Seismic tomography allows us to image the interior of the Earth. In general, to determine the nature of seismic anomalies, constraints on more than one seismic parameter are required. For example, the ratio $R$ between perturbations in $v_s$ and $v_p$ ($\mathrm{dln}{v}_s$ and $\mathrm{dln}{v}_p$, respectively) is studied extensively in the lowermost mantle and interpreted in terms of thermal and/or chemical anomalies. However, to jointly interpret tomographic models of variations in $v_s$ and $v_p$ or their ratio $R$, it is essential for them to share the same local resolution. Most existing models do not provide resolution information, and thus cannot guarantee to honour this condition. In addition, uncertainties are typically not provided, making it difficult to robustly interpret the ratio $R=\mathrm{dln}{v}_s/\mathrm{dln}{v}_p$. To overcome these issues, we utilise the recently developed SOLA tomographic method, a variant of the linear Backus–Gilbert inversion scheme. SOLA retrieves local-average model estimates, together with information on their uncertainties, whilst it also provides direct control on model resolution through target kernels. In this contribution, we apply SOLA to normal-mode data with sensitivity to both $v_s$ and $v_p$, as well as density throughout the mantle. Specifically, we aim to develop models of both $v_s$ and $v_p$ with the same local resolution. We test our methodology and approach using synthetic tests for various noise cases (random noise, data noise or also additional 3D Earth noise due to variations in other physical parameters than the one of interest). We find that the addition of the 3D noise increases the uncertainties in our model estimates significantly, only allowing us to find model estimates in six or four layers for $v_s$ and $v_p$, respectively. While the synthetic tests indicate that no satisfactory density models can be obtained, we easily manage to construct models of $\mathrm{dln}{v}_s$ and $\mathrm{dln}{v}_p$ with almost identical resolution, from which the ratio $R$ can be robustly inferred. The obtained values of $R$ in our synthetic experiments significantly depend on the noise case considered and the method used to calculate it, with the addition of 3D noise always leading to an overestimate of $R$. When applying our approach to real data, we obtain values of $R$ in the range of 2.5–4.0 in the lowest 600 km of the mantle, which are consistent with previous studies. Our model estimates with related resolving kernels and uncertainties can be used to test geodynamic model predictions to provide further insights into the temperature and composition of the mantle.

地震断层扫描使我们能够对地球内部进行成像。一般来说，为了确定地震异常的性质，需要对多个地震参数进行约束。例如，比率$右$之间的扰动$v_s$和$v_p$（$丁v_s$和$丁v_p$分别）在最下地幔中进行了广泛的研究，并根据热和/或化学异常进行了解释。然而，为了共同解释变化的断层扫描模型$v_s$和$v_p$或它们的比例$右$，它们必须共享相同的本地分辨率。大多数现有模型不提供分辨率信息，因此不能保证满足此条件。此外，通常不提供不确定性，因此很难稳健地解释该比率$右=丁v_s/丁v_p$。为了克服这些问题，我们利用最近开发的 SOLA 层析成像方法，这是线性巴科斯-吉尔伯特反演方案的一种变体。SOLA 检索局部平均模型估计及其不确定性信息，同时它还通过目标内核提供对模型分辨率的直接控制。在此贡献中，我们将 SOLA 应用于正常模式数据，并对两者都敏感$v_s$和$v_p$，以及整个地幔的密度。具体来说，我们的目标是开发两者的模型$v_s$和$v_p$具有相同的本地分辨率。我们使用各种噪声情况的综合测试来测试我们的方法和方法（随机噪声、数据噪声或由于除感兴趣的物理参数之外的其他物理参数的变化而导致的额外3D 地球噪声）。我们发现，3D 噪声的添加显着增加了模型估计的不确定性，仅允许我们在六层或四层中找到模型估计$v_s$和$v_p$， 分别。虽然综合测试表明无法获得令人满意的密度模型，但我们可以轻松地构建模型$丁v_s$和$丁v_p$具有几乎相同的分辨率，其中比率$右$可以稳健地推断。获得的值$右$在我们的合成实验中，很大程度上取决于所考虑的噪声情况以及用于计算它的方法，添加 3D 噪声总是会导致高估$右$。当将我们的方法应用于实际数据时，我们获得的值$右$地幔最低 600 km 范围内为 2.5~4.0，这与前人的研究一致。我们的模型估计与相关的解析核和不确定性可用于测试地球动力学模型预测，以进一步了解地幔的温度和成分。