WRF OBS_Nudging

Formula

where q is the quantity being nudged (e.g., water vapor mixing ratio), µ is the dry hydrostatic pressure, Fq represents the physical tendency terms of q (for water vapor this includes advection, diffusion, conversion from water vapor to cloud water, etc.), Gq is the nudging strength for q, N is the total number of observations, i is the index to the current observation, Wq is the spatiotemporal weighting function based on the temporal and spatial separation between the observation and the current model location, qo is the observed value of q, and qm(xi,yi,zi,t) is the model value of q interpolated to the observation location. 

note:

1.  Model values will be interpolated to the observation location to calculate the innovation. (Obs too few)
2.  The dry hydorstatic pressure here coresponds to the lhs fluxs form of q, which is the WRF feature.
3.  Because different variables have different scale(u-10m/s, q-0.1), in order to apply the same Wq to each variable beening nudged, Gq should be carefully choosed for each var. 
4.  Wq is the multiplication of the time weight, horizontal weight and vertical weight.

Data check

Any observations with a pressure lower than 80 hPa are discarded

Observations that are too far in the past to be used in nudging are immediately discarded

Also, any surface observation whose pressure falls outside the range 700 – 1050 hPa will have its pressure and temperature discarded.

calculate the innovations（Qo-Qb）

• note: Q is a general var being nudged. Qo means obeservation. Qb means Q background, ergo the NWP forcast.

The next step is to calculate the innovations (differences between the model and the observations; qo(i)−qm(xi,yi,zi,t) in equation 1). When the observation was read in the surface observation was assigned to the first vertical model level for purposes of calculating the innovation. At this point, for all observations not classified as surface observations the pressure is used to determine the placement of the observation in model vertical-level space. If the pressure of the observation is not available, it will first use observation height and model height and pressure to determine observation pressure. Note that the placement of the observation in model vertical-level space (and where applicable the determination of pressure based on height) is made when the observation is first used and is not recalculated at future times. This means that even if the pressure changes with time, the vertical location where the innovation is calculated will not be updated. The model value is then interpolated in 3D (i.e., 8 points) to the observation location and the model value is subtracted from the observed value to find the innovation. If we are dealing with a surface temperature or surface wind observation, the observation is multiplied by a factor to account for the difference in height between the lowest model level and the height of the observation. These factors are calculated based on the model relationship between the lowest prognostic level and 2- m/10-m diagnosed values where possible. Thus these factors are case and time specific and are based on WRF’s understanding of the meteorological conditions.

可以暂时按照GPT的来理解，具体原理得参考源码。

prevent neg_nudging 

因为仅仅是在观测点计算nudging, 其nudging倾向在观测点是最适用的。但是由于要更新的是模式点，模式点和观测点之间存在的一定距离会导致 观测点的nudging在模式点不精确，其中最糟糕的情况是导致模式点向 无物理意义的负值区去nudging。如一个观测点nudging结果是负数，说明模式预测结果偏高，因此这一点的nudging会拉低周围模式点的预测值以使其向观测真值靠近。但如果周围模式点中有一个小值，那么这个nudging的影响可能使该点结果变负，不符物理意义。

max函数中，第一个数 Qo-Qm(xi,yi,zi,t) 是观测点计算的值，第二个参数是模式点的值。这样做的好处是可以一定程度上缓解变负的情况。 例子如下：

i.g.:

For example, consider the case where the observed water vapor mixing ratio is 5 g kg−1 [qo(i)= 5 g kg−1 ], and the model value at that location is 10 g kg−1 [qm(xi,yi,zi,t) = 10 g kg−1 ]. The observed value is less than the corresponding model value, and so the innovation is negative [qo(i)−qm(xi,yi,zi,t) = −5 g kg−1 ] and the model is thus nudged towards drier conditions. However, if this innovation is spread to a location (x,y,z) where the modeled water vapor mixing ratio is 3 g kg−1 , this is less than the magnitude of the innovation (5 g kg−1 ); therefore at location (x,y,z) the model is being nudged towards −2 g kg−1 . Nudging towards negative water vapor mixing ratios can result in rapid drying of the model and can lead to a complete removal of water vapor at that location.

Weighting Functions

Overall Weighting Function

 Overall Weighting Function The overall weighting function (G in equation 1) is dependent only on the variable being observation nudged (obs_coef_wind, obs_coef_temp, and obs_coef_mois) and does not change in space nor time. As noted earlier, the inverse of G is the e-folding time of the difference between the model and the observation assuming the physical tendency terms are zero. Figure 1 illustrates the decrease in model error with time assuming the physical tendency terms are zero when nudging with various nudging weights ranging from 1e−4 s−1 to 128e−4 s−1.

Temporal Weighting

示意图如下，如图所示的坡度（ramp）主要是为了防止立刻引入或取消nudging会破坏模式在积分过程中形成的中小尺度气象结构，和引入噪声所导致的预报精度下降。

Temporal weighting的理解中三个量很重要。观测数据的时间分辨率；obs_twindo; nudging interval (nudging的时间间隔)。 twindo 的示意图如下，twindo即一个观测点的影响时间窗，可以影响到这个时间窗内的所有 nudging point，一般nudging interval 应当小于obs_twindo，不然观测点无法nudge到模式时刻，相当于没有nudging，结合下图理解。

Vertical Weighting

surface observations：

基本假设：边界层误差和自由大气误差相关性不强，nudging不互通。所以可根据obs资料的性质选择是否开启PBL nudging。 

obs_sfc_scheme_vert = 1

The vertical functions originally in WRF (obs_sfc_scheme_vert = 1) attempt to spread the influence of innovations calculated from surface observations throughout the PBL with decreasing weight with height. It uses a linear decrease in weight with vertical level number from the surface to the model level where the PBL top occurs with two exceptions. If the PBL top is below the 3rd model level or above the 25th model level, then the vertical spreading will be applied through the 3rd model level above the surface instead of to the PBL top. Note that the vertical placement of the 3rd and 25th model levels will vary significantly depending on the placement of vertical levels for a given case.

obs_sfc_scheme_vert = 0

PBL影响应当避免扩散到PBL以上。vert=0的时候，用户可以通过namelist自定义权重曲线，如下。

multiple-level observations

The innovations calculated at the observation location are vertically interpolated in pressure to the level at which they are being applied. The interpolation is done in log-pressure space.

This interpolation will not be done to model levels with pressures that are within a vertical gap in the observation greater than or equal to obs_max_sndng_gap. For example, consider a case with a sounding that has observations at 950, 850, 600, and 450 hPa, and the user has set obs_max_sndng_gap to 20 cb (200 hPa). The model will be horizontally and vertically interpolated to the observations and innovations will be calculated. The innovations at 950, 850, 600, and 450 hPa can then be vertically interpolated to model levels between 950 and 450 hPa. However, the gap between 850 and 600 hPa is 25 cb (250 hPa), which is greater than obs_max_sndng_gap, so no interpolation can be made to model levels with pressures between these two pressures. Therefore the innovations at 950 and 850 hPa will be interpolated to any model levels with pressures between 950 to 850 hPa where other weighting factors do not preclude nudging, and similarly between 600 and 450 hPa.

For single-level above-surface observations

the innovation is applied for model levels with pressures within 75 hPa of the observation.

The vertical weighting function linearly decreases with pressure from one at the pressure of the observation to zero   75 hPa above or below the observation.

Horizontal Weighting

RIN is the effective radius of influence for this observation (based on obs_rinxy and the vertical placement of the observation) and DIST is the distance between the observation and current model point where the innovation is being applied 

RIN由两个参数决定。

第一个是obs_rinxy，horizontal radius of influence， 水平影响半径。

第二个参数是 obs_sfcfacr，这个是地表因子，地表该值是1，从地表到500hpa逐步增长为2，500hpa到模式顶一直是2。该因子和 obs_rinxy 相乘 得到有效水平影响半径。

引入原因obs_sfcfacr 原因： 认为在近地表，任一变量的空间变化大，观测点位置处 innovation的倾向适用范围小，自由大气中相对空间变化小，一个观测对应的innovation较大，故引入一个变量来量化这种作用。  （即error correlation length）