First, we apply the continuous wavelet transform (WT) to the observed time series. The main steps and equations for the WT are provided here, though the reader is referred to Torrence and Compo (1998) and Liu et al. (2011) for more details.

Before applying the WT, a mother wavelet needs to be selected. In Torrence and Compo (1998), they discuss the key factors that should be considered when choosing the mother wavelet. There are four main considerations, including (i) orthogonal or nonorthogonal, (ii) complex or real, (iii) width, and (iv) shape. In this study, we follow Liu et al. (2011) in selecting the nonorthogonal and complex Morlet wavelet as follows: 1ψ(n)=π-1/4eiw0ne-n2/2, where w0 is the nondimensional frequency with a value of 6 (Torrence and Compo, 1998).

Once the mother wavelet is selected, the WT is applied to a time series, xn, in which n goes from n=0 to n=N-1 with a time step of δt. The WT is the convolution of the time series with the mother wavelet that has been scaled and normalized as follows: 2Wn(s)=∑n′=0N-1xn′ψ*(n′-nt)δts, where n′ is the localized time in [0, N-1], s is the scale parameter, and the asterisk indicates the complex conjugate of the wavelet function. The wavelet power is defined as |Wn2|, which represents the squared amplitude of an imaginary number when a complex wavelet is used as in this study. We use the bias-corrected wavelet power (Liu et al., 2007; Veleda et al., 2012), which ensures that the power is comparable across timescales. We also identify a maximum timescale a priori that corresponds to our application. We select 256 h (∼10 d), but this number could be higher or lower for other applications, and there are no real penalties for using too high a maximum (lower than the annual cycle).

The wavelet transform (WT) expands the dimensionality of the original time series by introducing the timescale (or period) dimension. Wavelet power is also a function of both time and timescale (e.g., Torrence and Compo, 1998). This is illustrated in Fig. 2. The streamflow time series (Fig. 2a) is expanded into a 2-dimensional (2-D) wavelet power spectrum (Fig. 2b). Wavelet analysis can detect localized signals in the time series (Daubechies, 1990), including hydrologic time series, which are often irregular or aperiodic (i.e., events may be isolated and do not regularly repeat) or nonstationary. We note that, in many wavelet applications, timescale is referred to as “period”, and this axis is indeed the Fourier period in our plots. However, to emphasize that our study is more focused on irregular events and less on periodic behavior of time series, we use the term timescale to denote the Fourier period (and not wavelet scale).

Because we are applying the WT to a finite time series, there are timescale-dependent errors at the beginning and end times of the power spectrum, where the entirety of the wavelet at each scale is not fully contained within the time series. This region of the WT is referred to as the cone of influence or COI (Torrence and Compo, 1998). Figure 2b illustrates the COI as the regions in which the colors are muted; we ignore all results within the COI in this study.

We make several additional notes on the wavelet power and its representation in the figures. The units of the wavelet power are those of the time series variance (m6 s-2 – meters to the sixth power per square second for streamflow), and it is natural to want to cast the power in a physical light or relate it to the time series variance. Indeed, the power is often normalized by the time series variance when presented graphically. However, it must be noted that the wavelet convolved with the time series frames the resulting power in terms of itself at a given scale. Wavelet power is a (normalized) measure of how well the wavelet and the time series match at a given time and scale. The power can only be compared to other values of power resulting from a similarly constructed WT. There are various transforms that can be applied to aid the graphical interpretation of the power (log and variance scaling), but the utility of these often depends on the nature of the individual time series analyzed. For simplicity, we plot the raw bias-rectified wavelet power in this paper.

In their seminal wavelet study, Torrence and Compo (1998) outline a method for objectively identifying statistical significance in the wavelet power by comparing the wavelet power spectra with a power spectra from a red noise process. Specifically, the observed time series is fitted with an order 1 autoregressive (AR1 or red noise) model, and the WT is applied to the AR1 time series. The power spectrum of the AR1 model provides the basis for the statistical significance testing. Significance is determined if the power spectra are statistically different using a chi-squared test.

Figure 2b shows significant (>=95 % confidence level) regions of wavelet power inside black contours. Statistical significance indicates wavelet power that falls outside the time series background statistical power based on an AR1 model of the time series. Statistical significance of the wavelet power can be thought of as events in the wavelet domain. We define events as regions of significant wavelet power outside the COI. Figure 2c displays the wavelet power for the events in this time series. We emphasize that events defined in this way are a function of both time and timescale and that, at a given time, events of different timescales can occur simultaneously.

Step 1b results in the identification of all events at all timescales and times. In this substep, the event space is sampled to suit the particular evaluation. Torrence and Compo (1998) offer the following two methods for smoothing the wavelet plot that can increase significance and confidence: (i) averaging in time (over timescale) or (ii) averaging in timescale (over time). Because the goal of this paper is to evaluate model timing errors over long simulation periods, we choose to sample the event space based on averaging in timescale. Although for some locations there may be physical reasons to expect certain timescales to be important (e.g., the seasonal cycle of snowmelt), the most important timescales at which hydrologic signals occur at a particular location are not necessarily known a priori. Averaging events in timescale can provide a useful diagnostic by identifying the dominant, or characteristic, timescales for a given time series. Averaging many events in a timescale can filter noise and help reveal the expected timescales of dominant variability corresponding to different processes or sets of processes.

In our analysis, we seek to uncover the dominant event timescales and to evaluate modeled timing errors on them. The following points articulate our methodological choices for summarizing the observed events:

Calculate the average event power in each timescale. Considering only the statistically significant areas of the observed wavelet spectrum, calculate the average power in each timescale (Fig. 2c, right panel). We point out that calculating the average power over events is different to what is found by averaging across all time points, which does not take statistical significance into consideration (Fig. 2b, right panel).

The cross-wavelet transform (XWT) is performed between the observed and synthetic time series. Given the WTs of an observed time series WnX(s) and a modeled time series WnY(s), the cross-wavelet spectrum can be defined as follows: 3WnXY(s)=WnX(s)WnY*(s), where the asterisk denotes the complex conjugate. The cross-wavelet power is defined as |WnXY(s)| and signifies the joint power of the two time series. The XWT between the Onion Creek observations and the synthetic 5 h offset time series is shown in Fig. 3b, with power represented by the color scale.

Similar to step 1b of the WT, we can also calculate areas of significance for the XWT power as shown by the black contour in Fig. 3b. For the XWT, significance is calculated with respect to the theoretical background wavelet spectra of each time series (Torrence and Compo, 1998). We define XWT events as points of significant XWT power outside the COI. XWT events indicate significant joint variability between the observed and modeled time series. Below, in step 2d, we employ XWT events as a basis for identifying hits and misses on observed events for which the timing errors are calculated. Figure 3c shows the observed events (colors) and the intersection between the observed and XWT events (dashed contour). As described later, this intersection (inside dashed contour) is a region of hits where timing errors are considered valid. Note that the early part of the observed events at shorter timescales is not in the XWT events. This is because the timing offset in the modeled time series misses the early part of the observed event for some timescales.

For complex wavelets, such as the Morlet used in this paper, the individual WTs include an imaginary component of the convolution. Together, the real and imaginary parts of the convolution describe the phase of each time series with respect to the wavelet. The cross-wavelet transform combines the WTs in conjugate, allowing the calculation of a phase difference or angle (radians), which can be computed as follows: 4ϕnXY(s)=tan⁡-1II〈s-1WnXY(s)〉RI〈s-1WnXY(s)〉, where I is the imaginary and R is the real component of WnXY(s). The arrows in Fig. 3b indicate the phase difference for our example case, which is used to calculate the timing errors. Note that these are calculated at all points in the wavelet domain.

The distance around the phase circle at each timescale is the Fourier period (hours). We convert the phase angle into the timing errors (hours) as in Liu et al. (2011) as follows: 5ΔtnXY(s)=ϕnXY(s)⋅T/2π, where T is the equivalent Fourier period of the wavelet. Note that the maximum timing error which can be represented at each timescale is half the Fourier period because the phase angle is in the interval (-π, π). In other words, only timescales greater than 2E can accurately represent a timing error E. Because the range of the arctan function is limited by ±π, true phase angles outside this range alias to angles inside this range. (For example, the phase angles 1.05⋅π and -0.95⋅π are both assigned to -0.95⋅π). Also note that, when the wavelet transforms are approximately antiphase, the computed phase differences and timing errors produce corresponding bimodal distributions given the noise in the data. Figure 3c shows phase aliasing in the negative timing errors at timescales less than 10 h, which is double the 5 h synthetic timing error we introduced. The bimodality of the phase and timing are also seen at the 10 h timescale when the timing errors abruptly change sign (or phase by 2π). We note the convention used is that the XWT produces timing errors that are interpreted as modeled minus observed, i.e., positive values mean the model occurs after the observed. Positive 5 h timing errors in Fig. 3c describe that the model is late compared to the observations as seen in the hydrographs in the top panel (Fig. 3a).

Step 2b results in an estimate of timing errors for all times and timescales in the cross-wavelet transform space. In our application, we are interested in the timing errors that correspond to the identified sample of observed events, especially for the maximum power events in each cluster for each characteristic timescale. In the synthetic Onion Creek example, the point of interest in the wavelet transform of the observed time series, used to sample the timing errors produced by the XWT, is shown by the gray asterisk in Fig. 3c.

The results for the synthetic Onion Creek example are summarized in Table 2. For the identified characteristic timescale of 22 h in the observed wavelet power (which had an average WT power of 555 700 m6 s-2; see Fig. 2c on the right), there was one event cluster, and the timing error at the cluster maximum was 5 h, and it occurred at hour 37 of the time series.

The premise of computing a timing error between the observed and modeled time series is that they share common events which can be meaningfully compared. In a two-way contingency analysis of events, a hit refers to when the modeled time series reproduces an observed event. When the modeled time series fails to reproduce an observed event, it is termed a miss. In the case of a miss, it does not make sense to include the timing error in the overall assessment. Once the characteristic timescales of the observed event spectrum are identified and event cluster maxima are located, timing errors are obtained at these locations in the XWT. In this step, the significance of the XWT on these event cluster maxima is used to decide if the model produced a hit or a miss for each point and to determine if the timing error is valid. As previewed above, Fig. 3c shows the observed events (colors), and the dashed contour shows the intersection between the observed and XWT events. Regions of intersection between observed events and XWT events are considered model hits, and observed events falling outside the XWT events are considered misses. Because we constrain our analysis to observed events in the wavelet power spectrum, we do not consider either of the remaining categories in a two-way analysis (false alarms and correct negatives). We note that a complete two-way event analysis could, alternatively, be constructed in the wavelet domain based on the Venn diagram of the observed and modeled events without necessarily using the XWT. We choose to use the XWT events because the XWT is the basis of the timing errors.

In the synthetic example of Onion Creek, a single characteristic timescale and event cluster yields a single cluster maximum, as shown by the asterisk in Fig. 3c. Because this asterisk falls both within the observed and XWT events, it is a hit, and the timing error at that point is valid (Table 2). For a longer time series, as seen in subsequent examples, a useful diagnostic and complement to the timing error statistics at each characteristic timescale is the percent hits. When summarizing timing error statistics for a timescale, we drop misses from the calculation and the percent hits indicates what portion of the time series was dropped (percent misses is equal to 100- percent hits). In our tables, we provided timing error statistics for hits only.

This example uses a 3-month time series from the Pemigewasset River, NH, to examine multiple peaks in the hydrograph (Fig. 4a). It is fairly straightforward to pick out three main peaks with the naked eye. From step 1 of our method, the wavelet transform is applied to the observations (Fig. 4b, left panel; Fig. 4c, left panel), revealing up to three event clusters, depending on the characteristic timescale examined (Fig. 4d). When we plot the average event power by timescale (Fig. 4c, right panel), we see that there are nine relative maxima (small gray dots); hence, there are nine characteristic scales for this example. The cluster maxima (gray asterisks) for each observed event cluster are shown in Fig. 4d.

Next, we compare the observed time series with the simulation from the NWM V1.2 (Fig. 5a) and follow step 2 of our method: (a) apply the cross-wavelet transform (Fig. 5b colors), (b) calculate the timing error for all observed events from the phase difference (Fig. 5b arrows), (c) subset the timing errors to the observed cluster maxima (Fig. 5c asterisks), and (d) retain only modeled hits (Fig. 5c asterisks within the dashed contours). Table 3 is ordered, by characteristic timescales, from highest to lowest average power; we only show the top five characteristic scales. The absolute maximum of the time average event spectrum has a timescale equal to 24.8 h; for cluster one, the model is nearly 11 h late, and cluster two is early (-3.5 h). Both are hits, and the average timing error is 3.5 h late. However, for the next timescale (=27.8 h), the third cluster maximum is a miss, so its timing error is reported as n/a (not applicable) and is not included in the average. This miss can be seen in Fig. 5c where the cluster 3 asterisk falls just outside the XWT events for the 27.8 h, timescale. Moreover, this miss can also be interpreted from the comparison of the hydrographs in Fig. 5a where the modeled third peak does not reasonably approximate the magnitude of the observed peak. Interestingly, while it is a narrow miss at the shorter timescale of 27.8 h, the associated (third) cluster maxima at the next most powerful characteristic timescale (33.1 h) is a hit. This reflects that the hydrograph is insufficiently peaked for this event but does have some of the observed, lower-frequency variability. Overall, the characteristic timescale of 33.1 h has timing results similar to the 27.8 h timescale, with the exception of the third cluster maximum. This raises the question of whether these are distinct characteristic timescales. In Sect. 5, we discuss smoothing the time average event power by timescale to address this issue.

The characteristic timescale with the fourth-highest time-averaged power occurs at 111 h, which is a different order of magnitude, suggesting that this may have a different physical process driving it. At this timescale, the model is late in both event clusters (10 and 16 h). Results are similar for the next timescale of 148 h. We do not show results for the remaining four characteristic timescales with lower average power, since they have similar characteristic timescale values and associated timing errors to what has already been shown.

In this example, we examine a 1-year time series from Taylor River, CO, that illustrates hydrograph peaks driven by different processes. The Taylor River is in a mountainous area where the spring hydrology is dominated by snowmelt runoff. Figure 6a shows the time series from Taylor River, CO, where we can see the snowmelt runoff in spring and also several peaks in summer, likely driven by summer rains. Figure 6b shows the WT and illustrates how missing data is handled. This results in additional COIs (muted colors) to account for the edge effects, and areas of the COI are ignored in our analyses.

From the statistically significant events in the WT, we see the peak in the characteristic timescales at 23.4 h (Fig. 6c, right), and there is another maxima at the 99 and 118 h timescales. The process-based shift in dominant timescales is evident in the wavelet power (Fig. 6b and c). The 23.4 h timescale is dominant before 1 July, during snowmelt runoff, and then shifts to the 99 and 118 h timescales, relating to flows from summer rains. In step 2, we compare the observed time series with the simulation from the NWM V1.2 (Fig. 7a); here, it is useful to magnify the spring melt season time series (Fig. 8), where we see that the amplitude of the diurnal signal is too high, but it is hard to visually tell much about the timing error. Next, the cross-wavelet transform (Fig. 7b) and timing errors are calculated (Fig. 7c). The results are summarized in Table 4. Starting with the dominant 23.4 h timescale, we see that there are 11 clusters, that 73 % (=8/11 cluster maxima) are hits, and that the model is generally early (the mean is 6 h early). For the 118 and 99 h timescales, there are no hits. This suggests that we are confident in the timing errors of the model for the diurnal snowmelt cycle, and these timing errors can be used as guidance for model performance and model improvements. However, the model does not successfully reproduce key variability during the summer, and timing errors are not valid at this timescale. This underscores the key point that timing errors are timescale dependent and can help diagnose which processes to target for improvements.