Bouchet's complementary relationship and the Budyko hypothesisare two classic frameworks that are inter-connected. To systematicallyinvestigate the connections between the two frameworks, we analyzeprecipitation, pan evaporation, and potential evaporation data at 259 weatherstations across the United States. The precipitation and pan evaporation dataare from field measurement and the potential evaporation data are collectedfrom a remote-sensing dataset. We use pan evaporation to representapparent potential evaporation, which is different from potentialevaporation. With these data, we study the correlations between precipitationand potential evaporation, and between precipitation and apparentpotential evaporation. The results show that 93 % of the study's weatherstations exhibit a negative correlation between precipitation andapparent potential evaporation. Also, the aggregated data cloud ofprecipitation vs. apparent potential evaporation with 5312 warm-season data pointsfrom 259 weather stations shows a negative trend in which apparentpotential evaporation decreases with increasing precipitation. On the otherhand, no significant correlation is found in the data cloud of precipitationvs. potential evaporation, indicating that precipitation and potentialevaporation are independent. We combine a Budyko-type expression, theTurc–Pike equation, with Bouchet's complementary relationship to deriveupper and lower Bouchet–Budyko curves, which display a complementaryrelationship between apparent potential evaporation and actualevaporation. The observed warm-season data follow the trend of theBouchet–Budyko curves. Our study shows the consistency between Budyko'sframework and Bouchet's complementary relationship, with the distinctionbetween potential evaporation and apparent potential evaporation. Theformulated complementary relationship can be used in quantitative modelingpractices.
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