishing Service, Canberra (link)

Pilgrim, D. H. and Cordery, I. (1993) Flood runoff. In: Maidment, D. R. (ed) Handbook of Hydrology.  McGraw Hill.

SMEC (1990) Papua New Guinea Flood Estimation Manual. Department of Environment and Conservation, Bureau of Water Resources. (link)

NZ (1980)A method for estimating design peak discharge: Technical Memorandum No. 61. New Zealand. Ministry of Works and Development. Water and Soil Division. Planning and Technical Services; National Water and Soil Conservation Organisation (N.Z). (1980)  (link to catalog entry) (link to document)

https://nicgreeneng.wordpress.com/2016/07/03/equal-area-slope-tool/

Time of concentration: Bransby Williams formula

The Bransby Williams formula for the time of concentration dates from 1922 when George Bransby Williams published a paper “Flood discharge and dimensions of spillways in India“.  This refers back to an earlier paper by “Mr Chamier” who noted that the:

The maximum flood will generally be produced by the greatest possible rainfall of a duration corresponding to the… time taken for the water to reach the point of discharge from the most distance point of the watershed.

Bransby Williams provides a formula for this “time taken…”, what we would now call the time of concentration.  In the original paper this is provided for a circular catchment as:

a = \sqrt{t_c^5h} \qquad \qquad \mathrm{(1)}

Where:

  • a is the catchment area in square miles
  • t_c is the time of concentration
  • h is the average number of feet fall per 100 ft from the edge of the watershed to the outfall.

For a non-circular catchment, t_c is increased by:

\frac{l}{d} \qquad \qquad \mathrm{(2)}

Where:

  • l is the greatest distance from the edge of the watershed to the outfall
  • d is the diameter of a circular catchment of the same area.

It is possible to combine equations (1) and (2) to derive the the usual Bransby Williams formula as presented in, for example, the Handbook of Hydrology (Equation 9.4.2, page 9.16).

For a circular catchment, area and diameter are related:

a = \frac{1}{4} \pi d^2 \qquad \qquad \mathrm{(3)}

Therefore:

d = \frac{2}{\sqrt{\pi}} a^{0.5} \qquad \qquad \mathrm{(4)}

Now, in equation (1), it is more common to use S, slope (ft/ft, or m/m) rather than h.

h = 100 S\qquad \qquad \mathrm{(5)}

From (1)

a = \sqrt{t_c^5 100 S}\qquad \qquad \mathrm{(6)}

a^2 = 100St_c^5\qquad \qquad \mathrm{(7)}

t_c = 100^{-0.2}S^{-0.2}a^{0.4} \qquad \qquad \mathrm{(8)}

For a non-circular catchment, multiply equation (8) by equation (2) and use equation (4) for d.

t_c = \frac{\sqrt{\pi}}{2 \times 100^{0.2} } l S^{-0.2}a^{-0.5}a^{0.4}\qquad \qquad \mathrm{(9)}

t_c = \frac{\sqrt{\pi}}{2 \times 100^{0.2} } l S^{-0.2}a^{-0.1}\qquad \qquad \mathrm{(10)}

t_c = 0.3528 \; l S^{-0.2}a^{-0.1}  \qquad \mathrm{(hours)} \qquad \qquad \mathrm{(11)}

t_c = 21.169 \; l S^{-0.2}a^{-0.1}  \qquad \mathrm{(minutes)} \qquad \qquad \mathrm{(12)}

In equation (12), area, a, is in square-miles and stream length, l, is in miles.

If area is in square-km, stream length in is km and slope in m/m then the formula is:

t_c = 14.467 \; l S^{-0.2}a^{-0.1}  \qquad \mathrm{(minutes)} \qquad \qquad \mathrm{(13)}

There are Bransby Williams calculators on the Internet e.g. here.

Bransby Williams doesn’t say how this formula was derived and doesn’t present any data.  He does note that: “This formula gives a somewhat more rapid concentration than actually takes place in most instances”.  He also states that his interest is Eastern India where: “a mean annual rainfall of less than 30 in. does not come into consideration”.

Despite the lack of supporting data and narrow geographic focus, the Bransby Williams formula has been widely used.  In the 1977 version of Australian Rainfall and Runoff it was recommended for the whole of Australia “until a better formula is developed” (equation 7.2, page 73).

The 1987 version of ARR (Book IV, Section 1.2) acknowledges that there is no: “universal method…applicable for the whole of Australia” but does describe the Bransby Williams formula as an “arbitrary but reasonable approach” where these is no specific formula available for a particular region. This recommendation was based on the work of French et al. (1974) who tested eight formulae for estimating the time of concentration and showed the Bransby Williams formula was least biased and straightforward to calculate.

The formula provided in ARR 1987 is:

t_c = \frac{58L}{A^0.1 S_e^{0.2}}

Where:

  • S_e is the equal area slope of the main stream projected to the catchment divide (m/km)
  • L is the main stream length measured to the catchment divide (km)
  • A is the area of the catchment (km2)

The equal area slope is: “the slope of a line drawn on a profile of a stream such that the line passes through the outlet and has the same area under it as the stream profile”.  (I’ve added a post on the calculation of the equal area slope).

ARR 1987 acknowledges that Bransby Williams used the average slope rather than the equal area slope but preferred the equal area slope and noted the reasonable fit to data in French et al. (1974), (although French et al. used the average slope in their analysis).   Other work suggested that the Bransby Williams formula wasn’t satisfactory as a measure of the characteristic response time of a catchment because it gave inconsistent values particularly when used with the equal area slope (McDermott and Pilgrim, 1982, p 33; Pilgrim and McDermott, 1982). Despite this, the Bransby Williams formula was recommended, in ARR1987, for the Northern Territory and northern and western semi-arid areas.

As of 2017, it is 30 years since ARR 1987 and close to 100 years since the publication of the Bransby Williams formula, yet the method is still occasionally used.  With the latest revision of the Australian Rainfall and Runoff, a new regional tool (the RFFE) has been developed which should provide better estimates of design floods without the need to calculate the time of concentration.

References

Bransby Williams, G. (1922). Flood discharge and the dimensions of spillways in India.  The Engineer (London) 121: 321-322. (link)

Chamier (1891) Minutes of Proceedings of the Institution of Civil Engineers Vol. cxxxiv., page 313 (as cited by Bransby Williams).

French, R., Pilgrim, D. H. And Laurenson, E. M. (1974) Experimental examination of the rational method for small rural catchments.  Civil Engineering Transactions CE16: 95-102.

McDermott, G. E. and Pilgrim, D. H. (1982) Design flood estimation for small catchments in New South Wales.  Research Project No. 78/104.  Australian Water Resources Council Technical Paper No. 73.  Department of National Development and Energy.  Australian Government Publishing Services (link)

Pilgrim, D. H. and Cordery, I. (1993) Flood runoff. In: Maidment, D. (ed) Handbook of Hydrology. McGraw-Hill.

Pilgrim, D. H. and McDermott, G. E. (1982) Design floods for small rural catchments in eastern New South Wales.  Civil Engineering Transactions. CE24: 226-234

Pattison, A. (1977) Australian Rainfall and Runoff: flood design and analysis.  The Institution of Engineers, Australia.  ISBN 0 85825 077 2