14. Measurement, inference, and uncertainty

测量、反演与不确定性

14.1 Define the observable before fitting

拟合前先定义可观测量

A silhouette radius $R_A=\sqrt{A_{\mathrm{image}}/\pi}$ is an area-equivalent radius. A volume-equivalent radius is $R_V=(3V/4\pi)^{1/3}$. These coincide for a sphere, but not for an arbitrary deformed bubble. For an axisymmetric shape with measured radius profile $r(z)$, a volume estimate is $V=\pi\int r(z)^2dz$. A single side view does not establish axisymmetry by itself.

轮廓半径$R_A=\sqrt{A_{\mathrm{image}}/\pi}$是面积等效半径；体积等效半径为$R_V=(3V/4\pi)^{1/3}$。二者对球体一致，但对任意变形气泡并不相同。对于具有实测径向轮廓$r(z)$的轴对称形状，可估计$V=\pi\int r(z)^2dz$。然而，一张侧视图本身并不能证明轴对称性。

Define $t=0$ explicitly: optical trigger, first visible nucleus, acoustic arrival, or maximum expansion are different choices. A fitted time offset may absorb an unmodeled nucleation delay, but it should not be interpreted automatically as a physical laser-to-bubble latency. Simultaneous bubble and jet imaging, as in [S412], helps distinguish stages that one signal alone would merge.

应明确定义$t=0$：光学触发、首次可见成核、声波到达或最大膨胀，是不同选择。拟合的时间偏移可能吸收未建模成核延迟，但不能自动解释为真实激光到气泡的延迟。类似[S412]中同时拍摄气泡与射流的做法，有助于区分单一信号会混合的不同阶段。

14.2 Derivatives amplify image noise

求导会放大图像噪声

If radius measurements have uncertainty $\delta R$ and spacing $\Delta t$, a naive first difference has noise scaling like $\delta R/\Delta t$, and a second difference like $\delta R/\Delta t^2$. Higher frame rate helps resolve motion, but raw numerical differentiation can still make inferred acceleration unstable. Fitting a physically constrained trajectory to radius data is often preferable to reconstructing pressure from noisy second differences.

如果半径测量不确定度为$\delta R$、采样间隔为$\Delta t$，简单一阶差分的噪声尺度约为$\delta R/\Delta t$，二阶差分约为$\delta R/\Delta t^2$。更高帧率有助于解析运动，但直接数值求导仍可能使推断加速度不稳定。通常，与其从带噪二阶差分反算压力，不如用具有物理约束的轨迹拟合半径数据。

A fitted model can be smooth and still wrong. Inspect residuals versus time: a systematic mismatch near the first collapse may indicate compressibility, shape change, or vapor physics rather than an optimizer failure. Report the observation model and smoothing choices, not just the final parameter values.

拟合模型即使十分平滑，也仍可能错误。应检查残差随时间的分布：第一次塌缩附近的系统偏差，可能意味着可压缩性、形状变化或蒸气过程缺失，而不只是优化器失败。报告时应说明观测模型与平滑选择，而不只是给出最终参数。

14.3 Identifiability: several stories can fit one curve

可辨识性：一条曲线可能对应多种解释

Let parameters be $\boldsymbol\theta=(R_0,p_{g0},\kappa,E_{\mathrm{eff}},t_0,\ldots)$ and observations be $R_i$. A weighted least-squares objective is

设参数为$\boldsymbol\theta=(R_0,p_{g0},\kappa,E_{\mathrm{eff}},t_0,\ldots)$，观测量为$R_i$。加权最小二乘目标可写为

The diagram locates the physical system; its key lists the symbols in the formula. Read the definition in the surrounding derivation when a letter has different roles.

图示标出物理体系，图例列出公式中的符号。若某字母有多种作用，应以邻近推导中的定义为准。

This formula does not make all parameters identifiable. In fact, $R_0$ and $p_{g0}$ may already be linked by the equilibrium relation. Treating linked quantities as independent fit parameters can create a redundant model. Different gas contents and thermal laws can sometimes fit a limited radius interval similarly while predicting different rebound behavior.

这个公式并不会使所有参数自动可辨识。事实上，$R_0$和$p_{g0}$可能已经通过平衡关系相联系。将彼此关联的量都作为独立拟合参数，可能造成模型冗余。不同气体含量和热学定律，有时可以相似地拟合一段有限半径历程，却预测出不同回弹行为。

Examine sensitivity columns $\partial R(t_i)/\partial\theta_j$. Nearly parallel columns indicate that parameters produce similar observational effects. Independent measurements—such as residual radius, absorbed energy, interface temperature, or acoustic arrival—can break some degeneracies. The detailed spherical laser-bubble analysis [S409] illustrates the importance of linking initialization and later oscillations rather than fitting arbitrary pressure spikes.

应检查灵敏度列$\partial R(t_i)/\partial\theta_j$。若不同列近乎平行，说明对应参数产生相似观测效应。残余半径、吸收能量、界面温度或声波到达等独立测量，可以解除部分退化。详尽的球形激光气泡研究[S409]说明，将初始化与后期振荡联系起来，比拟合任意压力尖峰更重要。

14.4 Pressure measurements are filtered observations

压力测量是经过滤波的观测

A sensor records a spatially and temporally filtered version of the physical field. An idealized representation is $p_{\mathrm{meas}}=h*p_{\mathrm{true}}+\epsilon$, where $h$ includes the instrument response and $\epsilon$ noise. A narrow collapse pulse may be strongly attenuated or broadened by bandwidth limitations. Its measured peak is therefore not automatically the local physical peak.

传感器记录的是经过空间和时间滤波的物理场。理想化表达为$p_{\mathrm{meas}}=h*p_{\mathrm{true}}+\epsilon$，其中$h$包含仪器响应，$\epsilon$为噪声。狭窄塌缩脉冲可能因带宽限制而显著衰减或展宽，因此实测峰值并不自动等于局部真实峰值。

Record sensor position, sensitive area, bandwidth, calibration, reflections, and uncertainty. Compare a model after applying the appropriate observation operator to the raw prediction. A high-resolution simulation compared directly with a bandwidth-limited trace can appear to disagree for purely instrumental reasons.

应记录传感器位置、敏感面积、带宽、标定、反射和不确定性。比较时应先对模型原始预测施加适当观测算子。若把高分辨率仿真直接与带宽受限信号比较，可能仅因仪器因素就出现看似明显的不一致。

14.5 Calibration is not validation

标定不是验证

Use one set of experiments to calibrate uncertain parameters and a separate set to test prediction. A useful holdout varies a parameter that the model claims to understand, such as pulse energy, ambient pressure, or stand-off distance. Matching the same trajectory used to tune the model is calibration evidence, not independent validation.

应使用一组实验标定不确定参数，再用另一组实验检验预测。有用的留出测试应改变模型声称能够解释的参数，例如脉冲能量、环境压力或离壁距离。与用于调参的同一条轨迹吻合，是标定证据，而不是独立验证。

For stochastic activation, repeat nominally identical trials and report event counts and confidence intervals. For deterministic-looking trajectories, also report run-to-run variation. A transfer process that succeeds once is different from a process with a measured success probability under stated conditions. [S405, S410]

对于随机激活，应重复名义上相同的试验，并报告事件次数与置信区间。对于看似确定性的轨迹，也应报告重复试验之间的变化。偶然成功一次的转印过程，不同于在明确条件下具有实测成功概率的过程。[S405, S410]