We analysed river level data from water level monitoring gauges with long time series (1–24 years) and high accuracy (1–3 mm, measured every 15–60 minutes) from USA, Honduras and Australia. The first selection of these sites was made by choosing the gauges having constant double-peaked daily evolution during 7 consecutive days in June-July 2011 (7 consecutive days represent the standard analysis window for fast graphical preview of the websites providing our data). Summer in the northern hemisphere is winter in the southern hemisphere. For example, the climates of Canberra (Australia) and San Antonio (Texas) are very different: the higher air temperatures and precipitation amounts occur during November-April in Canberra and during May-October in San Antonio. Also, in the same hemisphere, summer months mean different air temperatures and amounts of precipitation and greatly varying temporal distribution of the annual peaks of these parameters. The vast territories of USA, Australia and Honduras (used for selecting data) and the very large distances between the selected gauges provided a wide diversity of climates. Thus, the worldwide selection of sample gauges in a given month/group of months (without using discriminative/subjective criteria such as season, temperature, precipitation) is an unbiased sampling and must, in theory, prove the omnipresence of the semidiurnal signal if this is found to be worldwide spread. A number of 2000 USGS streamwater monitoring sites placed on non-tidal rivers in USA were searched in the states covering the following aquifers: High Plains, Edwards-Trinity, Mississippi Embayment-Texas Coastal Uplands, Coastal Lowlands, Piedmont and Blue Ridge. These representative aquifers were chosen in order to discover their possible tidal input in rivers. The streamwater level was also analysed for sites from New South Wales and Queensland (250 monitoring sites each one, from the Australian governmental monitoring network), especially in the Great Artesian Basin area. The gauge heights in Honduras (USGS network) were searched in order to observe if a different climatic influence on rivers affect the intensity of the possible semidiurnal signal. All analysed gauges had varying types of climate, from continental or humid temperate to subtropical and humid tropical.

139 gauges were preliminary selected as possibly having a constant M2 signal. Then, the gauges with persistent (120 days) double-peaked daily evolutions were selected for a Fast Fourier Transform (FFT) analysis (120 days of samples per station, one sample per hour, 30 minutes or 15 minutes, depending on gauge – we considered that 1/3 of a year is sufficiently long to provisionally decide if a signal is persistent or not to provide relevant data for further analyses). The FFT analysis permitted the identification of rivers with semidiurnal signal stronger than the surrounding red noise. Also, a satellitary analysis of each catchment was performed by using 2.5 m resolution SPOTImage imagery in order to discover and eliminate the rivers with major human interventions on river discharge. A final list containing 13 relevant gauges resulted (Table 1, no. 1–13). The selected river gauges were chosen as representative if they had strong semidiurnal signal and if the USGS site description was generally: “records well, no dam or deviation”. Even if half of the remaining gauges/catchments have some human intervention, we kept them because the intervention is not considered important and because the rivers showed relevant results at the later statistical analysis. Few remaining gauges have dams in their upstream catchments but, according to Zimmerman et al.20, the subdaily oscillations in rivers with dams do not greatly differ from the natural oscillations in rivers. Therefore, the human intervention is not considered important when the semidiurnal signal is very strong and the human influence (diversion, dam) is singular and not of large impact.

Table 1 Details of the primary gauges Full size table

In order to successfully detect the orthotidal behaviour in the selected rivers, the semidiurnal signal was compared to that of Hillsborough River, Florida (no. 0, control gauge), which is a representative tidal river. According to USGS, “the gauge height at this site is significantly affected by astronomical tides” and the river periodicities show a very strong and dominant M2, having typical semidiurnal tides, with two high and two low waters per day. All river gauges have 96 measurements per day (1 measurement per 15 minutes), excepting gauge no. 10 (1 measurement per 30 minutes). The majority of the selected time series ranges from Oct. 2007 to Feb. 2013, excepting no. 2 (Oct. 2009-Aug. 2011), no. 10 (June 1994-Oct. 2006), no. 11 (May 2009-July 2011) and no. 13 (Nov. 2012-Feb. 2013).

The continuous wavelet transform (CWT) analyses of the selected gauges showed semidiurnal periodicities with 0.95 confidence level against red noise for long time intervals in some time series when the analysis window was set for the entire length of data (Figure 1, a–f; Supplementary Fig. 1, a–h). On scalograms, the semidiurnal signal is to be found in the 0.3–0.6 days periodicities band of the vertical axis; on this band, the intensity of the signal (blue - minimum, red - maximum) varies regularly or irregularly from the beginning to the end of the time series depending on the importance of the generating and erasing factors. As it can be observed in Figure 1.d, the change in gauge height characteristics due to monitoring interventions has modified the accuracy of the semidiurnal signal. Moreover, for all full-length data scalograms, the consequences of high waters on smaller signals can be observed: the vertically elongated high-power and non-red noise areas, in opposition to the horizontally elongated high-power and non-red noise areas of the diurnal and semidiurnal signals.

Figure 1 CWT scalograms: (a–f) - of the time series no. 0–5 from Table 1 (the horizontal axes represent the number of consecutive measurements, 96 per day); (g–l) – of selected 3000 consecutive measurements from the time series no. 0–5 from Table 1; the thick black contours represent the 0.95 confidence level against AR1 red noise; the colours with lighter shade of the power spectrum represent temporal areas affected by edge effects. Full size image

We observed that the CWT analysis is more relevant for full time series without important episodes of high waters and for smaller time intervals, between high waters. When the CWT analysis is applied for shorter time series (3000 consecutive measurements per gauge), that do not include the high waters episodes and their disturbing effect, the power and the statistical relevance of the semidiurnal and diurnal periods increase. All selected gauges, when analysed for shorter time series, show semidiurnal signal with 0.95 confidence level against red noise. Much shorter time series will not have enough length to reveal the high statistical relevance of the semidiurnal signal. The CWT analyses of the shorter time series (Figure 1.g–l; Supplementary Fig. 1.i–p) include the most representative time intervals from the full-length data scalograms (the intervals with the greatest power of the semidiurnal signal). These scalograms show wider areas of the semidiurnal signal with 0.95 confidence against red noise (especially due to a diminished number of high waters episodes); these areas also have an enhanced power of the signal (red-shifted colours that indicates higher amplitudes and/or a more regular behaviour of the semidiurnal signal).

The global wavelet spectrum (GWS) is extracted from the scalograms of the shorter time series; on the resulted periodograms (Figure 2, Supplementary Fig. 2), the daily peak (diurnal oscillation) is visible as the main peak, while the semidiurnal peak is lower, having confidence levels from above 0.95 to under 0.67. The semidiurnal signal of inland rivers has the same wavelength as the semidiurnal signal of the tidal control gauge (M2): 12.42 hours (49.689 measurements).

Figure 2 Periodograms of the time series no. 0–5 from Table 1, extracted from Figure 1.g–l scalograms (the frequency represents cycles per number of measurements). Full size image

For inland rivers, the diurnal peak has a relatively stable hourly position, while the semidiurnal peaks are very mobile. The semidiurnal peaks often have repeating positions after a number of days or split the diurnal peak in 2 peaks with evolving and repeating shapes; sometimes, the M2 signal do not have enough power to create distinct peaks, but only to impose inflections on the ascending or descending slope of the diurnal peak (Figure 3.a–f; Supplementary Fig. 3.a–h). During and after strong rains, easily identified through the sudden increase of an asymmetrical high water peak, extending on multiple days, the semidiurnal signal is temporarily erased.

Figure 3 Selected short time series extracted from the time series no. 0–5 from Table 1 (a–f); periodograms of the GWS values extracted from PWC scalograms of the a–f time series (g–l) (the frequency represents cycles per number of measurements; *-examples of M2 inflections). Full size image

For inland groundwater, M2 is followed by the much weaker K1 and O1 lunar diurnals and the rivers do not show them; a solar semidiurnal S2 signal is, generally, the second strongest signal in groundwater18 and is caused by the solar tidal force. The GWS of the partial wavelet coherence (PWC) analyses (wavelet coherence between M2 or S2 sine waves and the river time series without S1 (solar diurnal) sine wave – S1 is extracted in order not to alter the results of the wavelet coherence analysis) clearly shows that the semidiurnal signal in the studied rivers is to be attributed to M2 (which has higher power), not to S2 (Figure 3.g–l, Supplementary Fig. 3.i–p).

The disturbing effect of high waters and red noise on M2 signal can be partly removed by obtaining a simple derivative (difference between neighbour values) from the water raw data. The new data can enhance signal detecting in simple plots (Figure 4) and scalograms (Figure 5) or can cause an important reduction in the confidence level against red noise (Figure 6) because of the increased data artificialization. Therefore, the results of the derivative method are to be used with precaution. In some cases, the differencing of the neighbour values alters the slope of the semidiurnal inflections when these are transformed into peaks by reducing their angles and this translates into weaker semidiurnal signal in the wavelet analysis.

Figure 4 Comparison of the M2 semidiurnal signal in raw data and simple derivative data of Bemboka River at Morans Crossing during the first half of October 2009 (the vertical axis represents normalised flow values for the river raw data and the difference between neighbour values *1000 for the simple derivative data). Full size image

Figure 5 The presence of the M2 signal in Leona River and Uvalde piezometer: (a) – the good graphical correlation between the peaks and valleys of the M2 signal in river and groundwater in August 2011; (b, c) – scalograms of the Uvalde piezometer during October 2007–August 2013 for raw data and, respectively, simple derivative data. Full size image

Figure 6 The presence of the M2 signal in Medina River and Edwards piezometer: (a, b) – scalograms of the Edwards piezometer during October 2007–August 2013 for raw data and, respectively, simple derivative data; (c) – the graphical correlation between the peaks and valleys of the M2 signal in river and groundwater in July 2013; (d, e) – scalograms of the raw and, respectively, simple derivative time series of Medina River during October 2007–August 2013; (f, g) - scalograms of the raw and, respectively, simple derivative time series of Medina River during November 2012–April 2013. Full size image

The M2 semidiurnal oscillation in the studied rivers is caused by the M2 signal in the regional aquifers. Examples are Leona River near Uvalde, where the semidiurnal oscillation is stronger when it is also stronger in the Mc Knight Formation below (for example, in the piezometer placed 10 km E of upper Leona streambed), part of Edwards-Trinity aquifer (Figure 5). This local behaviour of the Edwards-Trinity aquifer system is not an exception, being found in many other wells. The other aquifers related to the studied river gauges have M2 signal too, for example the Evangeline aquifer (part of the Gulf Coast aquifer, which is included in the Coastal lowlands aquifer system) (Supplementary Fig. 4) - the Evangeline aquifer provides water for Panther Brook at Gosling and the Conroe piezometer is placed 15 km N of the river gauge.

We searched if a strong and persistent M2 signal in local aquifers can be used as detector of rivers with a same type oscillation. For this case study, in the USGS water data network, we used only the Edwards-Trinity aquifer system. The amplitude of the M2 groundwater oscillation in USGS 295204099340201 AS-69-12-206 (upper Medina River catchment) is probably the most important in this aquifer. The M2 signal is much stronger than the diurnal one and has 2 peaks per synodic month, linked to the New and Full Moon (Figure 6.a,b). There is a good correlation between the groundwater in the mentioned piezometer and the Medina River at San Antonio gauge, the weaker or stronger semidiurnal signals appearing in groundwater-river pairs; however, it seems that the two M2 signals are in anti-phase (Figure 6.c). The M2 signal of Medina River is strongest at San Antonio gauge; it is weaker than the diurnal signal but, at a medium size analysis window (5 months) it shows 4 peaks of statistical power and 0.95 confidence level against red noise per synodic month (Figure 6.d–g). These peaks are probably related to the Moon phases.

The correlation between the river gauge height and groundwater level has many statistical relevancies depending on the indices and methods used. For example, the simple Pearson, Kendall and Spearman correlation coefficients between Medina River and Edwards piezometer show medium or weak correlations in June-August 2013 (2Table 3). A scaled correlation using the same types of correlation coefficients indicates a stronger correlation. The scaled correlation is obtained here by correlating parameters for each 24 hours consecutive window. The scaled correlation method generates daily correlation coefficients (Figure 7.a) which are averaged to obtain the scaled correlation coefficient (Table 3). The daily correlation coefficients generally show good negative correlations (Figure 7.a) and the scaled correlation coefficient is altered by the minority of days having weak or no correlations.

Table 2 Details of the secondary gauges Full size table

Table 3 Water level correlation coefficients of Medina River at San Antonio gauge and Edwards piezometer Full size table

Figure 7 Examples of correlations between Medina River at San Antonio gauge and Edwards piezometer (raw data): (a) – daily Pearson correlation coefficients; (b) – June 2013 WTC; (c) – July 2013 WTC; (d) – August 2013 WTC; (e) – January–December 2013 WTC. Full size image

A more reliable method to identify the strong and real correlation between 2 nonlinear signals in river/groundwater time series is the wavelet coherence analysis (WTC – method used to find similarities between signals/wavelets with changing time lags34). On scalograms in Figure 7.b–e the wavelet analysis identifies with a black line the areas where the two signals have a common evolution with a 0.95 confidence level against casual common evolution (by using a Monte Carlo test). The similarities between common signals in 2 time series is also indicated by the high/low power colours. The WTC scalograms of Medina River and Edwards piezometer show a good correlation of the semidiurnal signals. The same type good correlation exists between the semidiurnal signal in the other studied river-groundwater level pairs (Supplementary Fig. 5).

The good correlation between the presence/absence of the M2 signal in groundwater and rivers is valid for Australia, too. The Ranch piezometer in Bega catchment (which includes the previously studied Bemboka and Murrah rivers) shows good M2 signal (Supplementary Fig. 6). By searching the piezometers in the springs cluster area of the Great Artesian Basin, Macquarie River Basin, we discovered a very good M2 signal in Trangie piezometer (Figure 8.a,b). The Trangie piezometer is placed in Bogan River catchment. Bogan River has M2 signal, but only in the area near Trangie piezometer: the signal appears at Dandaloo gauge, becomes strongest at Neurie Plains gauge and fades at Nyngan gauge (Figure 8.c–j); the other upper and lower gauges, which are placed outside the springs cluster area, do not show the lunar semidiurnal signal. The WTC analysis of the river-groundwater pair in Bogan River catchment also show a good common evolution of the semidiurnal signal (Supplementary Fig. 7).

Figure 8 The presence of the M2 signal in Bogan River and Trangie at Dandaloo piezometer: (a, b) – scalograms of the Trangie at Dandaloo piezometer during 2012 for raw data and, respectively, simple derivative data; (c, d) – scalograms of the raw and, respectively, simple derivative time series of Bogan River at Dandaloo during 2012; (e, f) – scalograms of the raw and, respectively, simple derivative time series of Bogan River at Neurie Plains during 2012; (g, h) – scalograms of the raw and, respectively, simple derivative time series of Bogan River at Nyngan during 2012; (i, j) - scalograms of the raw and, respectively, simple derivative time series of Bogan River at Neurie Plains during December 2012. Full size image

For improving the statistical relevance of the M2 orthotidal signal in inland rivers, longer time series are necessary. Therefore, USGS archives data were added to the existing time series in order to obtain longer time series. The maximum length obtained per each gauge is equivalent to the period of the continuous 15–60 minutes measurements. The particular maximum length is of 24 years for Medina River. However, not all gauges time series were extended. The selected gauges on Leona and Comal rivers are newer than 2007, while the gauge on Middle Wichita River was discontinued in 2011. The extended time series are available in Figure 9 and Supplementary Fig. 8. Note that that the USGS archive was stored as river discharge, not river gauge height. Also, older data frequently have only hourly measurements (not for every 15 minutes). Therefore, the entire longer time series were transformed into hourly discharge data (by averaging, when necessary). For the Australian gauges (NSW Water Information), river data older than 3–4 years is very discontinuous and irregularly sampled and was not used to expand the existing time series. But, as it can be observed in Supplementary Fig. 9, for the same time period, the discharge data is less relevant than the gauge height. The most important factors contributing to that difference are: the diminishing importance of the very small water level oscillations in the used discharge formulae, the USGS data storing method (no decimal for discharges greater than 10 cubic feet per second, 1 decimal for discharges between 1 and 10 cubic feet per second and 2 decimals for discharges less than 1 cubic foot per second) and the frequently changing riverbed morphology (that continuously alters the discharge formulae, while the gauge height is altered discontinuously rather than in a continuous manner). Even if the USGS data is more continuously and regularly sampled than the Australian river data, the standard USGS gauge height increment is of only 0.01 feet (~3 mm), while the similar Australian increment is of 1 mm. This probably explains the smaller M2 signal amplitude found in Australian rivers when it is compared to the M2 signal in USGS rivers (higher accuracy means higher detection chances – Tables 1 and 2). Also, the hydrologists' custom of using river discharge instead of river water level can be an explanation of the late discovery of the M2 signal in inland rivers.