## 1 Introduction  
Olive trees were extended all over the Mediterranean countries by eastern civilization. Local wild trees were protected by families and tribes; thus, during many years, those olive trees that were well-adapted to environmental conditions were selected for cultivation. Consequently, olive trees are nowadays a traditional crop located in the Mediterranean basin, where originally wild olive trees existed. During the last decades, the demand for olive oil experienced a global increase; hence, it was necessary to increase its production using new intensification agronomic techniques [1]; one of these techniques was irrigation. This intensification produced an increase in tree growth and yield without affecting the quality of olive oil [1]. The three best-known olive tree varieties for super-high-density systems are \"Arbequina\", \"Arbosana\", and \"Koroneiki\". These cultivars have fast entry into production, tend to yield good annual productions, start bearing at an early age, and have excellent oil quality characteristics [2]. In Spain, olive orchards are nowadays one of the main irrigated crops (818,505 ha), only exceeded by cereals (889,411 ha) [3]. Water scarcity is one of the main issues all over the world, and it has a clear effect on agriculture. Regulated deficit irrigation (RDI) and sustained deficit irrigation (SDI) are some of the techniques that are being developed to confront this problem. RDI decreases the use of water during some specific growing states of olives, while SDI decreases the water applied in a uniform way during all the growing season [4]. Regarding olive oil quality, the European Union and the International Olive Council have regulations to classify olive oil according to their quality [5; 6]. With respect to nutritional and functional quality, it is well known that the lipid profile is one of the main contributors due to the high proportion of monounsaturated fatty acids (MUFAs) of this specific oil. Also, polyphenols have an important role, and, in fact, olive oil is the only food that has an authorized health claim [7]: \"_Olive oil polyphenols contribute to the protection of blood lipids from oxidative stress._\" Volatile compounds are essential to olive oil quality, with both main and minor compounds having important roles on flavor. The odor-active compounds are responsible for the oil aroma, while the minor ones, even when they are below the olfactory threshold, can be used as quality markers as they can be essential to understand degradation or formation reactions dealing with odor-active substances [8]. _HydroSOstainable_ products have been defined as fruits and vegetables cultivated under controlled deficit irrigation treatments, which give them differentiating characteristics that make them unique and environmentally friendly [9]. _HydroSOstainable_ products provide special characteristics to the final commercial commodities, which are richer in some bioactive compounds and have a higher intensity of key sensory attributes, making them attractive for consumers [9; 10]. Several products from olive trees under RDI are being studied as _hydroSOstainable_; for instance, \"Manzanilla\" table olives have been studied [11; 12; 13]. Arbequina olive oil under water deficit techniques have been previously studied [14; 15; 16; 17; 18], but there is not a clear trend on the effect of irrigation on its quality. At this time, there is not a systematic body of knowledge considering agronomic practices, phenological stage during RDI, climatic constraints, etc. to have a full understanding of the effects of RDI on \"Arbequina\" oil quality, and further studies looking for the best water saving technique are necessary. Therefore, the aim of this work was to study the effect of (i) two new regulated deficit irrigation (RDI) treatments applied during phase II (pit hardening phase) [19] and (ii) one sustained deficit irrigation (SDI) treatment on \"Arbequina\" olive oil composition and properties.