Keep the storm’s timing when you turn rainfall into runoff
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Full chapter index · Independent practice · Hydrology subject guide
Twelve millimetres of rain in an hour can mean gentle rain throughout the hour or a short intense burst. The total depth is the same. A model that depends on rainfall intensity or the timing of excess rainfall can nevertheless produce different runoff estimates.
This guide builds a hyetograph from interval depths, checks depth–intensity conversions and shows how aggregation can change a nonlinear loss calculation. You will also distinguish interpolation from recovering an unmeasured storm pattern. Every storm and catchment number is a fictional teaching example; no drainage design or real flood prediction is supplied.
Continue from The Gold Standard of Hydrology within the Gold Standard of Education and How X Works learning library.
Full chapter index
Define the water question · Chapters 1–2
Build the calculation · Chapters 3–4
Check the result · Chapters 5–6
Change the conditions · Chapters 7–8
Rainfall depth describes water volume per unit horizontal area, usually expressed in millimetres. Intensity is depth per time, such as millimetres per hour. Duration is the time interval over which the depth or intensity applies. The three quantities are connected but are not interchangeable.
For constant mean intensity over an interval, depth = intensity × duration. If a fifteen-minute interval contains 3 mm of rain, its mean intensity is 3/0.25 = 12 mm per hour. The interval depth remains 3 mm. Reporting 12 mm without the per-hour unit would multiply the water amount incorrectly.
A hyetograph describes the time distribution of rainfall, commonly with bars showing mean intensity or incremental depth for successive intervals. The vertical axis must say which quantity is plotted. Identical-looking bars can mean different water volumes when durations differ.
An hour total of 12 mm gives an hour-average intensity of 12 mm per hour. It does not establish that every minute had that intensity. Finer observations are needed to describe shorter bursts. The average is correct for its interval while remaining incomplete for a shorter-time question.
A useful first exercise is to write units beside every column before entering numbers. Time boundaries, duration, interval depth, mean intensity and cumulative depth each have a distinct role. Clear headings prevent a correct calculation from being used as the wrong input to the next model.
The checkpoint is verbal: this bar represents this depth over this duration, corresponding to this mean rate. If the sentence is unstable, repair the unit relationship before introducing infiltration, catchment routing or frequency labels.
Consider four consecutive fifteen-minute intervals with depths 1, 1, 9 and 1 mm. The storm total is 12 mm. The corresponding mean intensities are 4, 4, 36 and 4 mm per hour, because each interval lasts one quarter of an hour.
The cumulative depths at the end of the intervals are 1, 2, 11 and 12 mm. A cumulative record stores the running total; an incremental record stores the additional rain in each interval. Differencing consecutive cumulative values recovers interval depths when the record is valid and consistently defined.
For this example, the third interval contributes 11 − 2 = 9 mm. It would be wrong to enter the cumulative sequence 1, 2, 11 and 12 as four incremental depths: that would create a total of 26 mm instead of 12.
A cumulative rainfall series should generally not decrease within the uninterrupted accumulation period. A negative difference can indicate a reset, a corrected observation, missing-data treatment or a data problem. It should be investigated rather than automatically interpreted as negative rainfall.
Time labels also need a convention. A depth labelled 10:15 could describe the interval ending at 10:15 or an interval starting then, depending on the dataset. Shifting one series by an interval can change its alignment with runoff, even while all totals remain unchanged.
The rain-gauge consistency guide considers a different level of checking. Here the first audit is within one storm record: units, incremental versus cumulative representation, timestamps and the sum of interval depths.
Storm A contains 3 mm in each fifteen-minute interval. Storm B contains 1, 1, 9 and 1 mm. Both total 12 mm over one hour. Their hour-average intensities are identical, but their fifteen-minute intensity patterns differ.
Storm A has a constant mean intensity of 12 mm per hour. Storm B has a peak interval mean of 36 mm per hour, three times as large. This does not establish Storm B’s instantaneous maximum, which could be higher or lower within the measured interval. It establishes the maximum among the supplied interval averages.
If a calculation only asks for total rainfall volume over a fixed area, the two storms provide the same answer under a uniform-depth assumption. If the calculation responds to intensity thresholds or routes excess rainfall through time, the distinction can matter.
For a fixed 2 km² area with uniform 12 mm depth, rainfall volume is 0.012 m × 2,000,000 m² = 24,000 m³. That volume is identical for the two storms. The input time pattern remains different despite the volume agreement.
The important lesson is not that a total is useless. A total answers an amount question. A hyetograph adds the chronology needed for timing-sensitive questions. A competent explanation identifies which information the task requires before choosing a representation.
Connect this with flow-duration curves and hidden sequences. Summarising a series can preserve one useful feature while discarding another. The consequence depends on whether the next calculation needs the feature that was removed.
Use a deliberately simplified loss model with a constant infiltration-capacity rate f = 10 mm per hour. There is no initial abstraction, no stored unused capacity carried between intervals, and no other loss process. In each interval, loss depth is the smaller of rainfall depth and f × duration.
For a fifteen-minute interval, the available loss capacity is 10 × 0.25 = 2.5 mm. Excess depth is max(rainfall depth − 2.5, 0). This toy model makes the intensity-threshold effect visible without claiming to reproduce every infiltration process in a real catchment.
For Storm A’s four 3 mm intervals, each produces 0.5 mm of excess. Total excess is 2 mm. For Storm B, the 1 mm intervals produce no excess, while the 9 mm interval produces 6.5 mm. Total excess is therefore 6.5 mm.
The difference arises because unused capacity during a light-rain interval is not banked for a later burst. Storm B’s brief high intensity exceeds the available capacity substantially. The same total depth therefore yields different excess depth under the stated model.
Do not call this a universal claim that every intense storm with the same total always produces more runoff in every catchment. Other models include changing soil wetness, initial abstraction, saturation, surface storage and other processes. The example demonstrates a specific nonlinearity under explicit assumptions.
The checkpoint is to calculate each interval’s loss and excess separately before summing. If a spreadsheet subtracts a full hour’s capacity from a single hour total, it has changed the temporal representation of the calculation. Whether that change is acceptable depends on the model and the storm pattern.
| Interval | Rain depth (mm) | Intensity (mm/hour) | Loss (mm) | Excess (mm) |
|---|---|---|---|---|
| 1 | 1 | 4 | 1 | 0 |
| 2 | 1 | 4 | 1 | 0 |
| 3 | 9 | 36 | 2.5 | 6.5 |
| 4 | 1 | 4 | 1 | 0 |
| Total | 12 | Not additive | 5.5 | 6.5 |
Aggregate Storm B into one hour. The rainfall depth is 12 mm and the available capacity is 10 mm. Applying the same formula at the coarse interval gives excess = max(12 − 10, 0) = 2 mm.
At fifteen-minute resolution, the excess was 6.5 mm. Both calculations preserve the same rainfall total. They differ because averaging before a nonlinear threshold operation is not generally equivalent to applying the operation to the fine intervals and then summing.
The coarse result implicitly allows the hour’s total capacity to act against the hour’s total rain without resolving when rain was available. It cannot see the three low-intensity intervals where some capacity was unused. The missing chronology changes the result.
The arithmetic can be written as a comparison between max(sum of rainfall minus summed capacity, 0) and the sum of interval-wise maxima. The positive-part operation is nonlinear. One summary number cannot generally preserve all the interval-level threshold behaviour.
For Storm A, the hourly result and the fine result both equal 2 mm. This agreement occurs because all four interval intensities exceed the capacity by the same small amount. Agreement in one pattern does not validate aggregation for every pattern with the same total.
A useful audit changes the storm shape while holding its volume fixed. If a model’s answer changes under the fine representation but not the coarse one, the coarse input may be unable to express the mechanism being examined. This is an information question as well as a numerical one.
Excess rainfall depth is not yet a discharge hydrograph. A transformation model distributes that input through time according to the catchment response and its assumptions. Routing may alter the timing and magnitude of the output peak while conserving or otherwise accounting for volume.
The unit-hydrograph guide shows one simplified transformation using pulses of excess rainfall. Our present example concerns how those input pulses are constructed. Changing their depth or timing changes what is passed to the transformation.
For a uniform 2 km² catchment, excess depths of 2 mm and 6.5 mm correspond to 4,000 m³ and 13,000 m³ before further processes in the chosen model. Those are volumes. A peak discharge in cubic metres per second also requires the output time distribution.
Dividing a total volume by one hour gives a mean rate over that hour, not necessarily the hydrograph peak. Runoff may continue after the rainfall ends, and the output duration may exceed the input storm duration. Labelling that simple mean as a peak would add information the calculation does not contain.
Temporal ordering matters even when a linear transformation preserves total volume. A concentrated excess pulse and several spread-out pulses can produce different peaks and peak times. If losses are nonlinear as in our example, both volume and timing may change.
The checkpoint is to label each stage: rainfall depth, loss depth, excess depth, runoff volume and discharge through time. At every transition, name the model that connects the quantities. This keeps one correct unit conversion from being mistaken for a complete catchment prediction.
Suppose only an hourly total of 12 mm is available, but a model needs fifteen-minute inputs. Dividing equally gives four 3 mm intervals. That is one possible disaggregation pattern; it is not a recovery of the unknown observed storm shape.
The real storm could have followed Storm A, Storm B or many other patterns. All fit the same hourly total. The coarse observation does not identify which pattern occurred. A finer computational time step cannot create observational detail that was never measured.
Interpolation of a cumulative series can preserve the hour total while imposing a distribution between known points. The HEC-HMS Specified Hyetograph documentation explains its use of cumulative representation for interpolation. The distinction to retain is between a processing method and additional measured information.
If a design exercise specifies an adopted temporal pattern, document that pattern and its purpose. If a historical reconstruction uses a finer nearby gauge or radar product, describe the source and its limitations. Neither case should be labelled an exact observed fifteen-minute record merely because the model accepts it.
A sensitivity analysis can compare several plausible shapes with the same total. The resulting range reflects pattern sensitivity under the chosen model. It is not automatically a confidence interval or a probability distribution over storms without an additional justified statistical framework.
The transfer checkpoint is to ask what the data identify. An hourly total identifies an amount over an hour. A disaggregation rule supplies a possible distribution inside it. Keeping those roles separate makes a model easier to audit and its conclusions easier to qualify accurately.
A hyetograph at one gauge describes rainfall at that measurement location. A catchment-average hyetograph requires a spatial representation as well as a temporal one. A single gauge can miss a spatially concentrated storm even when its own clock and depth totals are correct.
The Thiessen rainfall guide develops area weighting across gauges. Apply compatible time intervals before combining depths, and state how missing gauge data or changing weights are handled. Spatial averaging and temporal aggregation solve different problems.
Time zones and timestamp conventions also matter when rainfall is compared with river measurements. A one-hour offset can create an apparent lag or obscure a genuine response. Check whether data use local time or UTC and whether labels mark interval starts, ends or instantaneous observations.
Different interval lengths require duration-weighted treatment. Averaging a fifteen-minute intensity and a forty-five-minute intensity without weights does not give the hour-average intensity unless their durations happen to be equal. The correct hour depth is the sum of each rate multiplied by its own duration.
For example, 24 mm per hour for fifteen minutes contributes 6 mm, while 8 mm per hour for forty-five minutes contributes another 6 mm. The hour total is 12 mm and the hour-average rate is 12 mm per hour. The unweighted mean of the two rates, 16, is incorrect for that duration mix.
The checkpoint is a small data contract: location or spatial support, units, time zone, interval duration, start/end convention and incremental/cumulative type. These details determine whether observations can be combined and whether the resulting model input describes the intended storm.
A rainfall depth associated with a specified annual exceedance probability is also associated with a duration and location or spatial definition. A short-duration depth and a long-duration depth answer different frequency questions. They should not be treated as interchangeable because they share a probability label.
A constructed design hyetograph combines adopted depth and timing information under a specified method. It is not necessarily one historically observed storm, and a rainfall-frequency label does not automatically become the same frequency label for the resulting river peak.
The HEC-HMS frequency-storm documentation describes a procedure for constructing a temporal pattern from duration-related depths. Read the method’s assumptions rather than inferring that every bar is an independent observed frequency event.
The existing flood return-period guide explains annual exceedance probability for a discharge threshold. Rainfall, excess rainfall and discharge are connected through models that can depend on catchment state and storm pattern.
For a fictional comparison, two storms with the same total depth can have different excess volumes under our constant-capacity example. Therefore the total-depth frequency alone cannot determine the output peak frequency without considering those other relationships.
The checkpoint is to read a probability label with its quantity: depth of what duration, over what support, under what reference conditions? That habit prevents an accurate rainfall statistic from supporting an unsupported claim about a drainage structure or a particular flood outcome.
Prove a useful direction check for the toy loss model
For non-negative rainfall increments Pi and capacities ci, summing max(Pi − ci, 0) cannot be smaller than max(sum Pi − sum ci, 0). Each positive-part term is at least Pi − ci, and their sum is also non-negative. Together those two properties establish the inequality.
This explains the direction of the discrepancy in our example. Applying the threshold only after aggregation can understate or match the fine-interval excess under this particular fixed-capacity model. The argument is mathematical and conditional on the defined model; it is not a universal theorem about every hydrological model.
Equality occurs in some patterns, including our uniformly above-capacity Storm A. A mixed sequence with unused capacity in one interval and excess in another can create a gap. Understanding the equality condition is more useful than memorising that one example happened to differ.
Audit the rainfall and excess volumes independently
For Storm B, interval rainfall depths sum to 12 mm. Interval losses in the toy model are 1, 1, 2.5 and 1 mm, totalling 5.5 mm. Excess is 6.5 mm. The balance 5.5 + 6.5 = 12 confirms that no rainfall has been created or lost from the accounting.
Over a uniform 2 km² catchment, these become 11,000 m³ of modelled loss and 13,000 m³ of excess. Their sum is 24,000 m³, matching the independently calculated rainfall volume. The balance does not validate the infiltration model, but it catches unit and bookkeeping mistakes.
The coarse calculation assigns 10 mm of loss and 2 mm of excess. It also balances to 12 mm. Therefore a correct total balance alone cannot prove that timing-sensitive processes were represented adequately. Conservation and process fidelity are separate checks.
Distinguish a missing interval from a dry interval
A zero rainfall increment is an observation or justified estimate of no rain for that interval. A missing increment means the amount is unknown. Replacing every missing value with zero can reduce storm depth and alter the pattern without acknowledging the uncertainty.
An accumulated total after a gap may constrain the combined missing amount while leaving its internal timing unresolved. For example, a cumulative rise of 10 mm across two unobserved intervals establishes their combined depth, but not whether they contained 5 and 5 mm, 0 and 10 mm or another compatible distribution.
If the next model responds nonlinearly to interval intensity, those possibilities can matter even though they preserve the known cumulative rise. State the gap-filling or disaggregation method and inspect sensitivity where appropriate. A complete-looking table is not evidence that every entry was measured.
The transferable audit asks two questions for each value: what is known about its amount, and what is known about its timing? Keeping those uncertainties separate helps readers understand both the strengths and the limits of the rainfall input.
Problem A gives 6 mm over thirty minutes. Its mean intensity is 6/0.5 = 12 mm per hour. If that rate continued for another thirty minutes, the hypothetical hour total would be 12 mm, but the original observation alone does not establish that it did continue.
Problem B gives three twenty-minute depths of 2, 8 and 2 mm. With f = 9 mm per hour in the same simple loss model, each interval has 3 mm of capacity. Excess depths are 0, 5 and 0, totalling 5 mm. Coarse hourly calculation gives max(12 − 9, 0) = 3 mm.
Problem C supplies cumulative depths 0, 2, 7 and 12 mm at equally spaced boundaries. The increments are 2, 5 and 5 mm. Do not sum the cumulative readings as if they were separate rainfall additions.
Now change Problem B to a purely linear rainfall-volume conversion over a fixed uniform area. Aggregating its depths preserves the total volume because addition and multiplication by a fixed area commute. The disagreement appears in the nonlinear loss calculation, not in the total-depth arithmetic.
This contrast is useful because it avoids the false lesson that coarse data are always wrong. Their adequacy depends on the question and model. A daily total can answer a daily amount question accurately while remaining insufficient for a short intense runoff response.
Diagnose errors by their relationship: rate–duration conversion, cumulative differencing, time alignment, nonlinear aggregation or output interpretation. Repair the relevant relationship before repeating a larger numerical problem. The aim is a learner who can choose the appropriate representation when the wording changes.
Does a shorter model time step improve the input data?
It can improve numerical resolution, but it does not recover an unmeasured storm pattern. The resampling rule determines how coarse observations are represented between known points. Numerical detail and observational information are different things.
Is the highest bar the instantaneous peak rainfall intensity?
Usually it is the highest interval-average intensity when that is what the bars represent. Shorter bursts within the interval may be unresolved. Read the axis and time resolution before naming the maximum.
Can equal storm totals produce different runoff?
Yes, under timing-sensitive or nonlinear models, and our explicit loss example shows how. The size and direction of a real difference depend on the catchment, state and model. A fictional threshold example is not a universal numerical rule.
Why interpolate cumulative rainfall?
It provides a way to distribute totals while retaining known cumulative amounts at their boundaries. The specific processing method still imposes assumptions inside the original intervals. Preserving volume does not prove that the reconstructed subinterval timing is observed truth.
What should accompany a hyetograph?
State the location or area, interval duration, units, timestamp convention, incremental/cumulative form, data source and any processing. For an adopted design pattern, state how it was selected. These details make the rainfall input reproducible.
A final check is to ask whether another reader can rebuild the total depth from the plotted bars. If not, the axis, durations or representation may be insufficiently explained. A clear plot is part of the quantitative argument, not simply decoration.
Primary technical references include the US Army Corps of Engineers Specified Hyetograph, Frequency Storm and Unit Hydrograph Basic Concepts. Our constant-capacity example is a deliberately simplified teaching model, with its assumptions stated before use.
Continue through The Gold Standard of Hydrology, area-weighted rainfall and unit hydrographs. Together they connect where rainfall is measured, when it falls and how a stated model transforms it.
The foundation-first approach illustrated by the Clementi mathematics guide suggests beginning with one depth–duration conversion. Then construct incremental and cumulative records, compare two equal-total storms and introduce one nonlinear threshold. Each added difficulty should preserve the meaning of the earlier quantities.
For retrieval, reconstruct the 1, 1, 9, 1 mm storm tomorrow. Explain why its fine-interval excess differs from the hourly calculation. Later, compare a volume conversion with a loss calculation and decide which information aggregation preserves for each.
This is the independent transfer described in the Gold Standard of Education: the learner sees which relationship survives a summary and which is lost. The total tells us how much water arrived; the time pattern tells us when the model had to respond.
Continue the connected explanation
Connect time-resolved rainfall with time-resolved water losses. Read The Gold Standard of Hydrology | Evaporation Depth, Changing Surface Area and Reservoir Water Balance.