## 1 Introduction
### Compositions of Nitrogen and Oxygen Stable Isotopes  
There are two naturally occurring stable isotopes of nitrogen, namely, \({}^{14}\)N and \({}^{15}\)N, and the corresponding natural abundances are 99.633% and 0.366%, respectively. Oxygen is composed of three stable isotopes, \({}^{16}\)O (99.757%), \({}^{17}\)O (0.038%), and \({}^{18}\)O (0.205%). Nitrogen and oxygen participate in various physical processes and chemical reactions in the natural environment. The stable isotope composition is usually expressed in delta (8) units and a per mil (%) notation relative to the respective international standards  
\[\delta=(R_{\rm sample}/R_{\rm standard}-1)\times 1000, \tag{1}\]  
where the positive value and negative value of \(\delta\) respectively represent the enrichment and impoverishment of heavy isotopes in test sample compared with a standard sample. The nitrogen and oxygen isotopic ratios (\(R\)) are reported as the per mil deviation from the \({}^{15}\)N/\({}^{14}\)N or \({}^{18}\)O/\({}^{16}\)O ratios relative to N\({}_{2}\) (air) and Vienna Standard Mean Ocean Water (V-SMOW). The overall error of measurement is below 1%. The \(\delta^{15}\)N and \(\delta^{18}\)O compositions of nitrate differ in different potential sources. \(\delta^{15}\)N is light in precipitation (\(\delta^{15}\)N value ranges from \(-10\%\) to 8%) and chemical fertilizer (\(\delta^{15}\)N value ranges from \(0\%\) to \(3\%\)). After the nitrification of microorganisms, the \(\delta^{15}\)N value of soil nitrogen ranges from \(-3\%\) to \(5\%\). However, \(\delta^{15}\)N is heavy in manure and sewage (\(\delta^{15}\)N value ranges from \(7\%\) to \(20\%\) or is larger than \(20\%\)) [5;8]. \(\delta^{15}\)N plays an important role in identifying the nitrate from ammonium nitrogen fertilizer, soil nitrogen, manure, and sewage. The \(\delta^{15}\)N values of nitrate from precipitation, soil, chemical fertilizer, manure, and sewage partly overlap (Figure 1). The synthetic fertilizer or ammonia fertilizer do not contribute to the increase of \(\delta^{15}\)N as they are both from atmospheric N\({}_{2}\)[9], so the \(\delta^{15}\)N of precipitation is not significantly different from that of synthetic fertilizer. The application of \(\delta^{15}\)N is not enough for the accurate identify of nitrate source. Mostly, it is combined with \(\delta^{18}\)O, another indicator for identifying nitrate source, so as to reduce the uncertainties of nitrogen isotope in identification [10]. The \(\delta^{18}\)O value of precipitation is high and usually ranges between +20\(\%\) and +70\(\%\). The \(\delta^{18}\)O value of synthetic nitrate nitrogen fertilizer is 22\(\%\)\(\pm\) 3\(\%\), while the \(\delta^{18}\)O value of soil nitrogen transformed from in-situ ammonium nitrogen by nitrification is between \(-5\%\) and \(5\%\). The final production of \(\delta^{18}\)O varies with the nitrification ways as well as the \(\delta^{18}\)O ratio of \(\delta^{18}\)O-O\({}_{2}\) in atmosphere to \(\delta^{18}\)O-H\({}_{2}\)O in water. The values of \(\delta^{18}\)O in manure and sewage are lower than 15\(\%\)[11]. The major sources of nitrate are preliminarily determined by comparing dual stable isotopes (\(\delta^{15}\)N and \(\delta^{18}\)O) of potential sources with those of surface water. Due to the influence of regional climate, source emissions, topography, geology, etc., the isotopic compositions from different nitrogen sources vary widely in different regions and time [12; 13]. For example, the typical range of \(\delta^{15}\)N in the precipitation is from \(-3\%\) to 7\(\%\) in China [14; 15; 16], while that is from \(-6\%\) to \(11\%\) in other countries and regions [11; 17]. The typical value range of \(\delta^{15}\)N of soil organic nitrogen is from 3\(\%\) to 8% [18; 19], while that is reported to be from 0 to 8% in other countries and regions [11]. The typical value range of 815N of Chinese chemical fertilizer is between \(-\)6% and 5% [4], which is smaller than that of American fertilizer (between \(-\)4% and 8%) [20].