Seasonality is ubiquitous in nature, and it is closely linked to water quality, ecology, hydrological extremes, and water resources management. Hydrological signatures aim at extracting information about certain aspects of hydrological behaviour. Commonly used seasonal hydro-climatological signatures consider climate or streamflow seasonality, but they do not consider how climate seasonality translates into streamflow seasonality. In order to analyse the translation of seasonal climate input (precipitation minus potential evapotranspiration) into seasonal catchment output (streamflow), we represent the two time series by their seasonal (annual) Fourier mode, i.e. by sine waves. A catchment alters the input sine wave by reducing its amplitude and by shifting its phase. We propose to use these quantities, the amplitude ratio and the phase shift, as seasonal hydrological signatures. We present analytical solutions describing the response of linear reservoirs to periodic forcing to interpret the seasonal signatures in terms of configurations of linear reservoirs. Using data from the UK and the US, we show that the seasonal signatures exhibit hydrologically interpretable patterns and that they are a function of both climate and catchment attributes. Wet, rather impermeable catchments hardly attenuate the seasonal climate input. Drier catchments, especially if underlain by a productive aquifer, strongly attenuate the input sine wave leading to phase shifts up to several months. As an example application, we test whether two commonly used hydrological models (Identification of unit Hydrographs and Component flows from Rainfall, Evaporation and Streamflow – IHACRES; modèle du Génie Rural à 4 paramètres Journalier – GR4J) can reproduce the observed ranges of seasonal signatures in the UK. The results show that the seasonal signatures have the potential to be useful for catchment classification, predictions in ungauged catchments, and model building and evaluation. The use of potential evapotranspiration in the input restricts the applicability of the signatures to energy-limited (humid) catchments.

The annual course of the Earth around the Sun leads to seasonal cycles in climate in many places. Seasonal patterns in precipitation, evapotranspiration, and snowfall, as well as the characteristics of the catchment a stream drains, often result in a distinct seasonal streamflow regime

In this work we focus on the average seasonal hydrological response of snow-free catchments. We do not focus, for instance, on the seasonality of events (e.g. storms), noting, however, that the seasonal water balance can have an impact at event scales

Different aspects of hydrological behaviour, such as streamflow seasonality, can be quantified by summarising metrics now mostly called hydrological signatures

There are many hydrological signatures, and we therefore need guidelines for signature selection

There are a multitude of hydrological signatures focusing on seasonality. Climate seasonality is accounted for by (hydro-)climatic signatures such as the (co-)seasonality of precipitation and potential evapotranspiration

In this work, we propose the use of hydrological signatures based on how catchments attenuate the seasonal climate input (forcing). We approximate the input signal to a catchment (the forcing) by precipitation minus potential evapotranspiration and the output signal from a catchment by streamflow. We quantify the seasonal component of both signals by fitting sine waves to them; i.e. we extract their (annual) Fourier modes. As the period is fixed (1 year), the incoming sine wave and the outgoing sine wave differ only in their amplitude, their phase, and their mean. As the mean is rather a measure of the annual water balance, we are primarily interested in amplitude and phase. The differences in amplitude and phase are used as signatures describing the steady-state response of a catchment to periodic forcing.
This idea is similar to the approach of

While there are other methods that quantify input–output relations, we propose the use of the seasonal signatures for several reasons. The seasonal signatures can be related to conceptual linear reservoirs (this will be outlined in Sect.

In the following, we will first define the seasonal signatures, and we will present analytical solutions describing the response of linear reservoirs to periodic forcing (Sect.

To analyse the periodic components (Fourier modes) of time series, we first need to quantify these components. While we could investigate the whole frequency spectrum of our time series and see how this is altered by a catchment

Since we know the period

Once we have extracted the seasonal components from our time series (precipitation minus potential evapotranspiration, streamflow), we can quantify how the outgoing sine wave

The derivations presented here all rely on the assumption of a linear time-invariant system

A linear reservoir is described by

If we approximate the seasonal input to a linear reservoir by a sine wave of period

From Eqs. (

The amplitude ratio

Amplitude ratio against phase shift for a single linear reservoir for varying time constants

Note the similarity of Fig.

Linear systems

Linear reservoirs in series can be conceptualised as follows. Every outflow is the inflow to the next reservoir. Hence, if the

Figure

Amplitude ratio against phase shift for two linear reservoirs in series. Each line corresponds to a fixed time constant for the first reservoir (

Amplitude ratio against phase shift for two linear reservoirs in parallel.

Linear reservoirs in parallel result in a “mixture” of the outflows from each reservoir. The resulting flow is a combination of sine waves of the same period, weighted by the fraction

Figure

As an example, Fig.

We use two conceptual rainfall–runoff models and we test whether the seasonal signatures can be used as a diagnostic tool to assess model performance

The first model is the IHACRES model. It is conceptually relatively similar to the considerations in Sect.

To test which ranges of seasonal signatures the two models can reproduce, we run a Monte Carlo sampling experiment. We sample parameter sets for both models using Latin hypercube sampling, an efficient sampling method

We use catchment data from the UK and the US. The data for the UK are obtained from different sources. Daily streamflow data, catchment characteristics, and catchment boundaries are obtained from the National River Flow Archive

We calculate different hydrological signatures, and we use different catchment attributes, all summarised in Table

Hydrological signatures and catchment attributes used in this study.

Climate input (

First, we extract seasonal components from

To visualise the seasonal signatures, we plot the amplitude ratios and phase shifts in a similar way as in Figs.

Figure

Figure

Amplitude ratio against phase shift for UK catchments. The grey solid line indicates a single linear reservoir, and the grey dashed line indicates the outer envelope for two reservoirs in parallel. Colours indicate

Pearson and Spearman correlation coefficients between seasonal signatures and catchment attributes for UK catchments.

Figure

Amplitude ratio against phase shift for CAMELS catchments. Catchments with a snow fraction (

Pearson and Spearman correlation coefficients between seasonal signatures and catchment attributes for CAMELS catchments.

In a similar fashion as for the observed catchment data, we now investigate the model runs using IHACRES and GR4J. Fig.

Amplitude ratio against phase shift for 40 catchments in the UK using 20 000 parameter sets each for

IHACRES (Fig.

Figure

Distributions of different hydrological signatures resulting from the modelling experiment. Each line stands for one of the 40 catchments, and the colours indicate the corresponding moisture index. The distributions of the modelled signatures are indicated by box-whisker-type plots. The thick line spans from the 25th to the 75th percentile. The thin line spans from the
1st (75th) to the 25th (99th) percentile. The dotted line indicates values below (above) the 1st (99th) percentile. The circles indicate the observed signature values, while filled circles indicate that the observed signature is inside the modelled signature space and vice versa. Panels

Figure

A sine wave is a simple way of describing the seasonality of a signal. The results suggest that for most of the catchments investigated here, this approach is reasonable and efficient. Figure

The UK catchments and most of the US catchments exhibit a relatively strong unimodal (climate) seasonality

The results, in particular Figs.

Figure

The variability among UK catchments that cannot be explained by catchment wetness can mostly be explained by subsurface properties and the associated response time of a catchment. Catchments with high BFIs and thus large baseflow components show lower amplitude ratios and larger phase shifts, that is a more damped and lagged response

Many models frequently used (and some of them developed) in the UK have a parallel flow structure, and catchments are usually conceptualised as having a fast and a slow component. While parametrisations and model structures vary between models, an overall parallel flow structure following a soil moisture module can be found in the probability distributed model

In summary, the first control on the attenuation of the seasonal signal in the UK is the partitioning between fast flow and slow flow. More saturated catchments partition more rainfall into fast flow and hence lead to a higher amplitude ratio and to a smaller phase shift. The second control are catchment subsurface properties, which determine the available storage and how slowly water leaves the system. The slower the catchment responds, the larger the phase shift is, and the lower the amplitude ratio is. The Chalk catchments in the UK might be seen as an extreme case where almost all the water infiltrates, and hence the response time of a single slow reservoir (or perhaps two reservoirs in series) is the main control on the propagation of a periodic signal. On the other end of the spectrum, there are fully saturated, very responsive catchments mostly along the west coast of the UK, which behave almost like a single fast reservoir. Using conceptual reservoirs is only one way to interpret the seasonal signatures. It is useful as many hydrological models are built in that way. There might be, however, other possible ways of interpretation which we do not consider here.

From Fig.

Some of the rather arid catchments in the US plot outside the area that can be modelled by two reservoirs in series or in parallel (Fig.

The ensemble of IHACRES simulations covers the observed range of amplitude ratios and phase shifts, although one catchment sits just at the boundary of the point cloud (Figs.

If one of the parallel reservoirs in IHACRES receives very little water (due to an extremely high or low fraction

If the soil moisture reservoir transmits water relatively quickly without much attenuation, the whole system acts like two reservoirs in parallel.
In summary, IHACRES is very similar to the idealised arrangement we introduced in Sect.

The ensemble of GR4J simulations covers most of the amplitude ratios and phase shifts observed in the UK (Figs.

Figure

This analysis is necessarily incomplete for (at least) two reasons. First, we only looked at 40 catchments in the UK to limit the computational demand. Therefore, the conclusions are not necessarily transferable to catchments outside the UK. More arid catchments (e.g. in the US) might show a different behaviour (e.g. the catchments showing phase shifts larger than 182 d; see Fig.

We have tested seasonal hydrological signatures aimed at representing how climate seasonality is translated into streamflow seasonality, both approximated by sine waves. The damping (the amplitude ratio) and the phase shift of the incoming sine wave have been used to quantify how catchments respond to seasonal forcing. The presented signatures follow the guidelines of

We have found that the propagation of the seasonal input through a catchment depends both on climate and catchment form. Climate aridity and seasonality, and corresponding annual and seasonal catchment wetness, drive the partitioning of the incoming forcing into fast flow and slow flow. Catchment form, such as subsurface properties, influences how strongly the seasonal input gets attenuated. This is particularly visible in the UK, where the hydrogeological classification available (fraction of highly productive aquifer) can explain the very slow response of some catchments. The seemingly more dominant (and less clear) role of climate in the US highlights that scaling from a regional to a continental (or global) scale is not straightforward and requires thoughtful, ideally process-based, approaches. Or in the words of

The model evaluation has shown that the signatures have the potential to be used as a diagnostic tool. GR4J could not reproduce the observed combinations of phase shift and BFI, pointing towards structural deficiencies of the model for certain catchments. As the seasonal signatures are relatable to conceptual model structures (arrangements of reservoirs), we could – given sufficient data – also build models based on inference from observed values of the signatures and not just test existing model structures. This could be done in a stepwise fashion, starting with the seasonal timescale and then adding more complexity if needed

The idea of exploring a model's signature space

A repository with MATLAB code used for the analysis and the resulting data is available from

The supplement related to this article is available online at:

SJG, NJKH, and RAW conceptualised the research project. SJG performed the formal analysis. SJG prepared the paper with contributions from all co-authors.

The authors declare that they have no conflict of interest.

Parts of this work were carried out using the computational facilities of the Advanced Computing Research Centre of the University of Bristol (

This work is funded as part of the Water Informatics Science and Engineering Centre for Doctoral Training (WISE CDT) under a grant from the Engineering and Physical Sciences Research Council (EPSRC; grant no. EP/L016214/1).

This paper was edited by Martijn Westhoff and reviewed by two anonymous referees.