【背景】對于地球重力場的應(yīng)用問題，我們需求高時(shí)空分辨率和信噪比，而GRACE較低的空間分辨率成為限制其發(fā)展的一個(gè)重要因素。由萬有引力定律可知，引力隨著距離的平方衰減。這種衰減還依賴于波長，即隨著距離的增加，短波引力異常比長波場衰減得更為迅速（The National Council, 1997）。軌道高度上的衛(wèi)星觀測得到的信號相比于地表的重力信號，其經(jīng)過衰減并且含有各類誤差和噪聲。為了恢復(fù)真實(shí)的地表重力信號，我們需要把衛(wèi)星觀測信號向下延拓，一般乘上衰減因子的倒數(shù)進(jìn)行信號放大，以得到地表真實(shí)的重力信號。但是如此一來也放大了觀測誤差，導(dǎo)致高階信號被污染。因此需要對GRACE球諧產(chǎn)品進(jìn)行后續(xù)的反演恢復(fù)。

Chen et al.(2013)在nature geoscience中發(fā)表文章《Contribution of ice sheet and mountain glacier melt to recent sea level rise》，文章內(nèi)容如下：

The global forward modelling can more effectively reduce the leakage effect between terrestrial and ocean signals, and help reconstruct true mass rates of the ocean, polar ice sheets, mountain glaciers and TWS at the same time. Full details of the forward modelling approach are provided in the Supplementary Information. A brief synopsis is as follows: on a global 1? ×1 ? grid, we first assign initial trial mass rates (for example, GRACE apparent mass rates after filtering and smoothing) to each grid point over land, and a uniform layer of water is added to the oceans to conserve total mass. The trial mass rate grid is converted into spherical harmonics (with the same resolution as GRACE data); we then convert the spherical harmonics of the trial mass rate grid back to a spatial grid after applying GRACE data processing procedures to the spherical harmonic representation of the trial mass field (including truncation and smoothing); we compare the processed trial spatial grid with the GRACE apparent mass rate grid and adjust the trial mass rates accordingly. These three steps are repeated iteratively, until the processed trial mass rate grid matches the GRACE apparent mass rate grid. This provides a single mass rate estimate for the oceans, which conserves mass globally, because terrestrial rates are estimated simultaneously. Averages for specific regions (such as Antarctica) can also be calculated.

簡單而言，該方法的原理是在迭代擬合的每一步中，觀察信號和模型信號之間的差值，并在下一步中加入模型中。步驟如下：

（1）平滑后的1度分辨率的等效水高作為輸入模型；

（2）將輸入模型球諧展開，采用與GRACE一樣的濾波處理方法，得到恢復(fù)信號；

（3）計(jì)算恢復(fù)信號與原始信號的差值，并回加到原始輸入模型；

（4）重復(fù)第2、3步，直至收斂。

下面我們展示利用迭代法恢復(fù)的信號對比。我們可以看到，經(jīng)過迭代之后的信號振幅得到恢復(fù)，而經(jīng)過球諧展開和濾波得到的結(jié)果與原結(jié)果類似。

參考文獻(xiàn)

Chen, J., Wilson, C. & Tapley, B. Contribution of ice sheet and mountain glacier melt to recent sea level rise.?Nature Geosci?6, 549–552 (2013). https://doi.org/10.1038/ngeo1829