PULSE, A Waveform Model for River Isotopic and Chemical Time-series Data
Authors: Hussein Mohamed Fahmy , Zouari Kamal , Anter Mohamed , Nosser Mariam 
Published: August 2020

Abstract
The isotopic compositions, electrical conductivity (EC), and ion concentrations frequently change with time in river water worldwide, on various time scales. Periodicity is also known for precipitation and other aqueous inputs. The time-series follow up of such temporal variations is usually to proceed by many hydrologists. The major purpose is to gain a thorough understanding of the underlying phenomena and to plot the data accurately. However, there is no appropriate formula for the graphic illustration and mathematical expression of such time-dependent fluctuations other than the complicated Fourier Series. Also, the use of the standard trigonometric sinusoidal function is not possible due to constraining the output of the values between +1 and -1. In contrast, the isotopic compositions require negative and positive values on the y-axis corresponding to time on the -x-axis, while the hydrochemical concentrations only admit positive values. These constraints are entirely void in the flexible waveform PULSE model proposed in this work. In this model, we introduce a modified sinusoidal formula that has the exciting capability of freely controlling the graphic waveform, in a highly accommodating way, for plotting the time-series isotopic and hydrochemical pulses in conformity with field observations. Three main parameters, and two optional secondary parameters, are to use in our model, open to modification. The model is to use in Excel®, whose SOLVER built-in macro may give the approximate values for the main parameters. Such values are then to improve to get the waveform best visual fit manually. We applied PULSE on EC, Cl, and δ18O values for Nile water, Cairo, and progressively improved the parameters' values as new data was to obtain. The sexagesimal angles are to handle for plotting the sampling dates on the x-axis. The angle that corresponds to one day = 360°/365.24 day-1 = 0.9856° day-1 = 0.0172 rad day-1. The standard Excel® time and date-models are also to use to assign the time-series sampling dates. The two optional parameters, β and γ, are to use only to damp or expand a decayed or stretched future pulses; otherwise, the values of those secondary parameters should be zero. The far parent of our fundamental PULSE formula is a three-parameter flickering medical equation (Sinusoidal Amplitude-modulated Flicker Model) used in the optical-fitness experiments run for testing the human vision adaptation to light luminance, using the appropriate electrophysiological devices. PULSE stands alone in its hydrological category. This model offers a unique quantitative definition for the isotopic and hydrochemical pulses in successive waveforms, with adjustable values for its parameters, in response to the involved variable and the sampling dates. Measurements on Nile water, using daily river water sampling for several years, were to carry out. PULSE revealed its practical merits for an extensive Nile water data set. Such data are for δ18O/V-SMOW ‰, EC dS m-1, and Cl mg l-1, PULSE works fine for such a riverine system. This application included a rare event of exceptional runoff suddenly imposed on Cairo Nile water composition by a scarce flash thunderstorm, where unusual waveforms were to assign to the isotopic and chemical trails of such abnormal runoff in the Sahara. Such a rare event was to use to get a backward look to the paleo-hydrology of the Nile water composition in Egypt.