tokens: Galactic latitude. The significance of this result can be assessed by a straightforward reduced χ 2 statistic: where ( x Bes , x MOA ) refer to the (model, observed) microlensing quantity. σ is the observational uncertainty within the field, and N fld is number of fields. The observational uncertainty within the field is calculated using formula from Han Gould ( 1995 ). The reduced χ 2 contribution of each MOA-II field are shown in Table 5 and Fig. 11 . The gb21 field is excluded due to the limit of the Besançon extinction maps. The model optical depth is in agreement with MOA-II data within 3σ MOA for most fields. The reduced χ 2 of resolved source and DIA sources optical depth are 2.4 and 2.0, respectively. Figure 11. The reduced χ 2 contribution of optical depth (a) and microlensing event rate per star (b) of resolved sources (left) and DIA sources (right). Table 5. Field-by-field contributions to the reduced χ 2 () between the Besançon model and the MOA-II data for the optical depth and microlensing event rate per star. The event rates show higher reduced chi-squared contribution than the optical depths. The Besançon resolved sources and DIA source results have values of 2.6 and 2.2, respectively. The low Galactic latitude area ( b 3°) of both parameters provide the bulk of the disagreement (see Section 5.1 ). In field gb1, there is a long time-scale event (gb1-3-1, t E = 157.6 d) which contributes more than half of optical depth in that field. This event provides a hotspot in MOA-II optical depth and average time-scale maps (Section 5.2 ). In order to check the reliability of reduced χ 2 test, the reduced χ 2 of field gb1 without gb1-3-1, gb1 Cut , is calculated. The result in Table 5 shows that the gb1 Cut field provide a better reduced than original gb1 field. Finally, we cut the events which have crossing time longer than 100 d which locate in 5 field; gb1, gb9, gb10, gb13 and gb 14. The new reduced chi-squared, , of optical depth (2.3 for resolved sources and 1.8 for DIA sources) and event rate per star (2.2 for resolved sources and 2.0 for DIA sources) is reduced. However, fields which locate around inner bulge except field gb6 still show high reduced chi-squared compare to high Galactic latitude field. Therefore, high reduced chi-squared region around inner bulge do not effect by long time-scale event, but show the mismatch of optical depth and event rate per star between the Besançon model and the MOA-II data at low Galactic latitude area. 6 MICROLENSING MODEL PARAMETRIZATION The MOA-II team parametrize the observed spatial microlensing distribution using a polynomial function. We can do likewise for our simulated maps. We model the structure of the optical depth, average time-scale and event rate maps shown in Fig. 5 using a 10-parameter cubic polynomial fit in l and b . The model function can be written as where x is the microlensing observable (rate, time-scale or optical depth). The best-fitting models are shown in Fig. 12 and
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